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leq","display":"\\leq","kind":"use","occurrences":[3578]},{"id":"symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"use","occurrences":[3449]},{"id":"symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"use","occurrences":[1367,2601,2607,2870]},{"id":"symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"use","occurrences":[29,2495,2628,3667,3797,3846,3930,3933,3942,3957,3971,3972]},{"id":"symbol-3c62641491514988","tex":"\\bigoplus_i","aliases":"⊕_i bigoplus","display":"\\bigoplus_i","kind":"use","occurrences":[3667]},{"id":"symbol-8817e97bab58b0ec","tex":"\\bigoplus_{k=0}^{K}","aliases":"⊕_k=0^K bigoplus","display":"\\bigoplus_{k=0}^{K}","kind":"use","occurrences":[3797]},{"id":"symbol-dde487846d5c46d3","tex":"\\bigoplus_{k=0}^n","aliases":"⊕_k=0^n bigoplus","display":"\\bigoplus_{k=0}^n","kind":"use","occurrences":[3846]},{"id":"symbol-d98af3ffc3344961","tex":"\\bigoplus_{L\\in p}","aliases":"⊕_L∈ p bigoplus in","display":"\\bigoplus_{L\\in p}","kind":"use","occurrences":[29]},{"id":"symbol-73d4349c6a5148b3","tex":"\\bigoplus_q","aliases":"⊕_q bigoplus","display":"\\bigoplus_q","kind":"use","occurrences":[2628]},{"id":"symbol-6e42b29f036c3ac2","tex":"\\bigoplus_{S\\subseteq[n]}","aliases":"⊕_S⊆[n] bigoplus subseteq","display":"\\bigoplus_{S\\subseteq[n]}","kind":"use","occurrences":[2495]},{"id":"symbol-ca12baf923be071e","tex":"\\bigoplus_\\lambda","aliases":"⊕_λ bigoplus lambda","display":"\\bigoplus_\\lambda","kind":"use","occurrences":[3933,3957,3971,3972]},{"id":"symbol-9134e99ccc88c15b","tex":"\\bigoplus_{\\lambda\\vdash n,\\,\\ell(\\lambda)\\le d}","aliases":"⊕_λvdash n, ℓ(λ)≤ d bigoplus lambda vdash ell lambda le","display":null,"kind":"use","occurrences":[3930,3942]},{"id":"symbol-036096cd7e9ecc74","tex":"\\bigcup","aliases":"⋃ bigcup","display":"\\bigcup","kind":"use","occurrences":[1968,1970]},{"id":"symbol-869d7204e96d9910","tex":"\\bigcup_{s<t}","aliases":"⋃_s<t bigcup","display":"\\bigcup_{s<t}","kind":"use","occurrences":[1970]},{"id":"symbol-89033592672e0222","tex":"\\bigcup_{t\\ne s}","aliases":"⋃_t≠ s bigcup ne","display":"\\bigcup_{t\\ne s}","kind":"use","occurrences":[1970]},{"id":"symbol-51f354fe46d1388b","tex":"\\bigcup_{\\sigma\\in\\{(t,s):t\\ne s\\}^L}","aliases":"⋃_σ∈{(t,s):t≠ s}^L bigcup sigma in ne","display":"\\bigcup_{\\sigma\\in\\{(t,s):t\\ne s\\}^L}","kind":"use","occurrences":[1968]},{"id":"symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","occurrences":[1024,1092,1095,1100,1101,1344,1692,1700,1707,2689,2706,2710,2745,2747,2788,2972,3112,3119,3210,3217,3248,3249,3313,3322,3540,3817,3825]},{"id":"symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","occurrences":[1024,1092,1095,1100,1101,1344,1692,1700,1707,2689,2706,2710,2745,2747,2788,2972,3112,3119,3210,3217,3225,3226,3228,3231,3242,3248,3249,3313,3322,3540,3817,3825]},{"id":"symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"use","occurrences":[1371,1965,3654,3760]},{"id":"symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","occurrences":[134,1028,1173,1965,2389,2498,2592,2595,2610,2611,2908,4022,4103]}],"contexts":["The guide does not supply a new toolchain, an independent checker or independent\nhuman review. It preserves the original rights record and the original release\nfiles. There is no new publication-novelty or unrestricted hardness claim.\n\\clearpage\n\\section*{The formal endpoint}\n\\addcontentsline{toc}{section}{The formal endpoint}\nLet $c$ be a finite legal classical NOT/CNOT/Toffoli circuit. For $n$ input bits and $a$ zero-initialized ancillas, assume for every $x$ and both initial target values $b$ that\n\\[\n\\operatorname{run}(c)(x,0^a,b)=(x,0^a,b\\x f(x)).\n\\]\nWrite $T=\\TC(c)$ for its number of Toffoli gates. The Block 02 joint endpoint is\n\\begin{statusbox}{Degree and complete-cover bound}\n\\[\n\\boxed{\\Deg(T+1,f)\\quad\\text{and}\\quad 2\\beta^*(f)\\le T+3.}\n\\]\n\\pathref{CircuitObstructions.restored_degree_and_cover_bound}\n\\end{statusbox}\nThe first conclusion is an explicit certificate that $f$ has a Boolean polynomial representation of degree at most $T+1$. The second is the affine-optimized complete safe-cover lower bound, equivale","The guide does not supply a new toolchain, an independent checker or independent\nhuman review. It preserves the original rights record and the original release\nfiles. There is no new publication-novelty or unrestricted hardness claim.\n\\clearpage\n\\section*{The formal endpoint}\n\\addcontentsline{toc}{section}{The formal endpoint}\nLet $c$ be a finite legal classical NOT/CNOT/Toffoli circuit. For $n$ input bits and $a$ zero-initialized ancillas, assume for every $x$ and both initial target values $b$ that\n\\[\n\\operatorname{run}(c)(x,0^a,b)=(x,0^a,b\\x f(x)).\n\\]\nWrite $T=\\TC(c)$ for its number of Toffoli gates. The Block 02 joint endpoint is\n\\begin{statusbox}{Degree and complete-cover bound}\n\\[\n\\boxed{\\Deg(T+1,f)\\quad\\text{and}\\quad 2\\beta^*(f)\\le T+3.}\n\\]\n\\pathref{CircuitObstructions.restored_degree_and_cover_bound}\n\\end{statusbox}\nThe first conclusion is an explicit certificate that $f$ has a Boolean polynomial representation of degree at most $T+1$. The second is the affine-optimized complete safe-cover lower bound, equivalently $T\\ge\\max\\{0,2\\beta^*(f)-3\\}$.","not supply a new toolchain, an independent checker or independent\nhuman review. It preserves the original rights record and the original release\nfiles. There is no new publication-novelty or unrestricted hardness claim.\n\\clearpage\n\\section*{The formal endpoint}\n\\addcontentsline{toc}{section}{The formal endpoint}\nLet $c$ be a finite legal classical NOT/CNOT/Toffoli circuit. For $n$ input bits and $a$ zero-initialized ancillas, assume for every $x$ and both initial target values $b$ that\n\\[\n\\operatorname{run}(c)(x,0^a,b)=(x,0^a,b\\x f(x)).\n\\]\nWrite $T=\\TC(c)$ for its number of Toffoli gates. The Block 02 joint endpoint is\n\\begin{statusbox}{Degree and complete-cover bound}\n\\[\n\\boxed{\\Deg(T+1,f)\\quad\\text{and}\\quad 2\\beta^*(f)\\le T+3.}\n\\]\n\\pathref{CircuitObstructions.restored_degree_and_cover_bound}\n\\end{statusbox}\nThe first conclusion is an explicit certificate that $f$ has a Boolean polynomial representation of degree at most $T+1$. The second is the affine-optimized complete safe-cover lower bound, equivalently $T\\ge\\max\\{0,2\\beta^*(f)-3\\}$.","er or independent\nhuman review. It preserves the original rights record and the original release\nfiles. There is no new publication-novelty or unrestricted hardness claim.\n\\clearpage\n\\section*{The formal endpoint}\n\\addcontentsline{toc}{section}{The formal endpoint}\nLet $c$ be a finite legal classical NOT/CNOT/Toffoli circuit. For $n$ input bits and $a$ zero-initialized ancillas, assume for every $x$ and both initial target values $b$ that\n\\[\n\\operatorname{run}(c)(x,0^a,b)=(x,0^a,b\\x f(x)).\n\\]\nWrite $T=\\TC(c)$ for its number of Toffoli gates. The Block 02 joint endpoint is\n\\begin{statusbox}{Degree and complete-cover bound}\n\\[\n\\boxed{\\Deg(T+1,f)\\quad\\text{and}\\quad 2\\beta^*(f)\\le T+3.}\n\\]\n\\pathref{CircuitObstructions.restored_degree_and_cover_bound}\n\\end{statusbox}\nThe first conclusion is an explicit certificate that $f$ has a Boolean polynomial representation of degree at most $T+1$. The second is the affine-optimized complete safe-cover lower bound, equivalently $T\\ge\\max\\{0,2\\beta^*(f)-3\\}$.","preserves the original rights record and the original release\nfiles. There is no new publication-novelty or unrestricted hardness claim.\n\\clearpage\n\\section*{The formal endpoint}\n\\addcontentsline{toc}{section}{The formal endpoint}\nLet $c$ be a finite legal classical NOT/CNOT/Toffoli circuit. For $n$ input bits and $a$ zero-initialized ancillas, assume for every $x$ and both initial target values $b$ that\n\\[\n\\operatorname{run}(c)(x,0^a,b)=(x,0^a,b\\x f(x)).\n\\]\nWrite $T=\\TC(c)$ for its number of Toffoli gates. The Block 02 joint endpoint is\n\\begin{statusbox}{Degree and complete-cover bound}\n\\[\n\\boxed{\\Deg(T+1,f)\\quad\\text{and}\\quad 2\\beta^*(f)\\le T+3.}\n\\]\n\\pathref{CircuitObstructions.restored_degree_and_cover_bound}\n\\end{statusbox}\nThe first conclusion is an explicit certificate that $f$ has a Boolean polynomial representation of degree at most $T+1$. The second is the affine-optimized complete safe-cover lower bound, equivalently $T\\ge\\max\\{0,2\\beta^*(f)-3\\}$.","the original rights record and the original release\nfiles. There is no new publication-novelty or unrestricted hardness claim.\n\\clearpage\n\\section*{The formal endpoint}\n\\addcontentsline{toc}{section}{The formal endpoint}\nLet $c$ be a finite legal classical NOT/CNOT/Toffoli circuit. For $n$ input bits and $a$ zero-initialized ancillas, assume for every $x$ and both initial target values $b$ that\n\\[\n\\operatorname{run}(c)(x,0^a,b)=(x,0^a,b\\x f(x)).\n\\]\nWrite $T=\\TC(c)$ for its number of Toffoli gates. The Block 02 joint endpoint is\n\\begin{statusbox}{Degree and complete-cover bound}\n\\[\n\\boxed{\\Deg(T+1,f)\\quad\\text{and}\\quad 2\\beta^*(f)\\le T+3.}\n\\]\n\\pathref{CircuitObstructions.restored_degree_and_cover_bound}\n\\end{statusbox}\nThe first conclusion is an explicit certificate that $f$ has a Boolean polynomial representation of degree at most $T+1$. The second is the affine-optimized complete safe-cover lower bound, equivalently $T\\ge\\max\\{0,2\\beta^*(f)-3\\}$.","here is no new publication-novelty or unrestricted hardness claim.\n\\clearpage\n\\section*{The formal endpoint}\n\\addcontentsline{toc}{section}{The formal endpoint}\nLet $c$ be a finite legal classical NOT/CNOT/Toffoli circuit. For $n$ input bits and $a$ zero-initialized ancillas, assume for every $x$ and both initial target values $b$ that\n\\[\n\\operatorname{run}(c)(x,0^a,b)=(x,0^a,b\\x f(x)).\n\\]\nWrite $T=\\TC(c)$ for its number of Toffoli gates. The Block 02 joint endpoint is\n\\begin{statusbox}{Degree and complete-cover bound}\n\\[\n\\boxed{\\Deg(T+1,f)\\quad\\text{and}\\quad 2\\beta^*(f)\\le T+3.}\n\\]\n\\pathref{CircuitObstructions.restored_degree_and_cover_bound}\n\\end{statusbox}\nThe first conclusion is an explicit certificate that $f$ has a Boolean polynomial representation of degree at most $T+1$. The second is the affine-optimized complete safe-cover lower bound, equivalently $T\\ge\\max\\{0,2\\beta^*(f)-3\\}$.","oc}{section}{The formal endpoint}\nLet $c$ be a finite legal classical NOT/CNOT/Toffoli circuit. For $n$ input bits and $a$ zero-initialized ancillas, assume for every $x$ and both initial target values $b$ that\n\\[\n\\operatorname{run}(c)(x,0^a,b)=(x,0^a,b\\x f(x)).\n\\]\nWrite $T=\\TC(c)$ for its number of Toffoli gates. The Block 02 joint endpoint is\n\\begin{statusbox}{Degree and complete-cover bound}\n\\[\n\\boxed{\\Deg(T+1,f)\\quad\\text{and}\\quad 2\\beta^*(f)\\le T+3.}\n\\]\n\\pathref{CircuitObstructions.restored_degree_and_cover_bound}\n\\end{statusbox}\nThe first conclusion is an explicit certificate that $f$ has a Boolean polynomial representation of degree at most $T+1$. The second is the affine-optimized complete safe-cover lower bound, equivalently $T\\ge\\max\\{0,2\\beta^*(f)-3\\}$.","alues $b$ that\n\\[\n\\operatorname{run}(c)(x,0^a,b)=(x,0^a,b\\x f(x)).\n\\]\nWrite $T=\\TC(c)$ for its number of Toffoli gates. The Block 02 joint endpoint is\n\\begin{statusbox}{Degree and complete-cover bound}\n\\[\n\\boxed{\\Deg(T+1,f)\\quad\\text{and}\\quad 2\\beta^*(f)\\le T+3.}\n\\]\n\\pathref{CircuitObstructions.restored_degree_and_cover_bound}\n\\end{statusbox}\nThe first conclusion is an explicit certificate that $f$ has a Boolean polynomial representation of degree at most $T+1$. The second is the affine-optimized complete safe-cover lower bound, equivalently $T\\ge\\max\\{0,2\\beta^*(f)-3\\}$.","x)).\n\\]\nWrite $T=\\TC(c)$ for its number of Toffoli gates. The Block 02 joint endpoint is\n\\begin{statusbox}{Degree and complete-cover bound}\n\\[\n\\boxed{\\Deg(T+1,f)\\quad\\text{and}\\quad 2\\beta^*(f)\\le T+3.}\n\\]\n\\pathref{CircuitObstructions.restored_degree_and_cover_bound}\n\\end{statusbox}\nThe first conclusion is an explicit certificate that $f$ has a Boolean polynomial representation of degree at most $T+1$. The second is the affine-optimized complete safe-cover lower bound, equivalently $T\\ge\\max\\{0,2\\beta^*(f)-3\\}$.","\\begin{statusbox}{Degree and complete-cover bound}\n\\[\n\\boxed{\\Deg(T+1,f)\\quad\\text{and}\\quad 2\\beta^*(f)\\le T+3.}\n\\]\n\\pathref{CircuitObstructions.restored_degree_and_cover_bound}\n\\end{statusbox}\nThe first conclusion is an explicit certificate that $f$ has a Boolean polynomial representation of degree at most $T+1$. The second is the affine-optimized complete safe-cover lower bound, equivalently $T\\ge\\max\\{0,2\\beta^*(f)-3\\}$.","\\subsection*{What did not change}\nThe mathematical scope remains the eight maintained claim groups. C02 is still partial as a group: its dual-certificate theorem is not included. The formal source also proves\n\\[\n\\beta(f)\\le\\beta^*(f)\\le n.\n\\]\nThis ceiling prevents the degree parameter alone from producing a superpolynomial lower bound. No new best-known bound or publication-novelty claim is made.","\\pagehead{Model and definitions}{THE ASSUMPTIONS ARE PART OF THE THEOREM}\nA general register state is a Boolean function $W\\to\\mathrm{Bool}$ on any wire type with decidable equality. C01 does not assume that $W$ is finite. The canonical degree model has a finite wire set with $n$ input wires, $a$ ancillas, and one target. A NOT complements a bit. A CNOT adds one control to a distinct target. A Toffoli adds the product of two distinct controls to a third distinct wire. Gate constructors carry the required inequalities.","All arithmetic is over $\\mathbb F_2$, implemented using Boolean XOR and AND. Each gate is its own inverse. Circuit reversal, restoration, and every cut are inherited from the unchanged Block 01 source.","midrule\n$T$ & Toffoli count. NOT and CNOT have cost zero in this count only.","$\\Deg(d,f)$ & $f$ is an XOR of finitely many monomials, each containing at most $d$ variable occurrences.","Safe detector & A Boolean $h$ with $h(x)=1\\Rightarrow f(x)=1$.","Complete coverage & Every positive $v$ has a safe detector with $h(v)=1$.","$\\beta(f)$ & Least degree bound sufficient for complete safe coverage.","$\\beta^*(f)$ & Least upper bound on $\\beta(f\\x g)$ over all degree-one $g$.","\\subsection*{Polynomial certificates are not postulated degree oracles}\nThe inductive certificate has three constructors: zero, a bounded-length monomial, and XOR of two certified functions. It is tied to the concrete evaluator\n\\[\n\\operatorname{polyEval}(p,x)=\\bigoplus_{L\\in p}\\prod_{i\\in L}x_i\n\\]\nby a proved equivalence. A certificate yields an explicit finite monomial list, and such a list yields a certificate.","\\pagehead{Why sharing does not break the degree bound}{A STRONGER INVARIANT THAN SINGLE-REGISTER DEGREE}\nFor a current state $s(x)$ after $t$ Toffolis, the invariant is\n\\[\n\\forall L,\\qquad\n\\Deg\\!\\left(t+|L|,\\;x\\mapsto\\prod_{i\\in L}s_i(x)\\right).\n\\]\n$L$ may be any finite list, including repeated registers. Initially each coordinate is a variable or a constant, so a product of $r$ coordinates has degree at most $r$.","sharing does not break the degree bound}{A STRONGER INVARIANT THAN SINGLE-REGISTER DEGREE}\nFor a current state $s(x)$ after $t$ Toffolis, the invariant is\n\\[\n\\forall L,\\qquad\n\\Deg\\!\\left(t+|L|,\\;x\\mapsto\\prod_{i\\in L}s_i(x)\\right).\n\\]\n$L$ may be any finite list, including repeated registers. Initially each coordinate is a variable or a constant, so a product of $r$ coordinates has degree at most $r$.","Suppose the next gate toggles target $u$ by the product of a control list of length $q$. A product that omits $u$ is unchanged. If $u$ occurs, idempotence removes all its repetitions. With $R$ the remaining list, the updated product is\n\\[\n\\left(s_u\\prod_{i\\in R}s_i\\right)\n\\x\n\\left(\\prod_{j\\in\\mathrm{controls}}s_j\\prod_{i\\in R}s_i\\right).\n\\]\nThe two old-state products have certified degrees at most\n\\[\nt+1+|R|\\qquad\\text{and}\\qquad t+q+|R|.\n\\]\nSince $|R|+1\\le |L|$, the invariant's budget rises by at most $\\max\\{q-1,0\\}$: zero for NOT and CNOT, one for Toffoli.","pose the next gate toggles target $u$ by the product of a control list of length $q$. A product that omits $u$ is unchanged. If $u$ occurs, idempotence removes all its repetitions. With $R$ the remaining list, the updated product is\n\\[\n\\left(s_u\\prod_{i\\in R}s_i\\right)\n\\x\n\\left(\\prod_{j\\in\\mathrm{controls}}s_j\\prod_{i\\in R}s_i\\right).\n\\]\nThe two old-state products have certified degrees at most\n\\[\nt+1+|R|\\qquad\\text{and}\\qquad t+q+|R|.\n\\]\nSince $|R|+1\\le |L|$, the invariant's budget rises by at most $\\max\\{q-1,0\\}$: zero for NOT and CNOT, one for Toffoli.","ct of a control list of length $q$. A product that omits $u$ is unchanged. If $u$ occurs, idempotence removes all its repetitions. With $R$ the remaining list, the updated product is\n\\[\n\\left(s_u\\prod_{i\\in R}s_i\\right)\n\\x\n\\left(\\prod_{j\\in\\mathrm{controls}}s_j\\prod_{i\\in R}s_i\\right).\n\\]\nThe two old-state products have certified degrees at most\n\\[\nt+1+|R|\\qquad\\text{and}\\qquad t+q+|R|.\n\\]\nSince $|R|+1\\le |L|$, the invariant's budget rises by at most $\\max\\{q-1,0\\}$: zero for NOT and CNOT, one for Toffoli.","$u$ is unchanged. If $u$ occurs, idempotence removes all its repetitions. With $R$ the remaining list, the updated product is\n\\[\n\\left(s_u\\prod_{i\\in R}s_i\\right)\n\\x\n\\left(\\prod_{j\\in\\mathrm{controls}}s_j\\prod_{i\\in R}s_i\\right).\n\\]\nThe two old-state products have certified degrees at most\n\\[\nt+1+|R|\\qquad\\text{and}\\qquad t+q+|R|.\n\\]\nSince $|R|+1\\le |L|$, the invariant's budget rises by at most $\\max\\{q-1,0\\}$: zero for NOT and CNOT, one for Toffoli.","Induction along the actual gate list gives\n\\[\n\\Deg\\!\\left(T+r,\\;x\\mapsto\\prod_{i\\in L}\\operatorname{run}(c)(s(x))_i\\right),\n\\qquad r=|L|.\n\\]\nWith $r=1$, every register has degree at most $T+1$.","\\pagehead{From the cut to a numerical lower bound}{EXISTENCE, BALANCE, AND COMPLETE COVERAGE}\nWrite $c=p\\mathbin{++}q$. At this cut, Block 01 defines\n\\[\nH_i(x)=\\operatorname{run}(p)(x,0^a,0)_i\\x\n       \\operatorname{run}(q^{-1})(x,0^a,0)_i\n\\]\nand proves\n\\[\nH_i\\le f,\\qquad f(x)=1\\iff\\exists i\\;H_i(x)=1.\n\\]\nReversal does not change Toffoli count. The new coordinate degree theorem therefore gives\n\\[\n\\deg(H_i)\\le\\max\\{\\TC(p),\\TC(q)\\}+1.\n\\]","\\subsection*{The least safe degree exists}\nFor a positive $v$, take the point indicator\n\\[\n\\delta_v(x)=\\prod_{i=1}^{n}\n\\begin{cases}\nx_i,&v_i=1,","\\subsection*{The least safe degree exists}\nFor a positive $v$, take the point indicator\n\\[\n\\delta_v(x)=\\prod_{i=1}^{n}\n\\begin{cases}\nx_i,&v_i=1,\\\\\n1+x_i,&v_i=0.\n\\end{cases}\n\\]\nIt is safe, detects $v$, and has a degree-$n$ certificate. Hence complete safe coverage exists at degree $n$ for every function. Strong induction proves the existence of a least admissible natural degree. The definition \\pathref{beta} chooses this bound and its specification and minimality are proved.","1+x_i,&v_i=0.\n\\end{cases}\n\\]\nIt is safe, detects $v$, and has a degree-$n$ certificate. Hence complete safe coverage exists at degree $n$ for every function. Strong induction proves the existence of a least admissible natural degree. The definition \\pathref{beta} chooses this bound and its specification and minimality are proved.","\\subsection*{The balanced cut is constructed}\nBecause each gate has cost zero or one, a prefix can realize every Toffoli count from zero to $T$. The source proves this fact and extracts a cut with\n\\[\n\\max\\{\\TC(p),\\TC(q)\\}\\le\\left\\lceil T/2\\right\\rceil.\n\\]\nThe complete safe cut cover then proves\n\\[\n\\beta(f)\\le\\left\\lceil T/2\\right\\rceil+1,\n\\qquad\\boxed{2\\beta(f)\\le T+3.}\n\\]\nNatural-number arithmetic retains the floor/ceiling and small-$T$ cases. No balance assumption is added to the final theorem.","ucted}\nBecause each gate has cost zero or one, a prefix can realize every Toffoli count from zero to $T$. The source proves this fact and extracts a cut with\n\\[\n\\max\\{\\TC(p),\\TC(q)\\}\\le\\left\\lceil T/2\\right\\rceil.\n\\]\nThe complete safe cut cover then proves\n\\[\n\\beta(f)\\le\\left\\lceil T/2\\right\\rceil+1,\n\\qquad\\boxed{2\\beta(f)\\le T+3.}\n\\]\nNatural-number arithmetic retains the floor/ceiling and small-$T$ cases. No balance assumption is added to the final theorem.","\\pagehead{Affine output adjustment is an actual circuit}{THE OPTIMIZATION DOES NOT ASSUME A CLEAN NORMAL FORM}\nFor every degree-one Boolean function $g$, the source constructs a restored implementation with zero Toffolis.","If $c$ restores $f$ and $d$ restores $g$, their concatenation restores $f\\x g$. For the affine implementation,\n\\[\n\\TC(c\\mathbin{++}d)=\\TC(c).\n\\]\nThe unoptimized bound applies to every affine $g$ at the same $T$. Define $\\beta^*(f)$ as the least upper bound of those $\\beta(f\\x g)$ values. Existence follows from their common upper bound $n$.","Therefore\n\\[\n\\boxed{2\\beta^*(f)\\le T+3,\\qquad \\beta(f)\\le\\beta^*(f)\\le n.}\n\\]\nThe final output at initial target zero is exactly $f$, so the forward coordinate theorem also gives $\\Deg(T+1,f)$. The joint endpoint combines these two proved conclusions.","A degree-one complete safe cover for AND would include a detector of its sole accepting input. Safety forces that detector to be AND itself, contradicting the corner-parity lemma. The degree-two monomial is a valid detector. The source proves\n\\[\n\\beta(\\mathrm{AND}_2)=\\beta^*(\\mathrm{AND}_2)=2\n\\]\nand therefore every restored implementation of AND$_2$ uses at least one Toffoli.","This is a theorem about all implementations in the model, not an exhaustive search through circuits. It does not formalize the general $k$-input monomial optimum.","\\pagehead{Reading the formal model literally}{QUANTIFIERS AND ENCODING CARRY THE HYPOTHESES}\nThe generic cut theory uses a logical-input type $X$, a wire type $W$ with\ndecidable equality, and a boundary structure. Its essential fields are an\nencoding $\\mathrm{encode}:X\\to\\mathrm{Bool}\\to(W\\to\\mathrm{Bool})$, a target\nwire, and the proof\n\\[\n \\mathrm{encode}(x,b)(\\mathrm{target})=b.\n\\]\nThis target equation separates the two target branches for each fixed input.\nIt excludes an encoding that silently ignores its supplied target bit.\nThe source proves \\code{Boundary.target_separated}; it does not assume an\ninjective encoding of every logical-input value as an additional hypothesis.","The restoration predicate is\n\\[\n \\forall x\\,b,\\quad \\mathrm{run}(c)(\\mathrm{encode}(x,b))\n =\\mathrm{encode}(x,b+f(x)).\n\\]\nIt is a hypothesis of the conditional theorem. The code does not assert that\nevery gate list realizes a restored oracle. The canonical encoding sets each\ninput wire to its input bit, each ancillary wire to zero, and the target to $b$.\nIts wire type is\n\\begin{lstlisting}\nSum (Fin n) (Sum (Fin a) Unit)\n\\end{lstlisting}\nwhich has exactly $n+a+1$ wires.","run}(c)(\\mathrm{encode}(x,b))\n =\\mathrm{encode}(x,b+f(x)).\n\\]\nIt is a hypothesis of the conditional theorem. The code does not assert that\nevery gate list realizes a restored oracle. The canonical encoding sets each\ninput wire to its input bit, each ancillary wire to zero, and the target to $b$.\nIts wire type is\n\\begin{lstlisting}\nSum (Fin n) (Sum (Fin a) Unit)\n\\end{lstlisting}\nwhich has exactly $n+a+1$ wires.","In the generic C01 theorem, completeness is the existential statement\n\\[\n f(x)=1\\quad\\Longleftrightarrow\\quad\\exists i:W,\\ H_i(x)=1.\n\\]\nNo finite disjunction is needed there. Specializing to the canonical finite\nwires yields the finite complete family used in the main paper. Completeness\ndoes not say that the XOR of all packets reconstructs $f$; detecting packets\ncan cancel under XOR. It states existence of a detecting coordinate at every\npositive input.","\\pagehead{Certificates, semantic minima and natural arithmetic}{WHAT THE FORMAL OBJECTS DO AND DO NOT COMPUTE}\n\\code{DegreeLE d f} is an inductive representation predicate. Its constructors\nbuild zero, a monomial represented by a variable-occurrence list of length\nat most $d$, and XOR of certified functions. The polynomial evaluator XORs\nthe monomial-list values. The source proves the equivalence between a degree\ncertificate and such a finite representation.","Repeated indices are permitted. Boolean idempotence reduces their repetitions,\nso the representation gives the usual multilinear degree bound. The formal\nblock supplies no separate canonical ANF normalization algorithm and no theorem\nthat extracting homogeneous grades is computationally free. Zero has a\ndegree-zero certificate. The Python degree helper's $-1$ convention for zero\nis a mathematical convention outside the natural-valued formal bound.","\\code{CompleteSafeDegreeLE f d} says that every positive point has some degree-$d$\nsafe detector. It does not require one detector to cover all positives, and\nit places no efficiency bound on finding those detectors. Point indicators\nshow that $d=n$ suffices. A least-bound existence theorem then permits the\nnoncomputable definition \\code{beta}. Its specification supplies complete\ncoverage; its minimality compares it with every sufficient natural bound.","\\code{AffineSafeDegreeLE f d} universally quantifies every degree-one output\nfunction $g$ and requires $\\beta(f+g)\\le d$. The definition \\code{betaStar}\nchooses its least common upper bound. There are finitely many Boolean functions\non $n$ inputs, so this least bound equals the main paper's maximum over affine\noutput masks. It is a maximum over mask values, not a minimum over masks.","The source proves $\\beta(f)\\le\\beta^*(f)\\le n$. The zero mask supplies the\nfirst inequality. The point-indicator bound supplies the second. Classical\nchoice in the definitions is not a polynomial-time algorithm for computing\nthese parameters.","All counters and degree bounds in this block are natural numbers. In\n\\code{2 * betaStar f - 3}, subtraction saturates at zero. Thus\n\\[\n 2\\beta^*(f)-3\\le T\n\\]\nin the formal natural-number statement is the integer inequality\n$\\max\\{0,2\\beta^*(f)-3\\}\\le T$. The nonsubtractive endpoint\n$2\\beta^*(f)\\le T+3$ avoids ambiguity. Rounding at odd $T$ is retained;\nthe balanced cut is chosen by Toffoli count, not by half the gate-list length.","Cut transport & Inverse suffix meets the appropriate forward target branch; $H_i=fD_i$ gives safety and pointwise completeness","Shared register products & Products of $r$ current registers have budget $t+r$ after $t$ Toffolis; repeats and feedback are allowed","Balanced cut & An attainable prefix cost gives both sides cost at most $\\lceil T/2\\rceil$; the complete family gives the beta bound","Joint endpoint & The forward target at initial zero supplies $f$; degree and optimized-cover conclusions are conjoined","The general run theorem needs degree-one certificates for the initial\nregister functions. The canonical boundary supplies these certificates;\nrestoration is used later to identify the final target with $f$. The affine\nexistence theorem constructs gate lists inside its proof. It does not expose\na separate executable compiler function or assert minimum total gate count.","\\chapter{Scope, notation, and computational model}\n\\label{c:ch:foundations}\n\\section{Boolean and algebraic conventions}\nA Boolean function is a map $f:\\B^n\\to\\B$. Unless a different field is explicitly stated, polynomials are the unique multilinear representatives in\n\\[\n\\mathcal R_n=\\F[x_1,\\ldots,x_n]/(x_i^2-x_i:1\\le i\\le n).\n\\]\nAddition in this ring is XOR. Multiplication is AND. Boolean OR is written $\\lor$ and satisfies $a\\lor b=a+b+ab$. A statement $h\\le f$ is pointwise and means that $h$ has no positive value at a negative input of $f$.","r{Scope, notation, and computational model}\n\\label{c:ch:foundations}\n\\section{Boolean and algebraic conventions}\nA Boolean function is a map $f:\\B^n\\to\\B$. Unless a different field is explicitly stated, polynomials are the unique multilinear representatives in\n\\[\n\\mathcal R_n=\\F[x_1,\\ldots,x_n]/(x_i^2-x_i:1\\le i\\le n).\n\\]\nAddition in this ring is XOR. Multiplication is AND. Boolean OR is written $\\lor$ and satisfies $a\\lor b=a+b+ab$. A statement $h\\le f$ is pointwise and means that $h$ has no positive value at a negative input of $f$.","d computational model}\n\\label{c:ch:foundations}\n\\section{Boolean and algebraic conventions}\nA Boolean function is a map $f:\\B^n\\to\\B$. Unless a different field is explicitly stated, polynomials are the unique multilinear representatives in\n\\[\n\\mathcal R_n=\\F[x_1,\\ldots,x_n]/(x_i^2-x_i:1\\le i\\le n).\n\\]\nAddition in this ring is XOR. Multiplication is AND. Boolean OR is written $\\lor$ and satisfies $a\\lor b=a+b+ab$. A statement $h\\le f$ is pointwise and means that $h$ has no positive value at a negative input of $f$.","c:ch:foundations}\n\\section{Boolean and algebraic conventions}\nA Boolean function is a map $f:\\B^n\\to\\B$. Unless a different field is explicitly stated, polynomials are the unique multilinear representatives in\n\\[\n\\mathcal R_n=\\F[x_1,\\ldots,x_n]/(x_i^2-x_i:1\\le i\\le n).\n\\]\nAddition in this ring is XOR. Multiplication is AND. Boolean OR is written $\\lor$ and satisfies $a\\lor b=a+b+ab$. A statement $h\\le f$ is pointwise and means that $h$ has no positive value at a negative input of $f$.","d algebraic conventions}\nA Boolean function is a map $f:\\B^n\\to\\B$. Unless a different field is explicitly stated, polynomials are the unique multilinear representatives in\n\\[\n\\mathcal R_n=\\F[x_1,\\ldots,x_n]/(x_i^2-x_i:1\\le i\\le n).\n\\]\nAddition in this ring is XOR. Multiplication is AND. Boolean OR is written $\\lor$ and satisfies $a\\lor b=a+b+ab$. A statement $h\\le f$ is pointwise and means that $h$ has no positive value at a negative input of $f$.","map $f:\\B^n\\to\\B$. Unless a different field is explicitly stated, polynomials are the unique multilinear representatives in\n\\[\n\\mathcal R_n=\\F[x_1,\\ldots,x_n]/(x_i^2-x_i:1\\le i\\le n).\n\\]\nAddition in this ring is XOR. Multiplication is AND. Boolean OR is written $\\lor$ and satisfies $a\\lor b=a+b+ab$. A statement $h\\le f$ is pointwise and means that $h$ has no positive value at a negative input of $f$.","The degree of a nonzero function is its multilinear algebraic degree. A nonzero constant has degree zero. For mathematical statements involving a constant-zero function we use $\\deg0=-\\infty$; stored profile tables and Python helpers use their documented sentinels instead. Those implementation conventions do not change the inequalities.","For $S\\subseteq[n]$, write $x^S=\\prod_{i\\in S}x_i$ and let $1_S$ be the input with support $S$. A minterm of a monotone function means an inclusion-minimal positive support, not an arbitrary positive input.","midrule\\endhead\n$C(f)$ & Minimum fan-in-two AND/OR/NOT gate count, allowing fan-out.","$L(f)$ & Minimum total NOT/CNOT/Toffoli count of a restored implementation.","$T$ & Number of Toffoli gates in a specified implementation.","$\\KTrest(f)$ & Minimum such Toffoli count, allowing any finite zero workspace.","$M(f)$ & Minimum AND count in an XOR--AND computation, with affine operations uncharged.","$I_f$ & Boolean functions supported only on positive inputs of $f$.","$\\alpha_1(f)$ & Minimum degree of one nonzero member of $I_f$.","$\\beta_v(f)$ & Minimum degree of a member of $I_f$ that is one at $v$.","$\\beta(f),\\beta^*(f)$ & Complete-cover degree and its affine-output optimization.","$[P]_2$ & Homogeneous degree-two component of the multilinear polynomial $P$ in the specified coordinates.","A restored oracle is a circuit satisfying\n\\begin{equation}\\label{c:eq:oracle}\n(x,0^a,y)\\longmapsto(x,0^a,y+f(x))\n\\end{equation}\nfor every logical input $x$ and both initial target values $y\\in\\B$. The zero-workspace condition is imposed at the boundary, not at intermediate times. The circuit may disturb logical inputs, use output feedback, reuse intermediate values, and make arbitrary reversible excursions before restoring the required registers.","NOT and CNOT are free only when counting $T$. They are charged in $L$. This distinction is essential in the projection example. There are no measurements, quantum phases, discarded work registers, or unverified ancilla initializations in this model.","\\section{Comparison with ordinary circuits}\n\\begin{theorem}[Classical/restored simulation comparison]\\label{c:thm:simulation}\nFor every Boolean function $f$,\n\\[\n\\frac{C(f)}5\\le L(f)\\le6C(f)+1.\n\\]\n\\end{theorem}\n\\begin{proof}\nCompute every Boolean gate onto a fresh zero wire. An AND costs one Toffoli. A NOT of an existing signal is obtained with one NOT and one CNOT. An OR is computed as $a+b+ab$, using two CNOTs and one Toffoli. Repeated operands are simplified to a copy, not encoded as an illegal repeated-control gate. Compute the output, copy it to $y$, and reverse the first computation. This uses at most $6C(f)+1$ gates and restores every work register.","\\[\n\\frac{C(f)}5\\le L(f)\\le6C(f)+1.\n\\]\n\\end{theorem}\n\\begin{proof}\nCompute every Boolean gate onto a fresh zero wire. An AND costs one Toffoli. A NOT of an existing signal is obtained with one NOT and one CNOT. An OR is computed as $a+b+ab$, using two CNOTs and one Toffoli. Repeated operands are simplified to a copy, not encoded as an illegal repeated-control gate. Compute the output, copy it to $y$, and reverse the first computation. This uses at most $6C(f)+1$ gates and restores every work register.","n{proof}\nCompute every Boolean gate onto a fresh zero wire. An AND costs one Toffoli. A NOT of an existing signal is obtained with one NOT and one CNOT. An OR is computed as $a+b+ab$, using two CNOTs and one Toffoli. Repeated operands are simplified to a copy, not encoded as an illegal repeated-control gate. Compute the output, copy it to $y$, and reverse the first computation. This uses at most $6C(f)+1$ gates and restores every work register.","Conversely, express each reversible gate as an ordinary Boolean update. XOR has a four-gate AND/OR/NOT realization. A Toffoli needs that XOR plus one AND, hence at most five gates. Substitute zero for initial ancillas and $y=0$, then read the final target. The resulting Boolean circuit has at most $5L(f)$ gates.\n\\end{proof}","\\section{The degree bound used throughout}\n\\begin{lemma}[Degree per multiplication]\\label{c:lem:degree}\nAn XOR--AND computation with $t$ AND gates has output degree at most $t+1$, allowing constants and arbitrary sharing. The same bound holds for each coordinate of a reversible computation containing $t$ Toffoli gates.\n\\end{lemma}\n\\begin{proof}\nIntroduce a formal Boolean variable for each AND output. Its defining expression is a product of two affine forms in the logical inputs and earlier gate variables. The final output is affine in the inputs and all gate variables.","Substitute the gate variables in reverse topological order. A multilinear monomial contains the variable being replaced at most once. Replacing it by a quadratic expression raises degree by at most one; multilinear reduction cannot raise it. After $t$ substitutions the degree is at most $t+1$.","\\begin{theorem}[Register-product degree invariant]\\label{c:thm:register-products}\nLet $s_i(x)$ be the contents of a finite register after $t$ Toffoli gates, with arbitrary intervening NOT and CNOT gates. Initial register contents are affine functions of $x$. For every finite list $L$ of $r$ register indices, including repeated indices,\n\\[\n \\deg\\left(\\prod_{i\\in L}s_i(x)\\right)\\le t+r.\n\\]\nThe zero-function convention is as above. In the formal source the conclusion is the concrete representation certificate \\texttt{DegreeLE}.\n\\end{theorem}\n\\begin{proof}\nInitially a product of $r$ affine functions has degree at most $r$. Suppose a gate toggles register $u$ by a product of $q$ controls, with $q=0,1,2$ for NOT, CNOT, and Toffoli respectively. If $u$ does not occur in $L$, the product is","\\begin{theorem}[Register-product degree invariant]\\label{c:thm:register-products}\nLet $s_i(x)$ be the contents of a finite register after $t$ Toffoli gates, with arbitrary intervening NOT and CNOT gates. Initial register contents are affine functions of $x$. For every finite list $L$ of $r$ register indices, including repeated indices,\n\\[\n \\deg\\left(\\prod_{i\\in L}s_i(x)\\right)\\le t+r.\n\\]\nThe zero-function convention is as above. In the formal source the conclusion is the concrete representation certificate \\texttt{DegreeLE}.\n\\end{theorem}\n\\begin{proof}\nInitially a product of $r$ affine functions has degree at most $r$. Suppose a gate toggles register $u$ by a product of $q$ controls, with $q=0,1,2$ for NOT, CNOT, and Toffoli respectively. If $u$ does not occur in $L$, the product is unchanged. If it occurs, Boolean idempotence r","\\begin{theorem}[Register-product degree invariant]\\label{c:thm:register-products}\nLet $s_i(x)$ be the contents of a finite register after $t$ Toffoli gates, with arbitrary intervening NOT and CNOT gates. Initial register contents are affine functions of $x$. For every finite list $L$ of $r$ register indices, including repeated indices,\n\\[\n \\deg\\left(\\prod_{i\\in L}s_i(x)\\right)\\le t+r.\n\\]\nThe zero-function convention is as above. In the formal source the conclusion is the concrete representation certificate \\texttt{DegreeLE}.\n\\end{theorem}\n\\begin{proof}\nInitially a product of $r$ affine functions has degree at most $r$. Suppose a gate toggles register $u$ by a product of $q$ controls, with $q=0,1,2$ for NOT, CNOT, and Toffoli respectively. If $u$ does not occur in $L$, the product is unchanged. If it occurs, Boolean idempotence removes all its repetitions. Let $R$ be the remaining list, so $|R|+1\\le r$. The updated product is\n\\[\n s_u\\prod_{i\\i","\\begin{theorem}[Register-product degree invariant]\\label{c:thm:register-products}\nLet $s_i(x)$ be the contents of a finite register after $t$ Toffoli gates, with arbitrary intervening NOT and CNOT gates. Initial register contents are affine functions of $x$. For every finite list $L$ of $r$ register indices, including repeated indices,\n\\[\n \\deg\\left(\\prod_{i\\in L}s_i(x)\\right)\\le t+r.\n\\]\nThe zero-function convention is as above. In the formal source the conclusion is the concrete representation certificate \\texttt{DegreeLE}.\n\\end{theorem}\n\\begin{proof}\nInitially a product of $r$ affine functions has degree at most $r$. Suppose a gate toggles register $u$ by a product of $q$ controls, with $q=0,1,2$ for NOT, CNOT, and Toffoli respectively. If $u$ does not occur in $L$, the product is unchanged. If it occurs, Boolean idempotence removes all its repetitions. Let $R$ be the remaining list, so $|R|+1\\le r$. The updated product is\n\\[\n s_u\\prod_{i\\in R}s_i+\n \\left(\\prod_{j\\in","\\begin{theorem}[Register-product degree invariant]\\label{c:thm:register-products}\nLet $s_i(x)$ be the contents of a finite register after $t$ Toffoli gates, with arbitrary intervening NOT and CNOT gates. Initial register contents are affine functions of $x$. For every finite list $L$ of $r$ register indices, including repeated indices,\n\\[\n \\deg\\left(\\prod_{i\\in L}s_i(x)\\right)\\le t+r.\n\\]\nThe zero-function convention is as above. In the formal source the conclusion is the concrete representation certificate \\texttt{DegreeLE}.\n\\end{theorem}\n\\begin{proof}\nInitially a product of $r$ affine functions has degree at most $r$. Suppose a gate toggles register $u$ by a product of $q$ controls, with $q=0,1,2$ for NOT, CNOT, and Toffoli respectively. If $u$ does not occur in $L$, the product is unchanged. If it occurs, Boolean idempotence removes all its repetitions. Let $R$ be the remaining list, so $|R|+1\\le r$. The updated product is\n\\[\n s_u\\prod_{i\\in R}s_i+\n \\left(\\prod_{j\\in\\mathrm","\\begin{theorem}[Register-product degree invariant]\\label{c:thm:register-products}\nLet $s_i(x)$ be the contents of a finite register after $t$ Toffoli gates, with arbitrary intervening NOT and CNOT gates. Initial register contents are affine functions of $x$. For every finite list $L$ of $r$ register indices, including repeated indices,\n\\[\n \\deg\\left(\\prod_{i\\in L}s_i(x)\\right)\\le t+r.\n\\]\nThe zero-function convention is as above. In the formal source the conclusion is the concrete representation certificate \\texttt{DegreeLE}.\n\\end{theorem}\n\\begin{proof}\nInitially a product of $r$ affine functions has degree at most $r$. Suppose a gate toggles register $u$ by a product of $q$ controls, with $q=0,1,2$ for NOT, CNOT, and Toffoli respectively. If $u$ does not occur in $L$, the product is unchanged. If it occurs, Boolean idempotence removes all its repetitions. Let $R$ be the remaining list, so $|R|+1\\le r$. The updated product is\n\\[\n s_u\\prod_{i\\in R}s_i+\n \\left(\\prod_{j\\in\\mathrm{controls}}s_j\\right)\\prod_{i\\in R}s_i.\n\\]\nBy the old invariant, these two summands have degrees a","NOT and CNOT gates. Initial register contents are affine functions of $x$. For every finite list $L$ of $r$ register indices, including repeated indices,\n\\[\n \\deg\\left(\\prod_{i\\in L}s_i(x)\\right)\\le t+r.\n\\]\nThe zero-function convention is as above. In the formal source the conclusion is the concrete representation certificate \\texttt{DegreeLE}.\n\\end{theorem}\n\\begin{proof}\nInitially a product of $r$ affine functions has degree at most $r$. Suppose a gate toggles register $u$ by a product of $q$ controls, with $q=0,1,2$ for NOT, CNOT, and Toffoli respectively. If $u$ does not occur in $L$, the product is unchanged. If it occurs, Boolean idempotence removes all its repetitions. Let $R$ be the remaining list, so $|R|+1\\le r$. The updated product is\n\\[\n s_u\\prod_{i\\in R}s_i+\n \\left(\\prod_{j\\in\\mathrm{controls}}s_j\\right)\\prod_{i\\in R}s_i.\n\\]\nBy the old invariant, these two summands have degrees at most $t+1+|R|$ and $t+q+|R|$. The latter is at most $t+r+\\max\\{q-1,0\\}$. NOT and CNOT therefore consume no degree budget; a Toffoli consumes at most one. Induction along the actual gate list pro","ntents are affine functions of $x$. For every finite list $L$ of $r$ register indices, including repeated indices,\n\\[\n \\deg\\left(\\prod_{i\\in L}s_i(x)\\right)\\le t+r.\n\\]\nThe zero-function convention is as above. In the formal source the conclusion is the concrete representation certificate \\texttt{DegreeLE}.\n\\end{theorem}\n\\begin{proof}\nInitially a product of $r$ affine functions has degree at most $r$. Suppose a gate toggles register $u$ by a product of $q$ controls, with $q=0,1,2$ for NOT, CNOT, and Toffoli respectively. If $u$ does not occur in $L$, the product is unchanged. If it occurs, Boolean idempotence removes all its repetitions. Let $R$ be the remaining list, so $|R|+1\\le r$. The updated product is\n\\[\n s_u\\prod_{i\\in R}s_i+\n \\left(\\prod_{j\\in\\mathrm{controls}}s_j\\right)\\prod_{i\\in R}s_i.\n\\]\nBy the old invariant, these two summands have degrees at most $t+1+|R|$ and $t+q+|R|$. The latter is at most $t+r+\\max\\{q-1,0\\}$. NOT and CNOT therefore consume no degree budget; a Toffoli consumes at most one. Induction along the actual gate list proves the claim. The empty list is the co…","or every finite list $L$ of $r$ register indices, including repeated indices,\n\\[\n \\deg\\left(\\prod_{i\\in L}s_i(x)\\right)\\le t+r.\n\\]\nThe zero-function convention is as above. In the formal source the conclusion is the concrete representation certificate \\texttt{DegreeLE}.\n\\end{theorem}\n\\begin{proof}\nInitially a product of $r$ affine functions has degree at most $r$. Suppose a gate toggles register $u$ by a product of $q$ controls, with $q=0,1,2$ for NOT, CNOT, and Toffoli respectively. If $u$ does not occur in $L$, the product is unchanged. If it occurs, Boolean idempotence removes all its repetitions. Let $R$ be the remaining list, so $|R|+1\\le r$. The updated product is\n\\[\n s_u\\prod_{i\\in R}s_i+\n \\left(\\prod_{j\\in\\mathrm{controls}}s_j\\right)\\prod_{i\\in R}s_i.\n\\]\nBy the old invariant, these two summands have degrees at most $t+1+|R|$ and $t+q+|R|$. The latter is at most $t+r+\\max\\{q-1,0\\}$. NOT and CNOT therefore consume no degree budget; a Toffoli consumes at most one. Induction along the actual gate list proves the claim. The empty list is the constant-one product and is also covere…","$L$ of $r$ register indices, including repeated indices,\n\\[\n \\deg\\left(\\prod_{i\\in L}s_i(x)\\right)\\le t+r.\n\\]\nThe zero-function convention is as above. In the formal source the conclusion is the concrete representation certificate \\texttt{DegreeLE}.\n\\end{theorem}\n\\begin{proof}\nInitially a product of $r$ affine functions has degree at most $r$. Suppose a gate toggles register $u$ by a product of $q$ controls, with $q=0,1,2$ for NOT, CNOT, and Toffoli respectively. If $u$ does not occur in $L$, the product is unchanged. If it occurs, Boolean idempotence removes all its repetitions. Let $R$ be the remaining list, so $|R|+1\\le r$. The updated product is\n\\[\n s_u\\prod_{i\\in R}s_i+\n \\left(\\prod_{j\\in\\mathrm{controls}}s_j\\right)\\prod_{i\\in R}s_i.\n\\]\nBy the old invariant, these two summands have degrees at most $t+1+|R|$ and $t+q+|R|$. The latter is at most $t+r+\\max\\{q-1,0\\}$. NOT and CNOT therefore consume no degree budget; a Toffoli consumes at most one. Induction along the actual gate list proves the claim. The empty list is the constant-one product and is also covered.\n\\end{proof}\nFor $r","r indices, including repeated indices,\n\\[\n \\deg\\left(\\prod_{i\\in L}s_i(x)\\right)\\le t+r.\n\\]\nThe zero-function convention is as above. In the formal source the conclusion is the concrete representation certificate \\texttt{DegreeLE}.\n\\end{theorem}\n\\begin{proof}\nInitially a product of $r$ affine functions has degree at most $r$. Suppose a gate toggles register $u$ by a product of $q$ controls, with $q=0,1,2$ for NOT, CNOT, and Toffoli respectively. If $u$ does not occur in $L$, the product is unchanged. If it occurs, Boolean idempotence removes all its repetitions. Let $R$ be the remaining list, so $|R|+1\\le r$. The updated product is\n\\[\n s_u\\prod_{i\\in R}s_i+\n \\left(\\prod_{j\\in\\mathrm{controls}}s_j\\right)\\prod_{i\\in R}s_i.\n\\]\nBy the old invariant, these two summands have degrees at most $t+1+|R|$ and $t+q+|R|$. The latter is at most $t+r+\\max\\{q-1,0\\}$. NOT and CNOT therefore consume no degree budget; a Toffoli consumes at most one. Induction along the actual gate list proves the claim. The empty list is the constant-one product and is also covered.\n\\end{proof}\nFor $r=1$ this recovers …","prod_{i\\in L}s_i(x)\\right)\\le t+r.\n\\]\nThe zero-function convention is as above. In the formal source the conclusion is the concrete representation certificate \\texttt{DegreeLE}.\n\\end{theorem}\n\\begin{proof}\nInitially a product of $r$ affine functions has degree at most $r$. Suppose a gate toggles register $u$ by a product of $q$ controls, with $q=0,1,2$ for NOT, CNOT, and Toffoli respectively. If $u$ does not occur in $L$, the product is unchanged. If it occurs, Boolean idempotence removes all its repetitions. Let $R$ be the remaining list, so $|R|+1\\le r$. The updated product is\n\\[\n s_u\\prod_{i\\in R}s_i+\n \\left(\\prod_{j\\in\\mathrm{controls}}s_j\\right)\\prod_{i\\in R}s_i.\n\\]\nBy the old invariant, these two summands have degrees at most $t+1+|R|$ and $t+q+|R|$. The latter is at most $t+r+\\max\\{q-1,0\\}$. NOT and CNOT therefore consume no degree budget; a Toffoli consumes at most one. Induction along the actual gate list proves the claim. The empty list is the constant-one product and is also covered.\n\\end{proof}\nFor $r=1$ this recovers the coordinate bound. The initial target can be fixed …","ght)\\le t+r.\n\\]\nThe zero-function convention is as above. In the formal source the conclusion is the concrete representation certificate \\texttt{DegreeLE}.\n\\end{theorem}\n\\begin{proof}\nInitially a product of $r$ affine functions has degree at most $r$. Suppose a gate toggles register $u$ by a product of $q$ controls, with $q=0,1,2$ for NOT, CNOT, and Toffoli respectively. If $u$ does not occur in $L$, the product is unchanged. If it occurs, Boolean idempotence removes all its repetitions. Let $R$ be the remaining list, so $|R|+1\\le r$. The updated product is\n\\[\n s_u\\prod_{i\\in R}s_i+\n \\left(\\prod_{j\\in\\mathrm{controls}}s_j\\right)\\prod_{i\\in R}s_i.\n\\]\nBy the old invariant, these two summands have degrees at most $t+1+|R|$ and $t+q+|R|$. The latter is at most $t+r+\\max\\{q-1,0\\}$. NOT and CNOT therefore consume no degree budget; a Toffoli consumes at most one. Induction along the actual gate list proves the claim. The empty list is the constant-one product and is also covered.\n\\end{proof}\nFor $r=1$ this recovers the coordinate bound. The initial target can be fixed to either boundary bit","he concrete representation certificate \\texttt{DegreeLE}.\n\\end{theorem}\n\\begin{proof}\nInitially a product of $r$ affine functions has degree at most $r$. Suppose a gate toggles register $u$ by a product of $q$ controls, with $q=0,1,2$ for NOT, CNOT, and Toffoli respectively. If $u$ does not occur in $L$, the product is unchanged. If it occurs, Boolean idempotence removes all its repetitions. Let $R$ be the remaining list, so $|R|+1\\le r$. The updated product is\n\\[\n s_u\\prod_{i\\in R}s_i+\n \\left(\\prod_{j\\in\\mathrm{controls}}s_j\\right)\\prod_{i\\in R}s_i.\n\\]\nBy the old invariant, these two summands have degrees at most $t+1+|R|$ and $t+q+|R|$. The latter is at most $t+r+\\max\\{q-1,0\\}$. NOT and CNOT therefore consume no degree budget; a Toffoli consumes at most one. Induction along the actual gate list proves the claim. The empty list is the constant-one product and is also covered.\n\\end{proof}\nFor $r=1$ this recovers the coordinate bound. The initial target can be fixed to either boundary bit while the inputs vary. Temporary input disturbance, dirty intermediate workspace, output controls…","tificate \\texttt{DegreeLE}.\n\\end{theorem}\n\\begin{proof}\nInitially a product of $r$ affine functions has degree at most $r$. Suppose a gate toggles register $u$ by a product of $q$ controls, with $q=0,1,2$ for NOT, CNOT, and Toffoli respectively. If $u$ does not occur in $L$, the product is unchanged. If it occurs, Boolean idempotence removes all its repetitions. Let $R$ be the remaining list, so $|R|+1\\le r$. The updated product is\n\\[\n s_u\\prod_{i\\in R}s_i+\n \\left(\\prod_{j\\in\\mathrm{controls}}s_j\\right)\\prod_{i\\in R}s_i.\n\\]\nBy the old invariant, these two summands have degrees at most $t+1+|R|$ and $t+q+|R|$. The latter is at most $t+r+\\max\\{q-1,0\\}$. NOT and CNOT therefore consume no degree budget; a Toffoli consumes at most one. Induction along the actual gate list proves the claim. The empty list is the constant-one product and is also covered.\n\\end{proof}\nFor $r=1$ this recovers the coordinate bound. The initial target can be fixed to either boundary bit while the inputs vary. Temporary input disturbance, dirty intermediate workspace, output controls, and repeated use of earlier …","em}\n\\begin{proof}\nInitially a product of $r$ affine functions has degree at most $r$. Suppose a gate toggles register $u$ by a product of $q$ controls, with $q=0,1,2$ for NOT, CNOT, and Toffoli respectively. If $u$ does not occur in $L$, the product is unchanged. If it occurs, Boolean idempotence removes all its repetitions. Let $R$ be the remaining list, so $|R|+1\\le r$. The updated product is\n\\[\n s_u\\prod_{i\\in R}s_i+\n \\left(\\prod_{j\\in\\mathrm{controls}}s_j\\right)\\prod_{i\\in R}s_i.\n\\]\nBy the old invariant, these two summands have degrees at most $t+1+|R|$ and $t+q+|R|$. The latter is at most $t+r+\\max\\{q-1,0\\}$. NOT and CNOT therefore consume no degree budget; a Toffoli consumes at most one. Induction along the actual gate list proves the claim. The empty list is the constant-one product and is also covered.\n\\end{proof}\nFor $r=1$ this recovers the coordinate bound. The initial target can be fixed to either boundary bit while the inputs vary. Temporary input disturbance, dirty intermediate workspace, output controls, and repeated use of earlier products are all allowed. The formal l…","h $q=0,1,2$ for NOT, CNOT, and Toffoli respectively. If $u$ does not occur in $L$, the product is unchanged. If it occurs, Boolean idempotence removes all its repetitions. Let $R$ be the remaining list, so $|R|+1\\le r$. The updated product is\n\\[\n s_u\\prod_{i\\in R}s_i+\n \\left(\\prod_{j\\in\\mathrm{controls}}s_j\\right)\\prod_{i\\in R}s_i.\n\\]\nBy the old invariant, these two summands have degrees at most $t+1+|R|$ and $t+q+|R|$. The latter is at most $t+r+\\max\\{q-1,0\\}$. NOT and CNOT therefore consume no degree budget; a Toffoli consumes at most one. Induction along the actual gate list proves the claim. The empty list is the constant-one product and is also covered.\n\\end{proof}\nFor $r=1$ this recovers the coordinate bound. The initial target can be fixed to either boundary bit while the inputs vary. Temporary input disturbance, dirty intermediate workspace, output controls, and repeated use of earlier products are all allowed. The formal locators are \\texttt{workspaceDegree\\_run} and \\texttt{canonical\\_run\\_coordinate\\_degree}.","r NOT, CNOT, and Toffoli respectively. If $u$ does not occur in $L$, the product is unchanged. If it occurs, Boolean idempotence removes all its repetitions. Let $R$ be the remaining list, so $|R|+1\\le r$. The updated product is\n\\[\n s_u\\prod_{i\\in R}s_i+\n \\left(\\prod_{j\\in\\mathrm{controls}}s_j\\right)\\prod_{i\\in R}s_i.\n\\]\nBy the old invariant, these two summands have degrees at most $t+1+|R|$ and $t+q+|R|$. The latter is at most $t+r+\\max\\{q-1,0\\}$. NOT and CNOT therefore consume no degree budget; a Toffoli consumes at most one. Induction along the actual gate list proves the claim. The empty list is the constant-one product and is also covered.\n\\end{proof}\nFor $r=1$ this recovers the coordinate bound. The initial target can be fixed to either boundary bit while the inputs vary. Temporary input disturbance, dirty intermediate workspace, output controls, and repeated use of earlier products are all allowed. The formal locators are \\texttt{workspaceDegree\\_run} and \\texttt{canonical\\_run\\_coordinate\\_degree}.","vely. If $u$ does not occur in $L$, the product is unchanged. If it occurs, Boolean idempotence removes all its repetitions. Let $R$ be the remaining list, so $|R|+1\\le r$. The updated product is\n\\[\n s_u\\prod_{i\\in R}s_i+\n \\left(\\prod_{j\\in\\mathrm{controls}}s_j\\right)\\prod_{i\\in R}s_i.\n\\]\nBy the old invariant, these two summands have degrees at most $t+1+|R|$ and $t+q+|R|$. The latter is at most $t+r+\\max\\{q-1,0\\}$. NOT and CNOT therefore consume no degree budget; a Toffoli consumes at most one. Induction along the actual gate list proves the claim. The empty list is the constant-one product and is also covered.\n\\end{proof}\nFor $r=1$ this recovers the coordinate bound. The initial target can be fixed to either boundary bit while the inputs vary. Temporary input disturbance, dirty intermediate workspace, output controls, and repeated use of earlier products are all allowed. The formal locators are \\texttt{workspaceDegree\\_run} and \\texttt{canonical\\_run\\_coordinate\\_degree}.","\\in\\mathrm{controls}}s_j\\right)\\prod_{i\\in R}s_i.\n\\]\nBy the old invariant, these two summands have degrees at most $t+1+|R|$ and $t+q+|R|$. The latter is at most $t+r+\\max\\{q-1,0\\}$. NOT and CNOT therefore consume no degree budget; a Toffoli consumes at most one. Induction along the actual gate list proves the claim. The empty list is the constant-one product and is also covered.\n\\end{proof}\nFor $r=1$ this recovers the coordinate bound. The initial target can be fixed to either boundary bit while the inputs vary. Temporary input disturbance, dirty intermediate workspace, output controls, and repeated use of earlier products are all allowed. The formal locators are \\texttt{workspaceDegree\\_run} and \\texttt{canonical\\_run\\_coordinate\\_degree}.","\\begin{circuitscopebox}{The representation is part of the theorem}\n\\texttt{DegreeLE(d,f)} is tied by proved equivalences to an explicit finite XOR of monomials with length at most $d$. It is not an assumed degree oracle. Canonical algebraic-normal-form extraction and a separate generic XOR--AND simulation compiler are outside this formal block.\n\\end{circuitscopebox}","\\chapter{Exact cut comparison and safe discrepancies}\n\\label{c:ch:cut}\\claim{C01}\n\\begin{theorem}[Cut-comparison identity and complete coverage]\\label{c:thm:cut}\nLet a restored oracle be written $\\mathcal C=Q\\circ P$, and put $R=Q^{-1}$. Suppress zero ancillas in boundary arguments and define\n\\[\n A(x)=P(x,0),\\qquad B(x)=R(x,0),\\qquad\n D(x)=P(x,1)+P(x,0).\n\\]\nThese are vectors indexed by all registers. Then\n\\begin{equation}\\label{c:eq:cut}\n A(x)+B(x)=f(x)D(x),\\qquad D(x)\\ne0.\n\\end{equation}\nConsequently, the coordinate discrepancies $H_i=A_i+B_i$ satisfy\n\\begin{equation}\\label{c:eq:cover}\n H_i\\le f\\quad\\hbox{for every }i,\\qquad\n \\bigvee_iH_i=f.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor a boundary state with target $z$, correctness of the full circuit gives\n\\[\n R(x,z)=P(x,z+f(x)).\n\\]\nWhen $f(x)=0$, this says $B(x)=A(x)$. When $f(x)=1$, it says $B(x)=P(x,1)$, and the discrepancy is $D(x)$. Both cases","\\chapter{Exact cut comparison and safe discrepancies}\n\\label{c:ch:cut}\\claim{C01}\n\\begin{theorem}[Cut-comparison identity and complete coverage]\\label{c:thm:cut}\nLet a restored oracle be written $\\mathcal C=Q\\circ P$, and put $R=Q^{-1}$. Suppress zero ancillas in boundary arguments and define\n\\[\n A(x)=P(x,0),\\qquad B(x)=R(x,0),\\qquad\n D(x)=P(x,1)+P(x,0).\n\\]\nThese are vectors indexed by all registers. Then\n\\begin{equation}\\label{c:eq:cut}\n A(x)+B(x)=f(x)D(x),\\qquad D(x)\\ne0.\n\\end{equation}\nConsequently, the coordinate discrepancies $H_i=A_i+B_i$ satisfy\n\\begin{equation}\\label{c:eq:cover}\n H_i\\le f\\quad\\hbox{for every }i,\\qquad\n \\bigvee_iH_i=f.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor a boundary state with target $z$, correctness of the full circuit gives\n\\[\n R(x,z)=P(x,z+f(x)).\n\\]\nWhen $f(x)=0$, this says $B(x)=A(x)$. When $f(x)=1$, it says $B(x)=P(x,1)$, and the discrepancy is $D(x)$. Both cases are summarized by \\e","\\chapter{Exact cut comparison and safe discrepancies}\n\\label{c:ch:cut}\\claim{C01}\n\\begin{theorem}[Cut-comparison identity and complete coverage]\\label{c:thm:cut}\nLet a restored oracle be written $\\mathcal C=Q\\circ P$, and put $R=Q^{-1}$. Suppress zero ancillas in boundary arguments and define\n\\[\n A(x)=P(x,0),\\qquad B(x)=R(x,0),\\qquad\n D(x)=P(x,1)+P(x,0).\n\\]\nThese are vectors indexed by all registers. Then\n\\begin{equation}\\label{c:eq:cut}\n A(x)+B(x)=f(x)D(x),\\qquad D(x)\\ne0.\n\\end{equation}\nConsequently, the coordinate discrepancies $H_i=A_i+B_i$ satisfy\n\\begin{equation}\\label{c:eq:cover}\n H_i\\le f\\quad\\hbox{for every }i,\\qquad\n \\bigvee_iH_i=f.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor a boundary state with target $z$, correctness of the full circuit gives\n\\[\n R(x,z)=P(x,z+f(x)).\n\\]\nWhen $f(x)=0$, this says $B(x)=A(x)$. When $f(x)=1$, it says $B(x)=P(x,1)$, and the discrepancy is $D(x)$. Both cases are summarized by \\eqref{c:eq:cut}.","ison and safe discrepancies}\n\\label{c:ch:cut}\\claim{C01}\n\\begin{theorem}[Cut-comparison identity and complete coverage]\\label{c:thm:cut}\nLet a restored oracle be written $\\mathcal C=Q\\circ P$, and put $R=Q^{-1}$. Suppress zero ancillas in boundary arguments and define\n\\[\n A(x)=P(x,0),\\qquad B(x)=R(x,0),\\qquad\n D(x)=P(x,1)+P(x,0).\n\\]\nThese are vectors indexed by all registers. Then\n\\begin{equation}\\label{c:eq:cut}\n A(x)+B(x)=f(x)D(x),\\qquad D(x)\\ne0.\n\\end{equation}\nConsequently, the coordinate discrepancies $H_i=A_i+B_i$ satisfy\n\\begin{equation}\\label{c:eq:cover}\n H_i\\le f\\quad\\hbox{for every }i,\\qquad\n \\bigvee_iH_i=f.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor a boundary state with target $z$, correctness of the full circuit gives\n\\[\n R(x,z)=P(x,z+f(x)).\n\\]\nWhen $f(x)=0$, this says $B(x)=A(x)$. When $f(x)=1$, it says $B(x)=P(x,1)$, and the discrepancy is $D(x)$. Both cases are summarized by \\eqref{c:eq:cut}.","erage]\\label{c:thm:cut}\nLet a restored oracle be written $\\mathcal C=Q\\circ P$, and put $R=Q^{-1}$. Suppress zero ancillas in boundary arguments and define\n\\[\n A(x)=P(x,0),\\qquad B(x)=R(x,0),\\qquad\n D(x)=P(x,1)+P(x,0).\n\\]\nThese are vectors indexed by all registers. Then\n\\begin{equation}\\label{c:eq:cut}\n A(x)+B(x)=f(x)D(x),\\qquad D(x)\\ne0.\n\\end{equation}\nConsequently, the coordinate discrepancies $H_i=A_i+B_i$ satisfy\n\\begin{equation}\\label{c:eq:cover}\n H_i\\le f\\quad\\hbox{for every }i,\\qquad\n \\bigvee_iH_i=f.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor a boundary state with target $z$, correctness of the full circuit gives\n\\[\n R(x,z)=P(x,z+f(x)).\n\\]\nWhen $f(x)=0$, this says $B(x)=A(x)$. When $f(x)=1$, it says $B(x)=P(x,1)$, and the discrepancy is $D(x)$. Both cases are summarized by \\eqref{c:eq:cut}.","d oracle be written $\\mathcal C=Q\\circ P$, and put $R=Q^{-1}$. Suppress zero ancillas in boundary arguments and define\n\\[\n A(x)=P(x,0),\\qquad B(x)=R(x,0),\\qquad\n D(x)=P(x,1)+P(x,0).\n\\]\nThese are vectors indexed by all registers. Then\n\\begin{equation}\\label{c:eq:cut}\n A(x)+B(x)=f(x)D(x),\\qquad D(x)\\ne0.\n\\end{equation}\nConsequently, the coordinate discrepancies $H_i=A_i+B_i$ satisfy\n\\begin{equation}\\label{c:eq:cover}\n H_i\\le f\\quad\\hbox{for every }i,\\qquad\n \\bigvee_iH_i=f.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor a boundary state with target $z$, correctness of the full circuit gives\n\\[\n R(x,z)=P(x,z+f(x)).\n\\]\nWhen $f(x)=0$, this says $B(x)=A(x)$. When $f(x)=1$, it says $B(x)=P(x,1)$, and the discrepancy is $D(x)$. Both cases are summarized by \\eqref{c:eq:cut}.",",\\qquad\n D(x)=P(x,1)+P(x,0).\n\\]\nThese are vectors indexed by all registers. Then\n\\begin{equation}\\label{c:eq:cut}\n A(x)+B(x)=f(x)D(x),\\qquad D(x)\\ne0.\n\\end{equation}\nConsequently, the coordinate discrepancies $H_i=A_i+B_i$ satisfy\n\\begin{equation}\\label{c:eq:cover}\n H_i\\le f\\quad\\hbox{for every }i,\\qquad\n \\bigvee_iH_i=f.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor a boundary state with target $z$, correctness of the full circuit gives\n\\[\n R(x,z)=P(x,z+f(x)).\n\\]\nWhen $f(x)=0$, this says $B(x)=A(x)$. When $f(x)=1$, it says $B(x)=P(x,1)$, and the discrepancy is $D(x)$. Both cases are summarized by \\eqref{c:eq:cut}.","ctors indexed by all registers. Then\n\\begin{equation}\\label{c:eq:cut}\n A(x)+B(x)=f(x)D(x),\\qquad D(x)\\ne0.\n\\end{equation}\nConsequently, the coordinate discrepancies $H_i=A_i+B_i$ satisfy\n\\begin{equation}\\label{c:eq:cover}\n H_i\\le f\\quad\\hbox{for every }i,\\qquad\n \\bigvee_iH_i=f.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor a boundary state with target $z$, correctness of the full circuit gives\n\\[\n R(x,z)=P(x,z+f(x)).\n\\]\nWhen $f(x)=0$, this says $B(x)=A(x)$. When $f(x)=1$, it says $B(x)=P(x,1)$, and the discrepancy is $D(x)$. Both cases are summarized by \\eqref{c:eq:cut}.","Then\n\\begin{equation}\\label{c:eq:cut}\n A(x)+B(x)=f(x)D(x),\\qquad D(x)\\ne0.\n\\end{equation}\nConsequently, the coordinate discrepancies $H_i=A_i+B_i$ satisfy\n\\begin{equation}\\label{c:eq:cover}\n H_i\\le f\\quad\\hbox{for every }i,\\qquad\n \\bigvee_iH_i=f.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor a boundary state with target $z$, correctness of the full circuit gives\n\\[\n R(x,z)=P(x,z+f(x)).\n\\]\nWhen $f(x)=0$, this says $B(x)=A(x)$. When $f(x)=1$, it says $B(x)=P(x,1)$, and the discrepancy is $D(x)$. Both cases are summarized by \\eqref{c:eq:cut}.","n}\\label{c:eq:cut}\n A(x)+B(x)=f(x)D(x),\\qquad D(x)\\ne0.\n\\end{equation}\nConsequently, the coordinate discrepancies $H_i=A_i+B_i$ satisfy\n\\begin{equation}\\label{c:eq:cover}\n H_i\\le f\\quad\\hbox{for every }i,\\qquad\n \\bigvee_iH_i=f.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor a boundary state with target $z$, correctness of the full circuit gives\n\\[\n R(x,z)=P(x,z+f(x)).\n\\]\nWhen $f(x)=0$, this says $B(x)=A(x)$. When $f(x)=1$, it says $B(x)=P(x,1)$, and the discrepancy is $D(x)$. Both cases are summarized by \\eqref{c:eq:cut}.","A(x)+B(x)=f(x)D(x),\\qquad D(x)\\ne0.\n\\end{equation}\nConsequently, the coordinate discrepancies $H_i=A_i+B_i$ satisfy\n\\begin{equation}\\label{c:eq:cover}\n H_i\\le f\\quad\\hbox{for every }i,\\qquad\n \\bigvee_iH_i=f.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor a boundary state with target $z$, correctness of the full circuit gives\n\\[\n R(x,z)=P(x,z+f(x)).\n\\]\nWhen $f(x)=0$, this says $B(x)=A(x)$. When $f(x)=1$, it says $B(x)=P(x,1)$, and the discrepancy is $D(x)$. Both cases are summarized by \\eqref{c:eq:cut}.","x),\\qquad D(x)\\ne0.\n\\end{equation}\nConsequently, the coordinate discrepancies $H_i=A_i+B_i$ satisfy\n\\begin{equation}\\label{c:eq:cover}\n H_i\\le f\\quad\\hbox{for every }i,\\qquad\n \\bigvee_iH_i=f.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor a boundary state with target $z$, correctness of the full circuit gives\n\\[\n R(x,z)=P(x,z+f(x)).\n\\]\nWhen $f(x)=0$, this says $B(x)=A(x)$. When $f(x)=1$, it says $B(x)=P(x,1)$, and the discrepancy is $D(x)$. Both cases are summarized by \\eqref{c:eq:cut}.","sequently, the coordinate discrepancies $H_i=A_i+B_i$ satisfy\n\\begin{equation}\\label{c:eq:cover}\n H_i\\le f\\quad\\hbox{for every }i,\\qquad\n \\bigvee_iH_i=f.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor a boundary state with target $z$, correctness of the full circuit gives\n\\[\n R(x,z)=P(x,z+f(x)).\n\\]\nWhen $f(x)=0$, this says $B(x)=A(x)$. When $f(x)=1$, it says $B(x)=P(x,1)$, and the discrepancy is $D(x)$. Both cases are summarized by \\eqref{c:eq:cut}.","The prefix $P$ is a permutation. Its images of the two distinct boundary states $(x,0)$ and $(x,1)$ are different, so $D(x)$ is a nonzero vector.","If $f(x)=0$, every discrepancy is zero. If $f(x)=1$, at least one discrepancy is one. These are precisely the safety and complete-coverage assertions in \\eqref{c:eq:cover}.\n\\end{proof}","\\section{What the comparison does and does not require}\nThe suffix need not reverse the prefix. Output feedback and a target direction distributed among many registers are permitted. Only boundary correctness and reversibility are used. Equivalently,\n\\[\n f(x)=1\\quad\\Longleftrightarrow\\quad P(x,0)\\ne Q^{-1}(x,0).\n\\]\nThis is a necessary representation, not a circuit lower bound from packet count. Expanded monomial counts, individual degrees, and algebra dimensions remain distinct from the shared cost of constructing the complete family. The declared finite cut checks are reported in Section~\\ref{c:app:evidence}.","\\chapter{Complete safe-packet degree}\n\\label{c:ch:degree}\\claim{C02}\n\\section{Pointwise coverage rather than one easy packet}\nLet\n\\[\n I_f=\\{h\\in\\mathcal R_n:h(1+f)=0\\}.\n\\]\nThis is the ideal of functions supported only on the accepting set of $f$. For nonzero $f$, the one-packet parameter is\n\\[\n \\alpha_1(f)=\\min\\{\\deg h:0\\ne h\\in I_f\\}.\n\\]\nIt asks for one safe nonzero polynomial, which may detect only an easy portion of the accepting set. One-sided polynomial immunity is established terminology in this setting; see the background reference \\cite{c:beckli}.","For a positive input $v$, define instead\n\\[\n \\beta_v(f)=\\min\\{\\deg h:h\\in I_f,\\ h(v)=1\\},\\qquad\n \\beta(f)=\\max_{v:f(v)=1}\\beta_v(f).\n\\]\nSet $\\beta(0)=0$. Thus $\\beta(f)$ is the degree needed before safe polynomials collectively detect \\emph{every} positive input. Finally, put\n\\[\n \\beta^*(f)=\\max_{\\ell\\in\\Aff_n}\\beta(f+\\ell),\n\\]\nwhere $\\Aff_n$ is the set of affine Boolean functions on the logical inputs.","\\begin{proposition}[Basic size bounds]\\label{c:prop:beta-bounds}\nFor nonzero $f$ on $n$ inputs,\n\\[\n \\alpha_1(f)\\le\\beta(f)\\le n.\n\\]\nIf $f$ has at least two positive inputs, then $\\beta(f)\\le n-1$. In particular $\\beta^*(f)\\le n$.\n\\end{proposition}\n\\begin{proof}\nAny polynomial detecting a specified positive input is nonzero, giving the first inequality. For $v\\in\\B^n$, its point indicator is\n\\[\n \\delta_v(x)=\\prod_{i:v_i=1}x_i\\prod_{i:v_i=0}(1+x_i).\n\\]\nIf $f(v)=1$, then $\\delta_v\\in I_f$ and has degree $n$, proving the upper bound.","unds}\nFor nonzero $f$ on $n$ inputs,\n\\[\n \\alpha_1(f)\\le\\beta(f)\\le n.\n\\]\nIf $f$ has at least two positive inputs, then $\\beta(f)\\le n-1$. In particular $\\beta^*(f)\\le n$.\n\\end{proposition}\n\\begin{proof}\nAny polynomial detecting a specified positive input is nonzero, giving the first inequality. For $v\\in\\B^n$, its point indicator is\n\\[\n \\delta_v(x)=\\prod_{i:v_i=1}x_i\\prod_{i:v_i=0}(1+x_i).\n\\]\nIf $f(v)=1$, then $\\delta_v\\in I_f$ and has degree $n$, proving the upper bound.","ro $f$ on $n$ inputs,\n\\[\n \\alpha_1(f)\\le\\beta(f)\\le n.\n\\]\nIf $f$ has at least two positive inputs, then $\\beta(f)\\le n-1$. In particular $\\beta^*(f)\\le n$.\n\\end{proposition}\n\\begin{proof}\nAny polynomial detecting a specified positive input is nonzero, giving the first inequality. For $v\\in\\B^n$, its point indicator is\n\\[\n \\delta_v(x)=\\prod_{i:v_i=1}x_i\\prod_{i:v_i=0}(1+x_i).\n\\]\nIf $f(v)=1$, then $\\delta_v\\in I_f$ and has degree $n$, proving the upper bound.","1(f)\\le\\beta(f)\\le n.\n\\]\nIf $f$ has at least two positive inputs, then $\\beta(f)\\le n-1$. In particular $\\beta^*(f)\\le n$.\n\\end{proposition}\n\\begin{proof}\nAny polynomial detecting a specified positive input is nonzero, giving the first inequality. For $v\\in\\B^n$, its point indicator is\n\\[\n \\delta_v(x)=\\prod_{i:v_i=1}x_i\\prod_{i:v_i=0}(1+x_i).\n\\]\nIf $f(v)=1$, then $\\delta_v\\in I_f$ and has degree $n$, proving the upper bound.","If $v$ and $w$ are distinct positive inputs, $\\delta_v+\\delta_w$ is safe and equals one at $v$. The degree-$n$ coefficients of its two summands are both one, so they cancel. Its degree is at most $n-1$. Do this for each positive $v$. The last assertion applies the general bound to every affine adjustment, with the stated zero convention.\n\\end{proof}","\\begin{proposition}[Affine input invariance]\\label{c:prop:affine-invariance}\nInvertible affine changes of logical input coordinates preserve $\\alpha_1$, $\\beta$, and $\\beta^*$, with the pointwise parameters transported by the corresponding bijection of inputs.\n\\end{proposition}\n\\begin{proof}\nComposition with an affine map cannot increase multilinear degree. Composition with its affine inverse proves equality of degrees. Such composition bijects the accepting and rejecting inputs, hence maps safe polynomials bijectively to safe polynomials and preserves their pointwise detecting property. It also permutes the affine output masks used in the definition of $\\beta^*$.\n\\end{proof}","nd{proposition}\n\\begin{proof}\nComposition with an affine map cannot increase multilinear degree. Composition with its affine inverse proves equality of degrees. Such composition bijects the accepting and rejecting inputs, hence maps safe polynomials bijectively to safe polynomials and preserves their pointwise detecting property. It also permutes the affine output masks used in the definition of $\\beta^*$.\n\\end{proof}","\\section{The restored lower bound}\n\\begin{theorem}[Affine-optimized complete-cover bound]\\label{c:thm:complete-bound}\nEvery restored implementation of $f$ with $T$ Toffoli gates satisfies\n\\begin{equation}\\label{c:eq:complete-bound}\n T\\ge\\max\\{0,\\ \\deg f-1,\\ 2\\beta^*(f)-3\\}.\n\\end{equation}\nIn particular,\n\\[\n T\\ge\\max\\{0,\\ \\deg f-1,\\ 2\\beta(f)-3,\\ 2\\beta(1+f)-3\\}.\n\\]\n\\end{theorem}\n\\begin{proof}\nCut the implementation after $\\lfloor T/2\\rfloor$ Toffolis. The forward prefix and the inverse suffix use at most $\\lceil T/2\\rceil$ Toffolis each. By Lemma~\\ref{c:lem:degree}, each of their coordinates, and therefore each discrepancy $H_i$, has degree at most\n\\[\n \\left\\lceil\\frac T2\\right\\rceil+1.\n\\]\nTheorem~\\ref{c:thm:cut} gives $H_i\\le f$ and a complete cover of the positive inputs. Every positive $v$ is therefore detected by a safe polynomial within","\\section{The restored lower bound}\n\\begin{theorem}[Affine-optimized complete-cover bound]\\label{c:thm:complete-bound}\nEvery restored implementation of $f$ with $T$ Toffoli gates satisfies\n\\begin{equation}\\label{c:eq:complete-bound}\n T\\ge\\max\\{0,\\ \\deg f-1,\\ 2\\beta^*(f)-3\\}.\n\\end{equation}\nIn particular,\n\\[\n T\\ge\\max\\{0,\\ \\deg f-1,\\ 2\\beta(f)-3,\\ 2\\beta(1+f)-3\\}.\n\\]\n\\end{theorem}\n\\begin{proof}\nCut the implementation after $\\lfloor T/2\\rfloor$ Toffolis. The forward prefix and the inverse suffix use at most $\\lceil T/2\\rceil$ Toffolis each. By Lemma~\\ref{c:lem:degree}, each of their coordinates, and therefore each discrepancy $H_i$, has degree at most\n\\[\n \\left\\lceil\\frac T2\\right\\rceil+1.\n\\]\nTheorem~\\ref{c:thm:cut} gives $H_i\\le f$ and a complete cover of the positive inputs. Every positive $v$ is therefore detected by a safe polynomial within this deg","\\section{The restored lower bound}\n\\begin{theorem}[Affine-optimized complete-cover bound]\\label{c:thm:complete-bound}\nEvery restored implementation of $f$ with $T$ Toffoli gates satisfies\n\\begin{equation}\\label{c:eq:complete-bound}\n T\\ge\\max\\{0,\\ \\deg f-1,\\ 2\\beta^*(f)-3\\}.\n\\end{equation}\nIn particular,\n\\[\n T\\ge\\max\\{0,\\ \\deg f-1,\\ 2\\beta(f)-3,\\ 2\\beta(1+f)-3\\}.\n\\]\n\\end{theorem}\n\\begin{proof}\nCut the implementation after $\\lfloor T/2\\rfloor$ Toffolis. The forward prefix and the inverse suffix use at most $\\lceil T/2\\rceil$ Toffolis each. By Lemma~\\ref{c:lem:degree}, each of their coordinates, and therefore each discrepancy $H_i$, has degree at most\n\\[\n \\left\\lceil\\frac T2\\right\\rceil+1.\n\\]\nTheorem~\\ref{c:thm:cut} gives $H_i\\le f$ and a complete cover of the positive inputs. Every positive $v$ is therefore detected by a safe polynomial within this degree bound, so\n\\[\n \\beta(f)\\le\\left\\lceil\\frac T2\\right\\rceil+1.\n\\]\nInteger rounding gives $T\\ge2\\beta(f)-3$.","r bound}\n\\begin{theorem}[Affine-optimized complete-cover bound]\\label{c:thm:complete-bound}\nEvery restored implementation of $f$ with $T$ Toffoli gates satisfies\n\\begin{equation}\\label{c:eq:complete-bound}\n T\\ge\\max\\{0,\\ \\deg f-1,\\ 2\\beta^*(f)-3\\}.\n\\end{equation}\nIn particular,\n\\[\n T\\ge\\max\\{0,\\ \\deg f-1,\\ 2\\beta(f)-3,\\ 2\\beta(1+f)-3\\}.\n\\]\n\\end{theorem}\n\\begin{proof}\nCut the implementation after $\\lfloor T/2\\rfloor$ Toffolis. The forward prefix and the inverse suffix use at most $\\lceil T/2\\rceil$ Toffolis each. By Lemma~\\ref{c:lem:degree}, each of their coordinates, and therefore each discrepancy $H_i$, has degree at most\n\\[\n \\left\\lceil\\frac T2\\right\\rceil+1.\n\\]\nTheorem~\\ref{c:thm:cut} gives $H_i\\le f$ and a complete cover of the positive inputs. Every positive $v$ is therefore detected by a safe polynomial within this degree bound, so\n\\[\n \\beta(f)\\le\\left\\lceil\\frac T2\\right\\rceil+1.\n\\]\nInteger rounding gives $T\\ge2\\beta(f)-3$.","bound}\nEvery restored implementation of $f$ with $T$ Toffoli gates satisfies\n\\begin{equation}\\label{c:eq:complete-bound}\n T\\ge\\max\\{0,\\ \\deg f-1,\\ 2\\beta^*(f)-3\\}.\n\\end{equation}\nIn particular,\n\\[\n T\\ge\\max\\{0,\\ \\deg f-1,\\ 2\\beta(f)-3,\\ 2\\beta(1+f)-3\\}.\n\\]\n\\end{theorem}\n\\begin{proof}\nCut the implementation after $\\lfloor T/2\\rfloor$ Toffolis. The forward prefix and the inverse suffix use at most $\\lceil T/2\\rceil$ Toffolis each. By Lemma~\\ref{c:lem:degree}, each of their coordinates, and therefore each discrepancy $H_i$, has degree at most\n\\[\n \\left\\lceil\\frac T2\\right\\rceil+1.\n\\]\nTheorem~\\ref{c:thm:cut} gives $H_i\\le f$ and a complete cover of the positive inputs. Every positive $v$ is therefore detected by a safe polynomial within this degree bound, so\n\\[\n \\beta(f)\\le\\left\\lceil\\frac T2\\right\\rceil+1.\n\\]\nInteger rounding gives $T\\ge2\\beta(f)-3$.","T\\ge\\max\\{0,\\ \\deg f-1,\\ 2\\beta^*(f)-3\\}.\n\\end{equation}\nIn particular,\n\\[\n T\\ge\\max\\{0,\\ \\deg f-1,\\ 2\\beta(f)-3,\\ 2\\beta(1+f)-3\\}.\n\\]\n\\end{theorem}\n\\begin{proof}\nCut the implementation after $\\lfloor T/2\\rfloor$ Toffolis. The forward prefix and the inverse suffix use at most $\\lceil T/2\\rceil$ Toffolis each. By Lemma~\\ref{c:lem:degree}, each of their coordinates, and therefore each discrepancy $H_i$, has degree at most\n\\[\n \\left\\lceil\\frac T2\\right\\rceil+1.\n\\]\nTheorem~\\ref{c:thm:cut} gives $H_i\\le f$ and a complete cover of the positive inputs. Every positive $v$ is therefore detected by a safe polynomial within this degree bound, so\n\\[\n \\beta(f)\\le\\left\\lceil\\frac T2\\right\\rceil+1.\n\\]\nInteger rounding gives $T\\ge2\\beta(f)-3$.","\\beta^*(f)-3\\}.\n\\end{equation}\nIn particular,\n\\[\n T\\ge\\max\\{0,\\ \\deg f-1,\\ 2\\beta(f)-3,\\ 2\\beta(1+f)-3\\}.\n\\]\n\\end{theorem}\n\\begin{proof}\nCut the implementation after $\\lfloor T/2\\rfloor$ Toffolis. The forward prefix and the inverse suffix use at most $\\lceil T/2\\rceil$ Toffolis each. By Lemma~\\ref{c:lem:degree}, each of their coordinates, and therefore each discrepancy $H_i$, has degree at most\n\\[\n \\left\\lceil\\frac T2\\right\\rceil+1.\n\\]\nTheorem~\\ref{c:thm:cut} gives $H_i\\le f$ and a complete cover of the positive inputs. Every positive $v$ is therefore detected by a safe polynomial within this degree bound, so\n\\[\n \\beta(f)\\le\\left\\lceil\\frac T2\\right\\rceil+1.\n\\]\nInteger rounding gives $T\\ge2\\beta(f)-3$.","1,\\ 2\\beta(f)-3,\\ 2\\beta(1+f)-3\\}.\n\\]\n\\end{theorem}\n\\begin{proof}\nCut the implementation after $\\lfloor T/2\\rfloor$ Toffolis. The forward prefix and the inverse suffix use at most $\\lceil T/2\\rceil$ Toffolis each. By Lemma~\\ref{c:lem:degree}, each of their coordinates, and therefore each discrepancy $H_i$, has degree at most\n\\[\n \\left\\lceil\\frac T2\\right\\rceil+1.\n\\]\nTheorem~\\ref{c:thm:cut} gives $H_i\\le f$ and a complete cover of the positive inputs. Every positive $v$ is therefore detected by a safe polynomial within this degree bound, so\n\\[\n \\beta(f)\\le\\left\\lceil\\frac T2\\right\\rceil+1.\n\\]\nInteger rounding gives $T\\ge2\\beta(f)-3$.","he implementation after $\\lfloor T/2\\rfloor$ Toffolis. The forward prefix and the inverse suffix use at most $\\lceil T/2\\rceil$ Toffolis each. By Lemma~\\ref{c:lem:degree}, each of their coordinates, and therefore each discrepancy $H_i$, has degree at most\n\\[\n \\left\\lceil\\frac T2\\right\\rceil+1.\n\\]\nTheorem~\\ref{c:thm:cut} gives $H_i\\le f$ and a complete cover of the positive inputs. Every positive $v$ is therefore detected by a safe polynomial within this degree bound, so\n\\[\n \\beta(f)\\le\\left\\lceil\\frac T2\\right\\rceil+1.\n\\]\nInteger rounding gives $T\\ge2\\beta(f)-3$.","the inverse suffix use at most $\\lceil T/2\\rceil$ Toffolis each. By Lemma~\\ref{c:lem:degree}, each of their coordinates, and therefore each discrepancy $H_i$, has degree at most\n\\[\n \\left\\lceil\\frac T2\\right\\rceil+1.\n\\]\nTheorem~\\ref{c:thm:cut} gives $H_i\\le f$ and a complete cover of the positive inputs. Every positive $v$ is therefore detected by a safe polynomial within this degree bound, so\n\\[\n \\beta(f)\\le\\left\\lceil\\frac T2\\right\\rceil+1.\n\\]\nInteger rounding gives $T\\ge2\\beta(f)-3$.","\\ref{c:lem:degree}, each of their coordinates, and therefore each discrepancy $H_i$, has degree at most\n\\[\n \\left\\lceil\\frac T2\\right\\rceil+1.\n\\]\nTheorem~\\ref{c:thm:cut} gives $H_i\\le f$ and a complete cover of the positive inputs. Every positive $v$ is therefore detected by a safe polynomial within this degree bound, so\n\\[\n \\beta(f)\\le\\left\\lceil\\frac T2\\right\\rceil+1.\n\\]\nInteger rounding gives $T\\ge2\\beta(f)-3$.","At the end of a restored oracle the logical inputs are available unchanged. CNOTs from those inputs and, when needed, a target NOT append any affine $\\ell$ to the target at no Toffoli cost. Apply the same inequality to each $f+\\ell$ and maximize. Finally, reading the target with initial value zero computes $f$ using $T$ multiplications, so Lemma~\\ref{c:lem:degree} gives $T\\ge\\deg f-1$.\n\\end{proof}","\\begin{circuitscopebox}{Scope of the lower bound}\nThe circuit may use output feedback and arbitrary dirty intermediate states, but it must satisfy the restored classical oracle convention. The inequality does not apply automatically to a model with discarded garbage or measurement. Because $\\beta^*(f)\\le n$, this parameter alone cannot prove a superpolynomial lower bound in the actual input length.\n\\end{circuitscopebox}","\\section{Exact evaluation-span certificates}\nFor an integer $d\\ge0$, form the evaluation vector\n\\[\n v_d(x)=\\bigl(x^S\\bigr)_{S\\subseteq[n],\\ |S|\\le d},\n\\]\nand write $U=f^{-1}(0)$.","\\begin{theorem}[Dual certificate for failure of safe detection]\\label{c:thm:dual}\nFor every positive input $v$,\n\\begin{equation}\\label{c:eq:dual}\n \\beta_v(f)>d\n \\quad\\Longleftrightarrow\\quad\n v_d(v)\\in\\Span_{\\F}\\{v_d(u):u\\in U\\}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nA degree-at-most-$d$ polynomial is a coefficient covector on the evaluation space. It is safe exactly when it annihilates every negative evaluation vector, equivalently their span. Such a covector can equal one on $v_d(v)$ exactly when $v_d(v)$ lies outside that span. This is finite-dimensional linear duality over $\\F$.\n\\end{proof}","ery positive input $v$,\n\\begin{equation}\\label{c:eq:dual}\n \\beta_v(f)>d\n \\quad\\Longleftrightarrow\\quad\n v_d(v)\\in\\Span_{\\F}\\{v_d(u):u\\in U\\}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nA degree-at-most-$d$ polynomial is a coefficient covector on the evaluation space. It is safe exactly when it annihilates every negative evaluation vector, equivalently their span. Such a covector can equal one on $v_d(v)$ exactly when $v_d(v)$ lies outside that span. This is finite-dimensional linear duality over $\\F$.\n\\end{proof}",",\n\\begin{equation}\\label{c:eq:dual}\n \\beta_v(f)>d\n \\quad\\Longleftrightarrow\\quad\n v_d(v)\\in\\Span_{\\F}\\{v_d(u):u\\in U\\}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nA degree-at-most-$d$ polynomial is a coefficient covector on the evaluation space. It is safe exactly when it annihilates every negative evaluation vector, equivalently their span. Such a covector can equal one on $v_d(v)$ exactly when $v_d(v)$ lies outside that span. This is finite-dimensional linear duality over $\\F$.\n\\end{proof}","v_d(v)\\in\\Span_{\\F}\\{v_d(u):u\\in U\\}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nA degree-at-most-$d$ polynomial is a coefficient covector on the evaluation space. It is safe exactly when it annihilates every negative evaluation vector, equivalently their span. Such a covector can equal one on $v_d(v)$ exactly when $v_d(v)$ lies outside that span. This is finite-dimensional linear duality over $\\F$.\n\\end{proof}","\\begin{corollary}[Finite support of a dual witness]\\label{c:cor:dual-witness}\nWhen the right side of \\eqref{c:eq:dual} holds, there is $W\\subseteq U$ with\n\\[\n v_d(v)=\\sum_{u\\in W}v_d(u),\\qquad\n |W|\\le\\sum_{j=0}^{\\min(d,n)}\\binom nj.\n\\]\nThe cardinality $|W|$ is odd.\n\\end{corollary}\n\\begin{proof}\nChoose a basis of the span from negative evaluation vectors and express $v_d(v)$ in that basis. Over $\\F$, the nonzero coefficients select a subset $W$ of size at most the ambient dimension. The constant-monomial coordinate of every evaluation vector is one; comparing it on both sides shows that $|W|$ is odd.\n\\end{proof}\nSuch a relation is replayable by exact linear arithmetic. It does not provide an efficient way to enumerate all negatives of an arbitrary SAT verifier. The theorem assumes that the negative vectors or their proved structure are a","\\begin{corollary}[Finite support of a dual witness]\\label{c:cor:dual-witness}\nWhen the right side of \\eqref{c:eq:dual} holds, there is $W\\subseteq U$ with\n\\[\n v_d(v)=\\sum_{u\\in W}v_d(u),\\qquad\n |W|\\le\\sum_{j=0}^{\\min(d,n)}\\binom nj.\n\\]\nThe cardinality $|W|$ is odd.\n\\end{corollary}\n\\begin{proof}\nChoose a basis of the span from negative evaluation vectors and express $v_d(v)$ in that basis. Over $\\F$, the nonzero coefficients select a subset $W$ of size at most the ambient dimension. The constant-monomial coordinate of every evaluation vector is one; comparing it on both sides shows that $|W|$ is odd.\n\\end{proof}\nSuch a relation is replayable by exact linear arithmetic. It does not provide an efficient way to enumerate all negatives of an arbitrary SAT verifier. The theorem assumes that the negative vectors or their proved structure are available.","tness]\\label{c:cor:dual-witness}\nWhen the right side of \\eqref{c:eq:dual} holds, there is $W\\subseteq U$ with\n\\[\n v_d(v)=\\sum_{u\\in W}v_d(u),\\qquad\n |W|\\le\\sum_{j=0}^{\\min(d,n)}\\binom nj.\n\\]\nThe cardinality $|W|$ is odd.\n\\end{corollary}\n\\begin{proof}\nChoose a basis of the span from negative evaluation vectors and express $v_d(v)$ in that basis. Over $\\F$, the nonzero coefficients select a subset $W$ of size at most the ambient dimension. The constant-monomial coordinate of every evaluation vector is one; comparing it on both sides shows that $|W|$ is odd.\n\\end{proof}\nSuch a relation is replayable by exact linear arithmetic. It does not provide an efficient way to enumerate all negatives of an arbitrary SAT verifier. The theorem assumes that the negative vectors or their proved structure are available.","|W|\\le\\sum_{j=0}^{\\min(d,n)}\\binom nj.\n\\]\nThe cardinality $|W|$ is odd.\n\\end{corollary}\n\\begin{proof}\nChoose a basis of the span from negative evaluation vectors and express $v_d(v)$ in that basis. Over $\\F$, the nonzero coefficients select a subset $W$ of size at most the ambient dimension. The constant-monomial coordinate of every evaluation vector is one; comparing it on both sides shows that $|W|$ is odd.\n\\end{proof}\nSuch a relation is replayable by exact linear arithmetic. It does not provide an efficient way to enumerate all negatives of an arbitrary SAT verifier. The theorem assumes that the negative vectors or their proved structure are available.","\\section{Monotone specialization}\n\\begin{theorem}[Safe monomial ideal for a monotone function]\\label{c:thm:monotone}\nIf $f$ is monotone, $I_f$ has a monomial basis indexed by the positive supports of $f$. For each positive $v$,\n\\[\n \\beta_v(f)=\\min\\{|S|: S\\subseteq\\supp(v),\\ f(1_S)=1\\}.\n\\]\nConsequently, for nonzero $f$,\n\\[\n \\alpha_1(f)=\\text{minimum minterm size},\\qquad\n \\beta(f)=\\text{maximum minterm size}.\n\\]\n\\end{theorem}\n\\begin{proof}\nFor any Boolean polynomial $h$, its coefficient on $x^S$ is\n\\[\n [x^S]h=\\sum_{T\\subseteq S}h(1_T)\n\\]\nover $\\F$. If $S$ is negative for monotone $f$, every subset $T$ is negative. A safe $h$ vanishes at each $1_T$, so its $x^S$ coefficient is zero. Conversely, a monomial on a positive support is safe by monotonicity. This proves the basis assertion.","for a monotone function]\\label{c:thm:monotone}\nIf $f$ is monotone, $I_f$ has a monomial basis indexed by the positive supports of $f$. For each positive $v$,\n\\[\n \\beta_v(f)=\\min\\{|S|: S\\subseteq\\supp(v),\\ f(1_S)=1\\}.\n\\]\nConsequently, for nonzero $f$,\n\\[\n \\alpha_1(f)=\\text{minimum minterm size},\\qquad\n \\beta(f)=\\text{maximum minterm size}.\n\\]\n\\end{theorem}\n\\begin{proof}\nFor any Boolean polynomial $h$, its coefficient on $x^S$ is\n\\[\n [x^S]h=\\sum_{T\\subseteq S}h(1_T)\n\\]\nover $\\F$. If $S$ is negative for monotone $f$, every subset $T$ is negative. A safe $h$ vanishes at each $1_T$, so its $x^S$ coefficient is zero. Conversely, a monomial on a positive support is safe by monotonicity. This proves the basis assertion.","\\label{c:thm:monotone}\nIf $f$ is monotone, $I_f$ has a monomial basis indexed by the positive supports of $f$. For each positive $v$,\n\\[\n \\beta_v(f)=\\min\\{|S|: S\\subseteq\\supp(v),\\ f(1_S)=1\\}.\n\\]\nConsequently, for nonzero $f$,\n\\[\n \\alpha_1(f)=\\text{minimum minterm size},\\qquad\n \\beta(f)=\\text{maximum minterm size}.\n\\]\n\\end{theorem}\n\\begin{proof}\nFor any Boolean polynomial $h$, its coefficient on $x^S$ is\n\\[\n [x^S]h=\\sum_{T\\subseteq S}h(1_T)\n\\]\nover $\\F$. If $S$ is negative for monotone $f$, every subset $T$ is negative. A safe $h$ vanishes at each $1_T$, so its $x^S$ coefficient is zero. Conversely, a monomial on a positive support is safe by monotonicity. This proves the basis assertion.","hm:monotone}\nIf $f$ is monotone, $I_f$ has a monomial basis indexed by the positive supports of $f$. For each positive $v$,\n\\[\n \\beta_v(f)=\\min\\{|S|: S\\subseteq\\supp(v),\\ f(1_S)=1\\}.\n\\]\nConsequently, for nonzero $f$,\n\\[\n \\alpha_1(f)=\\text{minimum minterm size},\\qquad\n \\beta(f)=\\text{maximum minterm size}.\n\\]\n\\end{theorem}\n\\begin{proof}\nFor any Boolean polynomial $h$, its coefficient on $x^S$ is\n\\[\n [x^S]h=\\sum_{T\\subseteq S}h(1_T)\n\\]\nover $\\F$. If $S$ is negative for monotone $f$, every subset $T$ is negative. A safe $h$ vanishes at each $1_T$, so its $x^S$ coefficient is zero. Conversely, a monomial on a positive support is safe by monotonicity. This proves the basis assertion.","monomial basis indexed by the positive supports of $f$. For each positive $v$,\n\\[\n \\beta_v(f)=\\min\\{|S|: S\\subseteq\\supp(v),\\ f(1_S)=1\\}.\n\\]\nConsequently, for nonzero $f$,\n\\[\n \\alpha_1(f)=\\text{minimum minterm size},\\qquad\n \\beta(f)=\\text{maximum minterm size}.\n\\]\n\\end{theorem}\n\\begin{proof}\nFor any Boolean polynomial $h$, its coefficient on $x^S$ is\n\\[\n [x^S]h=\\sum_{T\\subseteq S}h(1_T)\n\\]\nover $\\F$. If $S$ is negative for monotone $f$, every subset $T$ is negative. A safe $h$ vanishes at each $1_T$, so its $x^S$ coefficient is zero. Conversely, a monomial on a positive support is safe by monotonicity. This proves the basis assertion.","basis indexed by the positive supports of $f$. For each positive $v$,\n\\[\n \\beta_v(f)=\\min\\{|S|: S\\subseteq\\supp(v),\\ f(1_S)=1\\}.\n\\]\nConsequently, for nonzero $f$,\n\\[\n \\alpha_1(f)=\\text{minimum minterm size},\\qquad\n \\beta(f)=\\text{maximum minterm size}.\n\\]\n\\end{theorem}\n\\begin{proof}\nFor any Boolean polynomial $h$, its coefficient on $x^S$ is\n\\[\n [x^S]h=\\sum_{T\\subseteq S}h(1_T)\n\\]\nover $\\F$. If $S$ is negative for monotone $f$, every subset $T$ is negative. A safe $h$ vanishes at each $1_T$, so its $x^S$ coefficient is zero. Conversely, a monomial on a positive support is safe by monotonicity. This proves the basis assertion.","e supports of $f$. For each positive $v$,\n\\[\n \\beta_v(f)=\\min\\{|S|: S\\subseteq\\supp(v),\\ f(1_S)=1\\}.\n\\]\nConsequently, for nonzero $f$,\n\\[\n \\alpha_1(f)=\\text{minimum minterm size},\\qquad\n \\beta(f)=\\text{maximum minterm size}.\n\\]\n\\end{theorem}\n\\begin{proof}\nFor any Boolean polynomial $h$, its coefficient on $x^S$ is\n\\[\n [x^S]h=\\sum_{T\\subseteq S}h(1_T)\n\\]\nover $\\F$. If $S$ is negative for monotone $f$, every subset $T$ is negative. A safe $h$ vanishes at each $1_T$, so its $x^S$ coefficient is zero. Conversely, a monomial on a positive support is safe by monotonicity. This proves the basis assertion.","For each positive $v$,\n\\[\n \\beta_v(f)=\\min\\{|S|: S\\subseteq\\supp(v),\\ f(1_S)=1\\}.\n\\]\nConsequently, for nonzero $f$,\n\\[\n \\alpha_1(f)=\\text{minimum minterm size},\\qquad\n \\beta(f)=\\text{maximum minterm size}.\n\\]\n\\end{theorem}\n\\begin{proof}\nFor any Boolean polynomial $h$, its coefficient on $x^S$ is\n\\[\n [x^S]h=\\sum_{T\\subseteq S}h(1_T)\n\\]\nover $\\F$. If $S$ is negative for monotone $f$, every subset $T$ is negative. A safe $h$ vanishes at each $1_T$, so its $x^S$ coefficient is zero. Conversely, a monomial on a positive support is safe by monotonicity. This proves the basis assertion.","\\[\n \\beta_v(f)=\\min\\{|S|: S\\subseteq\\supp(v),\\ f(1_S)=1\\}.\n\\]\nConsequently, for nonzero $f$,\n\\[\n \\alpha_1(f)=\\text{minimum minterm size},\\qquad\n \\beta(f)=\\text{maximum minterm size}.\n\\]\n\\end{theorem}\n\\begin{proof}\nFor any Boolean polynomial $h$, its coefficient on $x^S$ is\n\\[\n [x^S]h=\\sum_{T\\subseteq S}h(1_T)\n\\]\nover $\\F$. If $S$ is negative for monotone $f$, every subset $T$ is negative. A safe $h$ vanishes at each $1_T$, so its $x^S$ coefficient is zero. Conversely, a monomial on a positive support is safe by monotonicity. This proves the basis assertion.","|S|: S\\subseteq\\supp(v),\\ f(1_S)=1\\}.\n\\]\nConsequently, for nonzero $f$,\n\\[\n \\alpha_1(f)=\\text{minimum minterm size},\\qquad\n \\beta(f)=\\text{maximum minterm size}.\n\\]\n\\end{theorem}\n\\begin{proof}\nFor any Boolean polynomial $h$, its coefficient on $x^S$ is\n\\[\n [x^S]h=\\sum_{T\\subseteq S}h(1_T)\n\\]\nover $\\F$. If $S$ is negative for monotone $f$, every subset $T$ is negative. A safe $h$ vanishes at each $1_T$, so its $x^S$ coefficient is zero. Conversely, a monomial on a positive support is safe by monotonicity. This proves the basis assertion.","q\\supp(v),\\ f(1_S)=1\\}.\n\\]\nConsequently, for nonzero $f$,\n\\[\n \\alpha_1(f)=\\text{minimum minterm size},\\qquad\n \\beta(f)=\\text{maximum minterm size}.\n\\]\n\\end{theorem}\n\\begin{proof}\nFor any Boolean polynomial $h$, its coefficient on $x^S$ is\n\\[\n [x^S]h=\\sum_{T\\subseteq S}h(1_T)\n\\]\nover $\\F$. If $S$ is negative for monotone $f$, every subset $T$ is negative. A safe $h$ vanishes at each $1_T$, so its $x^S$ coefficient is zero. Conversely, a monomial on a positive support is safe by monotonicity. This proves the basis assertion.","If a safe $h$ satisfies $h(v)=1$, at least one of its nonzero monomials is supported inside $\\supp(v)$. That support is positive and has size at most $\\deg h$. Conversely, a smallest such positive-support monomial detects $v$. This proves the pointwise formula. Taking the appropriate minima and maxima gives the minterm statements: a minimal positive support cannot contain a smaller positive one, and every positive input contains a minterm.\n\\end{proof}","\\begin{corollary}[Exact monomial oracle cost]\\label{c:cor:monomial}\nFor $k\\ge2$,\n\\[\n \\KTrest(x_1x_2\\cdots x_k)=2k-3.\n\\]\n\\end{corollary}\n\\begin{proof}\nThe unique positive input of the $k$-variable monomial needs a safe polynomial of degree $k$. The complete-cover lower bound gives $2k-3$. For the matching construction, successively compute $x_1x_2$, then $x_1x_2x_3$, and so on up to a $(k-1)$-fold product on clean work wires. Toggle the output using that product and $x_k$, then uncompute the $k-2$ work products. The count is $(k-2)+1+(k-2)=2k-3$. For $k=2$, a single Toffoli suffices.\n\\end{proof}","$k\\ge2$,\n\\[\n \\KTrest(x_1x_2\\cdots x_k)=2k-3.\n\\]\n\\end{corollary}\n\\begin{proof}\nThe unique positive input of the $k$-variable monomial needs a safe polynomial of degree $k$. The complete-cover lower bound gives $2k-3$. For the matching construction, successively compute $x_1x_2$, then $x_1x_2x_3$, and so on up to a $(k-1)$-fold product on clean work wires. Toggle the output using that product and $x_k$, then uncompute the $k-2$ work products. The count is $(k-2)+1+(k-2)=2k-3$. For $k=2$, a single Toffoli suffices.\n\\end{proof}","_2\\cdots x_k)=2k-3.\n\\]\n\\end{corollary}\n\\begin{proof}\nThe unique positive input of the $k$-variable monomial needs a safe polynomial of degree $k$. The complete-cover lower bound gives $2k-3$. For the matching construction, successively compute $x_1x_2$, then $x_1x_2x_3$, and so on up to a $(k-1)$-fold product on clean work wires. Toggle the output using that product and $x_k$, then uncompute the $k-2$ work products. The count is $(k-2)+1+(k-2)=2k-3$. For $k=2$, a single Toffoli suffices.\n\\end{proof}","ary}\n\\begin{proof}\nThe unique positive input of the $k$-variable monomial needs a safe polynomial of degree $k$. The complete-cover lower bound gives $2k-3$. For the matching construction, successively compute $x_1x_2$, then $x_1x_2x_3$, and so on up to a $(k-1)$-fold product on clean work wires. Toggle the output using that product and $x_k$, then uncompute the $k-2$ work products. The count is $(k-2)+1+(k-2)=2k-3$. For $k=2$, a single Toffoli suffices.\n\\end{proof}","que positive input of the $k$-variable monomial needs a safe polynomial of degree $k$. The complete-cover lower bound gives $2k-3$. For the matching construction, successively compute $x_1x_2$, then $x_1x_2x_3$, and so on up to a $(k-1)$-fold product on clean work wires. Toggle the output using that product and $x_k$, then uncompute the $k-2$ work products. The count is $(k-2)+1+(k-2)=2k-3$. For $k=2$, a single Toffoli suffices.\n\\end{proof}","\\chapter{Calibration families and exact small oracles}\n\\label{c:ch:calibration}\\claim{C03 and C04}\n\\section{An unbalanced disjunction}\nFor $k\\ge2$ and $n=k+2$, define\n\\[\n B=z_1\\cdots z_k,\\qquad\n F_k(a,b,z)=(ab)\\lor B=ab+B+abB,\\qquad G_k=F_k+a.\n\\]\nThis is a deliberately easy family with two differently sized accepting regions. It separates the cost of finding one safe packet from the degree needed for complete coverage.","\\begin{proposition}[Raw calibration parameters]\\label{c:prop:raw-family}\nFor this family,\n\\[\n \\alpha_1(F_k)=2,\\qquad\\beta(F_k)=k,\\qquad\\deg F_k=k+2.\n\\]\n\\end{proposition}\n\\begin{proof}\nThe minterms have supports $\\{a,b\\}$ and $\\{z_1,\\ldots,z_k\\}$. The first two assertions follow from Theorem~\\ref{c:thm:monotone}. The monomial $abB$ has degree $k+2$ and does not cancel with either lower-degree term in the displayed polynomial.\n\\end{proof}","\\begin{theorem}[Affine optimization of the one-packet parameter]\\label{c:thm:family-alpha}\nFor every $k\\ge2$,\n\\[\n \\max_{\\ell\\in\\Aff_n}\\alpha_1(F_k+\\ell)=3.\n\\]\n\\end{theorem}\n\\begin{proof}\nFirst restrict to the face $z_1=0$. There $F_k+\\ell$ becomes $ab+\\ell|_{z_1=0}$, a nonzero quadratic. If the restricted affine term contains a remaining $z$ variable, set $a=0$ and set the remaining affine form to one. This gives an accepting affine subspace of codimension at most two inside the face. If it contains no remaining $z$ variable, fix $a,b$ to an accepting pair instead. The indicator of the resulting affine subspace in the full input space has degree at most three and is safe. Thus every affine adjustment has a nonzero safe polynomial of degree at most three.","{\\ell\\in\\Aff_n}\\alpha_1(F_k+\\ell)=3.\n\\]\n\\end{theorem}\n\\begin{proof}\nFirst restrict to the face $z_1=0$. There $F_k+\\ell$ becomes $ab+\\ell|_{z_1=0}$, a nonzero quadratic. If the restricted affine term contains a remaining $z$ variable, set $a=0$ and set the remaining affine form to one. This gives an accepting affine subspace of codimension at most two inside the face. If it contains no remaining $z$ variable, fix $a,b$ to an accepting pair instead. The indicator of the resulting affine subspace in the full input space has degree at most three and is safe. Thus every affine adjustment has a nonzero safe polynomial of degree at most three.","pha_1(F_k+\\ell)=3.\n\\]\n\\end{theorem}\n\\begin{proof}\nFirst restrict to the face $z_1=0$. There $F_k+\\ell$ becomes $ab+\\ell|_{z_1=0}$, a nonzero quadratic. If the restricted affine term contains a remaining $z$ variable, set $a=0$ and set the remaining affine form to one. This gives an accepting affine subspace of codimension at most two inside the face. If it contains no remaining $z$ variable, fix $a,b$ to an accepting pair instead. The indicator of the resulting affine subspace in the full input space has degree at most three and is safe. Thus every affine adjustment has a nonzero safe polynomial of degree at most three.","For equality, take $\\ell=a$. The degree-three polynomial\n\\[\n a(1+b)(1+z_1)\n\\]\nis safe for $G_k$. Suppose a safe $h$ of degree at most two existed. On $a=0$, acceptance requires $B=1$. For $k\\ge3$, the monotone-ideal argument forces $h|_{a=0}=0$, hence $h=aL$ with $L$ affine in the remaining variables. The negative face $a=b=1$ forces $L(1,z)=0$, so $L=c(1+b)$. The negative point $a=1,b=0,z=(1,\\ldots,1)$ then forces $c=0$.","\\ell=a$. The degree-three polynomial\n\\[\n a(1+b)(1+z_1)\n\\]\nis safe for $G_k$. Suppose a safe $h$ of degree at most two existed. On $a=0$, acceptance requires $B=1$. For $k\\ge3$, the monotone-ideal argument forces $h|_{a=0}=0$, hence $h=aL$ with $L$ affine in the remaining variables. The negative face $a=b=1$ forces $L(1,z)=0$, so $L=c(1+b)$. The negative point $a=1,b=0,z=(1,\\ldots,1)$ then forces $c=0$.","When $k=2$, the restriction can initially be $h|_{a=0}=cB$. Write $h=cB+aL$. On the same negative face $a=b=1$, the quadratic term $cB$ cannot be cancelled by affine $L$, so $c=0$. The previous argument then applies. There is no nonzero safe polynomial of degree at most two, proving equality.\n\\end{proof}","\\begin{theorem}[Affine optimization of complete coverage]\\label{c:thm:family-beta}\nFor every $k\\ge2$,\n\\[\n \\beta^*(F_k)=k+1.\n\\]\n\\end{theorem}\n\\begin{proof}\nRestrict $G_k=F_k+a$ to $b=1$. Then\n\\[\n G_k|_{b=1}=(1+a)B.\n\\]\nThis is the indicator of one point on $k+1$ variables. At the positive input $a=0,b=1,z=(1,\\ldots,1)$, a safe detecting polynomial must restrict to that singleton indicator and therefore have degree at least $k+1$. Thus $\\beta(G_k)\\ge k+1$.","imization of complete coverage]\\label{c:thm:family-beta}\nFor every $k\\ge2$,\n\\[\n \\beta^*(F_k)=k+1.\n\\]\n\\end{theorem}\n\\begin{proof}\nRestrict $G_k=F_k+a$ to $b=1$. Then\n\\[\n G_k|_{b=1}=(1+a)B.\n\\]\nThis is the indicator of one point on $k+1$ variables. At the positive input $a=0,b=1,z=(1,\\ldots,1)$, a safe detecting polynomial must restrict to that singleton indicator and therefore have degree at least $k+1$. Thus $\\beta(G_k)\\ge k+1$.","complete coverage]\\label{c:thm:family-beta}\nFor every $k\\ge2$,\n\\[\n \\beta^*(F_k)=k+1.\n\\]\n\\end{theorem}\n\\begin{proof}\nRestrict $G_k=F_k+a$ to $b=1$. Then\n\\[\n G_k|_{b=1}=(1+a)B.\n\\]\nThis is the indicator of one point on $k+1$ variables. At the positive input $a=0,b=1,z=(1,\\ldots,1)$, a safe detecting polynomial must restrict to that singleton indicator and therefore have degree at least $k+1$. Thus $\\beta(G_k)\\ge k+1$.","For the upper bound, Theorem~\\ref{c:thm:family-alpha} shows that no $F_k+\\ell$ is a singleton indicator: a singleton would have one-packet degree $n=k+2>3$. These functions are also nonzero because $F_k$ is not affine. Each has at least two positive inputs, so Proposition~\\ref{c:prop:beta-bounds} gives $\\beta(F_k+\\ell)\\le n-1=k+1$.\n\\end{proof}","\\begin{corollary}[Restored cost interval for the calibration family]\\label{c:cor:family-cost}\nFor $n=k+2$,\n\\[\n 2n-5\\le\\KTrest(F_k)\\le2n-3.\n\\]\n\\end{corollary}\n\\begin{proof}\nThe lower bound follows from Theorems~\\ref{c:thm:complete-bound} and \\ref{c:thm:family-beta}. For the upper bound, compute $A=ab$ with one multiplication and $B$ using $k-1$ multiplications. Toggle the output by $A+B+AB$, requiring one output Toffoli and affine additions. Uncompute $A$ and $B$. The total Toffoli count is $k+1+k=2k+1=2n-3$.\n\\end{proof}","tion family]\\label{c:cor:family-cost}\nFor $n=k+2$,\n\\[\n 2n-5\\le\\KTrest(F_k)\\le2n-3.\n\\]\n\\end{corollary}\n\\begin{proof}\nThe lower bound follows from Theorems~\\ref{c:thm:complete-bound} and \\ref{c:thm:family-beta}. For the upper bound, compute $A=ab$ with one multiplication and $B$ using $k-1$ multiplications. Toggle the output by $A+B+AB$, requiring one output Toffoli and affine additions. Uncompute $A$ and $B$. The total Toffoli count is $k+1+k=2k+1=2n-3$.\n\\end{proof}","ily]\\label{c:cor:family-cost}\nFor $n=k+2$,\n\\[\n 2n-5\\le\\KTrest(F_k)\\le2n-3.\n\\]\n\\end{corollary}\n\\begin{proof}\nThe lower bound follows from Theorems~\\ref{c:thm:complete-bound} and \\ref{c:thm:family-beta}. For the upper bound, compute $A=ab$ with one multiplication and $B$ using $k-1$ multiplications. Toggle the output by $A+B+AB$, requiring one output Toffoli and affine additions. Uncompute $A$ and $B$. The total Toffoli count is $k+1+k=2k+1=2n-3$.\n\\end{proof}","r $n=k+2$,\n\\[\n 2n-5\\le\\KTrest(F_k)\\le2n-3.\n\\]\n\\end{corollary}\n\\begin{proof}\nThe lower bound follows from Theorems~\\ref{c:thm:complete-bound} and \\ref{c:thm:family-beta}. For the upper bound, compute $A=ab$ with one multiplication and $B$ using $k-1$ multiplications. Toggle the output by $A+B+AB$, requiring one output Toffoli and affine additions. Uncompute $A$ and $B$. The total Toffoli count is $k+1+k=2k+1=2n-3$.\n\\end{proof}","The preceding combined bound $\\max\\{0,\\deg F_k-1,2\\max_{\\ell}\\alpha_1(F_k+\\ell)-3\\}$ gives $n-1$ for this family; its immunity-only term is the constant $3$. The complete-cover bound gives $2n-5$. The stronger estimate can also be recognized by restriction to $b=1$ followed by a free affine output adjustment. No exact optimum is claimed when the displayed endpoints differ.","\\begin{center}\n\\begin{tabular}{@{}rrrrr@{}}\n\\toprule\nInputs $n$ & $\\max_\\ell\\alpha_1$ & $\\beta^*$ & One-packet bound & Complete-cover bound","\\bottomrule\n\\end{tabular}\n\\end{center}\nThe formulas are proved for all $k\\ge2$. The source evaluates the designated family truth tables through nine inputs, but checks every affine mask only through eight. The ninth-input affine maximum is justified by the proof, not by a stored exhaustive mask search.","\\section{An explicit moment certificate}\nOn the face $b=1$, every point except $v=(a=0,z=(1,\\ldots,1))$ is negative for $G_k$. A polynomial of degree at most $k$ sums to zero over the $(k+1)$-dimensional cube: each monomial omits some coordinate and contributes an even number of ones. Therefore\n\\[\n v_k(v)=\\sum_{u\\ne v\\text{ on the face}}v_k(u).\n\\]\nThis is the certificate from Theorem~\\ref{c:thm:dual}, proving that no safe degree-$k$ polynomial detects $v$.","icate}\nOn the face $b=1$, every point except $v=(a=0,z=(1,\\ldots,1))$ is negative for $G_k$. A polynomial of degree at most $k$ sums to zero over the $(k+1)$-dimensional cube: each monomial omits some coordinate and contributes an even number of ones. Therefore\n\\[\n v_k(v)=\\sum_{u\\ne v\\text{ on the face}}v_k(u).\n\\]\nThis is the certificate from Theorem~\\ref{c:thm:dual}, proving that no safe degree-$k$ polynomial detects $v$.","$, every point except $v=(a=0,z=(1,\\ldots,1))$ is negative for $G_k$. A polynomial of degree at most $k$ sums to zero over the $(k+1)$-dimensional cube: each monomial omits some coordinate and contributes an even number of ones. Therefore\n\\[\n v_k(v)=\\sum_{u\\ne v\\text{ on the face}}v_k(u).\n\\]\nThis is the certificate from Theorem~\\ref{c:thm:dual}, proving that no safe degree-$k$ polynomial detects $v$.","The finite four-input computation found $\\max_\\ell\\alpha_1(f+\\ell)=\\beta^*(f)$ for every function in that domain. The family above disproves extrapolation of that equality: it already fails at five inputs, with values three and four, and the difference grows without bound.","A cubic on three variables has the form\n\\[\n x_1x_2x_3+q(x)+\\ell(x),\n\\]\nwith $q$ quadratic and $\\ell$ affine. Translating $x_i\\mapsto x_i+c_i$ adjusts independently the coefficients of $x_jx_k$, so all quadratic terms can be cancelled. The resulting function is a three-variable monomial plus an affine function. Affine input changes and affine output additions cost no Toffolis. Corollary~\\ref{c:cor:monomial} gives both the lower bound three and a matching restored construction.\n\\end{proof}","There are $4+16+256=276$ functions over the three input sizes. The distribution is\n\\[\n \\begin{array}{c|rrr}\n \\text{Optimum Toffolis}&0&1&3","There are $4+16+256=276$ functions over the three input sizes. The distribution is\n\\[\n \\begin{array}{c|rrr}\n \\text{Optimum Toffolis}&0&1&3\\\\\\hline\n \\text{Number of functions}&28&120&128.\n \\end{array}\n\\]\nThe source constructs and checks the actual circuits. Optimality is obtained from the written lower bounds and matching constructions, not from exhaustive enumeration of all reversible circuits.","\\section{Two supporting lower bounds}\n\\begin{lemma}[Affine subspace after multiplication restrictions]\\label{c:lem:affine-subspace}\nIf a Boolean function on $n$ inputs is computed with $t$ AND gates in an XOR--AND circuit, there is an affine subspace of dimension at least $n-t$ on which the output is affine.\n\\end{lemma}\n\\begin{proof}\nProcess the AND gates in topological order while maintaining an affine subspace on which all preceding product outputs are affine. The operands of the next gate are then affine on the current subspace. If an operand is constant, their product is already affine. Otherwise impose one nonempty affine equation setting that operand to zero. This loses at most one dimension and makes the product affine. Earlier affine outputs stay affine under restriction. After at most $t$ restrictions the output is affine on a subspace of","\\section{Two supporting lower bounds}\n\\begin{lemma}[Affine subspace after multiplication restrictions]\\label{c:lem:affine-subspace}\nIf a Boolean function on $n$ inputs is computed with $t$ AND gates in an XOR--AND circuit, there is an affine subspace of dimension at least $n-t$ on which the output is affine.\n\\end{lemma}\n\\begin{proof}\nProcess the AND gates in topological order while maintaining an affine subspace on which all preceding product outputs are affine. The operands of the next gate are then affine on the current subspace. If an operand is constant, their product is already affine. Otherwise impose one nonempty affine equation setting that operand to zero. This loses at most one dimension and makes the product affine. Earlier affine outputs stay affine under restriction. After at most $t$ restrictions the output is affine on a subspace of dimension at least $n-t$.\n\\","\\section{Two supporting lower bounds}\n\\begin{lemma}[Affine subspace after multiplication restrictions]\\label{c:lem:affine-subspace}\nIf a Boolean function on $n$ inputs is computed with $t$ AND gates in an XOR--AND circuit, there is an affine subspace of dimension at least $n-t$ on which the output is affine.\n\\end{lemma}\n\\begin{proof}\nProcess the AND gates in topological order while maintaining an affine subspace on which all preceding product outputs are affine. The operands of the next gate are then affine on the current subspace. If an operand is constant, their product is already affine. Otherwise impose one nonempty affine equation setting that operand to zero. This loses at most one dimension and makes the product affine. Earlier affine outputs stay affine under restriction. After at most $t$ restrictions the output is affine on a subspace of dimension at least $n-t$.\n\\end{proof}\nThis is the affine-restriction method used in the background literature on mult","subspace on which all preceding product outputs are affine. The operands of the next gate are then affine on the current subspace. If an operand is constant, their product is already affine. Otherwise impose one nonempty affine equation setting that operand to zero. This loses at most one dimension and makes the product affine. Earlier affine outputs stay affine under restriction. After at most $t$ restrictions the output is affine on a subspace of dimension at least $n-t$.\n\\end{proof}\nThis is the affine-restriction method used in the background literature on multiplicative complexity \\cite{c:findboyar}. The proof is reproduced here in the scalar convention needed below.","of the next gate are then affine on the current subspace. If an operand is constant, their product is already affine. Otherwise impose one nonempty affine equation setting that operand to zero. This loses at most one dimension and makes the product affine. Earlier affine outputs stay affine under restriction. After at most $t$ restrictions the output is affine on a subspace of dimension at least $n-t$.\n\\end{proof}\nThis is the affine-restriction method used in the background literature on multiplicative complexity \\cite{c:findboyar}. The proof is reproduced here in the scalar convention needed below.","\\begin{proposition}[Exact cost of a nondegenerate paired quadratic]\\label{c:prop:quadratic-cost}\nFor even $n$, let\n\\[\n Q(x)=x_1x_2+x_3x_4+\\cdots+x_{n-1}x_n.\n\\]\nThen\n\\[\n M(Q)=\\KTrest(Q)=n/2.\n\\]\n\\end{proposition}\n\\begin{proof}\nThe polar bilinear form of $Q$ is the full-rank alternating form made of $n/2$ disjoint $2\\times2$ blocks. If $Q$ is affine on an affine subspace with direction space $U$, its second differences vanish on $U$, so $U$ is totally isotropic. Nondegeneracy gives $\\dim U\\le n/2$.","of a nondegenerate paired quadratic]\\label{c:prop:quadratic-cost}\nFor even $n$, let\n\\[\n Q(x)=x_1x_2+x_3x_4+\\cdots+x_{n-1}x_n.\n\\]\nThen\n\\[\n M(Q)=\\KTrest(Q)=n/2.\n\\]\n\\end{proposition}\n\\begin{proof}\nThe polar bilinear form of $Q$ is the full-rank alternating form made of $n/2$ disjoint $2\\times2$ blocks. If $Q$ is affine on an affine subspace with direction space $U$, its second differences vanish on $U$, so $U$ is totally isotropic. Nondegeneracy gives $\\dim U\\le n/2$.","degenerate paired quadratic]\\label{c:prop:quadratic-cost}\nFor even $n$, let\n\\[\n Q(x)=x_1x_2+x_3x_4+\\cdots+x_{n-1}x_n.\n\\]\nThen\n\\[\n M(Q)=\\KTrest(Q)=n/2.\n\\]\n\\end{proposition}\n\\begin{proof}\nThe polar bilinear form of $Q$ is the full-rank alternating form made of $n/2$ disjoint $2\\times2$ blocks. If $Q$ is affine on an affine subspace with direction space $U$, its second differences vanish on $U$, so $U$ is totally isotropic. Nondegeneracy gives $\\dim U\\le n/2$.","ratic-cost}\nFor even $n$, let\n\\[\n Q(x)=x_1x_2+x_3x_4+\\cdots+x_{n-1}x_n.\n\\]\nThen\n\\[\n M(Q)=\\KTrest(Q)=n/2.\n\\]\n\\end{proposition}\n\\begin{proof}\nThe polar bilinear form of $Q$ is the full-rank alternating form made of $n/2$ disjoint $2\\times2$ blocks. If $Q$ is affine on an affine subspace with direction space $U$, its second differences vanish on $U$, so $U$ is totally isotropic. Nondegeneracy gives $\\dim U\\le n/2$.","Lemma~\\ref{c:lem:affine-subspace} therefore implies $n-t\\le n/2$, or $t\\ge n/2$, for any XOR--AND computation. Every restored Toffoli implementation gives such a computation by fixing its ancillary inputs and initial target, so the same lower bound applies to $\\KTrest$. Directly add the $n/2$ disjoint pair products to the target with one Toffoli each. This attains both bounds and requires no nonlinear work register.\n\\end{proof}","\\section{A compact whole function}\nLet $m\\ge2$, let $a=(1,\\ldots,1)\\in\\F^m$, and introduce one logical bit $x_v$ for each\n\\[\n v\\in V=\\F^m\\setminus\\{0,a\\}.\n\\]\nThere are $n=2^m-2$ such bits. Define\n\\[\n L_j(x)=\\sum_{v:v_j=1}x_v,\\qquad\n P_m(x)=\\prod_{j=1}^mL_j(x).\n\\]","\\begin{theorem}[Exact nonlinear cost of the whole function]\\label{c:thm:whole-cost}\nFor the family above,\n\\[\n M(P_m)=m-1,\\qquad\\KTrest(P_m)=2m-3.\n\\]\n\\end{theorem}\n\\begin{proof}\nThe linear map $x\\mapsto(L_1(x),\\ldots,L_m(x))$ has rank $m$, since its columns include the standard basis vectors. Extend these forms to an invertible system of linear coordinates. In those coordinates $P_m$ is the product of the first $m$ variables, with the other $n-m$ variables unused.","[Exact nonlinear cost of the whole function]\\label{c:thm:whole-cost}\nFor the family above,\n\\[\n M(P_m)=m-1,\\qquad\\KTrest(P_m)=2m-3.\n\\]\n\\end{theorem}\n\\begin{proof}\nThe linear map $x\\mapsto(L_1(x),\\ldots,L_m(x))$ has rank $m$, since its columns include the standard basis vectors. Extend these forms to an invertible system of linear coordinates. In those coordinates $P_m$ is the product of the first $m$ variables, with the other $n-m$ variables unused.","hole function]\\label{c:thm:whole-cost}\nFor the family above,\n\\[\n M(P_m)=m-1,\\qquad\\KTrest(P_m)=2m-3.\n\\]\n\\end{theorem}\n\\begin{proof}\nThe linear map $x\\mapsto(L_1(x),\\ldots,L_m(x))$ has rank $m$, since its columns include the standard basis vectors. Extend these forms to an invertible system of linear coordinates. In those coordinates $P_m$ is the product of the first $m$ variables, with the other $n-m$ variables unused.","Degree is invariant under an invertible affine change, so $\\deg P_m=m$ and Lemma~\\ref{c:lem:degree} gives $M(P_m)\\ge m-1$. A product chain attains it.","For the restored lower bound, the safe ideal of the product in these coordinates consists of polynomials divisible by the $m$-variable monomial. Any nonzero safe polynomial has degree at least $m$. Theorem~\\ref{c:thm:complete-bound} gives $2m-3$. Compute the $m$ parities onto clean registers by CNOT, apply the optimal monomial oracle there, and uncompute the parities. No additional Toffolis are needed.\n\\end{proof}","\\section{Extracting the quadratic grade}\n\\begin{theorem}[Exact cost of the extracted grade]\\label{c:thm:grade-cost}\nLet $Q_m=[P_m]_2$ denote the homogeneous quadratic part in the original $x_v$ coordinates. Then\n\\[\n Q_m=\\sum_{\\{v,v+a\\}\\subseteq V}x_vx_{v+a},\n\\]\nwhere every unordered pair is used once, and\n\\[\n M(Q_m)=\\KTrest(Q_m)=n/2=2^{m-1}-1.\n\\]\n\\end{theorem}\n\\begin{proof}\nThe excluded columns give $P_m(0)=P_m(e_v)=0$. For distinct $u,v$, Boolean coefficient inversion therefore gives\n\\[\n [x_ux_v]P_m=P_m(e_u+e_v)=[u+v=a].\n\\]\nThe map $v\\mapsto v+a$ is a fixed-point-free involution of $V$, so the displayed terms form $n/2$ disjoint pairs. Proposition~\\ref{c:prop:quadratic-cost} supplies the exact costs.\n\\end{proof}","tion{Extracting the quadratic grade}\n\\begin{theorem}[Exact cost of the extracted grade]\\label{c:thm:grade-cost}\nLet $Q_m=[P_m]_2$ denote the homogeneous quadratic part in the original $x_v$ coordinates. Then\n\\[\n Q_m=\\sum_{\\{v,v+a\\}\\subseteq V}x_vx_{v+a},\n\\]\nwhere every unordered pair is used once, and\n\\[\n M(Q_m)=\\KTrest(Q_m)=n/2=2^{m-1}-1.\n\\]\n\\end{theorem}\n\\begin{proof}\nThe excluded columns give $P_m(0)=P_m(e_v)=0$. For distinct $u,v$, Boolean coefficient inversion therefore gives\n\\[\n [x_ux_v]P_m=P_m(e_u+e_v)=[u+v=a].\n\\]\nThe map $v\\mapsto v+a$ is a fixed-point-free involution of $V$, so the displayed terms form $n/2$ disjoint pairs. Proposition~\\ref{c:prop:quadratic-cost} supplies the exact costs.\n\\end{proof}","e}\n\\begin{theorem}[Exact cost of the extracted grade]\\label{c:thm:grade-cost}\nLet $Q_m=[P_m]_2$ denote the homogeneous quadratic part in the original $x_v$ coordinates. Then\n\\[\n Q_m=\\sum_{\\{v,v+a\\}\\subseteq V}x_vx_{v+a},\n\\]\nwhere every unordered pair is used once, and\n\\[\n M(Q_m)=\\KTrest(Q_m)=n/2=2^{m-1}-1.\n\\]\n\\end{theorem}\n\\begin{proof}\nThe excluded columns give $P_m(0)=P_m(e_v)=0$. For distinct $u,v$, Boolean coefficient inversion therefore gives\n\\[\n [x_ux_v]P_m=P_m(e_u+e_v)=[u+v=a].\n\\]\nThe map $v\\mapsto v+a$ is a fixed-point-free involution of $V$, so the displayed terms form $n/2$ disjoint pairs. Proposition~\\ref{c:prop:quadratic-cost} supplies the exact costs.\n\\end{proof}","label{c:thm:grade-cost}\nLet $Q_m=[P_m]_2$ denote the homogeneous quadratic part in the original $x_v$ coordinates. Then\n\\[\n Q_m=\\sum_{\\{v,v+a\\}\\subseteq V}x_vx_{v+a},\n\\]\nwhere every unordered pair is used once, and\n\\[\n M(Q_m)=\\KTrest(Q_m)=n/2=2^{m-1}-1.\n\\]\n\\end{theorem}\n\\begin{proof}\nThe excluded columns give $P_m(0)=P_m(e_v)=0$. For distinct $u,v$, Boolean coefficient inversion therefore gives\n\\[\n [x_ux_v]P_m=P_m(e_u+e_v)=[u+v=a].\n\\]\nThe map $v\\mapsto v+a$ is a fixed-point-free involution of $V$, so the displayed terms form $n/2$ disjoint pairs. Proposition~\\ref{c:prop:quadratic-cost} supplies the exact costs.\n\\end{proof}","the homogeneous quadratic part in the original $x_v$ coordinates. Then\n\\[\n Q_m=\\sum_{\\{v,v+a\\}\\subseteq V}x_vx_{v+a},\n\\]\nwhere every unordered pair is used once, and\n\\[\n M(Q_m)=\\KTrest(Q_m)=n/2=2^{m-1}-1.\n\\]\n\\end{theorem}\n\\begin{proof}\nThe excluded columns give $P_m(0)=P_m(e_v)=0$. For distinct $u,v$, Boolean coefficient inversion therefore gives\n\\[\n [x_ux_v]P_m=P_m(e_u+e_v)=[u+v=a].\n\\]\nThe map $v\\mapsto v+a$ is a fixed-point-free involution of $V$, so the displayed terms form $n/2$ disjoint pairs. Proposition~\\ref{c:prop:quadratic-cost} supplies the exact costs.\n\\end{proof}","v$ coordinates. Then\n\\[\n Q_m=\\sum_{\\{v,v+a\\}\\subseteq V}x_vx_{v+a},\n\\]\nwhere every unordered pair is used once, and\n\\[\n M(Q_m)=\\KTrest(Q_m)=n/2=2^{m-1}-1.\n\\]\n\\end{theorem}\n\\begin{proof}\nThe excluded columns give $P_m(0)=P_m(e_v)=0$. For distinct $u,v$, Boolean coefficient inversion therefore gives\n\\[\n [x_ux_v]P_m=P_m(e_u+e_v)=[u+v=a].\n\\]\nThe map $v\\mapsto v+a$ is a fixed-point-free involution of $V$, so the displayed terms form $n/2$ disjoint pairs. Proposition~\\ref{c:prop:quadratic-cost} supplies the exact costs.\n\\end{proof}","_{\\{v,v+a\\}\\subseteq V}x_vx_{v+a},\n\\]\nwhere every unordered pair is used once, and\n\\[\n M(Q_m)=\\KTrest(Q_m)=n/2=2^{m-1}-1.\n\\]\n\\end{theorem}\n\\begin{proof}\nThe excluded columns give $P_m(0)=P_m(e_v)=0$. For distinct $u,v$, Boolean coefficient inversion therefore gives\n\\[\n [x_ux_v]P_m=P_m(e_u+e_v)=[u+v=a].\n\\]\nThe map $v\\mapsto v+a$ is a fixed-point-free involution of $V$, so the displayed terms form $n/2$ disjoint pairs. Proposition~\\ref{c:prop:quadratic-cost} supplies the exact costs.\n\\end{proof}","\\begin{counterexample}[Projection does not preserve safe support]\\label{c:ce:unsafe}\nThe quadratic extraction $[P_m]_2$ need not be supported on the positive set of $P_m$.\n\\end{counterexample}\n\\begin{proof}\nTake $m=3$, identify the columns with the integers $1,\\ldots,6$ using their three-bit expansions, and set $x_1=x_3=x_4=1$ with all other inputs zero. Their column sum is $1+3+4=6$ under bitwise XOR, not $a=7$, so $P_3=0$. Among the complementary pairs $(1,6),(2,5),(3,4)$, exactly $(3,4)$ has both inputs one. Thus $Q_3=1$. In the source ordering this is the logical assignment encoded by integer 13.\n\\end{proof}","terexample}[Projection does not preserve safe support]\\label{c:ce:unsafe}\nThe quadratic extraction $[P_m]_2$ need not be supported on the positive set of $P_m$.\n\\end{counterexample}\n\\begin{proof}\nTake $m=3$, identify the columns with the integers $1,\\ldots,6$ using their three-bit expansions, and set $x_1=x_3=x_4=1$ with all other inputs zero. Their column sum is $1+3+4=6$ under bitwise XOR, not $a=7$, so $P_3=0$. Among the complementary pairs $(1,6),(2,5),(3,4)$, exactly $(3,4)$ has both inputs one. Thus $Q_3=1$. In the source ordering this is the logical assignment encoded by integer 13.\n\\end{proof}","}[Projection does not preserve safe support]\\label{c:ce:unsafe}\nThe quadratic extraction $[P_m]_2$ need not be supported on the positive set of $P_m$.\n\\end{counterexample}\n\\begin{proof}\nTake $m=3$, identify the columns with the integers $1,\\ldots,6$ using their three-bit expansions, and set $x_1=x_3=x_4=1$ with all other inputs zero. Their column sum is $1+3+4=6$ under bitwise XOR, not $a=7$, so $P_3=0$. Among the complementary pairs $(1,6),(2,5),(3,4)$, exactly $(3,4)$ has both inputs one. Thus $Q_3=1$. In the source ordering this is the logical assignment encoded by integer 13.\n\\end{proof}","port]\\label{c:ce:unsafe}\nThe quadratic extraction $[P_m]_2$ need not be supported on the positive set of $P_m$.\n\\end{counterexample}\n\\begin{proof}\nTake $m=3$, identify the columns with the integers $1,\\ldots,6$ using their three-bit expansions, and set $x_1=x_3=x_4=1$ with all other inputs zero. Their column sum is $1+3+4=6$ under bitwise XOR, not $a=7$, so $P_3=0$. Among the complementary pairs $(1,6),(2,5),(3,4)$, exactly $(3,4)$ has both inputs one. Thus $Q_3=1$. In the source ordering this is the logical assignment encoded by integer 13.\n\\end{proof}","quadratic extraction $[P_m]_2$ need not be supported on the positive set of $P_m$.\n\\end{counterexample}\n\\begin{proof}\nTake $m=3$, identify the columns with the integers $1,\\ldots,6$ using their three-bit expansions, and set $x_1=x_3=x_4=1$ with all other inputs zero. Their column sum is $1+3+4=6$ under bitwise XOR, not $a=7$, so $P_3=0$. Among the complementary pairs $(1,6),(2,5),(3,4)$, exactly $(3,4)$ has both inputs one. Thus $Q_3=1$. In the source ordering this is the logical assignment encoded by integer 13.\n\\end{proof}","d not be supported on the positive set of $P_m$.\n\\end{counterexample}\n\\begin{proof}\nTake $m=3$, identify the columns with the integers $1,\\ldots,6$ using their three-bit expansions, and set $x_1=x_3=x_4=1$ with all other inputs zero. Their column sum is $1+3+4=6$ under bitwise XOR, not $a=7$, so $P_3=0$. Among the complementary pairs $(1,6),(2,5),(3,4)$, exactly $(3,4)$ has both inputs one. Thus $Q_3=1$. In the source ordering this is the logical assignment encoded by integer 13.\n\\end{proof}","\\begin{center}\n\\begin{tabular}{@{}rrrrr@{}}\n\\toprule\n$m$ & Actual inputs $n$ & $\\KTrest(P_m)$ & $\\KTrest(Q_m)$ & Constructed total gates for $P_m$","\\bottomrule\n\\end{tabular}\n\\end{center}\nThese are exact cost formulas. The large-input entries were not obtained by searching for optimal circuits. The full construction uses\n\\[\n 2m(2^{m-1}-1)\n\\]\nCNOTs as well as $2m-3$ Toffolis, giving total count $O(n\\log n)$. The projected quadratic has $n/2$ total gates.","\\begin{circuitscopebox}{What the counterexample excludes}\nThe nonlinear projection cost grows exponentially in $m$, but only linearly in actual input length $n$. The example does not give an exponential separation in ordinary total gate count. It rules out treating this grade extraction as having overhead bounded only by a polynomial in the original Toffoli count, independently of linear and input-size resources. It also shows that homogeneous extraction may destroy safety. It does not exclude a bound polynomial in $n+T$ or every possible graded invariant.\n\\end{circuitscopebox}\nThe source checks all pair coefficients and exact ranks for $2\\le m\\le10$. Full truth tables and complete restored evaluations are restricted to $m=2,3,4$. At $m=4$, the quadratic accepts 7,616 inputs that the whole function rej","\\begin{circuitscopebox}{What the counterexample excludes}\nThe nonlinear projection cost grows exponentially in $m$, but only linearly in actual input length $n$. The example does not give an exponential separation in ordinary total gate count. It rules out treating this grade extraction as having overhead bounded only by a polynomial in the original Toffoli count, independently of linear and input-size resources. It also shows that homogeneous extraction may destroy safety. It does not exclude a bound polynomial in $n+T$ or every possible graded invariant.\n\\end{circuitscopebox}\nThe source checks all pair coefficients and exact ranks for $2\\le m\\le10$. Full truth tables and complete restored evaluations are restricted to $m=2,3,4$. At $m=4$, the quadratic accepts 7,616 inputs that the whole function rejects; that count is finite evidence, distinct","ly linearly in actual input length $n$. The example does not give an exponential separation in ordinary total gate count. It rules out treating this grade extraction as having overhead bounded only by a polynomial in the original Toffoli count, independently of linear and input-size resources. It also shows that homogeneous extraction may destroy safety. It does not exclude a bound polynomial in $n+T$ or every possible graded invariant.\n\\end{circuitscopebox}\nThe source checks all pair coefficients and exact ranks for $2\\le m\\le10$. Full truth tables and complete restored evaluations are restricted to $m=2,3,4$. At $m=4$, the quadratic accepts 7,616 inputs that the whole function rejects; that count is finite evidence, distinct from the explicit universal mechanism above.","rules out treating this grade extraction as having overhead bounded only by a polynomial in the original Toffoli count, independently of linear and input-size resources. It also shows that homogeneous extraction may destroy safety. It does not exclude a bound polynomial in $n+T$ or every possible graded invariant.\n\\end{circuitscopebox}\nThe source checks all pair coefficients and exact ranks for $2\\le m\\le10$. Full truth tables and complete restored evaluations are restricted to $m=2,3,4$. At $m=4$, the quadratic accepts 7,616 inputs that the whole function rejects; that count is finite evidence, distinct from the explicit universal mechanism above.","mial in the original Toffoli count, independently of linear and input-size resources. It also shows that homogeneous extraction may destroy safety. It does not exclude a bound polynomial in $n+T$ or every possible graded invariant.\n\\end{circuitscopebox}\nThe source checks all pair coefficients and exact ranks for $2\\le m\\le10$. Full truth tables and complete restored evaluations are restricted to $m=2,3,4$. At $m=4$, the quadratic accepts 7,616 inputs that the whole function rejects; that count is finite evidence, distinct from the explicit universal mechanism above.","iginal Toffoli count, independently of linear and input-size resources. It also shows that homogeneous extraction may destroy safety. It does not exclude a bound polynomial in $n+T$ or every possible graded invariant.\n\\end{circuitscopebox}\nThe source checks all pair coefficients and exact ranks for $2\\le m\\le10$. Full truth tables and complete restored evaluations are restricted to $m=2,3,4$. At $m=4$, the quadratic accepts 7,616 inputs that the whole function rejects; that count is finite evidence, distinct from the explicit universal mechanism above.","\\section{The witness algebra}\nLet $K$ be a field and let $W=\\B^m$. The full witness algebra is\n\\[\n \\mathcal B=K^W,\n\\]\nwith pointwise addition and multiplication. For each witness $v$, let $\\delta_v$ be its indicator function. These form a basis of $\\mathcal B$ and satisfy\n\\[\n \\delta_v^2=\\delta_v,\\qquad\n \\delta_v\\delta_u=0\\quad(u\\ne v),\\qquad\n \\sum_{v\\in W}\\delta_v=1.\n\\]\nIn characteristic two, this is equivalently the Boolean quotient algebra\n\\[\n K[w_1,\\ldots,w_m]/(w_i^2-w_i:1\\le i\\le m).\n\\]","and let $W=\\B^m$. The full witness algebra is\n\\[\n \\mathcal B=K^W,\n\\]\nwith pointwise addition and multiplication. For each witness $v$, let $\\delta_v$ be its indicator function. These form a basis of $\\mathcal B$ and satisfy\n\\[\n \\delta_v^2=\\delta_v,\\qquad\n \\delta_v\\delta_u=0\\quad(u\\ne v),\\qquad\n \\sum_{v\\in W}\\delta_v=1.\n\\]\nIn characteristic two, this is equivalently the Boolean quotient algebra\n\\[\n K[w_1,\\ldots,w_m]/(w_i^2-w_i:1\\le i\\le m).\n\\]","\\begin{theorem}[Dimension of a faithful multiplicative summary]\\label{c:thm:gluing-dim}\nLet $\\Phi:\\mathcal B\\to A$ be a $K$-linear multiplicative map into a $K$-algebra. Suppose\n\\[\n \\Phi(\\delta_v)\\ne0\\qquad\\text{for every }v\\in W.\n\\]\nThen\n\\[\n \\dim_K A\\ge2^m.\n\\]\nNo unital hypothesis on $\\Phi$ is required.\n\\end{theorem}\n\\begin{proof}\nWrite $e_v=\\Phi(\\delta_v)$. Multiplicativity makes these nonzero orthogonal idempotents. If\n\\[\n \\sum_{v\\in W}c_ve_v=0,\n\\]\nmultiply the equality by $e_u$. Orthogonality annihilates every term except $c_ue_u$, which must be zero. Since $K$ is a field and $e_u\\ne0$, $c_u=0$. This holds for every $u$, so the $2^m$ vectors $e_v$ are linearly independent.\n\\end{proof}","a faithful multiplicative summary]\\label{c:thm:gluing-dim}\nLet $\\Phi:\\mathcal B\\to A$ be a $K$-linear multiplicative map into a $K$-algebra. Suppose\n\\[\n \\Phi(\\delta_v)\\ne0\\qquad\\text{for every }v\\in W.\n\\]\nThen\n\\[\n \\dim_K A\\ge2^m.\n\\]\nNo unital hypothesis on $\\Phi$ is required.\n\\end{theorem}\n\\begin{proof}\nWrite $e_v=\\Phi(\\delta_v)$. Multiplicativity makes these nonzero orthogonal idempotents. If\n\\[\n \\sum_{v\\in W}c_ve_v=0,\n\\]\nmultiply the equality by $e_u$. Orthogonality annihilates every term except $c_ue_u$, which must be zero. Since $K$ is a field and $e_u\\ne0$, $c_u=0$. This holds for every $u$, so the $2^m$ vectors $e_v$ are linearly independent.\n\\end{proof}","-dim}\nLet $\\Phi:\\mathcal B\\to A$ be a $K$-linear multiplicative map into a $K$-algebra. Suppose\n\\[\n \\Phi(\\delta_v)\\ne0\\qquad\\text{for every }v\\in W.\n\\]\nThen\n\\[\n \\dim_K A\\ge2^m.\n\\]\nNo unital hypothesis on $\\Phi$ is required.\n\\end{theorem}\n\\begin{proof}\nWrite $e_v=\\Phi(\\delta_v)$. Multiplicativity makes these nonzero orthogonal idempotents. If\n\\[\n \\sum_{v\\in W}c_ve_v=0,\n\\]\nmultiply the equality by $e_u$. Orthogonality annihilates every term except $c_ue_u$, which must be zero. Since $K$ is a field and $e_u\\ne0$, $c_u=0$. This holds for every $u$, so the $2^m$ vectors $e_v$ are linearly independent.\n\\end{proof}","ltiplicative map into a $K$-algebra. Suppose\n\\[\n \\Phi(\\delta_v)\\ne0\\qquad\\text{for every }v\\in W.\n\\]\nThen\n\\[\n \\dim_K A\\ge2^m.\n\\]\nNo unital hypothesis on $\\Phi$ is required.\n\\end{theorem}\n\\begin{proof}\nWrite $e_v=\\Phi(\\delta_v)$. Multiplicativity makes these nonzero orthogonal idempotents. If\n\\[\n \\sum_{v\\in W}c_ve_v=0,\n\\]\nmultiply the equality by $e_u$. Orthogonality annihilates every term except $c_ue_u$, which must be zero. Since $K$ is a field and $e_u\\ne0$, $c_u=0$. This holds for every $u$, so the $2^m$ vectors $e_v$ are linearly independent.\n\\end{proof}","Suppose\n\\[\n \\Phi(\\delta_v)\\ne0\\qquad\\text{for every }v\\in W.\n\\]\nThen\n\\[\n \\dim_K A\\ge2^m.\n\\]\nNo unital hypothesis on $\\Phi$ is required.\n\\end{theorem}\n\\begin{proof}\nWrite $e_v=\\Phi(\\delta_v)$. Multiplicativity makes these nonzero orthogonal idempotents. If\n\\[\n \\sum_{v\\in W}c_ve_v=0,\n\\]\nmultiply the equality by $e_u$. Orthogonality annihilates every term except $c_ue_u$, which must be zero. Since $K$ is a field and $e_u\\ne0$, $c_u=0$. This holds for every $u$, so the $2^m$ vectors $e_v$ are linearly independent.\n\\end{proof}","delta_v)\\ne0\\qquad\\text{for every }v\\in W.\n\\]\nThen\n\\[\n \\dim_K A\\ge2^m.\n\\]\nNo unital hypothesis on $\\Phi$ is required.\n\\end{theorem}\n\\begin{proof}\nWrite $e_v=\\Phi(\\delta_v)$. Multiplicativity makes these nonzero orthogonal idempotents. If\n\\[\n \\sum_{v\\in W}c_ve_v=0,\n\\]\nmultiply the equality by $e_u$. Orthogonality annihilates every term except $c_ue_u$, which must be zero. Since $K$ is a field and $e_u\\ne0$, $c_u=0$. This holds for every $u$, so the $2^m$ vectors $e_v$ are linearly independent.\n\\end{proof}","0\\qquad\\text{for every }v\\in W.\n\\]\nThen\n\\[\n \\dim_K A\\ge2^m.\n\\]\nNo unital hypothesis on $\\Phi$ is required.\n\\end{theorem}\n\\begin{proof}\nWrite $e_v=\\Phi(\\delta_v)$. Multiplicativity makes these nonzero orthogonal idempotents. If\n\\[\n \\sum_{v\\in W}c_ve_v=0,\n\\]\nmultiply the equality by $e_u$. Orthogonality annihilates every term except $c_ue_u$, which must be zero. Since $K$ is a field and $e_u\\ne0$, $c_u=0$. This holds for every $u$, so the $2^m$ vectors $e_v$ are linearly independent.\n\\end{proof}",".\n\\]\nThen\n\\[\n \\dim_K A\\ge2^m.\n\\]\nNo unital hypothesis on $\\Phi$ is required.\n\\end{theorem}\n\\begin{proof}\nWrite $e_v=\\Phi(\\delta_v)$. Multiplicativity makes these nonzero orthogonal idempotents. If\n\\[\n \\sum_{v\\in W}c_ve_v=0,\n\\]\nmultiply the equality by $e_u$. Orthogonality annihilates every term except $c_ue_u$, which must be zero. Since $K$ is a field and $e_u\\ne0$, $c_u=0$. This holds for every $u$, so the $2^m$ vectors $e_v$ are linearly independent.\n\\end{proof}","\\dim_K A\\ge2^m.\n\\]\nNo unital hypothesis on $\\Phi$ is required.\n\\end{theorem}\n\\begin{proof}\nWrite $e_v=\\Phi(\\delta_v)$. Multiplicativity makes these nonzero orthogonal idempotents. If\n\\[\n \\sum_{v\\in W}c_ve_v=0,\n\\]\nmultiply the equality by $e_u$. Orthogonality annihilates every term except $c_ue_u$, which must be zero. Since $K$ is a field and $e_u\\ne0$, $c_u=0$. This holds for every $u$, so the $2^m$ vectors $e_v$ are linearly independent.\n\\end{proof}","^m.\n\\]\nNo unital hypothesis on $\\Phi$ is required.\n\\end{theorem}\n\\begin{proof}\nWrite $e_v=\\Phi(\\delta_v)$. Multiplicativity makes these nonzero orthogonal idempotents. If\n\\[\n \\sum_{v\\in W}c_ve_v=0,\n\\]\nmultiply the equality by $e_u$. Orthogonality annihilates every term except $c_ue_u$, which must be zero. Since $K$ is a field and $e_u\\ne0$, $c_u=0$. This holds for every $u$, so the $2^m$ vectors $e_v$ are linearly independent.\n\\end{proof}","\\begin{corollary}[Scalar unital summaries are evaluations]\\label{c:cor:scalar-eval}\nEvery unital $K$-linear multiplicative map $\\Phi:K^W\\to K$ is evaluation at exactly one witness.\n\\end{corollary}\n\\begin{proof}\nAn idempotent scalar in a field is either zero or one. Orthogonality allows at most one $\\Phi(\\delta_v)$ to be one, and preservation of the unit requires exactly one. If it occurs at $v_0$, linearity gives\n\\[\n \\Phi(g)=\\Phi\\left(\\sum_v g(v)\\delta_v\\right)=g(v_0).\n\\]\n\\end{proof}\nFor $m\\ge1$, such a scalar summary loses at least one singleton witness other than $v_0$. At $m=0$, the unique empty witness is preserved by the identity map. The dimension theorem is an elementary product-algebra fact, not a claim of new algebra or a universal computational lower bound.","calar unital summaries are evaluations]\\label{c:cor:scalar-eval}\nEvery unital $K$-linear multiplicative map $\\Phi:K^W\\to K$ is evaluation at exactly one witness.\n\\end{corollary}\n\\begin{proof}\nAn idempotent scalar in a field is either zero or one. Orthogonality allows at most one $\\Phi(\\delta_v)$ to be one, and preservation of the unit requires exactly one. If it occurs at $v_0$, linearity gives\n\\[\n \\Phi(g)=\\Phi\\left(\\sum_v g(v)\\delta_v\\right)=g(v_0).\n\\]\n\\end{proof}\nFor $m\\ge1$, such a scalar summary loses at least one singleton witness other than $v_0$. At $m=0$, the unique empty witness is preserved by the identity map. The dimension theorem is an elementary product-algebra fact, not a claim of new algebra or a universal computational lower bound.","al $K$-linear multiplicative map $\\Phi:K^W\\to K$ is evaluation at exactly one witness.\n\\end{corollary}\n\\begin{proof}\nAn idempotent scalar in a field is either zero or one. Orthogonality allows at most one $\\Phi(\\delta_v)$ to be one, and preservation of the unit requires exactly one. If it occurs at $v_0$, linearity gives\n\\[\n \\Phi(g)=\\Phi\\left(\\sum_v g(v)\\delta_v\\right)=g(v_0).\n\\]\n\\end{proof}\nFor $m\\ge1$, such a scalar summary loses at least one singleton witness other than $v_0$. At $m=0$, the unique empty witness is preserved by the identity map. The dimension theorem is an elementary product-algebra fact, not a claim of new algebra or a universal computational lower bound.","itness.\n\\end{corollary}\n\\begin{proof}\nAn idempotent scalar in a field is either zero or one. Orthogonality allows at most one $\\Phi(\\delta_v)$ to be one, and preservation of the unit requires exactly one. If it occurs at $v_0$, linearity gives\n\\[\n \\Phi(g)=\\Phi\\left(\\sum_v g(v)\\delta_v\\right)=g(v_0).\n\\]\n\\end{proof}\nFor $m\\ge1$, such a scalar summary loses at least one singleton witness other than $v_0$. At $m=0$, the unique empty witness is preserved by the identity map. The dimension theorem is an elementary product-algebra fact, not a claim of new algebra or a universal computational lower bound.","nd{corollary}\n\\begin{proof}\nAn idempotent scalar in a field is either zero or one. Orthogonality allows at most one $\\Phi(\\delta_v)$ to be one, and preservation of the unit requires exactly one. If it occurs at $v_0$, linearity gives\n\\[\n \\Phi(g)=\\Phi\\left(\\sum_v g(v)\\delta_v\\right)=g(v_0).\n\\]\n\\end{proof}\nFor $m\\ge1$, such a scalar summary loses at least one singleton witness other than $v_0$. At $m=0$, the unique empty witness is preserved by the identity map. The dimension theorem is an elementary product-algebra fact, not a claim of new algebra or a universal computational lower bound.","\\section{The quantified model boundary}\nThe same map $\\Phi$ here serves \\emph{all} predicates on a fixed witness space. It is linear, multiplicative, and retains every primitive witness indicator. Under exactly these assumptions, a polynomial-dimensional summary is impossible as $m$ grows.","\\begin{circuitscopebox}{No dimension-to-circuit inference}\nThe theorem excludes a fixed faithful linear multiplicative compression of the full witness algebra into fewer than $2^m$ dimensions. It does not exclude every polynomial-time decision procedure, does not prove $P\\ne NP$, and does not justify treating the algebra's dimension as the circuit cost of an individual decision function.\n\\end{circuitscopebox}","The finite scalar controls enumerate 256 linear maps from a two-point witness algebra to $\\mathrm{GF}(16)$, 65,536 maps for four points, and 256 maps from an eight-point algebra to $\\F$. The respective counts of unital algebra homomorphisms are two, four, and eight, matching the evaluation classification. The all-size result rests on the proof above.","\\chapter{Weighted witness fingerprints}\n\\label{c:ch:fingerprint}\\claim{C07}\n\\section{A polynomial that distinguishes witnesses}\nLet $R:\\B^m\\to\\B$ be a Boolean verifier, and let $K$ be a characteristic-two field. Define the formal polynomial\n\\[\n P_R(z_1,\\ldots,z_m)=\n \\sum_{w\\in\\B^m}R(w)\\prod_{i=1}^m z_i^{w_i}\\in K[z_1,\\ldots,z_m].\n\\]\nThe variables $z_i$ in this expression are ordinary polynomial variables, not reduced by $z_i^2=z_i$.","fingerprints}\n\\label{c:ch:fingerprint}\\claim{C07}\n\\section{A polynomial that distinguishes witnesses}\nLet $R:\\B^m\\to\\B$ be a Boolean verifier, and let $K$ be a characteristic-two field. Define the formal polynomial\n\\[\n P_R(z_1,\\ldots,z_m)=\n \\sum_{w\\in\\B^m}R(w)\\prod_{i=1}^m z_i^{w_i}\\in K[z_1,\\ldots,z_m].\n\\]\nThe variables $z_i$ in this expression are ordinary polynomial variables, not reduced by $z_i^2=z_i$.","\\begin{theorem}[Exact witness fingerprint]\\label{c:thm:fingerprint}\nThe polynomial $P_R$ is identically zero if and only if $R$ has no accepting witness. Its total degree is at most $m$.\n\\end{theorem}\n\\begin{proof}\nDifferent Boolean witnesses have different exponent vectors, so their monomials are distinct. Each accepting witness contributes coefficient one to its own monomial and cannot cancel another witness's term. The polynomial is zero exactly when there are no such terms. Each monomial uses at most $m$ variables.\n\\end{proof}\nThis avoids the cancellation of an unweighted parity count. It does not, by itself, give a small circuit for evaluating the sum.","and only if $R$ has no accepting witness. Its total degree is at most $m$.\n\\end{theorem}\n\\begin{proof}\nDifferent Boolean witnesses have different exponent vectors, so their monomials are distinct. Each accepting witness contributes coefficient one to its own monomial and cannot cancel another witness's term. The polynomial is zero exactly when there are no such terms. Each monomial uses at most $m$ variables.\n\\end{proof}\nThis avoids the cancellation of an unweighted parity count. It does not, by itself, give a small circuit for evaluating the sum.","\\section{Exact top-coefficient extraction}\n\\begin{lemma}[Top coefficient as a Boolean-cube sum]\\label{c:lem:top-coeff}\nLet $G(w)$ be a multilinear polynomial over a characteristic-two field. Then\n\\[\n [w_1\\cdots w_m]G=\\sum_{v\\in\\B^m}G(v).\n\\]\n\\end{lemma}\n\\begin{proof}\nIt is enough to check a monomial. A monomial of proper support is one on an even number $2^{m-|S|}$ of Boolean inputs, so its cube sum is zero in characteristic two. The full-support monomial is one at exactly one input and has cube sum one. Linearity gives the identity.\n\\end{proof}","\\begin{theorem}[Weighted top-coefficient identity]\\label{c:thm:weighted-top}\nLet $\\KTop_w$ extract the coefficient of $w_1\\cdots w_m$ after reduction in the Boolean $w$ algebra. Then\n\\begin{equation}\\label{c:eq:weighted-top}\n P_R(z)=\\KTop_w\\left[\n R(w)\\prod_{i=1}^m\\bigl(1+(1+z_i)w_i\\bigr)\n \\right].\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nAt $w_i=0$, the $i$th factor is one. At $w_i=1$, it is $z_i$. Lemma~\\ref{c:lem:top-coeff} therefore converts the right side to exactly the weighted sum defining $P_R$.\n\\end{proof}","efficient of $w_1\\cdots w_m$ after reduction in the Boolean $w$ algebra. Then\n\\begin{equation}\\label{c:eq:weighted-top}\n P_R(z)=\\KTop_w\\left[\n R(w)\\prod_{i=1}^m\\bigl(1+(1+z_i)w_i\\bigr)\n \\right].\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nAt $w_i=0$, the $i$th factor is one. At $w_i=1$, it is $z_i$. Lemma~\\ref{c:lem:top-coeff} therefore converts the right side to exactly the weighted sum defining $P_R$.\n\\end{proof}","The expression inside $\\KTop_w$ has a circuit with $O(|R|+m)$ operations when the verifier has circuit size $|R|$. The cost of applying $\\KTop_w$ to that succinct expression is not bounded by writing down the operator. The source evaluator explicitly stores $2^m$ coefficients and remains exponential in witness length.","\\section{The conditional algorithmic consequence}\n\\begin{lemma}[Finite-field root bound]\\label{c:lem:roots}\nIf a nonzero polynomial $P\\in K[z_1,\\ldots,z_m]$ has total degree at most $d$, where $K$ is finite, then\n\\[\n \\Pr_{z\\text{ uniform in }K^m}[P(z)=0]\\le\\frac d{|K|}.\n\\]\n\\end{lemma}\n\\begin{proof}\nInduct on the number of variables. For one variable this is the elementary root bound. In the inductive step, write\n\\[\n P=\\sum_{j=0}^r P_j(z_1,\\ldots,z_{m-1})z_m^j,\n \\qquad P_r\\ne0.\n\\]\nThe leading coefficient has total degree at most $d-r$, so it vanishes at a random first $(m-1)$-tuple with probability at most $(d-r)/|K|$. Otherwise the remaining univariate polynomial is nonzero with at most $r$ roots. The union bound gives $(d-r+r)/|K|=d/|K|$.\n\\end{proof}","tional algorithmic consequence}\n\\begin{lemma}[Finite-field root bound]\\label{c:lem:roots}\nIf a nonzero polynomial $P\\in K[z_1,\\ldots,z_m]$ has total degree at most $d$, where $K$ is finite, then\n\\[\n \\Pr_{z\\text{ uniform in }K^m}[P(z)=0]\\le\\frac d{|K|}.\n\\]\n\\end{lemma}\n\\begin{proof}\nInduct on the number of variables. For one variable this is the elementary root bound. In the inductive step, write\n\\[\n P=\\sum_{j=0}^r P_j(z_1,\\ldots,z_{m-1})z_m^j,\n \\qquad P_r\\ne0.\n\\]\nThe leading coefficient has total degree at most $d-r$, so it vanishes at a random first $(m-1)$-tuple with probability at most $(d-r)/|K|$. Otherwise the remaining univariate polynomial is nonzero with at most $r$ roots. The union bound gives $(d-r+r)/|K|=d/|K|$.\n\\end{proof}","in K[z_1,\\ldots,z_m]$ has total degree at most $d$, where $K$ is finite, then\n\\[\n \\Pr_{z\\text{ uniform in }K^m}[P(z)=0]\\le\\frac d{|K|}.\n\\]\n\\end{lemma}\n\\begin{proof}\nInduct on the number of variables. For one variable this is the elementary root bound. In the inductive step, write\n\\[\n P=\\sum_{j=0}^r P_j(z_1,\\ldots,z_{m-1})z_m^j,\n \\qquad P_r\\ne0.\n\\]\nThe leading coefficient has total degree at most $d-r$, so it vanishes at a random first $(m-1)$-tuple with probability at most $(d-r)/|K|$. Otherwise the remaining univariate polynomial is nonzero with at most $r$ roots. The union bound gives $(d-r+r)/|K|=d/|K|$.\n\\end{proof}","t most $d$, where $K$ is finite, then\n\\[\n \\Pr_{z\\text{ uniform in }K^m}[P(z)=0]\\le\\frac d{|K|}.\n\\]\n\\end{lemma}\n\\begin{proof}\nInduct on the number of variables. For one variable this is the elementary root bound. In the inductive step, write\n\\[\n P=\\sum_{j=0}^r P_j(z_1,\\ldots,z_{m-1})z_m^j,\n \\qquad P_r\\ne0.\n\\]\nThe leading coefficient has total degree at most $d-r$, so it vanishes at a random first $(m-1)$-tuple with probability at most $(d-r)/|K|$. Otherwise the remaining univariate polynomial is nonzero with at most $r$ roots. The union bound gives $(d-r+r)/|K|=d/|K|$.\n\\end{proof}","[\n \\Pr_{z\\text{ uniform in }K^m}[P(z)=0]\\le\\frac d{|K|}.\n\\]\n\\end{lemma}\n\\begin{proof}\nInduct on the number of variables. For one variable this is the elementary root bound. In the inductive step, write\n\\[\n P=\\sum_{j=0}^r P_j(z_1,\\ldots,z_{m-1})z_m^j,\n \\qquad P_r\\ne0.\n\\]\nThe leading coefficient has total degree at most $d-r$, so it vanishes at a random first $(m-1)$-tuple with probability at most $(d-r)/|K|$. Otherwise the remaining univariate polynomial is nonzero with at most $r$ roots. The union bound gives $(d-r+r)/|K|=d/|K|$.\n\\end{proof}","f}\nInduct on the number of variables. For one variable this is the elementary root bound. In the inductive step, write\n\\[\n P=\\sum_{j=0}^r P_j(z_1,\\ldots,z_{m-1})z_m^j,\n \\qquad P_r\\ne0.\n\\]\nThe leading coefficient has total degree at most $d-r$, so it vanishes at a random first $(m-1)$-tuple with probability at most $(d-r)/|K|$. Otherwise the remaining univariate polynomial is nonzero with at most $r$ roots. The union bound gives $(d-r+r)/|K|=d/|K|$.\n\\end{proof}","les. For one variable this is the elementary root bound. In the inductive step, write\n\\[\n P=\\sum_{j=0}^r P_j(z_1,\\ldots,z_{m-1})z_m^j,\n \\qquad P_r\\ne0.\n\\]\nThe leading coefficient has total degree at most $d-r$, so it vanishes at a random first $(m-1)$-tuple with probability at most $(d-r)/|K|$. Otherwise the remaining univariate polynomial is nonzero with at most $r$ roots. The union bound gives $(d-r+r)/|K|=d/|K|$.\n\\end{proof}","\\begin{proposition}[Conditional randomized satisfiability test]\\label{c:prop:conditional-sat}\nSuppose the weighted expression in \\eqref{c:eq:weighted-top} can be evaluated exactly and uniformly in time polynomial in the verifier description and witness length, over explicitly represented characteristic-two fields of polynomial size. Then choosing $|K|\\ge4m$ gives a one-sided randomized polynomial-time satisfiability test. It never accepts an unsatisfiable verifier and accepts a satisfiable one with probability at least $3/4$ per evaluation.\n\\end{proposition}\n\\begin{proof}\nChoose a uniformly random $z\\in K^m$, evaluate $P_R(z)$ under the assumed algorithm, and accept when it is nonzero. If no witness accepts, Theorem~\\ref{c:thm:fingerprint} makes the polynomial zero identically, so there is no false acceptance. Otherwise Lemma~\\ref{c:lem:roots} bounds the rejection probability by $m/|K|\\le1/4$. A field of polynomial size uses $O(\\log m)$ bits per element, so the random input has $O(m\\log m)$ bits. Under the explicit efficient-field and evaluat","in \\eqref{c:eq:weighted-top} can be evaluated exactly and uniformly in time polynomial in the verifier description and witness length, over explicitly represented characteristic-two fields of polynomial size. Then choosing $|K|\\ge4m$ gives a one-sided randomized polynomial-time satisfiability test. It never accepts an unsatisfiable verifier and accepts a satisfiable one with probability at least $3/4$ per evaluation.\n\\end{proposition}\n\\begin{proof}\nChoose a uniformly random $z\\in K^m$, evaluate $P_R(z)$ under the assumed algorithm, and accept when it is nonzero. If no witness accepts, Theorem~\\ref{c:thm:fingerprint} makes the polynomial zero identically, so there is no false acceptance. Otherwise Lemma~\\ref{c:lem:roots} bounds the rejection probability by $m/|K|\\le1/4$. A field of polynomial size uses $O(\\log m)$ bits per element, so the random input has $O(m\\log m)$ bits. Under the explicit efficient-field and evaluator hypotheses, the procedure has polynomial running time.\n\\end{proof}","nomial in the verifier description and witness length, over explicitly represented characteristic-two fields of polynomial size. Then choosing $|K|\\ge4m$ gives a one-sided randomized polynomial-time satisfiability test. It never accepts an unsatisfiable verifier and accepts a satisfiable one with probability at least $3/4$ per evaluation.\n\\end{proposition}\n\\begin{proof}\nChoose a uniformly random $z\\in K^m$, evaluate $P_R(z)$ under the assumed algorithm, and accept when it is nonzero. If no witness accepts, Theorem~\\ref{c:thm:fingerprint} makes the polynomial zero identically, so there is no false acceptance. Otherwise Lemma~\\ref{c:lem:roots} bounds the rejection probability by $m/|K|\\le1/4$. A field of polynomial size uses $O(\\log m)$ bits per element, so the random input has $O(m\\log m)$ bits. Under the explicit efficient-field and evaluator hypotheses, the procedure has polynomial running time.\n\\end{proof}","r description and witness length, over explicitly represented characteristic-two fields of polynomial size. Then choosing $|K|\\ge4m$ gives a one-sided randomized polynomial-time satisfiability test. It never accepts an unsatisfiable verifier and accepts a satisfiable one with probability at least $3/4$ per evaluation.\n\\end{proposition}\n\\begin{proof}\nChoose a uniformly random $z\\in K^m$, evaluate $P_R(z)$ under the assumed algorithm, and accept when it is nonzero. If no witness accepts, Theorem~\\ref{c:thm:fingerprint} makes the polynomial zero identically, so there is no false acceptance. Otherwise Lemma~\\ref{c:lem:roots} bounds the rejection probability by $m/|K|\\le1/4$. A field of polynomial size uses $O(\\log m)$ bits per element, so the random input has $O(m\\log m)$ bits. Under the explicit efficient-field and evaluator hypotheses, the procedure has polynomial running time.\n\\end{proof}","e one with probability at least $3/4$ per evaluation.\n\\end{proposition}\n\\begin{proof}\nChoose a uniformly random $z\\in K^m$, evaluate $P_R(z)$ under the assumed algorithm, and accept when it is nonzero. If no witness accepts, Theorem~\\ref{c:thm:fingerprint} makes the polynomial zero identically, so there is no false acceptance. Otherwise Lemma~\\ref{c:lem:roots} bounds the rejection probability by $m/|K|\\le1/4$. A field of polynomial size uses $O(\\log m)$ bits per element, so the random input has $O(m\\log m)$ bits. Under the explicit efficient-field and evaluator hypotheses, the procedure has polynomial running time.\n\\end{proof}","ation.\n\\end{proposition}\n\\begin{proof}\nChoose a uniformly random $z\\in K^m$, evaluate $P_R(z)$ under the assumed algorithm, and accept when it is nonzero. If no witness accepts, Theorem~\\ref{c:thm:fingerprint} makes the polynomial zero identically, so there is no false acceptance. Otherwise Lemma~\\ref{c:lem:roots} bounds the rejection probability by $m/|K|\\le1/4$. A field of polynomial size uses $O(\\log m)$ bits per element, so the random input has $O(m\\log m)$ bits. Under the explicit efficient-field and evaluator hypotheses, the procedure has polynomial running time.\n\\end{proof}","mly random $z\\in K^m$, evaluate $P_R(z)$ under the assumed algorithm, and accept when it is nonzero. If no witness accepts, Theorem~\\ref{c:thm:fingerprint} makes the polynomial zero identically, so there is no false acceptance. Otherwise Lemma~\\ref{c:lem:roots} bounds the rejection probability by $m/|K|\\le1/4$. A field of polynomial size uses $O(\\log m)$ bits per element, so the random input has $O(m\\log m)$ bits. Under the explicit efficient-field and evaluator hypotheses, the procedure has polynomial running time.\n\\end{proof}","\\begin{circuitscopebox}{Conditional means the evaluator is still missing}\nThe proposition assumes a uniform efficient evaluator that has not been constructed here. The implemented evaluator is exponential. Even with the assumed evaluator, the stated conclusion is randomized with one-sided error, not a deterministic proof of $P=NP$.\n\\end{circuitscopebox}","The exact tests use $\\mathrm{GF}(16)=\\F[t]/(t^4+t+1)$. They evaluate every specified verifier through three witness bits at every field point, while the independent top-coefficient check uses all field points through two witness bits and the declared smaller coordinate grid at three bits. These domains must not be conflated.","\\begin{proposition}[Disjoint-variable factorization]\\label{c:prop:disjoint}\nIf predicates $A(u)$ and $B(v)$ use disjoint witness-variable sets, then their weighted fingerprints satisfy\n\\[\n P_{A\\land B}(z_u,z_v)=P_A(z_u)P_B(z_v).\n\\]\n\\end{proposition}\n\\begin{proof}\nExpand the left side as a sum over the Cartesian product of witnesses:\n\\[\n \\sum_{u,v}A(u)B(v)z_u^uz_v^v\n =\\left(\\sum_uA(u)z_u^u\\right)\n  \\left(\\sum_vB(v)z_v^v\\right).\n\\]\nDisjointness makes the exponent data independent and the factorization exact.\n\\end{proof}","\\begin{counterexample}[Shared-variable multiplication fails]\\label{c:ce:overlap}\nFor a single witness bit, define $\\KTr_z(A)=A(0)+zA(1)$. Then in the Boolean $w$ algebra,\n\\[\n \\KTr_z(w)=z,\\qquad \\KTr_z(1+w)=1,\n\\]\nbut\n\\[\n \\KTr_z\\bigl(w(1+w)\\bigr)=0\n \\ne z=\\KTr_z(w)\\KTr_z(1+w)\n\\]\nwhenever $z\\ne0$. Also,\n\\[\n \\KTr_z(w^2)=z,\\qquad\\KTr_z(w)^2=z^2,\n\\]\nwhich differ for field elements $z$ with $z^2\\ne z$.\n\\end{counterexample}\n\\begin{proof}\nEvaluate $w$ and $1+w$ at zero and one. In the Boolean quotient $w(1+w)=w+w^2=0$ and $w^2=w$. The displayed identities follow directly. In $\\mathrm{GF}(16)$, any element outside $\\F$ supplies the second inequality.\n\\end{proof}","plication fails]\\label{c:ce:overlap}\nFor a single witness bit, define $\\KTr_z(A)=A(0)+zA(1)$. Then in the Boolean $w$ algebra,\n\\[\n \\KTr_z(w)=z,\\qquad \\KTr_z(1+w)=1,\n\\]\nbut\n\\[\n \\KTr_z\\bigl(w(1+w)\\bigr)=0\n \\ne z=\\KTr_z(w)\\KTr_z(1+w)\n\\]\nwhenever $z\\ne0$. Also,\n\\[\n \\KTr_z(w^2)=z,\\qquad\\KTr_z(w)^2=z^2,\n\\]\nwhich differ for field elements $z$ with $z^2\\ne z$.\n\\end{counterexample}\n\\begin{proof}\nEvaluate $w$ and $1+w$ at zero and one. In the Boolean quotient $w(1+w)=w+w^2=0$ and $w^2=w$. The displayed identities follow directly. In $\\mathrm{GF}(16)$, any element outside $\\F$ supplies the second inequality.\n\\end{proof}","n fails]\\label{c:ce:overlap}\nFor a single witness bit, define $\\KTr_z(A)=A(0)+zA(1)$. Then in the Boolean $w$ algebra,\n\\[\n \\KTr_z(w)=z,\\qquad \\KTr_z(1+w)=1,\n\\]\nbut\n\\[\n \\KTr_z\\bigl(w(1+w)\\bigr)=0\n \\ne z=\\KTr_z(w)\\KTr_z(1+w)\n\\]\nwhenever $z\\ne0$. Also,\n\\[\n \\KTr_z(w^2)=z,\\qquad\\KTr_z(w)^2=z^2,\n\\]\nwhich differ for field elements $z$ with $z^2\\ne z$.\n\\end{counterexample}\n\\begin{proof}\nEvaluate $w$ and $1+w$ at zero and one. In the Boolean quotient $w(1+w)=w+w^2=0$ and $w^2=w$. The displayed identities follow directly. In $\\mathrm{GF}(16)$, any element outside $\\F$ supplies the second inequality.\n\\end{proof}","ss bit, define $\\KTr_z(A)=A(0)+zA(1)$. Then in the Boolean $w$ algebra,\n\\[\n \\KTr_z(w)=z,\\qquad \\KTr_z(1+w)=1,\n\\]\nbut\n\\[\n \\KTr_z\\bigl(w(1+w)\\bigr)=0\n \\ne z=\\KTr_z(w)\\KTr_z(1+w)\n\\]\nwhenever $z\\ne0$. Also,\n\\[\n \\KTr_z(w^2)=z,\\qquad\\KTr_z(w)^2=z^2,\n\\]\nwhich differ for field elements $z$ with $z^2\\ne z$.\n\\end{counterexample}\n\\begin{proof}\nEvaluate $w$ and $1+w$ at zero and one. In the Boolean quotient $w(1+w)=w+w^2=0$ and $w^2=w$. The displayed identities follow directly. In $\\mathrm{GF}(16)$, any element outside $\\F$ supplies the second inequality.\n\\end{proof}","z(A)=A(0)+zA(1)$. Then in the Boolean $w$ algebra,\n\\[\n \\KTr_z(w)=z,\\qquad \\KTr_z(1+w)=1,\n\\]\nbut\n\\[\n \\KTr_z\\bigl(w(1+w)\\bigr)=0\n \\ne z=\\KTr_z(w)\\KTr_z(1+w)\n\\]\nwhenever $z\\ne0$. Also,\n\\[\n \\KTr_z(w^2)=z,\\qquad\\KTr_z(w)^2=z^2,\n\\]\nwhich differ for field elements $z$ with $z^2\\ne z$.\n\\end{counterexample}\n\\begin{proof}\nEvaluate $w$ and $1+w$ at zero and one. In the Boolean quotient $w(1+w)=w+w^2=0$ and $w^2=w$. The displayed identities follow directly. In $\\mathrm{GF}(16)$, any element outside $\\F$ supplies the second inequality.\n\\end{proof}","\\KTr_z(w)=z,\\qquad \\KTr_z(1+w)=1,\n\\]\nbut\n\\[\n \\KTr_z\\bigl(w(1+w)\\bigr)=0\n \\ne z=\\KTr_z(w)\\KTr_z(1+w)\n\\]\nwhenever $z\\ne0$. Also,\n\\[\n \\KTr_z(w^2)=z,\\qquad\\KTr_z(w)^2=z^2,\n\\]\nwhich differ for field elements $z$ with $z^2\\ne z$.\n\\end{counterexample}\n\\begin{proof}\nEvaluate $w$ and $1+w$ at zero and one. In the Boolean quotient $w(1+w)=w+w^2=0$ and $w^2=w$. The displayed identities follow directly. In $\\mathrm{GF}(16)$, any element outside $\\F$ supplies the second inequality.\n\\end{proof}","ut\n\\[\n \\KTr_z\\bigl(w(1+w)\\bigr)=0\n \\ne z=\\KTr_z(w)\\KTr_z(1+w)\n\\]\nwhenever $z\\ne0$. Also,\n\\[\n \\KTr_z(w^2)=z,\\qquad\\KTr_z(w)^2=z^2,\n\\]\nwhich differ for field elements $z$ with $z^2\\ne z$.\n\\end{counterexample}\n\\begin{proof}\nEvaluate $w$ and $1+w$ at zero and one. In the Boolean quotient $w(1+w)=w+w^2=0$ and $w^2=w$. The displayed identities follow directly. In $\\mathrm{GF}(16)$, any element outside $\\F$ supplies the second inequality.\n\\end{proof}","Theorem~\\ref{c:thm:gluing-dim} explains a related all-size restriction: a fixed linear multiplicative summary retaining every witness cannot reduce the full witness algebra below dimension $2^m$. That statement does not rule out all context-dependent nonlinear evaluation methods. It rules out the displayed context-free multiplicative compiler under its exact hypotheses.","\\section{Dependency structure}\nThe main proof dependencies are:\n\\[\n\\begin{gathered}\n\\text{restored oracle convention}\\ \\Longrightarrow\\ \\text{cut comparison},\\\\\n\\text{cut comparison}+\\text{degree per multiplication}\\\n\\Longrightarrow\\ \\text{complete-cover lower bound},\\\\\n\\text{safe ideal}+\\text{finite linear duality}\\\n\\Longrightarrow\\ \\text{evaluation-span certificates},\\\\\n\\text{monotone ideal}+\\text{affine input/output changes}\\\n\\Longrightarrow\\ \\text{calibration formulas},\\\\\n\\text{affine restriction}+\\text{quadratic polar rank}\\\n\\Longrightarrow\\ \\text{projection-cost lower bound},\\\\\n\\text{orthogonal idempotents}\\ \\Longrightarrow\\ \\text{gluing dimension},\\\\\n\\text{Boolean coefficient identity}+\\text{root bound}\\\n\\Longrightarrow\\ \\text{conditional fingerprint test}.\n\\end{gathered}\n\\]\nThe gluing-dimension theorem and fingerprint construction are separate from the circuit-cut theorem. No proof step identifies algebra dimension with unrestricted circuit size.","Projection coefficients & 693,965 pair coefficients and nine exact rank cases, through 1,022 inputs & Full truth tables only at $m=2,3,4$","Independent coefficient extraction & 20,544 cases & All field points for $m\\le2$; restricted coordinate grid at $m=3$","The field used in the fingerprint checker is\n\\[\n \\mathrm{GF}(16)=\\F[t]/(t^4+t+1).\n\\]\nAll mathematical arithmetic in these tests is Boolean, exact integer, or finite-field arithmetic. Runtime is not presented as a complexity benchmark. Equality of checksums certifies matching bytes, not mathematical truth.","\\guidepage{The model: boundary restoration and Boolean degree}\nA state has logical input wires $x$, zero-initialized ancillary wires, and a\ntarget bit $y$. A restored oracle is a legal classical circuit satisfying\n\\[\n C(x,0^a,y)=(x,0^a,y+f(x))\n \\qquad\\text{for every }x\\text{ and both }y.\n\\]\nThe input and scratch restoration conditions hold at the boundary. Wires may\nchange internally, and the target may control an intermediate gate. Each\nprimitive gate is an involution:\n\\[\n y\\leftarrow y+1,\\quad y\\leftarrow y+x_i,\\quad\n y\\leftarrow y+x_i x_j.\n\\]\nIn a CNOT the control differs from the target. In a Toffoli all three wires\nare distinct. These are construction rules, not informal assumptions.","gical input wires $x$, zero-initialized ancillary wires, and a\ntarget bit $y$. A restored oracle is a legal classical circuit satisfying\n\\[\n C(x,0^a,y)=(x,0^a,y+f(x))\n \\qquad\\text{for every }x\\text{ and both }y.\n\\]\nThe input and scratch restoration conditions hold at the boundary. Wires may\nchange internally, and the target may control an intermediate gate. Each\nprimitive gate is an involution:\n\\[\n y\\leftarrow y+1,\\quad y\\leftarrow y+x_i,\\quad\n y\\leftarrow y+x_i x_j.\n\\]\nIn a CNOT the control differs from the target. In a Toffoli all three wires\nare distinct. These are construction rules, not informal assumptions.","All polynomial arithmetic in the Boolean state model is over $\\F_2$, with\n$x_i^2=x_i$. Every function has a unique multilinear algebraic normal form.\nIts degree is the largest support size with a nonzero coefficient; the zero\nfunction has mathematical degree $-1$ in the Python helpers. A formal degree\nbound uses a natural number, so zero has a degree-zero certificate.","midrule\n$T$ & Number of Toffoli gates; NOT and CNOT have zero cost in this quantity","$L$ & Total number of NOT, CNOT, and Toffoli gates","$M$ & Number of binary AND operations in an XOR--AND computation","$n$ & Actual number of input bits, including all variables in a displayed family","$\\dim_K A$ & Dimension of a specified linear summary algebra over $K$","\\guidepage{An arbitrary cut compares two reversible computations}\nWrite the circuit as $C=Q\\circ P$. The prefix $P$ runs forward; the inverse\nsuffix $Q^{-1}$ starts from the same zero-target boundary state. Define\n\\[\n A(x)=P(x,0),\\qquad B(x)=Q^{-1}(x,0),\\qquad\n D(x)=P(x,1)+P(x,0).\n\\]\nThe zero ancillas are suppressed here. The two executions generally arrive\nat different interior states, even though they start at matching boundaries.","Correctness on both target branches gives\n\\[\n Q^{-1}(x,b)=P(x,b+f(x)).\n\\]\nTo derive it, apply $C$ to $(x,b+f(x))$ and then apply $Q^{-1}$ to the result.\nFor $b=0$, $B=A$ when $f(x)=0$, while $B=P(x,1)$ when $f(x)=1$. Thus\n\\[\n H(x):=A(x)+B(x)=f(x)D(x).\n\\]\nThe two original states differ in their target. Since $P$ is injective,\n$D(x)\\ne0$. Each coordinate discrepancy is supported on positives, and\nsome coordinate discrepancy detects every positive input.","\\guidepage{Why completeness is pointwise, and why no decoder is needed}\nFor a coordinate $i$, let $H_i=A_i+B_i$. The cut identity says\n\\[\n H_i(x)=f(x)D_i(x),\\qquad H_i\\le f,\\qquad \\bigvee_i H_i=f.\n\\]\nThe last equation is pointwise. For every fixed accepted $x$, at least one\ncoordinate has $D_i(x)=1$. The detecting coordinate may depend on $x$ and\nmay be an input, scratch, or target wire. The argument does not require a\nsingle fixed coordinate to detect all positives.","At an empty prefix, the direction is simply the target unit vector. At a\nfull prefix, all non-target wires have restored and the target discrepancy\nis $f$. Interior cuts can spread the direction through the workspace.\nThese minimum cases verify the composition order and the role of the two\ntarget branches.","\\begin{quote}\\small\n\\textbf{Essential hypothesis.} The endpoint condition holds for both initial\ntarget values. The inverse-suffix derivation uses the branch $b+f(x)$.\nCorrectness only for initial target zero is not the stated model.\n\\end{quote}","The theorem also does not require scratch to remain zero at the cut. Imposing\nthat extra condition would exclude admitted restored circuits. Conversely,\ndropping target separation from the boundary encoding permits a vacuous\nendpoint equation: an encoding that ignores $y$ can make the empty circuit\nappear to compute a nonzero function. The formal \\code{Boundary.target_eq}\nproperty prevents precisely this degeneracy.","\\guidepage{A dirty scratch wire can be fully restored}\nConsider the four-wire oracle in the Lean examples. First compute $ab$ into\nscratch $s$, copy $s$ into the target, and uncompute $s$:\n\\[\n (a,b,s,y)\\xrightarrow{T(a,b;s)}(a,b,s+ab,y)\n \\xrightarrow{C(s;y)}(a,b,s+ab,y+s+ab)\n \\xrightarrow{T(a,b;s)}(a,b,s,y+s+ab).\n\\]\nThe displayed final expression specializes to the required oracle when\nthe boundary scratch is zero. On the positive input $a=b=1$, the scratch\nis dirty between the first and last gates.","ly restored}\nConsider the four-wire oracle in the Lean examples. First compute $ab$ into\nscratch $s$, copy $s$ into the target, and uncompute $s$:\n\\[\n (a,b,s,y)\\xrightarrow{T(a,b;s)}(a,b,s+ab,y)\n \\xrightarrow{C(s;y)}(a,b,s+ab,y+s+ab)\n \\xrightarrow{T(a,b;s)}(a,b,s,y+s+ab).\n\\]\nThe displayed final expression specializes to the required oracle when\nthe boundary scratch is zero. On the positive input $a=b=1$, the scratch\nis dirty between the first and last gates.","\\visualfigure{dirty_and_trace}{Exact states for the accepting input, in wire order $(a,b,s,y)$. White cells are zero; teal cells are one.}","This circuit uses two Toffolis. It is a useful restoration witness, not the\noptimal two-input AND oracle: the direct gate $T(a,b;y)$ computes that oracle\nwith one Toffoli and no scratch. The guide distinguishes an exact example\nfrom an optimization claim.","\\guidepage{The output direction can move away from the target}\nTwo data/target wires already suffice. A SWAP is implemented by three legal\nCNOTs. Let $P$ be that SWAP, and let $Q$ be NOT on the data wire followed\nby another SWAP. Then\n\\[\n (x,y)\\xrightarrow{P}(y,x)\n \\xrightarrow{Q}(x,1+y).\n\\]\nThe whole circuit is a restored constant-one oracle with zero Toffolis.\nAt the cut after the first SWAP,\n\\[\n A=P(x,0)=(0,x),\\qquad B=Q^{-1}(x,0)=(1,x),\n \\qquad H=(1,0).\n\\]\nThe target discrepancy is zero on every input, while the data discrepancy\nis one on every input. Reading only the target at this cut would reject\nthe constant-one function.","\\begin{center}\n\\begin{tabular}{@{}ccccc@{}}\\toprule\n$x$ & $A$ & $B$ & $H$ & named target discrepancy","midrule\n0 & $(0,0)$ & $(1,0)$ & $(1,0)$ & 0","1 & $(0,1)$ & $(1,1)$ & $(1,0)$ & 0","\\guidepage{Degree bounds must retain shared register products}\nTake four independent input bits. Compute\n\\[\n q_1=x_1x_2,\\qquad q_2=x_3x_4,\\qquad q_3=q_1q_2.\n\\]\nAfter two ANDs, each nonlinear register has degree two, but their product\nhas degree four. After the third AND, a coordinate of degree four appears.","The correct invariant tracks a product of $r$ current registers after $t$\nToffolis:\n\\[\n \\deg\\!\\left(\\prod_{i\\in L}s_i(x)\\right)\\le t+|L|=t+r.\n\\]\nAt $t=2$, the product $q_1q_2$ has $r=2$ and saturates $t+r=4$. At $t=3$,\nthe resulting coordinate has $r=1$ and saturates $t+1=4$.","\\guidepage{The invariant handles duplicates, NOT, and CNOT}\nSuppose a gate toggles target $u$ by a product of $c$ controls. In a product\nlist $L$ that contains $u$, remove all occurrences of $u$ to obtain $R$.\nBoolean idempotence leaves one updated target factor, and\n\\[\n \\prod_{i\\in L}s_i'=\n s_u\\prod_{i\\in R}s_i+\n \\left(\\prod_{j\\in\\mathrm{controls}}s_j\\right)\\prod_{i\\in R}s_i.\n\\]\nThe old invariant bounds these terms by $t+1+|R|$ and $t+c+|R|$.\nSince $|R|+1\\le |L|$, the budget rises by at most $\\max\\{c-1,0\\}$.","handles duplicates, NOT, and CNOT}\nSuppose a gate toggles target $u$ by a product of $c$ controls. In a product\nlist $L$ that contains $u$, remove all occurrences of $u$ to obtain $R$.\nBoolean idempotence leaves one updated target factor, and\n\\[\n \\prod_{i\\in L}s_i'=\n s_u\\prod_{i\\in R}s_i+\n \\left(\\prod_{j\\in\\mathrm{controls}}s_j\\right)\\prod_{i\\in R}s_i.\n\\]\nThe old invariant bounds these terms by $t+1+|R|$ and $t+c+|R|$.\nSince $|R|+1\\le |L|$, the budget rises by at most $\\max\\{c-1,0\\}$.","cates, NOT, and CNOT}\nSuppose a gate toggles target $u$ by a product of $c$ controls. In a product\nlist $L$ that contains $u$, remove all occurrences of $u$ to obtain $R$.\nBoolean idempotence leaves one updated target factor, and\n\\[\n \\prod_{i\\in L}s_i'=\n s_u\\prod_{i\\in R}s_i+\n \\left(\\prod_{j\\in\\mathrm{controls}}s_j\\right)\\prod_{i\\in R}s_i.\n\\]\nThe old invariant bounds these terms by $t+1+|R|$ and $t+c+|R|$.\nSince $|R|+1\\le |L|$, the budget rises by at most $\\max\\{c-1,0\\}$.","NOT}\nSuppose a gate toggles target $u$ by a product of $c$ controls. In a product\nlist $L$ that contains $u$, remove all occurrences of $u$ to obtain $R$.\nBoolean idempotence leaves one updated target factor, and\n\\[\n \\prod_{i\\in L}s_i'=\n s_u\\prod_{i\\in R}s_i+\n \\left(\\prod_{j\\in\\mathrm{controls}}s_j\\right)\\prod_{i\\in R}s_i.\n\\]\nThe old invariant bounds these terms by $t+1+|R|$ and $t+c+|R|$.\nSince $|R|+1\\le |L|$, the budget rises by at most $\\max\\{c-1,0\\}$.","product of $c$ controls. In a product\nlist $L$ that contains $u$, remove all occurrences of $u$ to obtain $R$.\nBoolean idempotence leaves one updated target factor, and\n\\[\n \\prod_{i\\in L}s_i'=\n s_u\\prod_{i\\in R}s_i+\n \\left(\\prod_{j\\in\\mathrm{controls}}s_j\\right)\\prod_{i\\in R}s_i.\n\\]\nThe old invariant bounds these terms by $t+1+|R|$ and $t+c+|R|$.\nSince $|R|+1\\le |L|$, the budget rises by at most $\\max\\{c-1,0\\}$.","\\begin{center}\n\\begin{tabular}{@{}lccc@{}}\\toprule\nGate & $c$ & toggle expression & extra budget","If the target is absent from $L$, the product is unchanged. An empty list\nis the constant-one product. Repeated target indices are removed using\nidempotence; repeated other indices can only reduce multilinear degree.\nControls already appearing in $R$ also cause no difficulty.","For general XOR--AND computations, the written proof introduces a Boolean\nvariable for each AND output. Every wire is affine in inputs and preceding\ngate variables. Substitute those variables in reverse topological order.\nA squarefree monomial contains each substituted variable at most once,\nso replacing it by a quadratic raises degree by at most one. After $t$\nsubstitutions, the affine output has degree at most $t+1$.","AND computations, the written proof introduces a Boolean\nvariable for each AND output. Every wire is affine in inputs and preceding\ngate variables. Substitute those variables in reverse topological order.\nA squarefree monomial contains each substituted variable at most once,\nso replacing it by a quadratic raises degree by at most one. After $t$\nsubstitutions, the affine output has degree at most $t+1$.","\\guidepage{One safe packet need not cover every accepting point}\nFor $k=3$ consider\n\\[\n F_3(a,b,z_1,z_2,z_3)=ab\\ \\mathrm{OR}\\ z_1z_2z_3.\n\\]\nThe packet $ab$ is safe and has degree two. It detects every teal point\nbelow. It misses three additional positives, which require the other\npacket $z_1z_2z_3$.","\\visualfigure{packet_coverage}{All 32 assignments. There are 11 positives: 8 in the teal row and 3 additional gold points. The coordinate labels use $(z_1,z_2,z_3)$ order.}","The one-packet parameter $\\alpha_1$ asks for any nonzero safe polynomial.\nThe complete parameter $\\beta$ asks for safe detection of every positive:\n\\[\n \\beta_v(f)=\\min\\{\\deg h:h\\le f,\\ h(v)=1\\},\\qquad\n \\beta(f)=\\max_{v:f(v)=1}\\beta_v(f).\n\\]\nFor this monotone function, its minimal positive supports have sizes two\nand three. Thus $\\alpha_1(F_3)=2$ while $\\beta(F_3)=3$. Selecting one\neasy accepting region loses the pointwise obstruction at the other region.","\\guidepage{Balanced cuts turn coverage into a Toffoli bound}\nChoose a cut after $\\lfloor T/2\\rfloor$ Toffolis. Both sides have at most\n$\\lceil T/2\\rceil$ Toffolis. Reversing the suffix preserves its count because\nevery gate is self-inverse. Each forward or inverse-suffix coordinate has\ndegree at most $\\lceil T/2\\rceil+1$, and their XOR has the same upper bound.","For every positive $v$, completeness chooses a discrepancy $H_i$ that\ndetects $v$. Its safety and degree bound imply\n\\[\n \\beta(f)\\le\\left\\lceil\\frac T2\\right\\rceil+1,\n \\qquad 2\\beta(f)\\le T+3.\n\\]\nThe integer rounding matters. If $T=2r$, the first bound gives\n$2\\beta\\le T+2$. If $T=2r+1$, it gives $2\\beta\\le T+3$.\nThe uniform bound uses the latter constant.","An affine output mask $\\ell(x)$ costs zero additional Toffolis: append\nCNOTs from restored inputs and, if needed, a target NOT. This constructs\na restored oracle for $f+\\ell$, so the bound applies to every affine mask.\nDefine\n\\[\n \\beta^*(f)=\\max_{\\ell\\text{ affine}}\\beta(f+\\ell).\n\\]\nTogether with the output degree bound,\n\\[\n \\boxed{T\\ge\\max\\{0,\\deg f-1,2\\beta^*(f)-3\\}.}\n\\]","\\guidepage{An affine shift reveals a harder accepting point}\nFor $k\\ge2$, let $B=z_1\\cdots z_k$ and $F_k=ab\\ \\mathrm{OR}\\ B$ on\n$n=k+2$ inputs. Add the affine mask $a$ to obtain $G_k=F_k+a$.\nOn the face $b=1$,\n\\[\n G_k|_{b=1}=(1+a)B.\n\\]\nThis is a singleton indicator on the $k+1$ variables $(a,z)$.\nAt $v=(a=0,b=1,z=1^k)$, every safe detecting polynomial must restrict to\nthat singleton indicator, so its degree is at least $k+1$.","ft reveals a harder accepting point}\nFor $k\\ge2$, let $B=z_1\\cdots z_k$ and $F_k=ab\\ \\mathrm{OR}\\ B$ on\n$n=k+2$ inputs. Add the affine mask $a$ to obtain $G_k=F_k+a$.\nOn the face $b=1$,\n\\[\n G_k|_{b=1}=(1+a)B.\n\\]\nThis is a singleton indicator on the $k+1$ variables $(a,z)$.\nAt $v=(a=0,b=1,z=1^k)$, every safe detecting polynomial must restrict to\nthat singleton indicator, so its degree is at least $k+1$.","The general two-positive indicator construction gives the matching upper\nbound $\\beta(G_k)\\le n-1=k+1$. The maintained family proves\n\\[\n \\max_\\ell\\alpha_1(F_k+\\ell)=3,\\qquad \\beta^*(F_k)=k+1.\n\\]\nThe gap between one-packet and complete degree therefore grows with $k$.","\\begin{center}\n\\begin{tabular}{@{}rrrrr@{}}\\toprule\n$k$ & $n$ & $\\beta^*$ & complete bound $2k-1$ & combined degree/one-packet bound","For $k=3$, all 64 affine masks were freshly checked and the optimized\nparameters are exactly $(3,4)$. The all-size family proof is not inferred\nfrom those 64 cases. Its restored construction has $T\\le2n-3$, so this\nremains a linear-size easy family. The exact optimum between the lower and\nupper constructions is not asserted when they differ.","\\guidepage{A degree-one dual certificate for two-input AND}\nLet $f(a,b)=ab$, with unique positive point $(1,1)$. For degree at most one,\nthe evaluation vector is $v_1(a,b)=(1,a,b)$. The three negative points\nalready span the positive vector.","Any affine polynomial is a coefficient covector $c=(c_0,c_a,c_b)$.\nSafety means that it pairs to zero with each negative row. The displayed\ndependence therefore forces its value at $(1,1)$ to be zero. No safe affine\npacket detects the positive point.","The degree-two polynomial $ab$ does detect it, so\n$\\beta_{11}(f)=\\beta(f)=2$. Since $\\beta^*\\le n=2$ and $\\beta^*\\ge\\beta$,\n$\\beta^*=2$. The complete bound gives $T\\ge1$, attained by one direct\nToffoli to the target.","\\guidepage{The finite evaluation-span dual in full}\nFix a positive $v$ and integer $d\\ge0$. Let\n\\[\n v_d(x)=(x^S)_{S\\subseteq[n],\\ |S|\\le d},\\qquad\n U=f^{-1}(0),\\qquad N=\\Span_{\\F_2}\\{v_d(u):u\\in U\\}.\n\\]\nThe constant monomial $S=\\varnothing$ is included. Degree-$d$ coefficient\ncovectors form the finite dual of this evaluation space. A covector is safe\nexactly when it annihilates $N$.","If $v_d(v)\\in N$, every safe covector vanishes at $v$. If $v_d(v)\\notin N$,\nextend a basis of $N$ by $v_d(v)$ and then to a basis of the full space.\nAssign zero to the $N$ basis and one to $v_d(v)$. The resulting covector is\na safe detector. Hence\n\\[\n \\boxed{\\beta_v(f)>d\\quad\\Longleftrightarrow\\quad v_d(v)\\in N.}\n\\]\nChoose a basis of $N$ from negative evaluation rows. A span representation\nof the positive row gives a subset $W\\subseteq U$ with\n\\[\n v_d(v)=\\sum_{u\\in W}v_d(u),\\quad |W|\\text{ odd},\\quad\n |W|\\le\\sum_{j=0}^{\\min(d,n)}\\binom nj.\n\\]\nIt is a replayable finite obstruction to every safe degree-$d$ detector.","covector vanishes at $v$. If $v_d(v)\\notin N$,\nextend a basis of $N$ by $v_d(v)$ and then to a basis of the full space.\nAssign zero to the $N$ basis and one to $v_d(v)$. The resulting covector is\na safe detector. Hence\n\\[\n \\boxed{\\beta_v(f)>d\\quad\\Longleftrightarrow\\quad v_d(v)\\in N.}\n\\]\nChoose a basis of $N$ from negative evaluation rows. A span representation\nof the positive row gives a subset $W\\subseteq U$ with\n\\[\n v_d(v)=\\sum_{u\\in W}v_d(u),\\quad |W|\\text{ odd},\\quad\n |W|\\le\\sum_{j=0}^{\\min(d,n)}\\binom nj.\n\\]\nIt is a replayable finite obstruction to every safe degree-$d$ detector.","$v$. If $v_d(v)\\notin N$,\nextend a basis of $N$ by $v_d(v)$ and then to a basis of the full space.\nAssign zero to the $N$ basis and one to $v_d(v)$. The resulting covector is\na safe detector. Hence\n\\[\n \\boxed{\\beta_v(f)>d\\quad\\Longleftrightarrow\\quad v_d(v)\\in N.}\n\\]\nChoose a basis of $N$ from negative evaluation rows. A span representation\nof the positive row gives a subset $W\\subseteq U$ with\n\\[\n v_d(v)=\\sum_{u\\in W}v_d(u),\\quad |W|\\text{ odd},\\quad\n |W|\\le\\sum_{j=0}^{\\min(d,n)}\\binom nj.\n\\]\nIt is a replayable finite obstruction to every safe degree-$d$ detector.","ing covector is\na safe detector. Hence\n\\[\n \\boxed{\\beta_v(f)>d\\quad\\Longleftrightarrow\\quad v_d(v)\\in N.}\n\\]\nChoose a basis of $N$ from negative evaluation rows. A span representation\nof the positive row gives a subset $W\\subseteq U$ with\n\\[\n v_d(v)=\\sum_{u\\in W}v_d(u),\\quad |W|\\text{ odd},\\quad\n |W|\\le\\sum_{j=0}^{\\min(d,n)}\\binom nj.\n\\]\nIt is a replayable finite obstruction to every safe degree-$d$ detector.","At $d=0$, a negative exists exactly when the constant-one detector is unsafe.\nAt $d\\ge n$, the full evaluation rows are independent and no positive row\nlies in the negative span. This matches $\\beta_v\\le n$ from point indicators.\nFor $f=0$, there is no positive $v$ and the separate convention is $\\beta(0)=0$.","Fresh enumeration checked all 278 functions on zero through three inputs,\nall positives, and every $d=0,\\ldots,n$: 4,201 primal/dual checks agree.\nThose bounded checks support the independently supplied universal proof.","\\visualfigure{small_oracle_counts}{Reconstructed source constructions for all 276 functions. Totals are 28 with $T=0$, 120 with $T=1$, and 128 with $T=3$.}","\\guidepage{Projection is defined in the original coordinates}\nFor $m\\ge2$, put $a=(1,\\ldots,1)$ and\n\\[\n V=\\F_2^m\\setminus\\{0,a\\},\\qquad n=2^m-2.\n\\]\nThere is one input bit $x_v$ for each $v\\in V$. Define\n\\[\n L_j(x)=\\sum_{v:v_j=1}x_v,\\qquad\n P_m(x)=\\prod_{j=1}^m L_j(x).\n\\]\nThe linear map $x\\mapsto L(x)$ has rank $m$, because the standard basis\ncolumns are present. In the $L$ coordinates, $P_m$ is a monomial in $m$\nindependent bits. Its restored nonlinear cost is $2m-3$.","ojection is defined in the original coordinates}\nFor $m\\ge2$, put $a=(1,\\ldots,1)$ and\n\\[\n V=\\F_2^m\\setminus\\{0,a\\},\\qquad n=2^m-2.\n\\]\nThere is one input bit $x_v$ for each $v\\in V$. Define\n\\[\n L_j(x)=\\sum_{v:v_j=1}x_v,\\qquad\n P_m(x)=\\prod_{j=1}^m L_j(x).\n\\]\nThe linear map $x\\mapsto L(x)$ has rank $m$, because the standard basis\ncolumns are present. In the $L$ coordinates, $P_m$ is a monomial in $m$\nindependent bits. Its restored nonlinear cost is $2m-3$.","$m\\ge2$, put $a=(1,\\ldots,1)$ and\n\\[\n V=\\F_2^m\\setminus\\{0,a\\},\\qquad n=2^m-2.\n\\]\nThere is one input bit $x_v$ for each $v\\in V$. Define\n\\[\n L_j(x)=\\sum_{v:v_j=1}x_v,\\qquad\n P_m(x)=\\prod_{j=1}^m L_j(x).\n\\]\nThe linear map $x\\mapsto L(x)$ has rank $m$, because the standard basis\ncolumns are present. In the $L$ coordinates, $P_m$ is a monomial in $m$\nindependent bits. Its restored nonlinear cost is $2m-3$.","Now extract the quadratic homogeneous ANF component $Q_m$ in the\n\\emph{original} $x_v$ coordinates. Since $P_m(0)=P_m(e_v)=0$, for distinct\n$u,v$ its quadratic coefficient is\n\\[\n [x_ux_v]P_m=P_m(e_u+e_v)=[u+v=a].\n\\]\nThe involution $v\\mapsto v+a$ pairs every variable with one other variable:\n\\[\n Q_m=\\sum_{\\{v,v+a\\}}x_vx_{v+a}.\n\\]\nIt is a quadratic inner product on $n/2$ disjoint pairs. Its alternating\npolar matrix has full rank $n$.","neous ANF component $Q_m$ in the\n\\emph{original} $x_v$ coordinates. Since $P_m(0)=P_m(e_v)=0$, for distinct\n$u,v$ its quadratic coefficient is\n\\[\n [x_ux_v]P_m=P_m(e_u+e_v)=[u+v=a].\n\\]\nThe involution $v\\mapsto v+a$ pairs every variable with one other variable:\n\\[\n Q_m=\\sum_{\\{v,v+a\\}}x_vx_{v+a}.\n\\]\nIt is a quadratic inner product on $n/2$ disjoint pairs. Its alternating\npolar matrix has full rank $n$.","The coordinate choice is essential. Homogeneous grade is not invariant\nunder arbitrary affine changes of variables. A cheap representation in\nthe $L$ coordinates does not imply cheap independent generation of the\nquadratic grade in the original coordinates. The comparison keeps the\ndefinition of projection fixed throughout.","\\guidepage{Exact nonlinear projection costs versus actual input length}\nThe source proves the all-size formulas\n\\[\n M(P_m)=m-1,\\quad T_{\\mathrm{rest}}(P_m)=2m-3,\\qquad\n M(Q_m)=T_{\\mathrm{rest}}(Q_m)=n/2.\n\\]\nThe chart uses actual input length $n=2^m-2$ on its horizontal axis.","\\visualfigure{projection_costs}{Exact formulas for $m=2,\\ldots,10$. Both axes are logarithmic; the plotted points are not optimal-circuit search results.}","For $Q_m$, each disjoint quadratic pair is accumulated directly into the\ntarget with one Toffoli. The lower bound allows nonlinear intermediates:\nany computation with $t$ multiplications is affine on an affine subspace\nof codimension at most $t$. A nondegenerate quadratic can be affine only\non a subspace with isotropic direction space, whose dimension is at most\n$n/2$. Thus $t\\ge n/2$.","The separation grows exponentially in $m$, but $n$ itself grows\nexponentially in $m$. The quadratic cost is linear in actual input length.\nThe conclusion is about nonlinear generation and a specified grade\nextraction operation, not a superpolynomial unrestricted circuit bound.","\\evidence{All-size written cost proofs; constructor gate counts freshly\nasserted through $m=10$. No full truth-table enumeration at $m>4$ is claimed.}","\\guidepage{At $m=10$, the total-gate accounting points the other way}\nHere $n=1,022$. The full function uses 17 Toffolis; its quadratic grade\nneeds 511. However, computing and uncomputing the $m$ parities also uses\n\\[\n 2m(2^{m-1}-1)=20\\cdot511=10,220\\quad\\text{CNOTs}.\n\\]\nThe explicit full construction has $10,237$ total gates. The quadratic\naccumulation uses only 511 total gates.","\\visualfigure{projection_m10}{Two different resource comparisons at the same input length. The right panel reports the totals of the stated constructions, not an asserted optimal total for $P_{10}$.}","The left panel is a valid nonlinear-cost separation. The right panel\nprevents interpreting it as an exponential ordinary total-gate separation.\nThe total full construction is $O(n\\log n)$, whereas the quadratic\nconstruction is $n/2$ gates.","A proposed invariant that charges an original computation for every\nseparately extracted grade would have to justify that extraction's cost.\nThese formulas refute an overhead bounded only by a polynomial in the\noriginal nonlinear count, independently of input and linear resources.\nThey do not exclude an overhead polynomial in $n+T$ or a different\nprecisely defined invariant.","\\guidepage{Quadratic projection can destroy safety}\nAt $m=3$, variables are ordered by source labels $v=1,2,3,4,5,6$.\nAssignment index $13$ is read least-significant-bit first:\n\\[\n (x_1,x_2,x_3,x_4,x_5,x_6)=(1,0,1,1,0,0).\n\\]\nThe active labels have syndrome $1+3+4=6$, not $a=7$.","\\visualfigure{unsafe_projection}{A concrete accepted point for $Q_3$ that is rejected by $P_3$. The variable ordering matches the source checker.}","The three linear forms evaluate to $L=(0,1,1)$, so $P_3=L_1L_2L_3=0$.\nThe quadratic grade has pairs $(1,6)$, $(2,5)$, $(3,4)$, so\n\\[\n Q_3=x_1x_6+x_2x_5+x_3x_4=0+0+1=1.\n\\]\nThus $Q_3\\notin I_{P_3}$: homogeneous extraction does not preserve the\npositive-support ideal. The whole polynomial is supported on its own\naccepting set, but its isolated homogeneous components need not be.","The new script checks all 64 inputs here, finds 24 false positives, and\nconfirms that 13 is the first in the stated integer ordering. The frozen\n$m=4$ control similarly has 7,616 false positives among its $2^{14}$\nassignments. These are separate finite statements with exact ranges.","\\evidence{Fresh complete $m=3$ truth-table check; baseline $m=4$ check was\nreproduced by the full source verifier. Safety failure is already established\nby the single valid assignment above.}","\\guidepage{Witness indicators are orthogonal idempotents}\nLet $K$ be a field and $W=\\{0,1\\}^m$. The full witness algebra is\n$B=K^W$ with pointwise addition and multiplication. For each witness $v$,\nthe indicator $\\delta_v$ is one at $v$ and zero elsewhere. Then\n\\[\n \\delta_v^2=\\delta_v,\\qquad \\delta_u\\delta_v=0\\ (u\\ne v),\n \\qquad \\sum_v\\delta_v=1.\n\\]","\\visualfigure{orthogonal_gluing}{The four-witness multiplication table at $m=2$. Each nonzero diagonal entry is its own point indicator.}","The four indicator functions form a basis of $K^4$. General $m$ has $2^m$\nsuch basis elements. A verifier is the sum of the indicators of its accepted\nwitnesses, and pointwise multiplication models intersection of accepting\nsets.","\\evidence{Exact $m=2$ multiplication table; general product-algebra identities\nhold over any field. No characteristic-two assumption is needed for the\ndimension proof.}","\\guidepage{Faithful linear multiplicative summaries need $2^m$ dimensions}\nSuppose $\\Phi:K^W\\to A$ is $K$-linear and multiplicative, and\n$\\Phi(\\delta_v)\\ne0$ for every $v\\in W$. Put $e_v=\\Phi(\\delta_v)$.\nThe images are nonzero orthogonal idempotents.","If $\\sum_v c_v e_v=0$, multiply by $e_u$. Orthogonality removes every other\nterm and idempotence gives $c_u e_u=0$. Since $K$ is a field and $e_u\\ne0$,\n$c_u=0$. Hence all $e_v$ are linearly independent and\n\\[\n \\boxed{\\dim_K A\\ge |W|=2^m.}\n\\]\nNo unital hypothesis is needed. At $m=0$, there is one empty witness and\nthe bound is simply $\\dim_K A\\ge1$.","For a unital scalar map to $K$, scalar idempotents are zero or one.\nOrthogonality permits at most one nonzero image, and preservation of one\nrequires exactly one. Such a map is evaluation at one witness. For $m\\ge1$\nit loses at least one other singleton verifier. For $m=0$, there is no other\nwitness to lose.","\\guidepage{Distinct witness monomials prevent formal cancellation}\nOver a characteristic-two field, define\n\\[\n P_R(z)=\\sum_{w\\in\\{0,1\\}^m}R(w)\\prod_i z_i^{w_i}.\n\\]\nThe $z_i$ are formal indeterminates, not Boolean variables. Different\nwitnesses produce different monomials, so $P_R=0$ exactly when there is\nno accepting witness.","\\visualfigure{witness_fingerprint}{An exact three-bit verifier accepts witness indices $1,2,5,6$ in least-significant-bit order. Four accepts have parity zero, but their monomials are distinct.}","Here\n\\[\n P_R=z_1+z_2+z_1z_3+z_2z_3=(z_1+z_2)(1+z_3)\\ne0.\n\\]\nThe ordinary unweighted characteristic-two sum of the four accepts is\nzero. The formal polynomial avoids this cancellation by retaining witness\nlabels as monomial supports.","Nonzero as a formal polynomial does not mean nonzero at every field point.\nFor instance, this example vanishes whenever $z_1=z_2$ or $z_3=1$.\nRandom evaluation is governed by a root bound, not by the formal identity\nalone. Evaluating the polynomial efficiently remains a separate task.","\\evidence{Exact support fixture; general no-cancellation argument is written.\nThe source finite evaluator retains $2^m$ coefficients.}","\\guidepage{A short expression inside a trace is not a computed trace}\nLet $\\tau_w$ extract the top multilinear coefficient in the Boolean witness\nring. For any $G(w)$ over a characteristic-two field,\n\\[\n \\tau_w(G)=\\sum_{v\\in\\{0,1\\}^m}G(v).\n\\]\nA proper-support monomial evaluates to one an even number of times; the\nfull-support monomial does so once. This proves the identity directly.","Applying it to a weighted expression gives\n\\[\n P_R(z)=\\tau_w\\!\\left[\n R(w)\\prod_{i=1}^m\\bigl(1+(1+z_i)w_i\\bigr)\\right].\n\\]\nAt $w_i=0$, a multiplier equals one. At $w_i=1$, it equals $z_i$.\nSumming evaluations therefore produces exactly the witness monomials.\nFor the preceding verifier, the four surviving products are\n$z_1,z_2,z_1z_3,z_2z_3$.","The circuit expression \\emph{inside} the trace adds only $O(m)$ operations\nto the verifier. That observation bounds the expression's construction,\nnot the cost of applying $\\tau_w$. The finite checker computes with a\ncomplete table of $2^m$ witness coefficients.","midrule\nSmall weighted circuit inside $\\tau$ & A polynomial-time evaluator for the trace","At $m=0$, the empty product is one, the top coefficient is the constant\nitself, and there is a single empty witness. The definitions remain valid.","\\guidepage{Disjoint-variable factorization uses each predicate's own block}\nSuppose $A$ depends on witness variables in block $S$ and $B$ on a disjoint\nblock $T$. Write $\\tau_{z_S}(A)=\\sum_s A(s)z_S^s$ for the weighted trace\non $S$, distinct from the unweighted top-coefficient notation $\\tau_w$.\nWeighted sum over the union factors as\n\\[\n \\sum_{s,t} A(s)B(t)z_S^s z_T^t\n =\\left(\\sum_s A(s)z_S^s\\right)\n  \\left(\\sum_t B(t)z_T^t\\right).\n\\]\nEach factor is traced on its own variable block. The union trace is then\ntheir product. This is an ordinary factorization of a finite double sum.","For $A=w_1$ and $B=w_2$,\n\\[\n \\tau_{z_1}(A)=z_1,\\qquad \\tau_{z_2}(B)=z_2,\\qquad\n \\tau_{z_1,z_2}(AB)=z_1z_2.\n\\]\nTracing both factors over the full two-bit ambient domain introduces unused\nvariable sums:\n\\[\n \\tau_{z_1,z_2}(A)=z_1(1+z_2),\\qquad\n \\tau_{z_1,z_2}(B)=z_2(1+z_1).\n\\]\nTheir product generally differs from the union trace. At $z_1=z_2=1$, each\nfull ambient factor is zero, while the trace of $AB$ is one. Disjoint\ndependence alone does not justify multiplying traces with mismatched domains.","The source checker tests all four one-bit predicates per block and every\npair of GF(16) weights: $4\\cdot4\\cdot16^2=4,096$ own-block factorization\ncases. It does not claim an ambient-domain product identity.","\\guidepage{Overlap breaks the attempted context-free product rule}\nOne shared witness bit already defeats multiplicativity. With\n$\\tau_z(P)=P(0)+zP(1)$,\n\\[\n \\tau_z(w)=z,\\qquad \\tau_z(1+w)=1,\n \\qquad \\tau_z(w(1+w))=0.\n\\]\nThe product of the separate summaries is $z$. Also $w^2=w$ in the Boolean\nring, so $\\tau_z(w^2)=z$, whereas $\\tau_z(w)^2=z^2$.","\\visualfigure{overlap_gf16}{A concrete field point $z=\\theta$ with $\\theta\\ne0,1$. The two columns are unequal in both rows.}","The field is $\\F_{16}=\\F_2[\\theta]/(\\theta^4+\\theta+1)$. The checker stores\nfield elements as four-bit coefficient vectors: $\\theta$ is integer 2 and\n$\\theta^2$ is integer 4. These integers encode field elements; they are not\nordinary integer multiplication. Exact field multiplication gives the\nreported mismatch.","The first row loses witness compatibility: $w$ and $1+w$ have no common\naccepting witness, yet their separate weighted summaries multiply to a\nnonzero value. The second row loses Boolean idempotence. This is a genuine\ncounterexample to a context-free multiplicative trace compiler, not a\nfloating-point effect or an implementation anomaly.","\\guidepage{Randomized detection is conditional on efficient evaluation}\nThe witness polynomial has total degree at most $m$. Over a finite field\n$K$, a nonzero polynomial of total degree at most $m$ vanishes on at most\nan $m/|K|$ fraction of independent uniform field points. The standard\ninductive proof separates the leading coefficient in the last variable\nand applies the univariate root bound.","If a uniform polynomial-time exact evaluator for the weighted trace were\navailable, with efficient field representation and arithmetic and\n$|K|\\ge4m$, one evaluation would accept a satisfiable verifier with\nprobability at least $3/4$. An unsatisfiable verifier has the zero polynomial,\nso it never produces a false acceptance. Repetition with independent samples\nreduces the false-rejection probability to at most $4^{-r}$ after $r$ trials.","lynomial-time exact evaluator for the weighted trace were\navailable, with efficient field representation and arithmetic and\n$|K|\\ge4m$, one evaluation would accept a satisfiable verifier with\nprobability at least $3/4$. An unsatisfiable verifier has the zero polynomial,\nso it never produces a false acceptance. Repetition with independent samples\nreduces the false-rejection probability to at most $4^{-r}$ after $r$ trials.","xact evaluator for the weighted trace were\navailable, with efficient field representation and arithmetic and\n$|K|\\ge4m$, one evaluation would accept a satisfiable verifier with\nprobability at least $3/4$. An unsatisfiable verifier has the zero polynomial,\nso it never produces a false acceptance. Repetition with independent samples\nreduces the false-rejection probability to at most $4^{-r}$ after $r$ trials.","These source tests enumerate every verifier at each displayed witness length\nand all field points for the polynomial evaluations. The independent\ncoefficient-extraction comparison has a narrower three-variable grid:\ncoordinates $0,1,2,3$, rather than all GF(16) points. Both scopes are recorded.","This is a conditional randomized algorithmic implication. The required\nefficient trace evaluator is not supplied. The existing exact evaluator is\nexponential in witness length, and no deterministic $P=NP$ conclusion is\ndrawn. Random bits and polynomial output description alone do not account\nfor the evaluation operation.","The separate dual sanity suite runs as\n\\begin{lstlisting}\npython scripts/core_dual_checks.py \\\n  --output reviews/core_dual_checks.json\n\\end{lstlisting}\nIt compares exhaustive degree-bounded safe truth tables with a separate\nGaussian-elimination calculation of the negative evaluation span. Its\nrange is all truth tables at $n=0,1,2,3$, not an extrapolation.","The source release, earlier companions, and archived evidence remain\nunchanged. Publication corrections clarify evidence labels, the nonzero\nmonotone minterm domain, the $m=0$ scalar-summary boundary, and trace-variable\ndomains. No new conjecture or extension to unrestricted hardness is added.","\\subsection{Row-transvection convention}\nMatrices act on column coordinate vectors, while row operations left multiply the current matrix.  A row transvection\n\\[\n\\tau_{t,s}(a)=I+aE_{t,s}\n\\]\nperforms\n\\[\nR_t\\leftarrow R_t+aR_s.\n\\]\nIn the binary CNOT interpretation, $s$ is the control and $t$ is the target.","The canonical six-row target is\n\\[\nC=\n\\begin{pmatrix}\n1&0&0&0&0&0\\\\\n1&1&0&0&0&0\\\\\n0&0&1&0&0&0\\\\\n0&1&1&1&0&0\\\\\n1&1&1&0&1&0\\\\\n1&1&1&0&0&1\n\\end{pmatrix}.\n\\]\nOne ambient word is\n\\[\n\\tau_{1,2}(1),\n\\tau_{3,1}(1),\n\\tau_{1,0}(1),\n\\tau_{4,1}(1),\n\\tau_{5,1}(1),\n\\tau_{1,2}(-1).\n\\]\nA restricted word of length seven is\n\\[\n\\tau_{1,0}(1),\n\\tau_{2,1}(1),\n\\tau_{3,0}(-1),\n\\tau_{3,2}(1),\n\\tau_{4,2}(1),\n\\tau_{5,2}(1),\n\\tau_{2,1}(-1).\n\\]","1&1&1&0&0&1\n\\end{pmatrix}.\n\\]\nOne ambient word is\n\\[\n\\tau_{1,2}(1),\n\\tau_{3,1}(1),\n\\tau_{1,0}(1),\n\\tau_{4,1}(1),\n\\tau_{5,1}(1),\n\\tau_{1,2}(-1).\n\\]\nA restricted word of length seven is\n\\[\n\\tau_{1,0}(1),\n\\tau_{2,1}(1),\n\\tau_{3,0}(-1),\n\\tau_{3,2}(1),\n\\tau_{4,2}(1),\n\\tau_{5,2}(1),\n\\tau_{2,1}(-1).\n\\]","\\subsection{Qudit SUM gate}\nOn computational basis labels in $\\Z_d$,\n\\[\n\\SUM^{(a)}_{s\\to t}\n\\ket{x_s,x_t}\n=\n\\ket{x_s,x_t+a x_s\\bmod d}.\n\\]\nIts classical linear action is the transvection $I+aE_{t,s}$ over $\\Z_d$.","\\subsection{Binary symplectic form}\nPauli exponents are ordered as\n\\[\n(x_0,\\ldots,x_{n-1},z_0,\\ldots,z_{n-1}).\n\\]\nThe symplectic form is\n\\[\nJ=\n\\begin{pmatrix}\n0&I_n","\\subsection{Binary symplectic form}\nPauli exponents are ordered as\n\\[\n(x_0,\\ldots,x_{n-1},z_0,\\ldots,z_{n-1}).\n\\]\nThe symplectic form is\n\\[\nJ=\n\\begin{pmatrix}\n0&I_n\\\\\nI_n&0\n\\end{pmatrix}\n\\]\nover $\\F_2$.  A matrix $M$ is symplectic when\n\\[\nM^TJM=J.\n\\]","I_n&0\n\\end{pmatrix}\n\\]\nover $\\F_2$.  A matrix $M$ is symplectic when\n\\[\nM^TJM=J.\n\\]","\\subsection{Clifford generator matrices}\nFor qubit $i$, the Hadamard swaps $x_i$ and $z_i$:\n\\[\nH_i:e_i\\leftrightarrow e_{n+i}.\n\\]\nThe phase gate has\n\\[\nz_i\\leftarrow z_i+x_i.\n\\]\nFor CNOT with control $c$ and target $t$,\n\\[\nx_t\\leftarrow x_t+x_c,\n\\qquad\nz_c\\leftarrow z_c+z_t.\n\\]\nFor CZ on $i,j$,\n\\[\nz_i\\leftarrow z_i+x_j,\n\\qquad\nz_j\\leftarrow z_j+x_i.\n\\]\nSWAP exchanges both $x$ and $z$ coordinates of the two qubits.","The parabolic subgroup used in the census is generated by $S_i$, $\\CZ_{ij}$, and $\\CX_{i\\to j}$.  It stabilizes the $Z$ Lagrangian\n\\[\nL_Z=\\Span(e_n,\\ldots,e_{2n-1}).\n\\]","\\subsection{Two-qubit Clifford witness}\nThe target symplectic matrix is\n\\[\nM_2=\n\\begin{pmatrix}\n0&1&0&0\\\\\n1&0&0&0\\\\\n1&1&0&1\\\\\n1&1&1&0\n\\end{pmatrix}.\n\\]\nA restricted path is\n\\[\n\\CX_{0\\to1},\\ S_1,\\ \\CX_{1\\to0},\\ \\CX_{0\\to1},\n\\]\nwith three charged entangling gates.  An ambient path is\n\\[\nH_1,\\ \\CX_{1\\to0},\\ \\CX_{0\\to1},\\ H_0,\\ S_1,\\ S_0,\n\\]\nwith two charged entangling gates when $H,S$ are free.","1&1&1&0\n\\end{pmatrix}.\n\\]\nA restricted path is\n\\[\n\\CX_{0\\to1},\\ S_1,\\ \\CX_{1\\to0},\\ \\CX_{0\\to1},\n\\]\nwith three charged entangling gates.  An ambient path is\n\\[\nH_1,\\ \\CX_{1\\to0},\\ \\CX_{0\\to1},\\ H_0,\\ S_1,\\ S_0,\n\\]\nwith two charged entangling gates when $H,S$ are free.","\\subsection{Three-qubit Clifford witness}\nThe target symplectic matrix is\n\\[\nM_3=\n\\begin{pmatrix}\n0&1&0&0&0&0\\\\\n0&0&1&0&0&0\\\\\n1&0&0&0&0&0\\\\\n1&0&1&0&1&0\\\\\n0&1&0&0&0&1\\\\\n0&1&0&1&0&0\n\\end{pmatrix}.\n\\]\nA restricted total-count path is\n\\[\n\\CX_{0\\to2},\\ \\CZ_{1,2},\\ \\CX_{2\\to0},\\ \\CX_{0\\to2},\n\\ \\CX_{0\\to1},\\ \\CX_{1\\to0},\\ \\CX_{0\\to1},\n\\]\nwhile an ambient path is\n\\[\nH_1,\\ \\CX_{1\\to2},\\ \\CX_{2\\to1},\\ \\CX_{2\\to0},\\ \\CX_{0\\to2},\\ H_0.\n\\]","0&1&0&1&0&0\n\\end{pmatrix}.\n\\]\nA restricted total-count path is\n\\[\n\\CX_{0\\to2},\\ \\CZ_{1,2},\\ \\CX_{2\\to0},\\ \\CX_{0\\to2},\n\\ \\CX_{0\\to1},\\ \\CX_{1\\to0},\\ \\CX_{0\\to1},\n\\]\nwhile an ambient path is\n\\[\nH_1,\\ \\CX_{1\\to2},\\ \\CX_{2\\to1},\\ \\CX_{2\\to0},\\ \\CX_{0\\to2},\\ H_0.\n\\]","\\subsection{Classical reversible gates}\nFor a bit string $x\\in\\F_2^n$:\n\\[\nX_i:x_i\\leftarrow x_i+1,\n\\]\n\\[\n\\CX_{c\\to t}:x_t\\leftarrow x_t+x_c,\n\\]\n\\[\n\\TOF_{a,b\\to t}:x_t\\leftarrow x_t+x_ax_b.\n\\]\nAll other coordinates are unchanged.  The first two generate $\\operatorname{AGL}(n,2)$.  On three bits, all three generate the full permutation group $S_8$.  For $n\\ge4$ without ancillas they are all even permutations, so the generated group is contained in $A_{2^n}$ and is not the full symmetric group.","tring $x\\in\\F_2^n$:\n\\[\nX_i:x_i\\leftarrow x_i+1,\n\\]\n\\[\n\\CX_{c\\to t}:x_t\\leftarrow x_t+x_c,\n\\]\n\\[\n\\TOF_{a,b\\to t}:x_t\\leftarrow x_t+x_ax_b.\n\\]\nAll other coordinates are unchanged.  The first two generate $\\operatorname{AGL}(n,2)$.  On three bits, all three generate the full permutation group $S_8$.  For $n\\ge4$ without ancillas they are all even permutations, so the generated group is contained in $A_{2^n}$ and is not the full symmetric group.","\\subsection{Bosonic symplectic embedding}\nFor $A\\in\\GL(n,\\R)$,\n\\[\n\\iota(A)=\\diag(A,A^{-T})\\in\\Sp(2n,\\R).\n\\]\nThis convention orders quadratures as $(q_1,\\ldots,q_n,p_1,\\ldots,p_n)$.","\\subsection{Pauli rotations}\nFor a Hermitian Pauli $P$,\n\\[\nR_P(\\theta)=e^{-i\\theta P/2}.\n\\]\nClifford conjugation gives\n\\[\nC R_P(\\theta) C^\\dagger=R_{CPC^\\dagger}(\\theta).\n\\]\nThe sign of $CPC^\\dagger$ can be absorbed by changing $\\theta$ to $-\\theta$.","\\subsection{Tensor and direct-sum conventions}\nFor linear maps, $A\\oplus B$ denotes a block-diagonal direct sum.  The Kronecker product $A\\otimes B$ uses lexicographic ordering.  In quantum chapters, $U\\otimes V$ denotes the physical tensor product on disjoint systems.  These operations have different dimension and cost normalizations and are not interchanged silently.","\\subsection{Clifford census summary}\n\\begin{longtable}{@{}ccccrrrr@{}}\n\\toprule\nQubits & Metric & SWAP & Restricted size & Ambient size & $\\operatorname{diam}_R$ & $\\operatorname{diam}_A$ & Defects","\\midrule\nTwo qubits, entangling & $3\\to2$ & 4","Three qubits, total & $7\\to6$ & 6","Three qubits, total & $8\\to7$ & 18","Three qubits, entangling & $3\\to2$ & 24","Three qubits, entangling & $4\\to3$ & 432","Three qubits, entangling & $5\\to4$ & 1,056","Three qubits, entangling & $6\\to4$ & 48","Three qubits, entangling & $6\\to5$ & 64","\\subsection{Core binary six-dimensional census}\nFor $U_6(\\F_2)$ there are $2^{15}=32{,}768$ targets.  The complete comparison is:\n\\begin{longtable}{@{}rrr@{}}\n\\toprule\nRestricted distance & Ambient distance & Number of targets","\\subsection{Endpoint Green coefficients for the canonical atom}\nRestricted word counts for degrees zero through ten are\n\\[\n0,0,0,0,0,0,0,543,25080,510703,10824940.\n\\]\nAmbient word counts for degrees zero through eight are\n\\[\n0,0,0,0,0,0,2,776,39454.\n\\]\nThus first arrival occurs at seven and six, respectively.","for modulus in (2, 3, 5, 8):\n    print(verify_canonical_atom(modulus))\n\\end{lstlisting}\nThe function evaluates the displayed restricted and ambient words over $\\Z/m\\Z$ and compares both endpoints with the canonical target.","print(exact_clifford_census(2, entangling_cost=True))\nprint(exact_clifford_census(2, entangling_cost=False))\n\\end{lstlisting}\nThe optional three-qubit Python census is slower.  The release includes an optimized C++ program for the full $1{,}451{,}520$-state group.","word = [\"H1\", \"CX12\", \"CX21\", \"CX20\", \"CX02\", \"H0\"]\nprint(gate_word_to_openqasm(word, 3, title=\"ambient witness\"))\n\\end{lstlisting}\nThe exporter supports the named $H$, $S$, $X$, CX, CZ, and SWAP gates used by the finite atlas and writes OpenQASM~3.  The supplied files in \\texttt{examples/} encode the two- and three-qubit restricted and ambient witness words.  The Clifford census is performed modulo global phase and Pauli translations, so equality of an exported pair refers to its binary symplectic action unless the ignored Pauli and phase data are separately reconstructed.","ng}[style=pythonstyle]\nfrom catalytic_gate_atlas.search import (\n    NamedGenerator,\n    breadth_first_distances,\n    breadth_first_with_parents,\n    zero_one_distances,\n    zero_one_with_parents,\n    dijkstra_distances,\n    reconstruct_path,\n    defect_distribution,\n)\n\\end{lstlisting}\nEvery state must be immutable and hashable.  The caller supplies a multiplication function with the convention\n\\[\n\\texttt{multiply(generator, state)}.\n\\]","\\midrule\nL1 & Canonical six-row transvection target has lengths $7$ and $6$ over nonzero commutative rings. & Proved & Independent external proof audit.","L2 & Dimensions below six are rigid over every coefficient field or ring. & Open & Symbolic bounded-skeleton elimination for $n\\le5$.","L3 & Maximum unitriangular defect is $\\Theta(n\\log n)$. & Open & Matching upper bound or larger interacting-catalyst family.","L4 & Multiplicative unitriangular distortion is bounded by $3/2$. & Open & Normal form beyond one clean carrier.","C1 & Two-qubit parabolic Clifford entangling model has four $3\\to2$ defects in one free-phase orbit. & Analytic costs and orbit; finite classification & Complete: Theorem~\\ref{g:thm:clifford-two-analytic} plus the exhaustive census.","C3 & Unit-cost SWAP eliminates all three-qubit total defects. & Computed & Determine whether rigidity persists for larger $n$.","P2 & Phase-polynomial $T$-count displays tensor activation. & Open & Joint synthesis of repeated phase targets.","R1 & A nonsimply laced root system has catalytic ratio exceeding $3/2$. & Open & Low-rank Chevalley word search and restricted lower bound.","\\subsection{Why a separate theory is needed}\nA target transformation often belongs to a visibly structured class even when the machine used to realize it is capable of much more.  A lower-triangular linear transformation may be synthesized by an unrestricted CNOT network.  A diagonal unitary may be realized by a non-diagonal Clifford+$T$ circuit.  A charge-preserving unitary may be compiled by gates that temporarily visit other charge sectors.  A Gaussian target may be realized by a circuit that briefly uses a non-Gaussian primitive.  These situations share a question that is not answered by ordinary exact synthesis:\n\\begin{quote}\nHow much does it cost to require the \\emph{entire path} of a computation to retain the structure possessed by its endpoint?\n\\end{quote}\nThe distinction is path dependent.  It is not determined only by membership of the final target in a subgroup, a subcategory, or a model class.","\\subsection{Exact gate models}\n\\begin{definition}[Gate model]\nAn \\emph{exact gate model} is a tuple\n\\[\n\\mathfrak M=(\\mathcal X,\\mathcal S,c,\\circ),\n\\]\nwhere $\\mathcal X$ is a set of states or morphisms, $\\mathcal S$ is a specified family of elementary generators, $c:\\mathcal S\\to\\R_{\\ge0}$ is a cost function, and $\\circ$ is the composition law.  A word\n\\[\nw=s_L\\cdots s_1\n\\]\nhas cost\n\\[\nc(w)=\\sum_{j=1}^L c(s_j).\n\\]\nFor a target $U\\in\\mathcal X$, the exact synthesis cost is\n\\[\n\\ell_{\\mathfrak M}(U)=\\inf\\{c(w):w=U\\}.\n\\]\nThe value is $+\\infty$ when the target has no realization.\n\\end{definition}","begin{definition}[Gate model]\nAn \\emph{exact gate model} is a tuple\n\\[\n\\mathfrak M=(\\mathcal X,\\mathcal S,c,\\circ),\n\\]\nwhere $\\mathcal X$ is a set of states or morphisms, $\\mathcal S$ is a specified family of elementary generators, $c:\\mathcal S\\to\\R_{\\ge0}$ is a cost function, and $\\circ$ is the composition law.  A word\n\\[\nw=s_L\\cdots s_1\n\\]\nhas cost\n\\[\nc(w)=\\sum_{j=1}^L c(s_j).\n\\]\nFor a target $U\\in\\mathcal X$, the exact synthesis cost is\n\\[\n\\ell_{\\mathfrak M}(U)=\\inf\\{c(w):w=U\\}.\n\\]\nThe value is $+\\infty$ when the target has no realization.\n\\end{definition}","te model} is a tuple\n\\[\n\\mathfrak M=(\\mathcal X,\\mathcal S,c,\\circ),\n\\]\nwhere $\\mathcal X$ is a set of states or morphisms, $\\mathcal S$ is a specified family of elementary generators, $c:\\mathcal S\\to\\R_{\\ge0}$ is a cost function, and $\\circ$ is the composition law.  A word\n\\[\nw=s_L\\cdots s_1\n\\]\nhas cost\n\\[\nc(w)=\\sum_{j=1}^L c(s_j).\n\\]\nFor a target $U\\in\\mathcal X$, the exact synthesis cost is\n\\[\n\\ell_{\\mathfrak M}(U)=\\inf\\{c(w):w=U\\}.\n\\]\nThe value is $+\\infty$ when the target has no realization.\n\\end{definition}",",c,\\circ),\n\\]\nwhere $\\mathcal X$ is a set of states or morphisms, $\\mathcal S$ is a specified family of elementary generators, $c:\\mathcal S\\to\\R_{\\ge0}$ is a cost function, and $\\circ$ is the composition law.  A word\n\\[\nw=s_L\\cdots s_1\n\\]\nhas cost\n\\[\nc(w)=\\sum_{j=1}^L c(s_j).\n\\]\nFor a target $U\\in\\mathcal X$, the exact synthesis cost is\n\\[\n\\ell_{\\mathfrak M}(U)=\\inf\\{c(w):w=U\\}.\n\\]\nThe value is $+\\infty$ when the target has no realization.\n\\end{definition}","\\begin{definition}[Nested realization pair]\nA \\emph{nested realization pair} consists of restricted and ambient models\n\\[\n\\mathfrak M_{\\mathrm{res}}\\subseteq\\mathfrak M_{\\mathrm{amb}}\n\\]\nwith a common target class $\\mathcal T$ of finite ambient cost and the same cost on common generators.  For $U\\in\\mathcal T$, define\n\\[\n\\ellRes(U)=\\ell_{\\mathfrak M_{\\mathrm{res}}}(U),\n\\qquad\n\\ellAmb(U)=\\ell_{\\mathfrak M_{\\mathrm{amb}}}(U).\n\\]\nThe additive defect is\n\\[\n\\Delta(U)=\\ellRes(U)-\\ellAmb(U).\n\\]\n\\end{definition}","ed realization pair} consists of restricted and ambient models\n\\[\n\\mathfrak M_{\\mathrm{res}}\\subseteq\\mathfrak M_{\\mathrm{amb}}\n\\]\nwith a common target class $\\mathcal T$ of finite ambient cost and the same cost on common generators.  For $U\\in\\mathcal T$, define\n\\[\n\\ellRes(U)=\\ell_{\\mathfrak M_{\\mathrm{res}}}(U),\n\\qquad\n\\ellAmb(U)=\\ell_{\\mathfrak M_{\\mathrm{amb}}}(U).\n\\]\nThe additive defect is\n\\[\n\\Delta(U)=\\ellRes(U)-\\ellAmb(U).\n\\]\n\\end{definition}","When both costs are finite, $\\Delta(U)$ is a metric or optimization defect.  If $\\ellRes(U)=+\\infty$ and $\\ellAmb(U)<+\\infty$, it is an existence defect.  This convention unifies finite gate savings with the failure of a restricted realization category.","\\begin{proposition}[Monotonicity]\nFor every target in the common domain,\n\\[\n\\ellAmb(U)\\le \\ellRes(U).\n\\]\nConsequently $\\Delta(U)\\ge0$ whenever both models use the same nonnegative cost on their common generators.\n\\end{proposition}\n\\begin{proof}\nEvery restricted word is also an ambient word.\n\\end{proof}","\\begin{definition}[Resource vector]\nLet\n\\[\n\\mathbf c(s)=(c_1(s),\\ldots,c_r(s))\\in\\R_{\\ge0}^r\n\\]\nbe the resource vector of a gate.  The achievable resource region of a target is\n\\[\n\\mathcal P_{\\mathfrak M}(U)\n=\n\\left\\{\n\\sum_{j=1}^L\\mathbf c(s_j):s_L\\cdots s_1=U\n\\right\\}.\n\\]\nIts nondominated points form the Pareto boundary.\n\\end{definition}","For a weight vector $\\lambda\\in\\R_{\\ge0}^r$, scalarization gives\n\\[\n\\ell_{\\mathfrak M}^{\\lambda}(U)\n=\n\\inf_{x\\in\\mathcal P_{\\mathfrak M}(U)}\\lambda\\cdot x.\n\\]\nThe scalar defect $\\Delta^\\lambda(U)$ can vary with $\\lambda$ even when the target and gate alphabets are fixed.","\\subsection{Worst-case and threshold functions}\nFor a size parameter $n$, let $\\mathcal T_n$ denote the target family.\n\\begin{definition}[Extremal distortion]\nDefine\n\\[\nD_n=\\sup_{U\\in\\mathcal T_n}\\Delta(U)\n\\]\nand, when $0<\\ellAmb(U)<\\infty$,\n\\[\nR_n=\\sup_{U\\in\\mathcal T_n}\n\\frac{\\ellRes(U)}{\\ellAmb(U)}.\n\\]\nThe first defective size is\n\\[\n\\tau=\\min\\{n:D_n>0\\}.\n\\]\nWe set a threshold to $+\\infty$ when its defining set is empty.\n\\end{definition}","Several threshold notions should not be conflated:\n\\begin{align*}\n\\tau_{\\mathrm{metric}}&=\\min\\{n:\\exists U,\\ 0<\\Delta(U)<\\infty\\},\\\\\n\\tau_{\\mathrm{exist}}&=\\min\\{n:\\exists U,\\ \\ellRes(U)=\\infty,\\ \\ellAmb(U)<\\infty\\},\\\\\n\\tau_{q}&=\\min\\{n:\\exists U,\\ \\ell_{\\le q}(U)<\\ell_{\\le0}(U)\\},\n\\end{align*}\nwhere $\\ell_{\\le q}$ permits at most $q$ ambient-only gates.  A model may have an early metric threshold and a later existence threshold, or no threshold at all.","\\tau_{q}&=\\min\\{n:\\exists U,\\ \\ell_{\\le q}(U)<\\ell_{\\le0}(U)\\},\n\\end{align*}\nwhere $\\ell_{\\le q}$ permits at most $q$ ambient-only gates.  A model may have an early metric threshold and a later existence threshold, or no threshold at all.","\\begin{definition}[Cost-preserving equivalence]\nTargets $U$ and $V$ are equivalent if an automorphism $\\varphi$ of the ambient realization system satisfies\n\\[\n\\varphi(U)=V,\n\\qquad\n\\varphi(\\mathcal S_{\\mathrm{res}})=\\mathcal S_{\\mathrm{res}},\n\\qquad\n\\varphi(\\mathcal S_{\\mathrm{amb}})=\\mathcal S_{\\mathrm{amb}},\n\\]\nand preserves the gate costs.\n\\end{definition}\nUnder this equivalence, both lengths and the defect are orbit invariants.  A useful threshold theorem should classify minimal defective orbits, not merely list raw matrices.","\\subsection{Approximate synthesis and border defects}\nFor continuous gate models, exact word length can differ from the minimum length achievable as a limit.\n\\begin{definition}[Border cost]\nLet $\\overline{\\mathcal W}_{L}$ be the topological or Zariski closure of targets realizable with cost at most $L$.  The border cost is\n\\[\n\\underline\\ell(U)=\\inf\\{L:U\\in\\overline{\\mathcal W}_{L}\\}.\n\\]\nThe border defect is\n\\[\n\\underline\\Delta(U)=\n\\underline\\ell_{\\mathrm{res}}(U)-\n\\underline\\ell_{\\mathrm{amb}}(U).\n\\]\n\\end{definition}\nThis is the circuit analogue of the distinction between tensor rank and border rank.  An exact defect can disappear under degeneration, persist, or become larger after approximation changes the relevant cost geometry.","pproximate synthesis and border defects}\nFor continuous gate models, exact word length can differ from the minimum length achievable as a limit.\n\\begin{definition}[Border cost]\nLet $\\overline{\\mathcal W}_{L}$ be the topological or Zariski closure of targets realizable with cost at most $L$.  The border cost is\n\\[\n\\underline\\ell(U)=\\inf\\{L:U\\in\\overline{\\mathcal W}_{L}\\}.\n\\]\nThe border defect is\n\\[\n\\underline\\Delta(U)=\n\\underline\\ell_{\\mathrm{res}}(U)-\n\\underline\\ell_{\\mathrm{amb}}(U).\n\\]\n\\end{definition}\nThis is the circuit analogue of the distinction between tensor rank and border rank.  An exact defect can disappear under degeneration, persist, or become larger after approximation changes the relevant cost geometry.","\\subsection{Metric and existence defects in one category}\nLet $\\mathcal C_{\\mathrm{res}}\\subseteq\\mathcal C_{\\mathrm{amb}}$ be categories with weighted generating morphisms.  For a morphism $f$ in the restricted target class, the same definition\n\\[\n\\Delta(f)=c_{\\mathrm{res}}(f)-c_{\\mathrm{amb}}(f)\n\\]\nincludes group-word metrics, adaptive quantum channels, tensor-network realizations, and orthogonality designs.  This categorical form is essential when the ambient model permits measurement, branching, postselection, or entangled objects that do not lie in a common group.","\\subsection{Subgroup metrics and Schreier loops}\nAssume first that the restricted and ambient models are groups\n\\[\nH=\\langle\\mathcal S_H\\rangle\n\\le\nG=\\langle\\mathcal S_G\\rangle.\n\\]\nFor $h\\in H$, the intrinsic and ambient word metrics are\n\\[\n\\ell_H(h)=|h|_{\\mathcal S_H},\n\\qquad\n\\ell_G(h)=|h|_{\\mathcal S_G}.\n\\]\nIn the group-metric statements, the generating alphabets are symmetric and a generator and its inverse have equal cost.  The quotient geometry is the left Schreier graph $G/H$, with edges $gH\\longmapsto sgH$.  For the base-fiber assertion below, assume also $\\mathcal S_H=\\mathcal S_G\\cap H$.  This last condition excludes additional ambient primitives that already lie in $H$ but have no equally cheap restricted realization.","and ambient models are groups\n\\[\nH=\\langle\\mathcal S_H\\rangle\n\\le\nG=\\langle\\mathcal S_G\\rangle.\n\\]\nFor $h\\in H$, the intrinsic and ambient word metrics are\n\\[\n\\ell_H(h)=|h|_{\\mathcal S_H},\n\\qquad\n\\ell_G(h)=|h|_{\\mathcal S_G}.\n\\]\nIn the group-metric statements, the generating alphabets are symmetric and a generator and its inverse have equal cost.  The quotient geometry is the left Schreier graph $G/H$, with edges $gH\\longmapsto sgH$.  For the base-fiber assertion below, assume also $\\mathcal S_H=\\mathcal S_G\\cap H$.  This last condition excludes additional ambient primitives that already lie in $H$ but have no equally cheap restricted realization.","are groups\n\\[\nH=\\langle\\mathcal S_H\\rangle\n\\le\nG=\\langle\\mathcal S_G\\rangle.\n\\]\nFor $h\\in H$, the intrinsic and ambient word metrics are\n\\[\n\\ell_H(h)=|h|_{\\mathcal S_H},\n\\qquad\n\\ell_G(h)=|h|_{\\mathcal S_G}.\n\\]\nIn the group-metric statements, the generating alphabets are symmetric and a generator and its inverse have equal cost.  The quotient geometry is the left Schreier graph $G/H$, with edges $gH\\longmapsto sgH$.  For the base-fiber assertion below, assume also $\\mathcal S_H=\\mathcal S_G\\cap H$.  This last condition excludes additional ambient primitives that already lie in $H$ but have no equally cheap restricted realization.","rangle.\n\\]\nFor $h\\in H$, the intrinsic and ambient word metrics are\n\\[\n\\ell_H(h)=|h|_{\\mathcal S_H},\n\\qquad\n\\ell_G(h)=|h|_{\\mathcal S_G}.\n\\]\nIn the group-metric statements, the generating alphabets are symmetric and a generator and its inverse have equal cost.  The quotient geometry is the left Schreier graph $G/H$, with edges $gH\\longmapsto sgH$.  For the base-fiber assertion below, assume also $\\mathcal S_H=\\mathcal S_G\\cap H$.  This last condition excludes additional ambient primitives that already lie in $H$ but have no equally cheap restricted realization.","h)=|h|_{\\mathcal S_G}.\n\\]\nIn the group-metric statements, the generating alphabets are symmetric and a generator and its inverse have equal cost.  The quotient geometry is the left Schreier graph $G/H$, with edges $gH\\longmapsto sgH$.  For the base-fiber assertion below, assume also $\\mathcal S_H=\\mathcal S_G\\cap H$.  This last condition excludes additional ambient primitives that already lie in $H$ but have no equally cheap restricted realization.","\\begin{theorem}[Holonomy-compression identity]\nLet every generator have unit cost.  For $h\\in H$, $\\ell_G(h)$ is the minimum length of a based loop in $G/H$ whose lifted word has holonomy $h$.  The intrinsic length $\\ell_H(h)$ is the minimum length among loops that remain in the base fiber.\n\\end{theorem}\n\\begin{proof}\nAn ambient word $s_k\\cdots s_1=h$ projects to the loop\n\\[\nH,\ns_1H,\ns_2s_1H,\n\\ldots,\nhH=H.\n\\]\nConversely, a based Schreier loop has product in $H$.  If every prefix lies in $H$, each edge label is the product of two elements of $H$ and belongs to $\\mathcal S_G\\cap H=\\mathcal S_H$.  Conversely every restricted word remains in the base fiber.  Minimizing the labeled words with endpoint $h$ gives the two asserted lengths.\n\\end{proof}","tor have unit cost.  For $h\\in H$, $\\ell_G(h)$ is the minimum length of a based loop in $G/H$ whose lifted word has holonomy $h$.  The intrinsic length $\\ell_H(h)$ is the minimum length among loops that remain in the base fiber.\n\\end{theorem}\n\\begin{proof}\nAn ambient word $s_k\\cdots s_1=h$ projects to the loop\n\\[\nH,\ns_1H,\ns_2s_1H,\n\\ldots,\nhH=H.\n\\]\nConversely, a based Schreier loop has product in $H$.  If every prefix lies in $H$, each edge label is the product of two elements of $H$ and belongs to $\\mathcal S_G\\cap H=\\mathcal S_H$.  Conversely every restricted word remains in the base fiber.  Minimizing the labeled words with endpoint $h$ gives the two asserted lengths.\n\\end{proof}","H$, $\\ell_G(h)$ is the minimum length of a based loop in $G/H$ whose lifted word has holonomy $h$.  The intrinsic length $\\ell_H(h)$ is the minimum length among loops that remain in the base fiber.\n\\end{theorem}\n\\begin{proof}\nAn ambient word $s_k\\cdots s_1=h$ projects to the loop\n\\[\nH,\ns_1H,\ns_2s_1H,\n\\ldots,\nhH=H.\n\\]\nConversely, a based Schreier loop has product in $H$.  If every prefix lies in $H$, each edge label is the product of two elements of $H$ and belongs to $\\mathcal S_G\\cap H=\\mathcal S_H$.  Conversely every restricted word remains in the base fiber.  Minimizing the labeled words with endpoint $h$ gives the two asserted lengths.\n\\end{proof}","in $G/H$ whose lifted word has holonomy $h$.  The intrinsic length $\\ell_H(h)$ is the minimum length among loops that remain in the base fiber.\n\\end{theorem}\n\\begin{proof}\nAn ambient word $s_k\\cdots s_1=h$ projects to the loop\n\\[\nH,\ns_1H,\ns_2s_1H,\n\\ldots,\nhH=H.\n\\]\nConversely, a based Schreier loop has product in $H$.  If every prefix lies in $H$, each edge label is the product of two elements of $H$ and belongs to $\\mathcal S_G\\cap H=\\mathcal S_H$.  Conversely every restricted word remains in the base fiber.  Minimizing the labeled words with endpoint $h$ gives the two asserted lengths.\n\\end{proof}","ed word has holonomy $h$.  The intrinsic length $\\ell_H(h)$ is the minimum length among loops that remain in the base fiber.\n\\end{theorem}\n\\begin{proof}\nAn ambient word $s_k\\cdots s_1=h$ projects to the loop\n\\[\nH,\ns_1H,\ns_2s_1H,\n\\ldots,\nhH=H.\n\\]\nConversely, a based Schreier loop has product in $H$.  If every prefix lies in $H$, each edge label is the product of two elements of $H$ and belongs to $\\mathcal S_G\\cap H=\\mathcal S_H$.  Conversely every restricted word remains in the base fiber.  Minimizing the labeled words with endpoint $h$ gives the two asserted lengths.\n\\end{proof}","begin{proof}\nAn ambient word $s_k\\cdots s_1=h$ projects to the loop\n\\[\nH,\ns_1H,\ns_2s_1H,\n\\ldots,\nhH=H.\n\\]\nConversely, a based Schreier loop has product in $H$.  If every prefix lies in $H$, each edge label is the product of two elements of $H$ and belongs to $\\mathcal S_G\\cap H=\\mathcal S_H$.  Conversely every restricted word remains in the base fiber.  Minimizing the labeled words with endpoint $h$ gives the two asserted lengths.\n\\end{proof}","Thus\n\\[\n\\Delta_{H\\hookrightarrow G}(h)\n=\n\\text{intrinsic fiber cost}\n-\n\\text{shortest holonomy loop cost}.\n\\]\nThis is the geometric content of a dirty-workspace shortcut.","\\subsection{The basic conjugation inequality}\nLet $g\\in G$ and $k\\in H$ satisfy $gkg^{-1}\\in H$.  Then\n\\[\n\\ell_G(gkg^{-1})\n\\le 2\\ell_G(g)+\\ell_H(k).\n\\]\nDefine the intrinsic conjugation stretch\n\\[\n\\operatorname{Stretch}_g(k)\n=\n\\ell_H(gkg^{-1})-\\ell_H(k).\n\\]","\\begin{proposition}[Catalytic stretch criterion]\nIf\n\\[\n\\operatorname{Stretch}_g(k)>2\\ell_G(g),\n\\]\nthen $gkg^{-1}$ has positive defect.\n\\end{proposition}\nThe opening and closing cost is paid once.  Every useful consumer inside the excursion contributes to the stretch.","\\subsection{Many consumers and catalytic excess}\nSuppose the middle word is\n\\[\nk=h_m\\cdots h_1,\n\\qquad h_i\\in H.\n\\]\nThe ambient word\n\\[\ngh_m\\cdots h_1g^{-1}\n\\]\nhas cost at most\n\\[\n2c(g)+\\sum_i c(h_i).\n\\]\nIf the restricted images $gh_ig^{-1}$ require costs $r_i$ and a lower-bound invariant proves that their obligations cannot be shared, then\n\\[\n\\ell_H(gkg^{-1})\\ge\\sum_i r_i.\n\\]\nA sufficient inequality is\n\\[\n\\sum_i(r_i-c(h_i))>2c(g).\n\\]\nThe left side is the gross catalytic benefit.  The right side is the restoration cost.","on{Many consumers and catalytic excess}\nSuppose the middle word is\n\\[\nk=h_m\\cdots h_1,\n\\qquad h_i\\in H.\n\\]\nThe ambient word\n\\[\ngh_m\\cdots h_1g^{-1}\n\\]\nhas cost at most\n\\[\n2c(g)+\\sum_i c(h_i).\n\\]\nIf the restricted images $gh_ig^{-1}$ require costs $r_i$ and a lower-bound invariant proves that their obligations cannot be shared, then\n\\[\n\\ell_H(gkg^{-1})\\ge\\sum_i r_i.\n\\]\nA sufficient inequality is\n\\[\n\\sum_i(r_i-c(h_i))>2c(g).\n\\]\nThe left side is the gross catalytic benefit.  The right side is the restoration cost.","\\begin{definition}[Catalytic excess]\nFor a normal form with $r$ matched excursions and $C$ useful cross interactions, define\n\\[\n\\mathcal E=C-2r\n\\]\nwhen each consumer contributes one unit and each opening or closing costs one unit.\n\\end{definition}\nPositive excess is necessary in the simplest transvection model, but it is not sufficient.  An all-cross configuration may admit a mirror word of equal restricted length.  Temporal asymmetry or phase holonomy is also required.","\\subsection{Excursion budgets}\nCounting ambient-only gates is only one measure of how far a circuit leaves its restricted model.\n\\begin{definition}[Excursion profile]\nFor a circuit path $X_0,X_1,\\ldots,X_L$, an excursion profile may include\n\\begin{enumerate}\n\\item the number $q$ of ambient-only gates,\n\\item the maximum Schreier distance $d(X_tH,H)$,\n\\item the rank of a forbidden block,\n\\item the bandwidth of a forbidden dependency,\n\\item the number of simultaneously open catalytic intervals,\n\\item the intersection graph of those intervals,\n\\item the maximum algebraic degree created temporarily,\n\\item the maximum ancilla dimension or branch count.\n\\end{enumerate}\n\\end{definition}\nThe established linear family shows that a rank-one, bandwidth-one excursion using exactly two forbidden gates can save linearly many operations.  Thus geometric excursion size need not control computational benefit.","\\subsection{Catalytic interval geometry}\nAssociate to every matched opening and closing operation an interval\n\\[\nI_j=[t_j^{\\mathrm{open}},t_j^{\\mathrm{close}}].\n\\]\nThe family of intervals can be disjoint, nested, laminar, or crossing.","Define the catalytic width\n\\[\nw_{\\mathrm{cat}}=\n\\max_t\n\\#\\{j:t\\in I_j\\}.\n\\]\nOne may then study\n\\[\nD_n^{(w)}\n=\n\\max_{U}\n\\left(\n\\ellRes(U)-\\ellAmb^{(w)}(U)\n\\right),\n\\]\nwhere the ambient circuit has width at most $w$.  The unitriangular snapshot family gives $D_n^{(1)}=\\Omega(n)$ in its corresponding interval model.","\\subsection{Four recurrent catalytic mechanisms}\n\\subsubsection{Basis-change catalysis}\nAn ambient gate $g$ changes the frame in which several restricted operations are cheap:\n\\[\nh_i\\mapsto gh_ig^{-1}.\n\\]\nHadamard and Fourier transforms are the basic quantum examples.  The frame is restored at the end.","\\subsubsection{Commutator catalysis}\nAn ambient commutator creates a new direction:\n\\[\n[g,h]=ghg^{-1}h^{-1}.\n\\]\nFor continuous controls,\n\\[\ne^{\\varepsilon X}e^{\\varepsilon Y}e^{-\\varepsilon X}e^{-\\varepsilon Y}\n=\ne^{\\varepsilon^2[X,Y]+O(\\varepsilon^3)}.\n\\]\nThe desired effective generator may lie in the restricted algebra even though the path does not.","\\subsection{When defects are impossible}\nA useful positive theorem is a criterion guaranteeing geodesic convexity.\n\\begin{proposition}[One-Lipschitz retraction criterion]\nSuppose there is a group homomorphism $\\pi:G\\to H$ with $\\pi|_H=\\operatorname{id}_H$ such that every ambient generator $s\\in\\mathcal S_G$ satisfies\n\\[\n\\ell_H(\\pi(s))\\le c(s).\n\\]\nThen\n\\[\n\\ell_H(h)\\le\\ell_G(h)\n\\]\nfor every $h\\in H$, hence the two metrics agree on $H$.\n\\end{proposition}\n\\begin{proof}\nFor an ambient word $h=s_L\\cdots s_1$, multiplicativity gives $h=\\pi(h)=\\pi(s_L)\\cdots\\pi(s_1)$.  Subadditivity and the generator bounds give $\\ell_H(h)\\le\\sum_j c(s_j)$.  Taking the infimum over ambient words proves the claim.\n\\end{proof}","sible}\nA useful positive theorem is a criterion guaranteeing geodesic convexity.\n\\begin{proposition}[One-Lipschitz retraction criterion]\nSuppose there is a group homomorphism $\\pi:G\\to H$ with $\\pi|_H=\\operatorname{id}_H$ such that every ambient generator $s\\in\\mathcal S_G$ satisfies\n\\[\n\\ell_H(\\pi(s))\\le c(s).\n\\]\nThen\n\\[\n\\ell_H(h)\\le\\ell_G(h)\n\\]\nfor every $h\\in H$, hence the two metrics agree on $H$.\n\\end{proposition}\n\\begin{proof}\nFor an ambient word $h=s_L\\cdots s_1$, multiplicativity gives $h=\\pi(h)=\\pi(s_L)\\cdots\\pi(s_1)$.  Subadditivity and the generator bounds give $\\ell_H(h)\\le\\sum_j c(s_j)$.  Taking the infimum over ambient words proves the claim.\n\\end{proof}","ing geodesic convexity.\n\\begin{proposition}[One-Lipschitz retraction criterion]\nSuppose there is a group homomorphism $\\pi:G\\to H$ with $\\pi|_H=\\operatorname{id}_H$ such that every ambient generator $s\\in\\mathcal S_G$ satisfies\n\\[\n\\ell_H(\\pi(s))\\le c(s).\n\\]\nThen\n\\[\n\\ell_H(h)\\le\\ell_G(h)\n\\]\nfor every $h\\in H$, hence the two metrics agree on $H$.\n\\end{proposition}\n\\begin{proof}\nFor an ambient word $h=s_L\\cdots s_1$, multiplicativity gives $h=\\pi(h)=\\pi(s_L)\\cdots\\pi(s_1)$.  Subadditivity and the generator bounds give $\\ell_H(h)\\le\\sum_j c(s_j)$.  Taking the infimum over ambient words proves the claim.\n\\end{proof}","}[One-Lipschitz retraction criterion]\nSuppose there is a group homomorphism $\\pi:G\\to H$ with $\\pi|_H=\\operatorname{id}_H$ such that every ambient generator $s\\in\\mathcal S_G$ satisfies\n\\[\n\\ell_H(\\pi(s))\\le c(s).\n\\]\nThen\n\\[\n\\ell_H(h)\\le\\ell_G(h)\n\\]\nfor every $h\\in H$, hence the two metrics agree on $H$.\n\\end{proposition}\n\\begin{proof}\nFor an ambient word $h=s_L\\cdots s_1$, multiplicativity gives $h=\\pi(h)=\\pi(s_L)\\cdots\\pi(s_1)$.  Subadditivity and the generator bounds give $\\ell_H(h)\\le\\sum_j c(s_j)$.  Taking the infimum over ambient words proves the claim.\n\\end{proof}",":G\\to H$ with $\\pi|_H=\\operatorname{id}_H$ such that every ambient generator $s\\in\\mathcal S_G$ satisfies\n\\[\n\\ell_H(\\pi(s))\\le c(s).\n\\]\nThen\n\\[\n\\ell_H(h)\\le\\ell_G(h)\n\\]\nfor every $h\\in H$, hence the two metrics agree on $H$.\n\\end{proposition}\n\\begin{proof}\nFor an ambient word $h=s_L\\cdots s_1$, multiplicativity gives $h=\\pi(h)=\\pi(s_L)\\cdots\\pi(s_1)$.  Subadditivity and the generator bounds give $\\ell_H(h)\\le\\sum_j c(s_j)$.  Taking the infimum over ambient words proves the claim.\n\\end{proof}","\\subsection{Weighted restoration and phase transitions}\nLet every restricted gate cost one and every ambient-only gate cost $\\lambda$.  For a catalyst used by $v$ consumers with unit benefit, the displayed construction has net gain at least\n\\[\nv-2\\lambda.\n\\]\nThe critical weight is therefore target and mechanism dependent.","\\begin{definition}[Critical catalyst penalty]\nFor a target family $U_n$, define\n\\[\n\\lambda_c(U_n)\n=\n\\sup\\{\\lambda:\\ell_{\\lambda,\\mathrm{amb}}(U_n)<\\ellRes(U_n)\\}.\n\\]\n\\end{definition}\nDetermining $\\lambda_c$ exactly is a refined synthesis problem.  It can be more physically meaningful than unweighted gate count.","\\subsection{Word distance as first arrival}\nLet $G$ be finite and let $S=S^{-1}$ be a symmetric generating multiset.  In the right regular representation, define\n\\[\nA_S=\\sum_{s\\in S}\\rho(s).\n\\]\nThen\n\\[\n\\langle\\delta_g,A_S^k\\delta_e\\rangle\n\\]\ncounts length-$k$ words ending at $g$.","\\begin{theorem}[Endpoint Green valuation]\nThe word distance is\n\\[\nd_S(e,g)\n=\n\\ord_{z=0}\n\\left\\langle\n\\delta_g,(I-zA_S)^{-1}\\delta_e\n\\right\\rangle.\n\\]\n\\end{theorem}\n\\begin{proof}\nExpand the resolvent as a formal power series:\n\\[\n(I-zA_S)^{-1}=\\sum_{k\\ge0}z^kA_S^k.\n\\]\nThe first nonzero endpoint coefficient occurs at the shortest word length.\n\\end{proof}","For strictly positive integer costs, replace each generator $s$ by the monomial $z^{c(s)}\\rho(s)$.  Every coefficient then counts finitely many words of a given total cost, and the same formal inverse is defined.  Zero-cost cycles require separate treatment: they can create infinitely many words of cost zero, so the ordinary counting series need not exist.  The finite atlases with free gates use shortest-path search with zero-cost edges instead.  Multivariate variables can track several resources when the grading gives every generator positive total degree.","\\subsection{Why eigenvalues alone are insufficient}\nIf\n\\[\nA_S=\\sum_\\lambda\\lambda\\Pi_\\lambda,\n\\]\nthen\n\\[\n\\mathcal G_{S,g}(z)\n=\n\\sum_\\lambda\n\\frac{\\langle\\delta_g,\\Pi_\\lambda\\delta_e\\rangle}{1-\\lambda z}.\n\\]\nThe eigenvalues and multiplicities do not determine the endpoint coefficients\n\\[\n\\langle\\delta_g,\\Pi_\\lambda\\delta_e\\rangle.\n\\]\nTwo Cayley or Schreier systems can therefore be cospectral and still have different first-arrival behavior at distinguished targets.","\\subsection{Subgroup-holonomy Fourier packets}\nLet $H\\le G$.  Let $\\mu_k(h)$ count ambient words of length $k$ with product $h\\in H$.  For every irreducible representation $\\tau\\in\\widehat H$, define\n\\[\n\\widehat\\mu_k(\\tau)\n=\n\\sum_{h\\in H}\\mu_k(h)\\tau(h).\n\\]\nFourier inversion gives\n\\[\n\\mu_k(h)\n=\n\\frac1{|H|}\n\\sum_{\\tau\\in\\widehat H}\n\\dim(\\tau)\n\\operatorname{tr}\n\\bigl(\n\\tau(h^{-1})\\widehat\\mu_k(\\tau)\n\\bigr).\n\\]\nThe matrix-valued Green packet\n\\[\n\\widehat{\\mathcal G}_\\tau(z)\n=\n\\sum_{k\\ge0}\\widehat\\mu_k(\\tau)z^k\n\\]\nrecovers every endpoint series after Fourier inversion.","}\nLet $H\\le G$.  Let $\\mu_k(h)$ count ambient words of length $k$ with product $h\\in H$.  For every irreducible representation $\\tau\\in\\widehat H$, define\n\\[\n\\widehat\\mu_k(\\tau)\n=\n\\sum_{h\\in H}\\mu_k(h)\\tau(h).\n\\]\nFourier inversion gives\n\\[\n\\mu_k(h)\n=\n\\frac1{|H|}\n\\sum_{\\tau\\in\\widehat H}\n\\dim(\\tau)\n\\operatorname{tr}\n\\bigl(\n\\tau(h^{-1})\\widehat\\mu_k(\\tau)\n\\bigr).\n\\]\nThe matrix-valued Green packet\n\\[\n\\widehat{\\mathcal G}_\\tau(z)\n=\n\\sum_{k\\ge0}\\widehat\\mu_k(\\tau)z^k\n\\]\nrecovers every endpoint series after Fourier inversion.","The trivial representation records only the total number of loops returning to the subgroup.  Nontrivial $H$-types retain their holonomy.  This yields a hierarchy:\n\\[\n\\text{spherical return data}\n\\prec\n\\text{type-resolved holonomy packets}\n\\prec\n\\text{exact endpoint realization}.\n\\]","\\subsection{Word maps and product loci}\nLet $K$ be a field and let $X\\subseteq G$ be an algebraic gate family.  For a fixed length $L$, multiplication defines\n\\[\n\\mu_L:X^L\\to G,\n\\qquad\n(x_L,\\ldots,x_1)\\mapsto x_L\\cdots x_1.\n\\]\nThe exact length-$L$ locus is $\\mu_L(X^L)$, and the length-at-most-$L$ locus is a finite union of such images when the gate family has finitely many components.","\\begin{theorem}[Dimension per gate]\nSuppose every irreducible component of $X$ has dimension at most $r$.  Then the Zariski closure of the locus reachable in at most $L$ gates has dimension at most $rL$.\n\\end{theorem}\n\\begin{proof}\nEvery component of $X^L$ has dimension at most $rL$.  A morphism cannot increase dimension.  A finite union has dimension equal to the maximum dimension of its components.\n\\end{proof}","For scaled transvections, $r=1$.  This immediately yields generic lower bounds in a target variety whose dimension exceeds $L$.","\\subsection{Generic rigidity}\nWork over an algebraically closed field.  Suppose the restricted target variety $V$ is irreducible of dimension $N$, every target has a restricted realization using at most $N$ unit-cost gates, and the ambient alphabet is an algebraic family with finitely many components of dimension at most one.  Then ambient realizations of length below $N$ occupy a proper closed subset of $V$.","on{Generic rigidity}\nWork over an algebraically closed field.  Suppose the restricted target variety $V$ is irreducible of dimension $N$, every target has a restricted realization using at most $N$ unit-cost gates, and the ambient alphabet is an algebraic family with finitely many components of dimension at most one.  Then ambient realizations of length below $N$ occupy a proper closed subset of $V$.","\\begin{theorem}[Abstract generic rigidity]\nUnder the preceding hypotheses, there is a nonempty Zariski-open subset $\\Omega\\subseteq V$ such that\n\\[\n\\ellRes(U)=\\ellAmb(U)=N\n\\qquad(U\\in\\Omega).\n\\]\n\\end{theorem}\nThe unitriangular theorem in the core chapter is a concrete instance with\n\\[\nN=\\binom n2.\n\\]","\\subsection{Defect codimension}\nIf every target has restricted cost at most $N$, then defect at least $d$ forces ambient cost at most $N-d$.  The dimension-per-gate theorem gives\n\\[\n\\codim_V\\overline{\\{U:\\Delta(U)\\ge d\\}}\n\\ge d\n\\]\nwhen each ambient gate carries one parameter.","\\subsection{Finite-field sparsity}\nA word-counting analogue applies over $\\F_q$.  If the ambient alphabet has at most $M(q-1)$ nontrivial scaled gates, then the number of targets reached in $L$ gates is at most\n\\[\n\\sum_{j=0}^L[M(q-1)]^j.\n\\]\nWhen the target space has $q^N$ points and defect $d$ implies ambient length at most $N-d$, the crude bound is\n\\[\n\\Prob[\\Delta\\ge d]\n=O_{N,M}(q^{-d}).\n\\]\nThe estimate can be sharpened by orbit counting, stabilizers, and the geometry of word-map fibers.","\\subsection{Direct sums and monoidal amplification}\nSuppose targets admit a direct sum $U\\oplus V$ and a homomorphism from the large restricted group onto the two factor groups sends $U\\oplus V$ to $(U,V)$.  Require the aggregate generator bound\n\\[\n\\ellRes(\\pi_1(s))+\\ellRes(\\pi_2(s))\\le c(s)\n\\]\nfor every large restricted generator $s$.  Assume that both restricted and ambient factor circuits embed into the corresponding large model without increasing cost.  Then\n\\[\n\\ellRes(U\\oplus V)\n=\n\\ellRes(U)+\\ellRes(V).\n\\]\nIndeed, projecting any word and summing the two factor bounds gives the lower bound, while concatenating the embedded factor circuits gives the upper bound.  Separate one-Lipschitz projections alone do not imply this sum inequality.\nUnder the ambient embedding assumption, the con","\\subsection{Direct sums and monoidal amplification}\nSuppose targets admit a direct sum $U\\oplus V$ and a homomorphism from the large restricted group onto the two factor groups sends $U\\oplus V$ to $(U,V)$.  Require the aggregate generator bound\n\\[\n\\ellRes(\\pi_1(s))+\\ellRes(\\pi_2(s))\\le c(s)\n\\]\nfor every large restricted generator $s$.  Assume that both restricted and ambient factor circuits embed into the corresponding large model without increasing cost.  Then\n\\[\n\\ellRes(U\\oplus V)\n=\n\\ellRes(U)+\\ellRes(V).\n\\]\nIndeed, projecting any word and summing the two factor bounds gives the lower bound, while concatenating the embedded factor circuits gives the upper bound.  Separate one-Lipschitz projections alone do not imply this sum inequality.\nUnder the ambient embedding assumption, the constructive upper bound is\n\\[\n\\ellAmb(U\\oplus V)\n\\le\n\\ellAmb(U)+\\ellAmb(V),\n\\]\nso\n\\[\n\\Delta(U\\oplu","\\subsection{Direct sums and monoidal amplification}\nSuppose targets admit a direct sum $U\\oplus V$ and a homomorphism from the large restricted group onto the two factor groups sends $U\\oplus V$ to $(U,V)$.  Require the aggregate generator bound\n\\[\n\\ellRes(\\pi_1(s))+\\ellRes(\\pi_2(s))\\le c(s)\n\\]\nfor every large restricted generator $s$.  Assume that both restricted and ambient factor circuits embed into the corresponding large model without increasing cost.  Then\n\\[\n\\ellRes(U\\oplus V)\n=\n\\ellRes(U)+\\ellRes(V).\n\\]\nIndeed, projecting any word and summing the two factor bounds gives the lower bound, while concatenating the embedded factor circuits gives the upper bound.  Separate one-Lipschitz projections alone do not imply this sum inequality.\nUnder the ambient embedding assumption, the constructive upper bound is\n\\[\n\\ellAmb(U\\oplus V)\n\\le\n\\ellAmb(U)+\\ellAmb(V),\n\\]\nso\n\\[\n\\Delta(U\\oplus V)\n\\ge\n\\D","\\subsection{Direct sums and monoidal amplification}\nSuppose targets admit a direct sum $U\\oplus V$ and a homomorphism from the large restricted group onto the two factor groups sends $U\\oplus V$ to $(U,V)$.  Require the aggregate generator bound\n\\[\n\\ellRes(\\pi_1(s))+\\ellRes(\\pi_2(s))\\le c(s)\n\\]\nfor every large restricted generator $s$.  Assume that both restricted and ambient factor circuits embed into the corresponding large model without increasing cost.  Then\n\\[\n\\ellRes(U\\oplus V)\n=\n\\ellRes(U)+\\ellRes(V).\n\\]\nIndeed, projecting any word and summing the two factor bounds gives the lower bound, while concatenating the embedded factor circuits gives the upper bound.  Separate one-Lipschitz projections alone do not imply this sum inequality.\nUnder the ambient embedding assumption, the constructive upper bound is\n\\[\n\\ellAmb(U\\oplus V)\n\\le\n\\ellAmb(U)+\\ellAmb(V),\n\\]\nso\n\\[\n\\Delta(U\\oplus V)\n\\ge\n\\Delta(U)+\\Delta(V).\n\\]\nOne finite defect therefore gives an unbounded additive family.","dmit a direct sum $U\\oplus V$ and a homomorphism from the large restricted group onto the two factor groups sends $U\\oplus V$ to $(U,V)$.  Require the aggregate generator bound\n\\[\n\\ellRes(\\pi_1(s))+\\ellRes(\\pi_2(s))\\le c(s)\n\\]\nfor every large restricted generator $s$.  Assume that both restricted and ambient factor circuits embed into the corresponding large model without increasing cost.  Then\n\\[\n\\ellRes(U\\oplus V)\n=\n\\ellRes(U)+\\ellRes(V).\n\\]\nIndeed, projecting any word and summing the two factor bounds gives the lower bound, while concatenating the embedded factor circuits gives the upper bound.  Separate one-Lipschitz projections alone do not imply this sum inequality.\nUnder the ambient embedding assumption, the constructive upper bound is\n\\[\n\\ellAmb(U\\oplus V)\n\\le\n\\ellAmb(U)+\\ellAmb(V),\n\\]\nso\n\\[\n\\Delta(U\\oplus V)\n\\ge\n\\Delta(U)+\\Delta(V).\n\\]\nOne finite defect therefore gives an unbounded additive family.","large model without increasing cost.  Then\n\\[\n\\ellRes(U\\oplus V)\n=\n\\ellRes(U)+\\ellRes(V).\n\\]\nIndeed, projecting any word and summing the two factor bounds gives the lower bound, while concatenating the embedded factor circuits gives the upper bound.  Separate one-Lipschitz projections alone do not imply this sum inequality.\nUnder the ambient embedding assumption, the constructive upper bound is\n\\[\n\\ellAmb(U\\oplus V)\n\\le\n\\ellAmb(U)+\\ellAmb(V),\n\\]\nso\n\\[\n\\Delta(U\\oplus V)\n\\ge\n\\Delta(U)+\\Delta(V).\n\\]\nOne finite defect therefore gives an unbounded additive family.","\\oplus V)\n=\n\\ellRes(U)+\\ellRes(V).\n\\]\nIndeed, projecting any word and summing the two factor bounds gives the lower bound, while concatenating the embedded factor circuits gives the upper bound.  Separate one-Lipschitz projections alone do not imply this sum inequality.\nUnder the ambient embedding assumption, the constructive upper bound is\n\\[\n\\ellAmb(U\\oplus V)\n\\le\n\\ellAmb(U)+\\ellAmb(V),\n\\]\nso\n\\[\n\\Delta(U\\oplus V)\n\\ge\n\\Delta(U)+\\Delta(V).\n\\]\nOne finite defect therefore gives an unbounded additive family.","\\subsection{Tensor products and normalized rates}\nLet $U$ have dimension $m$ and $V$ dimension $n$.  In linear transvection models with tree-tight certificates,\n\\[\n\\ellRes(U\\otimes V)\n=\nn\\ellRes(U)+m\\ellRes(V).\n\\]\nThe ambient model satisfies the corresponding inequality.  Normalizing by dimension gives an entropy-like law:\n\\[\n\\frac{\\ellRes(U\\otimes V)}{mn}\n=\n\\frac{\\ellRes(U)}m+\n\\frac{\\ellRes(V)}n.\n\\]\nThe normalized defect is superadditive under tensor products.","\\begin{definition}[Tensor synergy]\nDefine\n\\[\n\\Syn(U,V)\n=\nn\\ellAmb(U)+m\\ellAmb(V)-\\ellAmb(U\\otimes V).\n\\]\nPositive synergy means that the joint ambient realization improves on independent lifted circuits.\n\\end{definition}\nRestricted tree-tight metrics have zero tensor synergy.  Ambient synergy is a possible source of collective activation.","\\begin{definition}[Stable tensor cost]\nFor a size-$m$ target $U$, define\n\\[\n\\tau_\\otimes(U)\n=\n\\lim_{r\\to\\infty}\n\\frac{\\ellAmb(U^{\\otimes r})}{r m^{r-1}},\n\\]\nwhen the subadditivity argument applies.  Define\n\\[\n\\kappa_\\otimes(U)\n=\n\\ellRes(U)-\\tau_\\otimes(U).\n\\]\n\\end{definition}\nFekete's lemma gives the limit from the tensor subadditivity inequality.","\\begin{definition}[Tensor activation]\nA target is tensor activated if\n\\[\n\\Delta(U)=0\n\\quad\\text{but}\\quad\n\\kappa_\\otimes(U)>0.\n\\]\n\\end{definition}\nThis is analogous to regularized capacities and asymptotic ranks.  It asks whether collective synthesis can unlock a catalyst invisible in one copy.","\\subsection{Border catalysis}\nLet $X_{\\mathrm{res}}$ and $X_{\\mathrm{amb}}$ be algebraic gate varieties.  The exact product rank is\n\\[\n\\operatorname{rk}_{X}(U)\n=\n\\min\\{L:U\\in X^L\\},\n\\]\nwhile border product rank is\n\\[\n\\underline{\\operatorname{rk}}_{X}(U)\n=\n\\min\\{L:U\\in\\overline{X^L}\\}.\n\\]\nThe four possible comparisons are:\n\\begin{enumerate}\n\\item exact and border defects both positive,\n\\item exact defect positive but border defect zero,\n\\item one-copy defect zero but tensor-regularized defect positive,\n\\item ambient and restricted border ranks both smaller than exact ranks but by different amounts.\n\\end{enumerate}\nEach requires a separate certificate.  Numerical approximation alone does not distinguish them.","\\subsection{Intermediate states as a representable matroid}\nA linear circuit visits vectors that satisfy exact dependence relations.  If $\\mathcal V$ is the set of all row states appearing during a computation, then $\\mathcal V$ determines a representable matroid.  A transvection\n\\[\nu\\longmapsto u+a v\n\\]\ncreates a three-point projective line\n\\[\n[u],\\ [v],\\ [u+av].\n\\]\nThe ambient circuit chooses a tour through this projective configuration.  A flag-preserving circuit must lift the same dependencies through prescribed pivot positions.","\\begin{definition}[Flagged matroid tour]\nLet $M$ be a representable matroid with designated initial basis, target basis, and consumer states.  A \\emph{flagged tour} is a sequence of basis exchanges or transvections in which the $i$th carrier remains inside a specified filtration layer.  An ambient tour drops this constraint.\n\\end{definition}\nThe six-row atom realizes the graphic regular matroid $M(K_4)$.  Its ambient state tour is shorter than every compatible flag lift.  Regularity explains why the same dependence pattern survives every field and, through integer matrices, every nonzero commutative coefficient ring in the stated model.","\\begin{conjecturebox}\nMinimal field-universal catalytic atoms should be governed by regular matroids, while characteristic-dependent atoms should arise from representable matroids with excluded-characteristic minors.\n\\end{conjecturebox}\nThis conjecture is a programmatic guide.  The core paper establishes only the $M(K_4)$ instance.","\\subsection{Dynamic rank-one factorization}\nConsider a height-two block shear\n\\[\nE(C)=\n\\begin{pmatrix}\nI_s&0\\\\\nC&I_t\n\\end{pmatrix}.\n\\]\nA restricted circuit may change source and output bases while transmitting rank-one terms.  Let $P_r\\in U_s$ be the source basis before the $r$th transmission and let $Q_{>r}\\in U_t$ be the future output basis.  A cross gate contributes\n\\[\nQ_{>r}e_{i_r}e_{j_r}^{T}P_{<r}\n\\]\nto the final block.  Hence\n\\[\nC=\n\\sum_{r=1}^{c}\nQ_{>r}e_{i_r}e_{j_r}^{T}P_{<r},\n\\]\nwith closed basis paths\n\\[\nP_{\\mathrm{final}}=I_s,\n\\qquad\nQ_{\\mathrm{final}}=I_t.\n\\]","C&I_t\n\\end{pmatrix}.\n\\]\nA restricted circuit may change source and output bases while transmitting rank-one terms.  Let $P_r\\in U_s$ be the source basis before the $r$th transmission and let $Q_{>r}\\in U_t$ be the future output basis.  A cross gate contributes\n\\[\nQ_{>r}e_{i_r}e_{j_r}^{T}P_{<r}\n\\]\nto the final block.  Hence\n\\[\nC=\n\\sum_{r=1}^{c}\nQ_{>r}e_{i_r}e_{j_r}^{T}P_{<r},\n\\]\nwith closed basis paths\n\\[\nP_{\\mathrm{final}}=I_s,\n\\qquad\nQ_{\\mathrm{final}}=I_t.\n\\]","end{pmatrix}.\n\\]\nA restricted circuit may change source and output bases while transmitting rank-one terms.  Let $P_r\\in U_s$ be the source basis before the $r$th transmission and let $Q_{>r}\\in U_t$ be the future output basis.  A cross gate contributes\n\\[\nQ_{>r}e_{i_r}e_{j_r}^{T}P_{<r}\n\\]\nto the final block.  Hence\n\\[\nC=\n\\sum_{r=1}^{c}\nQ_{>r}e_{i_r}e_{j_r}^{T}P_{<r},\n\\]\nwith closed basis paths\n\\[\nP_{\\mathrm{final}}=I_s,\n\\qquad\nQ_{\\mathrm{final}}=I_t.\n\\]","\\begin{definition}[Dynamic rank-one cost]\nThe dynamic rank-one cost of $C$ is the minimum of\n\\[\n\\text{source basis motion}\n+\n\\text{output basis motion}\n+\n\\text{number of transmissions}\n\\]\nover such timed decompositions.\n\\end{definition}\nOrdinary rank retains only the number of transmissions after arbitrary bases are given for free.  Circuit complexity charges the closed paths needed to expose and restore those bases.","\\subsection{Quiver gate models}\nLet $Q=(Q_0,Q_1)$ be a quiver.  Associate a vector space $V_v$ to every vertex.  Internal gates change bases inside $V_v$, while edge gates transmit low-rank maps along arrows.  A target is a representation of the path algebra or an automorphism of a filtered quiver module.","\\subsection{Pattern groups and architecture constraints}\nFor a poset $P$, define the pattern group\n\\[\nU_P(K)\n=\n\\{I+A:A_{ij}=0\\text{ unless }j<_P i\\}.\n\\]\nWhen the allowed positions are transitively closed, $U_P$ is a group.  Compare the intrinsic metric generated by $P$-allowed transvections with the metric induced by all transvections.","The first threshold now depends on the forbidden-edge geometry of $P$.  One may ask whether a six-role catalytic motif embeds as a generator-preserving retract.  If it does, the larger architecture inherits a defect.","\\begin{proposition}[Retraction inheritance]\nSuppose a smaller nested pair $H\\le G$ embeds into $H'\\le G'$ by homomorphisms preserving the inclusion, the ambient embedding does not increase gate cost, and there is a homomorphic retraction $\\pi:H'\\to H$ onto the embedded copy.  Require $\\ell_H(\\pi(s))\\le c(s)$ for every restricted generator $s$ of $H'$.  Then every defect in $H\\le G$ gives a defect at least as large in $H'\\le G'$.\n\\end{proposition}\nThe proof projects a restricted word in the larger system and embeds the shorter ambient word from the smaller system.","traction inheritance]\nSuppose a smaller nested pair $H\\le G$ embeds into $H'\\le G'$ by homomorphisms preserving the inclusion, the ambient embedding does not increase gate cost, and there is a homomorphic retraction $\\pi:H'\\to H$ onto the embedded copy.  Require $\\ell_H(\\pi(s))\\le c(s)$ for every restricted generator $s$ of $H'$.  Then every defect in $H\\le G$ gives a defect at least as large in $H'\\le G'$.\n\\end{proposition}\nThe proof projects a restricted word in the larger system and embeds the shorter ambient word from the smaller system.","\\subsection{Root subgroups and Chevalley relations}\nLet $G$ be a split reductive group \\cite{g:Carter1972,g:Humphreys1975} with root system $\\Phi$.  Let $\\Psi\\subseteq\\Phi$ be a closed set of allowed roots and let $U_\\Psi$ be generated by the root groups $x_\\beta(t)$ for $\\beta\\in\\Psi$.  Define\n\\[\n\\ell_\\Psi(u)\n=\n\\min\\{L:u=x_{\\beta_L}(t_L)\\cdots x_{\\beta_1}(t_1),\\ \\beta_i\\in\\Psi\\},\n\\]\nand let $\\ell_\\Phi$ permit all roots.","The Chevalley conjugation formula has the schematic form\n\\[\nx_\\alpha(a)x_\\beta(b)x_\\alpha(a)^{-1}\n=\nx_\\beta(b)\n\\prod_{k>0}\nx_{\\beta+k\\alpha}(c_k a^k b).\n\\]\nA forbidden root $\\alpha$ can therefore create several allowed root effects before being removed.","\\subsection{Type \\texorpdfstring{$A$}{A} and beyond}\nRow transvections are root-group elements of type $A$.  The established unitriangular atom occurs in $A_5$.  In nonsimply laced systems, root strings can have length greater than two.  This raises the possibility that one ambient root can create multiple restricted effects per consumer.","\\begin{conjecture}[Long-root catalytic gain]\nThere exist gate alphabets in types $B$, $C$, or $G_2$ for which a clean one-catalyst multiplicative distortion exceeds the type-$A$ snapshot limit of $3/2$.\n\\end{conjecture}\nNo lower-bound certificate is presently supplied.  The conjecture identifies a concrete reason to test nonsimply laced root systems rather than merely repeating type $A$ computations.","\\subsection{Symplectic root systems and Clifford logic}\nThe binary Clifford group modulo phases acts through $\\Sp(2n,2)$.  A parabolic subgroup stabilizing a Lagrangian is the symplectic analogue of a Borel or flag-preserving circuit class.  Hadamard gates leave this parabolic, while $S$, $\\CZ$, and $\\CX$ preserve it.  The exact Clifford censuses in \\cref{g:ch:clifford} are therefore a first root-system extension of the linear theory.","\\subsection{Categorical realization defects}\nThe group framework does not cover measurement, feedforward, postselection, or entangled design objects.  Let\n\\[\n\\mathcal C_{\\mathrm{res}}\n\\subseteq\n\\mathcal C_{\\mathrm{amb}}\n\\]\nbe monoidal categories with costed generators.  A target morphism can have a finite metric defect or an infinite existence defect.","A CNOT circuit on $n$ labeled wires represents an element of $\\GL_n(\\F_2)=\\SL_n(\\F_2)$, and minimizing its gate count is the word-metric problem for transvection generators.  This formulation supports asymptotically efficient synthesis, architecture-aware elimination, computable lower bounds, and exact small-width optimization \\cite{g:SaeediMarkov2013,g:PatelMarkovHayes2008,g:JiangEtAl2020,g:GoubaultEtAl2021,g:Bataille2022,g:BuFanJoo2025,g:ChristensenEtAl2025,g:WebsterEtAl2025}.  Recent work has also produced stronger explicit lower-bound families and established NP-completeness of exact all-to-all CNOT distance \\cite{g:Jorgensen2026,g:GongYu2026,g:AcuavivaEtAl2026}.  The present paper studies a different compa","A CNOT circuit on $n$ labeled wires represents an element of $\\GL_n(\\F_2)=\\SL_n(\\F_2)$, and minimizing its gate count is the word-metric problem for transvection generators.  This formulation supports asymptotically efficient synthesis, architecture-aware elimination, computable lower bounds, and exact small-width optimization \\cite{g:SaeediMarkov2013,g:PatelMarkovHayes2008,g:JiangEtAl2020,g:GoubaultEtAl2021,g:Bataille2022,g:BuFanJoo2025,g:ChristensenEtAl2025,g:WebsterEtAl2025}.  Recent work has also produced stronger explicit lower-bound families and established NP-completeness of exact all-to-all CNOT distance \\cite{g:Jorgensen2026,g:GongYu2026,g:AcuavivaEtAl2026}.  The present paper studies a different comparison: the target is lower unitriangular, but the circuit is eith","Let $R$ be a nonzero commutative ring with identity.  A row transvection is\n\\[\n\\tau_{t,s}(a)=I+aE_{t,s},\n\\qquad t\\ne s,\n\\qquad a\\in R\\setminus\\{0\\},\n\\]\nacting by\n\\[\nR_t\\longleftarrow R_t+aR_s.\n\\]\nThe transvection is \\emph{lower directed} when $s<t$.  The lower-directed gates generate $\\U_n(R)$.  We compare\n\\[\n\\ellDown(A)\n =\\min\\{\\text{number of lower-directed transvections producing }A\\}\n\\]\nwith\n\\[\n\\ellAll(A)\n =\\min\\{\\text{number of arbitrary row transvections producing }A\\},\n\\qquad A\\in\\U_n(R).\n\\]\nFor a field $\\F$, the first metric is intrinsic to the unipotent radical and the second is induced from the elementary group $\\Egrp_n(\\F)=\\SL_n(\\F)$.","et $R$ be a nonzero commutative ring with identity.  A row transvection is\n\\[\n\\tau_{t,s}(a)=I+aE_{t,s},\n\\qquad t\\ne s,\n\\qquad a\\in R\\setminus\\{0\\},\n\\]\nacting by\n\\[\nR_t\\longleftarrow R_t+aR_s.\n\\]\nThe transvection is \\emph{lower directed} when $s<t$.  The lower-directed gates generate $\\U_n(R)$.  We compare\n\\[\n\\ellDown(A)\n =\\min\\{\\text{number of lower-directed transvections producing }A\\}\n\\]\nwith\n\\[\n\\ellAll(A)\n =\\min\\{\\text{number of arbitrary row transvections producing }A\\},\n\\qquad A\\in\\U_n(R).\n\\]\nFor a field $\\F$, the first metric is intrinsic to the unipotent radical and the second is induced from the elementary group $\\Egrp_n(\\F)=\\SL_n(\\F)$.","s,\n\\qquad a\\in R\\setminus\\{0\\},\n\\]\nacting by\n\\[\nR_t\\longleftarrow R_t+aR_s.\n\\]\nThe transvection is \\emph{lower directed} when $s<t$.  The lower-directed gates generate $\\U_n(R)$.  We compare\n\\[\n\\ellDown(A)\n =\\min\\{\\text{number of lower-directed transvections producing }A\\}\n\\]\nwith\n\\[\n\\ellAll(A)\n =\\min\\{\\text{number of arbitrary row transvections producing }A\\},\n\\qquad A\\in\\U_n(R).\n\\]\nFor a field $\\F$, the first metric is intrinsic to the unipotent radical and the second is induced from the elementary group $\\Egrp_n(\\F)=\\SL_n(\\F)$.","rected} when $s<t$.  The lower-directed gates generate $\\U_n(R)$.  We compare\n\\[\n\\ellDown(A)\n =\\min\\{\\text{number of lower-directed transvections producing }A\\}\n\\]\nwith\n\\[\n\\ellAll(A)\n =\\min\\{\\text{number of arbitrary row transvections producing }A\\},\n\\qquad A\\in\\U_n(R).\n\\]\nFor a field $\\F$, the first metric is intrinsic to the unipotent radical and the second is induced from the elementary group $\\Egrp_n(\\F)=\\SL_n(\\F)$.","The equality $\\ellDown(A)=\\ellAll(A)$ is tempting because the target and the obvious Gaussian-elimination paths are lower unitriangular.  It is false.  The failure first occurs at six wires over $\\F_2$ and $\\F_3$, and it is caused by a reusable temporary carrier rather than by a better ordering of the same lower gates.","\\begin{theorem}[Universal catalytic atom over rings]\\label{g:thm:universal-atom}\nFor every nonzero commutative ring $R$ with identity, let\n\\begin{equation}\\label{g:eq:canonical-C}\nC=\n\\begin{pmatrix}\n1&0&0&0&0&0","\\begin{theorem}[Universal catalytic atom over rings]\\label{g:thm:universal-atom}\nFor every nonzero commutative ring $R$ with identity, let\n\\begin{equation}\\label{g:eq:canonical-C}\nC=\n\\begin{pmatrix}\n1&0&0&0&0&0\\\\\n1&1&0&0&0&0\\\\\n0&0&1&0&0&0\\\\\n0&1&1&1&0&0\\\\\n1&1&1&0&1&0\\\\\n1&1&1&0&0&1\n\\end{pmatrix}\n\\in\\U_6(R).\n\\end{equation}\nThen\n\\[\n\\ellDown(C)=7,\n\\qquad\n\\ellAll(C)=6.\n\\]\n\\end{theorem}","1&1&1&0&0&1\n\\end{pmatrix}\n\\in\\U_6(R).\n\\end{equation}\nThen\n\\[\n\\ellDown(C)=7,\n\\qquad\n\\ellAll(C)=6.\n\\]\n\\end{theorem}","The six-gate ambient circuit uses row $1$ as a dirty workspace.  It first adds row $2$ into row $1$, exports the carrier $e_1+e_2$, changes the carrier to $e_0+e_1+e_2$, exports it twice, and finally removes the temporary $e_2$ component.  No extra wire is introduced.  The workspace is one of the target rows and is restored to its prescribed value.","\\begin{theorem}[Exact binary threshold and census]\\label{g:thm:binary-census}\nOver $\\F_2$:\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item $\\ellDown(A)=\\ellAll(A)$ for every $A\\in\\U_n(\\F_2)$ when $n\\le5$.\n\\item Among the $2^{15}=32768$ matrices in $\\U_6(\\F_2)$, exactly twenty satisfy $\\ellDown(A)>\\ellAll(A)$.\n\\item Every defective target satisfies\n\\[\n\\ellDown(A)=7,\n\\qquad\n\\ellAll(A)=6.\n\\]\nNo other distance pair is defective.\n\\end{enumerate}\n\\end{theorem}","\\begin{theorem}[Minimal catalytic classification]\\label{g:thm:motif-classification}\nThe twenty defective targets in $\\U_6(\\F_2)$ are exactly the targets generated by a simple one-catalyst word having three cross interactions, one phase interaction, and a nontrivial before/after split across the phase.  The numbers arising from the three possible left/right auxiliary splits are\n\\[\n6,\n\\qquad\n8,\n\\qquad\n6,\n\\]\nfor a total of twenty.\n\\end{theorem}","The field-independent atom and exact ternary search also give the same finite threshold over $\\F_3$.","\\begin{theorem}[Exact ternary threshold]\\label{g:thm:ternary-threshold}\nFor $n\\le5$ and every $A\\in\\U_n(\\F_3)$,\n\\[\n\\ellDown(A)=\\ellAll(A).\n\\]\nThe matrix $C\\in\\U_6(\\F_3)$ of \\cref{g:eq:canonical-C} has distances $(7,6)$.  Hence six is the first defective dimension over $\\F_3$.\n\\end{theorem}","\\begin{theorem}[Connected snapshot family]\\label{g:thm:snapshot-family}\nLet $u\\ge1$, let $1\\le v\\le u+1$, and choose distinct phases\n\\[\n0\\le s_1<\\cdots<s_v\\le u.\n\\]\nThere is a connected target\n\\[\nA(u;s_1,\\ldots,s_v)\\in\\U_{u+v+2}(\\F)\n\\]\nwith\n\\[\n\\ellDown\\bigl(A(u;s_1,\\ldots,s_v)\\bigr)=u+2v\n\\]\nand\n\\[\n\\ellAll\\bigl(A(u;s_1,\\ldots,s_v)\\bigr)\\le u+v+2.\n\\]\nHence its certified defect is at least $v-2$.\n\\end{theorem}","\\begin{corollary}[Connected defects in every dimension]\\label{g:cor:connected-gap}\nFor every field and every $n\\ge7$, there is a connected target $A_n\\in\\U_n(\\F)$ satisfying\n\\[\n\\ellDown(A_n)-\\ellAll(A_n)\n\\ge\n\\left\\lfloor\\frac{n-5}{2}\\right\\rfloor.\n\\]\n\\end{corollary}","\\begin{theorem}[Two forbidden gates and weighted penalties]\\label{g:thm:resource-robustness}\nFor every field and every $n\\ge7$, there is a connected target $A_n\\in\\U_n(\\F)$ such that\n\\[\n\\ellBudget{0}(A_n)-\\ellBudget{2}(A_n)\n\\ge\n\\left\\lfloor\\frac{n-5}{2}\\right\\rfloor,\n\\]\nwhere $\\ellBudget{q}$ permits at most $q$ upper-directed gates.  More generally, if lower gates have cost $1$ and upper gates have cost $\\lambda>0$, then\n\\[\n\\ellDown(A_n)-\\ellWeight{\\lambda}(A_n)\n\\ge\n\\left\\lfloor\\frac{n-1}{2}\\right\\rfloor-2\\lambda.\n\\]\nThus no fixed upper-gate budget of two, and no fixed finite upper-gate penalty, suppresses catalytic escape asymptotically.\n\\end{theorem}","heorem}[Two forbidden gates and weighted penalties]\\label{g:thm:resource-robustness}\nFor every field and every $n\\ge7$, there is a connected target $A_n\\in\\U_n(\\F)$ such that\n\\[\n\\ellBudget{0}(A_n)-\\ellBudget{2}(A_n)\n\\ge\n\\left\\lfloor\\frac{n-5}{2}\\right\\rfloor,\n\\]\nwhere $\\ellBudget{q}$ permits at most $q$ upper-directed gates.  More generally, if lower gates have cost $1$ and upper gates have cost $\\lambda>0$, then\n\\[\n\\ellDown(A_n)-\\ellWeight{\\lambda}(A_n)\n\\ge\n\\left\\lfloor\\frac{n-1}{2}\\right\\rfloor-2\\lambda.\n\\]\nThus no fixed upper-gate budget of two, and no fixed finite upper-gate penalty, suppresses catalytic escape asymptotically.\n\\end{theorem}","en gates and weighted penalties]\\label{g:thm:resource-robustness}\nFor every field and every $n\\ge7$, there is a connected target $A_n\\in\\U_n(\\F)$ such that\n\\[\n\\ellBudget{0}(A_n)-\\ellBudget{2}(A_n)\n\\ge\n\\left\\lfloor\\frac{n-5}{2}\\right\\rfloor,\n\\]\nwhere $\\ellBudget{q}$ permits at most $q$ upper-directed gates.  More generally, if lower gates have cost $1$ and upper gates have cost $\\lambda>0$, then\n\\[\n\\ellDown(A_n)-\\ellWeight{\\lambda}(A_n)\n\\ge\n\\left\\lfloor\\frac{n-1}{2}\\right\\rfloor-2\\lambda.\n\\]\nThus no fixed upper-gate budget of two, and no fixed finite upper-gate penalty, suppresses catalytic escape asymptotically.\n\\end{theorem}","The same paths remain extremely close to the restricted subgroup.  For a matrix $X$ over a field, let $X^+$ be its strict upper part, let $r^+(X)=\\rank X^+$, and let $b^+(X)$ be the largest upper displacement $j-i$ with $X_{ij}\\ne0$.  If $\\ellExc{r}{b}$ denotes the minimum length among circuits satisfying $r^+(X_t)\\le r$ and $b^+(X_t)\\le b$ at every prefix, then the connected family can be chosen so that\n\\[\n\\ellExc{0}{0}(A_n)-\\ellExc{1}{1}(A_n)\n\\ge\n\\left\\lfloor\\frac{n-5}{2}\\right\\rfloor.\n\\]\nThus a rank-one excursion across one adjacent flag layer can create a linear saving.","aths remain extremely close to the restricted subgroup.  For a matrix $X$ over a field, let $X^+$ be its strict upper part, let $r^+(X)=\\rank X^+$, and let $b^+(X)$ be the largest upper displacement $j-i$ with $X_{ij}\\ne0$.  If $\\ellExc{r}{b}$ denotes the minimum length among circuits satisfying $r^+(X_t)\\le r$ and $b^+(X_t)\\le b$ at every prefix, then the connected family can be chosen so that\n\\[\n\\ellExc{0}{0}(A_n)-\\ellExc{1}{1}(A_n)\n\\ge\n\\left\\lfloor\\frac{n-5}{2}\\right\\rfloor.\n\\]\nThus a rank-one excursion across one adjacent flag layer can create a linear saving.","\\begin{theorem}[Generic rigidity and defect codimension]\\label{g:thm:generic-rigidity-intro}\nLet $K$ be an infinite field and put $N=\\binom n2$.  There is a nonempty Zariski-open subset $\\Omega_n\\subseteq\\U_n$ with $\\Omega_n(K)\\ne\\varnothing$ such that\n\\[\n\\ellDown(A)=\\ellAll(A)=N\n\\qquad(A\\in\\Omega_n(K)).\n\\]\nMoreover, after base change to an algebraic closure $\\overline K$, for\n\\[\n\\mathcal D_{n,\\ge d}(\\overline K)\n=\\{A\\in\\U_n(\\overline K):\\DeltaDef(A)\\ge d\\},\n\\]\none has\n\\[\n\\codim_{(\\U_n)_{\\overline K}}\n\\overline{\\mathcal D_{n,\\ge d}(\\overline K)}\\ge d.\n\\]\n\\end{theorem}","-rigidity-intro}\nLet $K$ be an infinite field and put $N=\\binom n2$.  There is a nonempty Zariski-open subset $\\Omega_n\\subseteq\\U_n$ with $\\Omega_n(K)\\ne\\varnothing$ such that\n\\[\n\\ellDown(A)=\\ellAll(A)=N\n\\qquad(A\\in\\Omega_n(K)).\n\\]\nMoreover, after base change to an algebraic closure $\\overline K$, for\n\\[\n\\mathcal D_{n,\\ge d}(\\overline K)\n=\\{A\\in\\U_n(\\overline K):\\DeltaDef(A)\\ge d\\},\n\\]\none has\n\\[\n\\codim_{(\\U_n)_{\\overline K}}\n\\overline{\\mathcal D_{n,\\ge d}(\\overline K)}\\ge d.\n\\]\n\\end{theorem}","\\begin{corollary}[Finite-field sparsity]\\label{g:cor:finite-field-sparsity-intro}\nLet $N=\\binom n2$ and $M=n(n-1)$.  For $1\\le d\\le N$,\n\\[\n\\Prob_{A\\in\\U_n(\\F_q)}[\\DeltaDef(A)\\ge d]\n\\le\n\\min\\{1,M^{N-d}q^{-d}\\}.\n\\]\nFor fixed $n$, a uniformly random target therefore has defect zero with probability $1-O_n(q^{-1})$ as $q\\to\\infty$.\n\\end{corollary}","\\begin{theorem}[Tensor-amplified snapshot family]\\label{g:thm:tensor-main}\nLet $A=A(u;s_1,\\ldots,s_v)$ have dimension $m=u+v+2$.  For $N=m^r$,\n\\[\n\\ellDown(A^{\\otimes r})\n=r m^{r-1}(u+2v),\n\\]\nwhile\n\\[\n\\ellAll(A^{\\otimes r})\n\\le r m^r.\n\\]\nConsequently,\n\\[\n\\ellDown(A^{\\otimes r})-\\ellAll(A^{\\otimes r})\n\\ge\n\\frac{v-2}{m\\log_2m}N\\log_2N.\n\\]\n\\end{theorem}","\\begin{corollary}[Explicit $n\\log n$ distortion]\\label{g:cor:nlogn-gap}\nFor\n\\[\nD_n(\\F)\n=\n\\sup_{A\\in\\U_n(\\F)}\n\\bigl(\\ellDown(A)-\\ellAll(A)\\bigr),\n\\]\none has, for every field and every $n$,\n\\[\nD_n(\\F)\n\\ge\n\\frac{8}{21\\log_2 21}n\\log_2n\n-\n\\frac{82}{105}n.\n\\]\nThe coefficient is attained within the snapshot family by the twenty-one-dimensional seed\n\\[\nA(9;0,1,\\ldots,9).\n\\]\n\\end{corollary}","Over $\\F_2$, the carrier states used by the canonical word form six of the seven points of $\\operatorname{PG}(2,2)$.  The four projective lines supported on those six points form a Pasch configuration.  The same six vectors represent the graphic matroid $M(K_4)$ over every field; this regular-matroid structure isolates the characteristic-independent core of the atom.  At the spectral level, the distance to a target is the first nonzero coefficient of an endpoint Green kernel.  For $C$, the ambient kernel begins in degree six and the restricted kernel begins in degree seven.","of $\\operatorname{PG}(2,2)$.  The four projective lines supported on those six points form a Pasch configuration.  The same six vectors represent the graphic matroid $M(K_4)$ over every field; this regular-matroid structure isolates the characteristic-independent core of the atom.  At the spectral level, the distance to a target is the first nonzero coefficient of an endpoint Green kernel.  For $C$, the ambient kernel begins in degree six and the restricted kernel begins in degree seven.","support a linear saving, even if each forbidden gate carries any fixed finite penalty.  Third, tensor products add normalized restricted complexity while ambient complexity remains subadditive.  This converts one local carrier advantage into a superlinear global defect.  Finally, the algebraic-geometric bounds show that these amplifiable atoms are exceptional at fixed dimension: a defect of size $d$ imposes at least $d$ algebraic codimensions.  The resulting picture is one of algebraic rarity coupled with asymptotic amplification.","ng, even if each forbidden gate carries any fixed finite penalty.  Third, tensor products add normalized restricted complexity while ambient complexity remains subadditive.  This converts one local carrier advantage into a superlinear global defect.  Finally, the algebraic-geometric bounds show that these amplifiable atoms are exceptional at fixed dimension: a defect of size $d$ imposes at least $d$ algebraic codimensions.  The resulting picture is one of algebraic rarity coupled with asymptotic amplification.","ct cases through dimension nine, and connected families in every larger dimension.  \\Cref{g:sec:resources} proves bounded, weighted, and microscopic-excursion theorems.  \\Cref{g:sec:rarity} proves generic rigidity, defect-codimension bounds, and finite-field sparsity.  \\Cref{g:sec:amplification} proves block, recursive, and tensor amplification, introduces stable tensor capacity, and derives the $n\\log n$ extremal bound.  The appendices provide the exact target list, algorithms, and reproducibility data.","Fix a nonzero commutative ring $R$ with identity and an integer $n\\ge2$.  Rows and columns are indexed by\n\\[\n0,1,\\ldots,n-1.\n\\]\nFor $s\\ne t$ and $a\\in R\\setminus\\{0\\}$, set\n\\[\n\\tau_{t,s}(a)=I+aE_{t,s}.\n\\]\nLeft multiplication by $\\tau_{t,s}(a)$ performs\n\\[\nR_t\\longleftarrow R_t+aR_s.\n\\]\nWe call $s$ the source and $t$ the target.  A circuit word\n\\[\nw=(\\tau_{t_m,s_m}(a_m),\\ldots,\\tau_{t_1,s_1}(a_1))\n\\]\nproduces\n\\[\n[w]=\\tau_{t_m,s_m}(a_m)\\cdots\\tau_{t_1,s_1}(a_1).\n\\]","commutative ring $R$ with identity and an integer $n\\ge2$.  Rows and columns are indexed by\n\\[\n0,1,\\ldots,n-1.\n\\]\nFor $s\\ne t$ and $a\\in R\\setminus\\{0\\}$, set\n\\[\n\\tau_{t,s}(a)=I+aE_{t,s}.\n\\]\nLeft multiplication by $\\tau_{t,s}(a)$ performs\n\\[\nR_t\\longleftarrow R_t+aR_s.\n\\]\nWe call $s$ the source and $t$ the target.  A circuit word\n\\[\nw=(\\tau_{t_m,s_m}(a_m),\\ldots,\\tau_{t_1,s_1}(a_1))\n\\]\nproduces\n\\[\n[w]=\\tau_{t_m,s_m}(a_m)\\cdots\\tau_{t_1,s_1}(a_1).\n\\]","Let\n\\[\nS_n^\\downarrow(R)\n=\\{\\tau_{t,s}(a):0\\le s<t<n,\\ a\\in R\\setminus\\{0\\}\\}\n\\]\nand\n\\[\nS_n^*(R)\n=\\{\\tau_{t,s}(a):0\\le s,t<n,\\ s\\ne t,\\ a\\in R\\setminus\\{0\\}\\}.\n\\]\nBoth sets are symmetric because\n\\[\n\\tau_{t,s}(a)^{-1}=\\tau_{t,s}(-a).\n\\]\nLet\n\\[\n\\Egrp_n(R)=\\langle S_n^*(R)\\rangle.\n\\]\nWhen $R=\\F$ is a field and $n\\ge2$, elementary elimination gives\n\\[\n\\Egrp_n(\\F)=\\SL_n(\\F).\n\\]\nThe ambient metric below is therefore the transvection word metric in $\\SL_n(\\F)$, viewed inside $\\GL_n(\\F)$.  When $\\F=\\F_2$, one has $\\SL_n(\\F_2)=\\GL_n(\\F_2)$, every generator is an involution, and it is exactly a CNOT gate under the usual row-action convention.","s<t<n,\\ a\\in R\\setminus\\{0\\}\\}\n\\]\nand\n\\[\nS_n^*(R)\n=\\{\\tau_{t,s}(a):0\\le s,t<n,\\ s\\ne t,\\ a\\in R\\setminus\\{0\\}\\}.\n\\]\nBoth sets are symmetric because\n\\[\n\\tau_{t,s}(a)^{-1}=\\tau_{t,s}(-a).\n\\]\nLet\n\\[\n\\Egrp_n(R)=\\langle S_n^*(R)\\rangle.\n\\]\nWhen $R=\\F$ is a field and $n\\ge2$, elementary elimination gives\n\\[\n\\Egrp_n(\\F)=\\SL_n(\\F).\n\\]\nThe ambient metric below is therefore the transvection word metric in $\\SL_n(\\F)$, viewed inside $\\GL_n(\\F)$.  When $\\F=\\F_2$, one has $\\SL_n(\\F_2)=\\GL_n(\\F_2)$, every generator is an involution, and it is exactly a CNOT gate under the usual row-action convention.","}\\}\n\\]\nand\n\\[\nS_n^*(R)\n=\\{\\tau_{t,s}(a):0\\le s,t<n,\\ s\\ne t,\\ a\\in R\\setminus\\{0\\}\\}.\n\\]\nBoth sets are symmetric because\n\\[\n\\tau_{t,s}(a)^{-1}=\\tau_{t,s}(-a).\n\\]\nLet\n\\[\n\\Egrp_n(R)=\\langle S_n^*(R)\\rangle.\n\\]\nWhen $R=\\F$ is a field and $n\\ge2$, elementary elimination gives\n\\[\n\\Egrp_n(\\F)=\\SL_n(\\F).\n\\]\nThe ambient metric below is therefore the transvection word metric in $\\SL_n(\\F)$, viewed inside $\\GL_n(\\F)$.  When $\\F=\\F_2$, one has $\\SL_n(\\F_2)=\\GL_n(\\F_2)$, every generator is an involution, and it is exactly a CNOT gate under the usual row-action convention.","(R)\n=\\{\\tau_{t,s}(a):0\\le s,t<n,\\ s\\ne t,\\ a\\in R\\setminus\\{0\\}\\}.\n\\]\nBoth sets are symmetric because\n\\[\n\\tau_{t,s}(a)^{-1}=\\tau_{t,s}(-a).\n\\]\nLet\n\\[\n\\Egrp_n(R)=\\langle S_n^*(R)\\rangle.\n\\]\nWhen $R=\\F$ is a field and $n\\ge2$, elementary elimination gives\n\\[\n\\Egrp_n(\\F)=\\SL_n(\\F).\n\\]\nThe ambient metric below is therefore the transvection word metric in $\\SL_n(\\F)$, viewed inside $\\GL_n(\\F)$.  When $\\F=\\F_2$, one has $\\SL_n(\\F_2)=\\GL_n(\\F_2)$, every generator is an involution, and it is exactly a CNOT gate under the usual row-action convention.","):0\\le s,t<n,\\ s\\ne t,\\ a\\in R\\setminus\\{0\\}\\}.\n\\]\nBoth sets are symmetric because\n\\[\n\\tau_{t,s}(a)^{-1}=\\tau_{t,s}(-a).\n\\]\nLet\n\\[\n\\Egrp_n(R)=\\langle S_n^*(R)\\rangle.\n\\]\nWhen $R=\\F$ is a field and $n\\ge2$, elementary elimination gives\n\\[\n\\Egrp_n(\\F)=\\SL_n(\\F).\n\\]\nThe ambient metric below is therefore the transvection word metric in $\\SL_n(\\F)$, viewed inside $\\GL_n(\\F)$.  When $\\F=\\F_2$, one has $\\SL_n(\\F_2)=\\GL_n(\\F_2)$, every generator is an involution, and it is exactly a CNOT gate under the usual row-action convention.","\\begin{lemma}\\label{g:lem:lower-generates}\nThe set $S_n^\\downarrow(R)$ generates $\\U_n(R)$.  Every prefix of a lower-directed circuit belongs to $\\U_n(R)$.\n\\end{lemma}","\\begin{proof}\nA lower-directed transvection is lower unitriangular, and products of lower-unitriangular matrices remain lower unitriangular.  Conversely, ordinary elimination of the strict lower entries, processed in a compatible order, expresses every element of $\\U_n(R)$ as a product of lower-directed transvections.\n\\end{proof}","\\begin{definition}[Intrinsic and ambient lengths]\\label{g:def:lengths}\nFor $A\\in\\U_n(R)$, define\n\\[\n\\ellDown(A)=|A|_{S_n^\\downarrow(R)},\n\\qquad\n\\ellAll(A)=|A|_{S_n^*(R)}.\n\\]\nThe \\emph{catalytic defect} is\n\\[\n\\DeltaDef(A)=\\ellDown(A)-\\ellAll(A)\\ge0.\n\\]\n\\end{definition}","The inequality follows because $S_n^\\downarrow(R)\\subseteq S_n^*(R)$.  The central question is whether the inclusion can be strict on a lower-unitriangular target.","For the geometric interpretation in this subsection, assume $R=\\F$ is a field.  An invertible matrix is an ordered basis\n\\[\n(R_0,\\ldots,R_{n-1})\n\\]\nof $\\F^n$.  A transvection replaces one basis vector by its sum with a nonzero multiple of another.  Let\n\\[\nV_i=\\Span(e_0,\\ldots,e_i).\n\\]\nA matrix lies in $\\U_n(\\F)$ exactly when\n\\[\nR_i\\in e_i+V_{i-1}\n\\]\nfor every $i$.  Thus lower circuits move among frames compatible with the standard complete flag\n\\[\nV_0\\subset V_1\\subset\\cdots\\subset V_{n-1}.\n\\]\nAn upper-directed gate leaves this fiber of the flag map.  A strict defect means that the flag-compatible frame subgraph is not geodesically convex in the full transvection graph.",", assume $R=\\F$ is a field.  An invertible matrix is an ordered basis\n\\[\n(R_0,\\ldots,R_{n-1})\n\\]\nof $\\F^n$.  A transvection replaces one basis vector by its sum with a nonzero multiple of another.  Let\n\\[\nV_i=\\Span(e_0,\\ldots,e_i).\n\\]\nA matrix lies in $\\U_n(\\F)$ exactly when\n\\[\nR_i\\in e_i+V_{i-1}\n\\]\nfor every $i$.  Thus lower circuits move among frames compatible with the standard complete flag\n\\[\nV_0\\subset V_1\\subset\\cdots\\subset V_{n-1}.\n\\]\nAn upper-directed gate leaves this fiber of the flag map.  A strict defect means that the flag-compatible frame subgraph is not geodesically convex in the full transvection graph.","Set\n\\[\nG=\\Egrp_n(R),\n\\qquad\nH=\\U_n(R).\n\\]\nConsider the right Schreier graph on the left cosets $H\\backslash G$: a generator $s\\in S_n^*(R)$ gives an edge\n\\[\nHg\\longrightarrow Hgs.\n\\]\nA word representing $h\\in H$ traces a based loop at the coset $H$.  Its label product is the holonomy $h$.","\\begin{proposition}[Holonomy-compression interpretation]\\label{g:prop:schreier-holonomy}\nFor $h\\in H$, the ambient length $\\ellAll(h)$ is the least length of a labeled based loop in $H\\backslash G$ with holonomy $h$.  The restricted length $\\ellDown(h)$ is the least length among such words whose every prefix remains in $H$.  Consequently,\n\\[\n\\DeltaDef(h)\n=\n\\text{intrinsic flag-preserving cost}\n-\n\\text{shortest ambient holonomy-loop cost}.\n\\]\n\\end{proposition}","\\begin{proof}\nA word $w=s_1\\cdots s_k$ ends at the coset $Hw$.  It is a based loop exactly when $w\\in H$, and its label product is then the represented element of $H$.  Minimizing over all transvection words representing $h$ gives $\\ellAll(h)$.  Requiring every generator to be lower directed keeps every prefix in $H$ and gives $\\ellDown(h)$.\n\\end{proof}","\\begin{lemma}[Target count]\\label{g:lem:target-count}\nSuppose a circuit starts at the identity and finishes at $A$.\n\\begin{enumerate}[label=\\textup{(\\roman*)}]\n\\item If row $i$ of $A$ differs from $e_i$, then row $i$ is targeted at least once.\n\\item If row $i$ of $A$ equals $e_i$ and is targeted, then it is targeted at least twice.\n\\end{enumerate}\n\\end{lemma}","\\begin{proof}\nA row changes only when it is targeted.  Every row of an invertible matrix over $R$ is unimodular.  If $aR_s=0$ for such a row, then multiplying a unimodular relation for $R_s$ by $a$ gives $a=0$.  Hence a gate with nonzero coefficient changes its target by a nonzero vector, so one targeting cannot leave that row equal to its initial value.\n\\end{proof}","\\begin{lemma}[One-target source condition]\\label{g:lem:one-target-source}\nLet $W\\le R^n$.  Suppose row $t$ starts at $e_t$, is targeted exactly once, and finishes at\n\\[\ne_t+w,\n\\qquad w\\in W,\n\\]\nwhere at least one coordinate of $w$ is a unit of $R$.  At the moment of its unique targeting, the gate coefficient is a unit and the source row is a unit multiple of $w$.\n\\end{lemma}","\\begin{proof}\nIf the unique gate is $R_t\\leftarrow R_t+aR_s$, then $aR_s=w$.  Choose a coordinate $j$ for which $w_j$ is a unit.  Since $a(R_s)_j=w_j$, the factor $a$ is a unit in the commutative ring $R$.  Hence $R_s=a^{-1}w$.\n\\end{proof}","The second lemma is especially restrictive when $W$ is a low-dimensional source space and output rows carry unique pivot coordinates outside $W$.","Let $R$ be a nonzero commutative ring with identity, and let\n\\[\nv=e_1+e_2,\n\\qquad\nw=e_0+e_1+e_2,\n\\qquad\nW=\\Span(e_0,e_1,e_2).\n\\]\nThe target $C$ of \\cref{g:eq:canonical-C} has rows\n\\begin{align*}\nR_0&=e_0,\n& R_1&=e_1+e_0,\n& R_2&=e_2,","Let $R$ be a nonzero commutative ring with identity, and let\n\\[\nv=e_1+e_2,\n\\qquad\nw=e_0+e_1+e_2,\n\\qquad\nW=\\Span(e_0,e_1,e_2).\n\\]\nThe target $C$ of \\cref{g:eq:canonical-C} has rows\n\\begin{align*}\nR_0&=e_0,\n& R_1&=e_1+e_0,\n& R_2&=e_2,\\\\\nR_3&=e_3+v,\n& R_4&=e_4+w,\n& R_5&=e_5+w.\n\\end{align*}","\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.56\\linewidth]{assets/g_canonical_matrix.pdf}\n\\caption{The canonical target $C$.  Black cells are nonzero.  Rows $0$ and $2$ are restored exactly, while rows $3,4,5$ require two distinct additions $v$ and $w$ from the source space $W$.}\n\\label{g:fig:canonical-matrix}\n\\end{figure}","\\begin{proposition}\\label{g:prop:ambient-upper}\nFor every nonzero commutative ring $R$ with identity,\n\\[\n\\ellAll(C)\\le6.\n\\]\n\\end{proposition}","\\begin{proof}\nApply the following six gates:\n\\begin{align}\nR_1&\\leftarrow R_1+R_2,\\label{g:gate:a1}\\\\\nR_3&\\leftarrow R_3+R_1,\\label{g:gate:a2}\\\\\nR_1&\\leftarrow R_1+R_0,\\label{g:gate:a3}\\\\\nR_4&\\leftarrow R_4+R_1,\\label{g:gate:a4}\\\\\nR_5&\\leftarrow R_5+R_1,\\label{g:gate:a5}\\\\\nR_1&\\leftarrow R_1-R_2.\\label{g:gate:a6}\n\\end{align}\nAfter \\cref{g:gate:a1}, row $1$ contains $v$.  Gate \\cref{g:gate:a2} copies $v$ to row $3$.  Gate \\cref{g:gate:a3} changes the carrier to $w$, which is copied to rows $4$ and $5$ by \\cref{g:gate:a4,g:gate:a5}.  Gate \\cref{g:gate:a6} removes the temporary $e_2$ component, leaving $e_1+e_0$.  Every row now agrees with $C$.\n\\end{proof}","R_1&\\leftarrow R_1-R_2.\\label{g:gate:a6}\n\\end{align}\nAfter \\cref{g:gate:a1}, row $1$ contains $v$.  Gate \\cref{g:gate:a2} copies $v$ to row $3$.  Gate \\cref{g:gate:a3} changes the carrier to $w$, which is copied to rows $4$ and $5$ by \\cref{g:gate:a4,g:gate:a5}.  Gate \\cref{g:gate:a6} removes the temporary $e_2$ component, leaving $e_1+e_0$.  Every row now agrees with $C$.\n\\end{proof}","\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.98\\linewidth]{assets/g_carrier_timeline.pdf}\n\\caption{Carrier timeline for the six-gate ambient word.  The two upper-directed operations are the opening and closing gates on row $1$.  The state $e_1+e_2$ is exported once and $e_0+e_1+e_2$ is exported twice.}\n\\label{g:fig:carrier-timeline}\n\\end{figure}","\\begin{proposition}\\label{g:prop:lower-upper}\nFor every nonzero commutative ring $R$ with identity,\n\\[\n\\ellDown(C)\\le7.\n\\]\n\\end{proposition}","\\begin{proof}\nThe lower-directed word\n\\begin{align*}\nR_1&\\leftarrow R_1+R_0,\\\\\nR_2&\\leftarrow R_2+R_1,\\\\\nR_3&\\leftarrow R_3-R_0,\\\\\nR_3&\\leftarrow R_3+R_2,\\\\\nR_4&\\leftarrow R_4+R_2,\\\\\nR_5&\\leftarrow R_5+R_2,\\\\\nR_2&\\leftarrow R_2-R_1\n\\end{align*}\nproduces $C$.  After the first two gates, row $2$ contains $w$.  Rows $4$ and $5$ copy that state.  The two gates on row $3$ cancel the unwanted $e_0$ and leave $v$.  The final gate restores row $2$.\n\\end{proof}","R_2&\\leftarrow R_2-R_1\n\\end{align*}\nproduces $C$.  After the first two gates, row $2$ contains $w$.  Rows $4$ and $5$ copy that state.  The two gates on row $3$ cancel the unwanted $e_0$ and leave $v$.  The final gate restores row $2$.\n\\end{proof}","\\begin{proposition}\\label{g:prop:lower-lower-bound}\nFor every nonzero commutative ring $R$ with identity,\n\\[\n\\ellDown(C)\\ge7.\n\\]\n\\end{proposition}","\\begin{proof}\nRows $1,3,4,5$ differ from their initial values, so they must each be targeted.  Assume for contradiction that a lower-directed word of length at most six produces $C$.","First suppose row $2$ is never targeted.  Row $0$ remains $e_0$, row $2$ remains $e_2$, and every state ever assumed by row $1$ lies in $\\Span(e_0,e_1)$ because the only possible source below row $1$ is row $0$.  Every output row retains coefficient $1$ on its own pivot coordinate in a lower-directed circuit, so no output row can lie in $W$.  Hence no available source row is a unit multiple of either\n\\[\nv=e_1+e_2\n\\quad\\text{or}\\quad\nw=e_0+e_1+e_2.\n\\]\nBy \\cref{g:lem:one-target-source}, each of rows $3,4,5$ must be targeted at least twice.  Together with the required targeting of row $1$, this gives at least seven gates.","suppose row $2$ is never targeted.  Row $0$ remains $e_0$, row $2$ remains $e_2$, and every state ever assumed by row $1$ lies in $\\Span(e_0,e_1)$ because the only possible source below row $1$ is row $0$.  Every output row retains coefficient $1$ on its own pivot coordinate in a lower-directed circuit, so no output row can lie in $W$.  Hence no available source row is a unit multiple of either\n\\[\nv=e_1+e_2\n\\quad\\text{or}\\quad\nw=e_0+e_1+e_2.\n\\]\nBy \\cref{g:lem:one-target-source}, each of rows $3,4,5$ must be targeted at least twice.  Together with the required targeting of row $1$, this gives at least seven gates.","ever assumed by row $1$ lies in $\\Span(e_0,e_1)$ because the only possible source below row $1$ is row $0$.  Every output row retains coefficient $1$ on its own pivot coordinate in a lower-directed circuit, so no output row can lie in $W$.  Hence no available source row is a unit multiple of either\n\\[\nv=e_1+e_2\n\\quad\\text{or}\\quad\nw=e_0+e_1+e_2.\n\\]\nBy \\cref{g:lem:one-target-source}, each of rows $3,4,5$ must be targeted at least twice.  Together with the required targeting of row $1$, this gives at least seven gates.","w row $1$ is row $0$.  Every output row retains coefficient $1$ on its own pivot coordinate in a lower-directed circuit, so no output row can lie in $W$.  Hence no available source row is a unit multiple of either\n\\[\nv=e_1+e_2\n\\quad\\text{or}\\quad\nw=e_0+e_1+e_2.\n\\]\nBy \\cref{g:lem:one-target-source}, each of rows $3,4,5$ must be targeted at least twice.  Together with the required targeting of row $1$, this gives at least seven gates.","Now suppose row $2$ is targeted.  Since its final value is $e_2$, \\cref{g:lem:target-count} forces at least two targetings of row $2$.  A six-gate word must therefore target row $2$ exactly twice and each of rows $1,3,4,5$ exactly once.  Each output row $3,4,5$ must receive its complete addition in its unique targeting.  Row $0$ cannot supply $v$ or $w$, and row $1$ never contains an $e_2$ component.  Therefore row $2$ must supply all three outputs, up to unit scaling.  But two targetings of row $2$ expose at most one noninitial intermediate state before row $2$ returns to $e_2$, while the outputs require both distinct states $v$ and $w$.  This is impossible.\n\\end{proof}","is targeted.  Since its final value is $e_2$, \\cref{g:lem:target-count} forces at least two targetings of row $2$.  A six-gate word must therefore target row $2$ exactly twice and each of rows $1,3,4,5$ exactly once.  Each output row $3,4,5$ must receive its complete addition in its unique targeting.  Row $0$ cannot supply $v$ or $w$, and row $1$ never contains an $e_2$ component.  Therefore row $2$ must supply all three outputs, up to unit scaling.  But two targetings of row $2$ expose at most one noninitial intermediate state before row $2$ returns to $e_2$, while the outputs require both distinct states $v$ and $w$.  This is impossible.\n\\end{proof}","least two targetings of row $2$.  A six-gate word must therefore target row $2$ exactly twice and each of rows $1,3,4,5$ exactly once.  Each output row $3,4,5$ must receive its complete addition in its unique targeting.  Row $0$ cannot supply $v$ or $w$, and row $1$ never contains an $e_2$ component.  Therefore row $2$ must supply all three outputs, up to unit scaling.  But two targetings of row $2$ expose at most one noninitial intermediate state before row $2$ returns to $e_2$, while the outputs require both distinct states $v$ and $w$.  This is impossible.\n\\end{proof}","target row $2$ exactly twice and each of rows $1,3,4,5$ exactly once.  Each output row $3,4,5$ must receive its complete addition in its unique targeting.  Row $0$ cannot supply $v$ or $w$, and row $1$ never contains an $e_2$ component.  Therefore row $2$ must supply all three outputs, up to unit scaling.  But two targetings of row $2$ expose at most one noninitial intermediate state before row $2$ returns to $e_2$, while the outputs require both distinct states $v$ and $w$.  This is impossible.\n\\end{proof}","exactly twice and each of rows $1,3,4,5$ exactly once.  Each output row $3,4,5$ must receive its complete addition in its unique targeting.  Row $0$ cannot supply $v$ or $w$, and row $1$ never contains an $e_2$ component.  Therefore row $2$ must supply all three outputs, up to unit scaling.  But two targetings of row $2$ expose at most one noninitial intermediate state before row $2$ returns to $e_2$, while the outputs require both distinct states $v$ and $w$.  This is impossible.\n\\end{proof}",".  Each output row $3,4,5$ must receive its complete addition in its unique targeting.  Row $0$ cannot supply $v$ or $w$, and row $1$ never contains an $e_2$ component.  Therefore row $2$ must supply all three outputs, up to unit scaling.  But two targetings of row $2$ expose at most one noninitial intermediate state before row $2$ returns to $e_2$, while the outputs require both distinct states $v$ and $w$.  This is impossible.\n\\end{proof}","output row $3,4,5$ must receive its complete addition in its unique targeting.  Row $0$ cannot supply $v$ or $w$, and row $1$ never contains an $e_2$ component.  Therefore row $2$ must supply all three outputs, up to unit scaling.  But two targetings of row $2$ expose at most one noninitial intermediate state before row $2$ returns to $e_2$, while the outputs require both distinct states $v$ and $w$.  This is impossible.\n\\end{proof}","\\begin{proposition}\\label{g:prop:ambient-lower-bound}\nFor every nonzero commutative ring $R$ with identity,\n\\[\n\\ellAll(C)\\ge6.\n\\]\n\\end{proposition}","\\begin{proof}\nAssume that an ambient word of length at most five produces $C$.  Rows $1,3,4,5$ must each be targeted.  Rows $0$ and $2$ are unchanged.  If either were targeted, it would require at least two targetings, giving at least six gates in total.  Thus rows $0$ and $2$ are never targeted.  If all four changed rows are targeted once, each has only its initial and final state.  No such source is a unit multiple of $v=e_1+e_2$ or $w=e_0+e_1+e_2$, so the output rows cannot be completed.  In the only remaining case, exactly one of rows $1,3,4,5$ is targeted twice and the other three are targeted once.","an ambient word of length at most five produces $C$.  Rows $1,3,4,5$ must each be targeted.  Rows $0$ and $2$ are unchanged.  If either were targeted, it would require at least two targetings, giving at least six gates in total.  Thus rows $0$ and $2$ are never targeted.  If all four changed rows are targeted once, each has only its initial and final state.  No such source is a unit multiple of $v=e_1+e_2$ or $w=e_0+e_1+e_2$, so the output rows cannot be completed.  In the only remaining case, exactly one of rows $1,3,4,5$ is targeted twice and the other three are targeted once.","d of length at most five produces $C$.  Rows $1,3,4,5$ must each be targeted.  Rows $0$ and $2$ are unchanged.  If either were targeted, it would require at least two targetings, giving at least six gates in total.  Thus rows $0$ and $2$ are never targeted.  If all four changed rows are targeted once, each has only its initial and final state.  No such source is a unit multiple of $v=e_1+e_2$ or $w=e_0+e_1+e_2$, so the output rows cannot be completed.  In the only remaining case, exactly one of rows $1,3,4,5$ is targeted twice and the other three are targeted once.","ged.  If either were targeted, it would require at least two targetings, giving at least six gates in total.  Thus rows $0$ and $2$ are never targeted.  If all four changed rows are targeted once, each has only its initial and final state.  No such source is a unit multiple of $v=e_1+e_2$ or $w=e_0+e_1+e_2$, so the output rows cannot be completed.  In the only remaining case, exactly one of rows $1,3,4,5$ is targeted twice and the other three are targeted once.","Project the rows to the quotient module $R^6/W$.  Rows $0,1,2$ have zero image, while rows $3,4,5$ begin and end at the three distinct quotient basis vectors of the free quotient module.  An output row targeted exactly once must therefore use a source lying in $W$, by \\cref{g:lem:one-target-source}.","Suppose first that row $1$ is the twice-targeted row.  The three output rows are each targeted once.  Rows $0$ and $2$ supply only scalar multiples of $e_0$ and $e_2$.  Row $1$ begins at $e_1$, has one intermediate state, and ends at $e_1+e_0$.  Thus at most one of the two required unit-multiple classes of $v$ and $w$ can appear as an available source.  The three outputs require both, a contradiction.","Suppose instead that an output row, say row $r\\in\\{3,4,5\\}$, is targeted twice.  The other two outputs and row $1$ are targeted once.  We claim that row $r$ cannot lie in $W$ between its two targetings.  To cancel its unique quotient pivot $e_r$ at the first targeting, the source would already have to contain an $e_r$ component.  Before row $r$ is first targeted, the only way another row can acquire such a component is to be targeted using row $r$ as source.  That other row is targeted only once and would then retain an unwanted $e_r$ component in its final value.  Therefore row $r$ never becomes a $W$-valued carrier.  The only $W$-valued source states are rows $0$ and $2$ and the two states of singly targeted row $1$, namely its initial and final","Suppose instead that an output row, say row $r\\in\\{3,4,5\\}$, is targeted twice.  The other two outputs and row $1$ are targeted once.  We claim that row $r$ cannot lie in $W$ between its two targetings.  To cancel its unique quotient pivot $e_r$ at the first targeting, the source would already have to contain an $e_r$ component.  Before row $r$ is first targeted, the only way another row can acquire such a component is to be targeted using row $r$ as source.  That other row is targeted only once and would then retain an unwanted $e_r$ component in its final value.  Therefore row $r$ never becomes a $W$-valued carrier.  The only $W$-valued source states are rows $0$ and $2$ and the two states of singly targeted row $1$, namely its initial and final states.  None supplies both $v$ and $w$.  The two singl","Suppose instead that an output row, say row $r\\in\\{3,4,5\\}$, is targeted twice.  The other two outputs and row $1$ are targeted once.  We claim that row $r$ cannot lie in $W$ between its two targetings.  To cancel its unique quotient pivot $e_r$ at the first targeting, the source would already have to contain an $e_r$ component.  Before row $r$ is first targeted, the only way another row can acquire such a component is to be targeted using row $r$ as source.  That other row is targeted only once and would then retain an unwanted $e_r$ component in its final value.  Therefore row $r$ never becomes a $W$-valued carrier.  The only $W$-valued source states are rows $0$ and $2$ and the two states of singly targeted row $1$, namely its initial and final states.  None supplies both $v$ and $w$.  The two singly targeted outputs cannot be completed, ag","Suppose instead that an output row, say row $r\\in\\{3,4,5\\}$, is targeted twice.  The other two outputs and row $1$ are targeted once.  We claim that row $r$ cannot lie in $W$ between its two targetings.  To cancel its unique quotient pivot $e_r$ at the first targeting, the source would already have to contain an $e_r$ component.  Before row $r$ is first targeted, the only way another row can acquire such a component is to be targeted using row $r$ as source.  That other row is targeted only once and would then retain an unwanted $e_r$ component in its final value.  Therefore row $r$ never becomes a $W$-valued carrier.  The only $W$-valued source states are rows $0$ and $2$ and the two states of singly targeted row $1$, namely its initial and final states.  None supplies both $v$ and $w$.  The two singly targeted outputs cannot be completed, again a contradictio","Suppose instead that an output row, say row $r\\in\\{3,4,5\\}$, is targeted twice.  The other two outputs and row $1$ are targeted once.  We claim that row $r$ cannot lie in $W$ between its two targetings.  To cancel its unique quotient pivot $e_r$ at the first targeting, the source would already have to contain an $e_r$ component.  Before row $r$ is first targeted, the only way another row can acquire such a component is to be targeted using row $r$ as source.  That other row is targeted only once and would then retain an unwanted $e_r$ component in its final value.  Therefore row $r$ never becomes a $W$-valued carrier.  The only $W$-valued source states are rows $0$ and $2$ and the two states of singly targeted row $1$, namely its initial and final states.  None supplies both $v$ and $w$.  The two singly targeted outputs cannot be completed, again a contradiction.\n\\end{proof}","\\{3,4,5\\}$, is targeted twice.  The other two outputs and row $1$ are targeted once.  We claim that row $r$ cannot lie in $W$ between its two targetings.  To cancel its unique quotient pivot $e_r$ at the first targeting, the source would already have to contain an $e_r$ component.  Before row $r$ is first targeted, the only way another row can acquire such a component is to be targeted using row $r$ as source.  That other row is targeted only once and would then retain an unwanted $e_r$ component in its final value.  Therefore row $r$ never becomes a $W$-valued carrier.  The only $W$-valued source states are rows $0$ and $2$ and the two states of singly targeted row $1$, namely its initial and final states.  None supplies both $v$ and $w$.  The two singly targeted outputs cannot be completed, again a contradiction.\n\\end{proof}","e claim that row $r$ cannot lie in $W$ between its two targetings.  To cancel its unique quotient pivot $e_r$ at the first targeting, the source would already have to contain an $e_r$ component.  Before row $r$ is first targeted, the only way another row can acquire such a component is to be targeted using row $r$ as source.  That other row is targeted only once and would then retain an unwanted $e_r$ component in its final value.  Therefore row $r$ never becomes a $W$-valued carrier.  The only $W$-valued source states are rows $0$ and $2$ and the two states of singly targeted row $1$, namely its initial and final states.  None supplies both $v$ and $w$.  The two singly targeted outputs cannot be completed, again a contradiction.\n\\end{proof}","two targetings.  To cancel its unique quotient pivot $e_r$ at the first targeting, the source would already have to contain an $e_r$ component.  Before row $r$ is first targeted, the only way another row can acquire such a component is to be targeted using row $r$ as source.  That other row is targeted only once and would then retain an unwanted $e_r$ component in its final value.  Therefore row $r$ never becomes a $W$-valued carrier.  The only $W$-valued source states are rows $0$ and $2$ and the two states of singly targeted row $1$, namely its initial and final states.  None supplies both $v$ and $w$.  The two singly targeted outputs cannot be completed, again a contradiction.\n\\end{proof}","cancel its unique quotient pivot $e_r$ at the first targeting, the source would already have to contain an $e_r$ component.  Before row $r$ is first targeted, the only way another row can acquire such a component is to be targeted using row $r$ as source.  That other row is targeted only once and would then retain an unwanted $e_r$ component in its final value.  Therefore row $r$ never becomes a $W$-valued carrier.  The only $W$-valued source states are rows $0$ and $2$ and the two states of singly targeted row $1$, namely its initial and final states.  None supplies both $v$ and $w$.  The two singly targeted outputs cannot be completed, again a contradiction.\n\\end{proof}","ot $e_r$ at the first targeting, the source would already have to contain an $e_r$ component.  Before row $r$ is first targeted, the only way another row can acquire such a component is to be targeted using row $r$ as source.  That other row is targeted only once and would then retain an unwanted $e_r$ component in its final value.  Therefore row $r$ never becomes a $W$-valued carrier.  The only $W$-valued source states are rows $0$ and $2$ and the two states of singly targeted row $1$, namely its initial and final states.  None supplies both $v$ and $w$.  The two singly targeted outputs cannot be completed, again a contradiction.\n\\end{proof}","he source would already have to contain an $e_r$ component.  Before row $r$ is first targeted, the only way another row can acquire such a component is to be targeted using row $r$ as source.  That other row is targeted only once and would then retain an unwanted $e_r$ component in its final value.  Therefore row $r$ never becomes a $W$-valued carrier.  The only $W$-valued source states are rows $0$ and $2$ and the two states of singly targeted row $1$, namely its initial and final states.  None supplies both $v$ and $w$.  The two singly targeted outputs cannot be completed, again a contradiction.\n\\end{proof}","e would already have to contain an $e_r$ component.  Before row $r$ is first targeted, the only way another row can acquire such a component is to be targeted using row $r$ as source.  That other row is targeted only once and would then retain an unwanted $e_r$ component in its final value.  Therefore row $r$ never becomes a $W$-valued carrier.  The only $W$-valued source states are rows $0$ and $2$ and the two states of singly targeted row $1$, namely its initial and final states.  None supplies both $v$ and $w$.  The two singly targeted outputs cannot be completed, again a contradiction.\n\\end{proof}","nent.  Before row $r$ is first targeted, the only way another row can acquire such a component is to be targeted using row $r$ as source.  That other row is targeted only once and would then retain an unwanted $e_r$ component in its final value.  Therefore row $r$ never becomes a $W$-valued carrier.  The only $W$-valued source states are rows $0$ and $2$ and the two states of singly targeted row $1$, namely its initial and final states.  None supplies both $v$ and $w$.  The two singly targeted outputs cannot be completed, again a contradiction.\n\\end{proof}","row can acquire such a component is to be targeted using row $r$ as source.  That other row is targeted only once and would then retain an unwanted $e_r$ component in its final value.  Therefore row $r$ never becomes a $W$-valued carrier.  The only $W$-valued source states are rows $0$ and $2$ and the two states of singly targeted row $1$, namely its initial and final states.  None supplies both $v$ and $w$.  The two singly targeted outputs cannot be completed, again a contradiction.\n\\end{proof}","acquire such a component is to be targeted using row $r$ as source.  That other row is targeted only once and would then retain an unwanted $e_r$ component in its final value.  Therefore row $r$ never becomes a $W$-valued carrier.  The only $W$-valued source states are rows $0$ and $2$ and the two states of singly targeted row $1$, namely its initial and final states.  None supplies both $v$ and $w$.  The two singly targeted outputs cannot be completed, again a contradiction.\n\\end{proof}","\\begin{corollary}[Failure of geodesic convexity]\\label{g:cor:nonconvex}\nFor every field $\\F$ and every $n\\ge6$, the lower-unitriangular frame subgraph is not geodesically convex in the transvection Cayley graph of $\\Egrp_n(\\F)=\\SL_n(\\F)$.\n\\end{corollary}","\\begin{proof}\nEmbed $C$ as $C\\oplus I_{n-6}$.  A six-gate ambient word acts on the first six rows.  In a lower-directed circuit, rows with index at least six cannot be used as sources for targets among the first six rows, so the restricted lower bound of seven persists.\n\\end{proof}","For the remainder of the paper, unless explicitly stated otherwise, the coefficient domain is a field $\\F$.","Let $p<q<y$ and $x<p<q$.  Direct multiplication gives\n\\begin{equation}\\label{g:eq:read-conjugation}\n\\tau_{p,q}(-a)\\,\\tau_{y,p}(b)\\,\\tau_{p,q}(a)\n=\n\\tau_{y,p}(b)\\,\\tau_{y,q}(ab),\n\\end{equation}\nand\n\\begin{equation}\\label{g:eq:update-conjugation}\n\\tau_{p,q}(-a)\\,\\tau_{q,x}(b)\\,\\tau_{p,q}(a)\n=\n\\tau_{q,x}(b)\\,\\tau_{p,x}(-ab).\n\\end{equation}\nA single forbidden upper transvection can therefore turn one lower interaction into two lower effects.  Interactions of the forms $x\\to p$ and $q\\to y$ commute through the excursion without duplication.  We call the duplicated interactions \\emph{cross interactions} and the unduplicated interactions \\emph{phase interactions}.","\\label{g:eq:read-conjugation}\n\\tau_{p,q}(-a)\\,\\tau_{y,p}(b)\\,\\tau_{p,q}(a)\n=\n\\tau_{y,p}(b)\\,\\tau_{y,q}(ab),\n\\end{equation}\nand\n\\begin{equation}\\label{g:eq:update-conjugation}\n\\tau_{p,q}(-a)\\,\\tau_{q,x}(b)\\,\\tau_{p,q}(a)\n=\n\\tau_{q,x}(b)\\,\\tau_{p,x}(-ab).\n\\end{equation}\nA single forbidden upper transvection can therefore turn one lower interaction into two lower effects.  Interactions of the forms $x\\to p$ and $q\\to y$ commute through the excursion without duplication.  We call the duplicated interactions \\emph{cross interactions} and the unduplicated interactions \\emph{phase interactions}.","ead-conjugation}\n\\tau_{p,q}(-a)\\,\\tau_{y,p}(b)\\,\\tau_{p,q}(a)\n=\n\\tau_{y,p}(b)\\,\\tau_{y,q}(ab),\n\\end{equation}\nand\n\\begin{equation}\\label{g:eq:update-conjugation}\n\\tau_{p,q}(-a)\\,\\tau_{q,x}(b)\\,\\tau_{p,q}(a)\n=\n\\tau_{q,x}(b)\\,\\tau_{p,x}(-ab).\n\\end{equation}\nA single forbidden upper transvection can therefore turn one lower interaction into two lower effects.  Interactions of the forms $x\\to p$ and $q\\to y$ commute through the excursion without duplication.  We call the duplicated interactions \\emph{cross interactions} and the unduplicated interactions \\emph{phase interactions}.","\\begin{definition}[Simple one-catalyst word]\\label{g:def:simple-catalyst}\nA simple one-catalyst word has the form\n\\[\n\\tau_{p,q}(-a)\\,h_m\\cdots h_1\\,\\tau_{p,q}(a),\n\\qquad p<q,\n\\]\nwhere every $h_i$ is lower directed and is one of\n\\[\n\\tau_{q,x}(b),\\quad \\tau_{p,x}(b),\\quad\n\\tau_{y,p}(b),\\quad \\tau_{y,q}(b)\n\\]\nwith $x<p<q<y$.  Auxiliary rows are distinct unless stated otherwise.\n\\end{definition}","The circuit is read from right to left in the matrix product and from left to right as a gate list.  The opening and closing operations temporarily place the $q$ component in row $p$.","\\begin{proposition}[Cross-interaction accounting]\\label{g:prop:cross-accounting}\nLet a simple one-catalyst word have $m$ middle interactions, of which $c$ are cross interactions.  Its target admits a lower-directed realization of length at most\n\\[\nm+c.\n\\]\nConsequently, if the ambient word of length $m+2$ is strictly shorter than every lower word, then\n\\[\nc\\ge3.\n\\]\n\\end{proposition}","\\begin{proof}\nConjugate each middle generator using \\cref{g:eq:read-conjugation,g:eq:update-conjugation}.  A phase interaction remains one lower transvection, while a cross interaction becomes a product of two lower transvections.  This gives length at most $m+c$.  If $c\\le2$, then $m+c\\le m+2$, so the ambient excursion cannot be strictly better.\n\\end{proof}","\\begin{lemma}[All-cross mirror]\\label{g:lem:all-cross-mirror}\nSuppose a simple one-catalyst word contains only cross interactions, with left inputs $X$ and right outputs $Y$.  Then its target has a lower-directed realization with the same number of gates.\n\\end{lemma}","\\begin{proof}\nWrite the opening coefficient as $a\\in\\F^\\times$, the left-update coefficients as $b_x$, and the right-read coefficients as $c_y$.  Since the all-cross middle gates commute past one another, the ambient gate list may be written\n\\[\nR_p\\leftarrow R_p+aR_q;\n\\quad\n[R_q\\leftarrow R_q+b_xR_x]_{x\\in X};\n\\quad\n[R_y\\leftarrow R_y+c_yR_p]_{y\\in Y};\n\\quad\nR_p\\leftarrow R_p-aR_q.\n\\]\nPut $B=\\sum_{x\\in X}b_xe_x$.  Its final nontrivial rows are\n\\[\nR_p=e_p-aB,\n\\qquad\nR_q=e_q+B,\n\\qquad\nR_y=e_y+c_ye_p+ac_ye_q\n\\quad(y\\in Y).\n\\]\nThe following lower-directed gate list produces the same rows:\n\\[\nR_q\\leftarrow R_q+a^{-1}R_p;\n\\quad\n[R_y\\leftarrow R_y+ac_yR_q]_{y\\in Y};\n\\quad\n[R_p\\leftarrow R_p-ab_xR_x]_{x\\in X};\n\\quad\nR_q\\leftarrow R_q-a^{-1}R_p.\n\\]\nIndeed, aft","\\begin{proof}\nWrite the opening coefficient as $a\\in\\F^\\times$, the left-update coefficients as $b_x$, and the right-read coefficients as $c_y$.  Since the all-cross middle gates commute past one another, the ambient gate list may be written\n\\[\nR_p\\leftarrow R_p+aR_q;\n\\quad\n[R_q\\leftarrow R_q+b_xR_x]_{x\\in X};\n\\quad\n[R_y\\leftarrow R_y+c_yR_p]_{y\\in Y};\n\\quad\nR_p\\leftarrow R_p-aR_q.\n\\]\nPut $B=\\sum_{x\\in X}b_xe_x$.  Its final nontrivial rows are\n\\[\nR_p=e_p-aB,\n\\qquad\nR_q=e_q+B,\n\\qquad\nR_y=e_y+c_ye_p+ac_ye_q\n\\quad(y\\in Y).\n\\]\nThe following lower-directed gate list produces the same rows:\n\\[\nR_q\\leftarrow R_q+a^{-1}R_p;\n\\quad\n[R_y\\leftarrow R_y+ac_yR_q]_{y\\in Y};\n\\quad\n[R_p\\leftarrow R_p-ab_xR_x]_{x\\in X};\n\\quad\nR_q\\leftarrow R_q-a^{-1}R_p.\n\\]\nIndeed, after the first gate $R_q=e_q+a^{-1}e_p$,","\\begin{proof}\nWrite the opening coefficient as $a\\in\\F^\\times$, the left-update coefficients as $b_x$, and the right-read coefficients as $c_y$.  Since the all-cross middle gates commute past one another, the ambient gate list may be written\n\\[\nR_p\\leftarrow R_p+aR_q;\n\\quad\n[R_q\\leftarrow R_q+b_xR_x]_{x\\in X};\n\\quad\n[R_y\\leftarrow R_y+c_yR_p]_{y\\in Y};\n\\quad\nR_p\\leftarrow R_p-aR_q.\n\\]\nPut $B=\\sum_{x\\in X}b_xe_x$.  Its final nontrivial rows are\n\\[\nR_p=e_p-aB,\n\\qquad\nR_q=e_q+B,\n\\qquad\nR_y=e_y+c_ye_p+ac_ye_q\n\\quad(y\\in Y).\n\\]\nThe following lower-directed gate list produces the same rows:\n\\[\nR_q\\leftarrow R_q+a^{-1}R_p;\n\\quad\n[R_y\\leftarrow R_y+ac_yR_q]_{y\\in Y};\n\\quad\n[R_p\\leftarrow R_p-ab_xR_x]_{x\\in X};\n\\quad\nR_q\\leftarrow R_q-a^{-1}R_p.\n\\]\nIndeed, after the first gate $R_q=e_q+a^{-1}e_p$, so the output gates create exactly $c_ye_p","\\begin{proof}\nWrite the opening coefficient as $a\\in\\F^\\times$, the left-update coefficients as $b_x$, and the right-read coefficients as $c_y$.  Since the all-cross middle gates commute past one another, the ambient gate list may be written\n\\[\nR_p\\leftarrow R_p+aR_q;\n\\quad\n[R_q\\leftarrow R_q+b_xR_x]_{x\\in X};\n\\quad\n[R_y\\leftarrow R_y+c_yR_p]_{y\\in Y};\n\\quad\nR_p\\leftarrow R_p-aR_q.\n\\]\nPut $B=\\sum_{x\\in X}b_xe_x$.  Its final nontrivial rows are\n\\[\nR_p=e_p-aB,\n\\qquad\nR_q=e_q+B,\n\\qquad\nR_y=e_y+c_ye_p+ac_ye_q\n\\quad(y\\in Y).\n\\]\nThe following lower-directed gate list produces the same rows:\n\\[\nR_q\\leftarrow R_q+a^{-1}R_p;\n\\quad\n[R_y\\leftarrow R_y+ac_yR_q]_{y\\in Y};\n\\quad\n[R_p\\leftarrow R_p-ab_xR_x]_{x\\in X};\n\\quad\nR_q\\leftarrow R_q-a^{-1}R_p.\n\\]\nIndeed, after the first gate $R_q=e_q+a^{-1}e_p$, so the output gates create exactly $c_ye_p+ac_ye_q$.  The source updates then leave $R_p=e_p-aB$, and the final gate changes $R_q$ to $e_q+B$.  Both words have $|X|+|Y|+2$ gates.\n\\end{proof}","in\\F^\\times$, the left-update coefficients as $b_x$, and the right-read coefficients as $c_y$.  Since the all-cross middle gates commute past one another, the ambient gate list may be written\n\\[\nR_p\\leftarrow R_p+aR_q;\n\\quad\n[R_q\\leftarrow R_q+b_xR_x]_{x\\in X};\n\\quad\n[R_y\\leftarrow R_y+c_yR_p]_{y\\in Y};\n\\quad\nR_p\\leftarrow R_p-aR_q.\n\\]\nPut $B=\\sum_{x\\in X}b_xe_x$.  Its final nontrivial rows are\n\\[\nR_p=e_p-aB,\n\\qquad\nR_q=e_q+B,\n\\qquad\nR_y=e_y+c_ye_p+ac_ye_q\n\\quad(y\\in Y).\n\\]\nThe following lower-directed gate list produces the same rows:\n\\[\nR_q\\leftarrow R_q+a^{-1}R_p;\n\\quad\n[R_y\\leftarrow R_y+ac_yR_q]_{y\\in Y};\n\\quad\n[R_p\\leftarrow R_p-ab_xR_x]_{x\\in X};\n\\quad\nR_q\\leftarrow R_q-a^{-1}R_p.\n\\]\nIndeed, after the first gate $R_q=e_q+a^{-1}e_p$, so the output gates create exactly $c_ye_p+ac_ye_q$.  The source updates then leave $R_p=e_p-aB$, and the final gate changes $R_q$ to $e_q+B$.  Both words have $|X|+|Y|+2$ gates.\n\\end{proof}","e another, the ambient gate list may be written\n\\[\nR_p\\leftarrow R_p+aR_q;\n\\quad\n[R_q\\leftarrow R_q+b_xR_x]_{x\\in X};\n\\quad\n[R_y\\leftarrow R_y+c_yR_p]_{y\\in Y};\n\\quad\nR_p\\leftarrow R_p-aR_q.\n\\]\nPut $B=\\sum_{x\\in X}b_xe_x$.  Its final nontrivial rows are\n\\[\nR_p=e_p-aB,\n\\qquad\nR_q=e_q+B,\n\\qquad\nR_y=e_y+c_ye_p+ac_ye_q\n\\quad(y\\in Y).\n\\]\nThe following lower-directed gate list produces the same rows:\n\\[\nR_q\\leftarrow R_q+a^{-1}R_p;\n\\quad\n[R_y\\leftarrow R_y+ac_yR_q]_{y\\in Y};\n\\quad\n[R_p\\leftarrow R_p-ab_xR_x]_{x\\in X};\n\\quad\nR_q\\leftarrow R_q-a^{-1}R_p.\n\\]\nIndeed, after the first gate $R_q=e_q+a^{-1}e_p$, so the output gates create exactly $c_ye_p+ac_ye_q$.  The source updates then leave $R_p=e_p-aB$, and the final gate changes $R_q$ to $e_q+B$.  Both words have $|X|+|Y|+2$ gates.\n\\end{proof}","R_q.\n\\]\nPut $B=\\sum_{x\\in X}b_xe_x$.  Its final nontrivial rows are\n\\[\nR_p=e_p-aB,\n\\qquad\nR_q=e_q+B,\n\\qquad\nR_y=e_y+c_ye_p+ac_ye_q\n\\quad(y\\in Y).\n\\]\nThe following lower-directed gate list produces the same rows:\n\\[\nR_q\\leftarrow R_q+a^{-1}R_p;\n\\quad\n[R_y\\leftarrow R_y+ac_yR_q]_{y\\in Y};\n\\quad\n[R_p\\leftarrow R_p-ab_xR_x]_{x\\in X};\n\\quad\nR_q\\leftarrow R_q-a^{-1}R_p.\n\\]\nIndeed, after the first gate $R_q=e_q+a^{-1}e_p$, so the output gates create exactly $c_ye_p+ac_ye_q$.  The source updates then leave $R_p=e_p-aB$, and the final gate changes $R_q$ to $e_q+B$.  Both words have $|X|+|Y|+2$ gates.\n\\end{proof}","al rows are\n\\[\nR_p=e_p-aB,\n\\qquad\nR_q=e_q+B,\n\\qquad\nR_y=e_y+c_ye_p+ac_ye_q\n\\quad(y\\in Y).\n\\]\nThe following lower-directed gate list produces the same rows:\n\\[\nR_q\\leftarrow R_q+a^{-1}R_p;\n\\quad\n[R_y\\leftarrow R_y+ac_yR_q]_{y\\in Y};\n\\quad\n[R_p\\leftarrow R_p-ab_xR_x]_{x\\in X};\n\\quad\nR_q\\leftarrow R_q-a^{-1}R_p.\n\\]\nIndeed, after the first gate $R_q=e_q+a^{-1}e_p$, so the output gates create exactly $c_ye_p+ac_ye_q$.  The source updates then leave $R_p=e_p-aB$, and the final gate changes $R_q$ to $e_q+B$.  Both words have $|X|+|Y|+2$ gates.\n\\end{proof}","ad\nR_y=e_y+c_ye_p+ac_ye_q\n\\quad(y\\in Y).\n\\]\nThe following lower-directed gate list produces the same rows:\n\\[\nR_q\\leftarrow R_q+a^{-1}R_p;\n\\quad\n[R_y\\leftarrow R_y+ac_yR_q]_{y\\in Y};\n\\quad\n[R_p\\leftarrow R_p-ab_xR_x]_{x\\in X};\n\\quad\nR_q\\leftarrow R_q-a^{-1}R_p.\n\\]\nIndeed, after the first gate $R_q=e_q+a^{-1}e_p$, so the output gates create exactly $c_ye_p+ac_ye_q$.  The source updates then leave $R_p=e_p-aB$, and the final gate changes $R_q$ to $e_q+B$.  Both words have $|X|+|Y|+2$ gates.\n\\end{proof}","\\]\nThe following lower-directed gate list produces the same rows:\n\\[\nR_q\\leftarrow R_q+a^{-1}R_p;\n\\quad\n[R_y\\leftarrow R_y+ac_yR_q]_{y\\in Y};\n\\quad\n[R_p\\leftarrow R_p-ab_xR_x]_{x\\in X};\n\\quad\nR_q\\leftarrow R_q-a^{-1}R_p.\n\\]\nIndeed, after the first gate $R_q=e_q+a^{-1}e_p$, so the output gates create exactly $c_ye_p+ac_ye_q$.  The source updates then leave $R_p=e_p-aB$, and the final gate changes $R_q$ to $e_q+B$.  Both words have $|X|+|Y|+2$ gates.\n\\end{proof}","llowing lower-directed gate list produces the same rows:\n\\[\nR_q\\leftarrow R_q+a^{-1}R_p;\n\\quad\n[R_y\\leftarrow R_y+ac_yR_q]_{y\\in Y};\n\\quad\n[R_p\\leftarrow R_p-ab_xR_x]_{x\\in X};\n\\quad\nR_q\\leftarrow R_q-a^{-1}R_p.\n\\]\nIndeed, after the first gate $R_q=e_q+a^{-1}e_p$, so the output gates create exactly $c_ye_p+ac_ye_q$.  The source updates then leave $R_p=e_p-aB$, and the final gate changes $R_q$ to $e_q+B$.  Both words have $|X|+|Y|+2$ gates.\n\\end{proof}","e list produces the same rows:\n\\[\nR_q\\leftarrow R_q+a^{-1}R_p;\n\\quad\n[R_y\\leftarrow R_y+ac_yR_q]_{y\\in Y};\n\\quad\n[R_p\\leftarrow R_p-ab_xR_x]_{x\\in X};\n\\quad\nR_q\\leftarrow R_q-a^{-1}R_p.\n\\]\nIndeed, after the first gate $R_q=e_q+a^{-1}e_p$, so the output gates create exactly $c_ye_p+ac_ye_q$.  The source updates then leave $R_p=e_p-aB$, and the final gate changes $R_q$ to $e_q+B$.  Both words have $|X|+|Y|+2$ gates.\n\\end{proof}","\\begin{proposition}[Six-role threshold in the simple model]\\label{g:prop:six-role}\nA strict shortcut in the distinct-leaf simple one-catalyst model requires at least six row roles:\n\\[\n2\\text{ catalyst rows}\n+\n3\\text{ cross interactions}\n+\n1\\text{ phase interaction}.\n\\]\nThe bound is attained by the matrix $C$.\n\\end{proposition}","Fix six ordered rows.  The catalyst pair must be adjacent in a minimal six-role word.  Write\n\\[\np<q=p+1,\n\\]\nlet $L$ be the set of rows left of $p$, and let $R$ be the set of rows right of $q$.  Then\n\\[\n|L|+|R|=4,\n\\qquad\n|L|,|R|>0.\n\\]\nThere are two orientations.","\\paragraph{Left phase.}\nChoose a phase leaf $x\\in L$.  The remaining $|L|-1$ left leaves interact by $u\\to q$.  Every right leaf is a cross read $p\\to y$.  To make the phase effective, the right leaves are divided into a nonempty set read before $x\\to p$ and a nonempty set read after it.  The number of targets is\n\\[\n|L|\\bigl(2^{|R|}-2\\bigr).\n\\]","\\paragraph{Right phase.}\nChoose a phase leaf $y\\in R$.  Every other right leaf is a cross read.  The left updates into $q$ are divided nontrivially before and after the phase gate $q\\to y$.  The number is\n\\[\n|R|\\bigl(2^{|L|}-2\\bigr).\n\\]","\\begin{proposition}[Motif count]\\label{g:prop:motif-count}\nFor a split $(L,R)=(|L|,|R|)$ with $L+R=4$, the number of minimal phase-separated targets is\n\\[\nN_{L,R}\n=L(2^R-2)+R(2^L-2).\n\\]\nFor the three possible splits,\n\\[\nN_{1,3}=6,\n\\qquad\nN_{2,2}=8,\n\\qquad\nN_{3,1}=6.\n\\]\nHence there are twenty targets.\n\\end{proposition}","Over $\\F_2$, an element of $\\U_n(\\F_2)$ is determined by its $n(n-1)/2$ strict lower entries.  For $n=6$ there are $2^{15}=32768$ states.  Breadth-first search from the identity using the fifteen lower transvections gives every restricted distance and the restricted diameter.","For $n=6$, the restricted diameter is eight.  Therefore every lower-unitriangular target has ambient distance at most eight.  This makes a radius-four meet-in-the-middle search complete.","Let\n\\[\nB_4=\\{g\\in\\GL_6(\\F_2):\\ellAll(g)\\le4\\}.\n\\]\nExact breadth-first search gives\n\\[\n|B_4|=110626.\n\\]\nGroup $B_4$ by right cosets of $\\U_6(\\F_2)$.  If $a,b$ lie in the same right coset, then\n\\[\nb^{-1}a\\in\\U_6(\\F_2),\n\\]\nand\n\\[\n\\ellAll(b^{-1}a)\\le\\ellAll(b)+\\ellAll(a)\\le8.\n\\]\nThe implementation uses a canonical right-coset key obtained by reducing matrix columns from right to left.  Right multiplication by a lower-unitriangular matrix changes column $j$ only by a combination of columns with larger index, so this reduction is constant on right cosets and separates them.","t breadth-first search gives\n\\[\n|B_4|=110626.\n\\]\nGroup $B_4$ by right cosets of $\\U_6(\\F_2)$.  If $a,b$ lie in the same right coset, then\n\\[\nb^{-1}a\\in\\U_6(\\F_2),\n\\]\nand\n\\[\n\\ellAll(b^{-1}a)\\le\\ellAll(b)+\\ellAll(a)\\le8.\n\\]\nThe implementation uses a canonical right-coset key obtained by reducing matrix columns from right to left.  Right multiplication by a lower-unitriangular matrix changes column $j$ only by a combination of columns with larger index, so this reduction is constant on right cosets and separates them.","\\begin{lemma}[Meet-in-the-middle completeness]\\label{g:lem:mitm-complete}\nFor every $u\\in\\U_6(\\F_2)$,\n\\[\n\\ellAll(u)\n=\n\\min\\{\\ellAll(a)+\\ellAll(b):a,b\\in B_4,\\ b^{-1}a=u\\}.\n\\]\n\\end{lemma}","\\begin{proof}\nEvery pair on the right gives a word for $u$, so the right-hand side is at least $\\ellAll(u)$.  Conversely, $\\ellAll(u)\\le8$.  Take a shortest path from the identity to $u$ and split it at a midpoint.  The prefix endpoint $a$ has distance at most four.  The inverse of the remaining suffix is an element $b$ of distance at most four, and $u=b^{-1}a$.  Hence the minimum is at most $\\ellAll(u)$.\n\\end{proof}","The radius-four ball splits into $6948$ right-coset buckets.  The largest bucket has $3775$ elements.  The computation checks $45178326$ ordered pairs and recovers an ambient distance for every one of the $32768$ targets.","\\begin{table}[t]\n\\centering\n\\caption{Complete distance distribution in $\\U_6(\\F_2)$.}\n\\label{g:tab:census}\n\\begin{tabular}{@{}rrr@{}}\n\\toprule\n$\\ellDown$ & $\\ellAll$ & number of targets","\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.72\\linewidth]{assets/g_distance_census.pdf}\n\\caption{Complete binary dimension-six census.  The only off-diagonal cell contains the twenty targets with distance pair $(7,6)$.}\n\\label{g:fig:distance-census}\n\\end{figure}","\\begin{proof}[Proof of \\cref{g:thm:binary-census}]\nFor $n=2,3,4,5$, complete restricted breadth-first search and complete ambient meet-in-the-middle search return zero defects.  The state counts are respectively $2,8,64,1024$.  For $n=6$, \\cref{g:lem:mitm-complete} proves that the ambient calculation is exhaustive.  The resulting distribution is \\cref{g:tab:census}; its counts sum to $32768$, no target is missing, and the only strict pair is $(7,6)$ with multiplicity twenty.\n\\end{proof}","s}]\nFor $n=2,3,4,5$, complete restricted breadth-first search and complete ambient meet-in-the-middle search return zero defects.  The state counts are respectively $2,8,64,1024$.  For $n=6$, \\cref{g:lem:mitm-complete} proves that the ambient calculation is exhaustive.  The resulting distribution is \\cref{g:tab:census}; its counts sum to $32768$, no target is missing, and the only strict pair is $(7,6)$ with multiplicity twenty.\n\\end{proof}","\\begin{proof}[Proof of \\cref{g:thm:motif-classification}]\nThe analytic generator of \\cref{g:prop:motif-count} produces twenty distinct targets without reference to the distance census.  Exact set comparison shows that these twenty targets are precisely the twenty off-diagonal targets in \\cref{g:tab:census}.  Each therefore has distances $(7,6)$, and no other dimension-six target is defective.\n\\end{proof}","For $\\F_3$, every nonzero scaled transvection is included.  Complete searches give the data in \\cref{g:tab:small-fields}.","\\begin{table}[t]\n\\centering\n\\caption{Complete no-defect computations below dimension six.}\n\\label{g:tab:small-fields}\n\\begin{tabular}{@{}crrr@{}}\n\\toprule\nfield & $n$ & $|\\U_n(\\F)|$ & restricted diameter","\\midrule\n$\\F_2$&2&2&1","$\\F_2$&3&8&2","$\\F_2$&4&64&4","$\\F_2$&5&1024&6","\\addlinespace\n$\\F_3$&2&3&1","$\\F_3$&3&27&3","$\\F_3$&4&729&5","$\\F_3$&5&59049&7","The $n=5$ ternary computation uses radius-three and radius-four balls, checks $38415205$ right-coset pairs, and recovers all $59049$ targets.  It finds no defect.  The field-independent \\cref{g:thm:universal-atom} supplies a defect at $n=6$, proving \\cref{g:thm:ternary-threshold}.  Separate exact searches over $\\F_3$ and $\\F_5$ verify the distances of the canonical atom directly.  The $\\F_5$ calculation is an audit of the atom, not a complete threshold computation below dimension six.","inary-census,g:thm:motif-classification,g:thm:ternary-threshold} depend on exact finite enumeration.  The source archive contains:\n\\begin{itemize}\n\\item the C++ programs for every breadth-first and meet-in-the-middle search;\n\\item the Python program that independently generates the twenty motifs and compares sets;\n\\item symbolic checks for the bounded-resource formulas and the totally unimodular $M(K_4)$ representation;\n\\item exact stdout files and JSON packet data;\n\\item SHA-256 checksums;\n\\item build and reproduction commands.\n\\end{itemize}\nThe programs were rebuilt from source and rerun independently when this manuscript was assembled; the outputs matched byte for byte.","Over $\\F_2$, write\n\\[\nx=e_0,\n\\qquad\np=e_1,\n\\qquad\nq=e_2.\n\\]\nThe canonical ambient word uses the six nonzero vectors\n\\[\nx,\n\\quad p,\n\\quad q,\n\\quad p+q,\n\\quad p+q+x,\n\\quad p+x\n\\]\ninside $\\Span(x,p,q)\\cong\\F_2^3$.  The only missing nonzero vector is $x+q$.","In $\\operatorname{PG}(2,2)$, three distinct points form a line exactly when their vector representatives sum to zero.  The six points support the four lines\n\\[\n\\{x,p,x+p\\},\n\\quad\n\\{x,p+q,x+p+q\\},\n\\quad\n\\{p,q,p+q\\},\n\\quad\n\\{x+p,q,x+p+q\\}.\n\\]\nEvery point lies on exactly two of these lines.  This is the Pasch configuration, equivalently the Fano plane with one point and its three incident lines removed from the full incidence structure.","\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.65\\linewidth]{assets/g_pasch_configuration.pdf}\n\\caption{A real drawing of the four-line Pasch incidence structure.  The labels are the binary point encodings $1,2,3,4,6,7$; point $5$ is the missing Fano point.}\n\\label{g:fig:pasch}\n\\end{figure}","The configuration is flagged because the rows have fixed roles.  Points $x$ and $q$ are stationary sources, $p$ is the carrier, and the remaining points occur as carrier states.  Forgetting those roles loses the circuit ordering information.","\\subsection{The regular matroid \\texorpdfstring{$M(K_4)$}{M(K4)}}","The six carrier states admit a field-independent representation.  In the ordered basis $(x,p,q)$, place them as the columns of\n\\begin{equation}\\label{g:eq:k4-matrix}\nB=\n\\begin{pmatrix}\n1&0&0&1&0&1","The six carrier states admit a field-independent representation.  In the ordered basis $(x,p,q)$, place them as the columns of\n\\begin{equation}\\label{g:eq:k4-matrix}\nB=\n\\begin{pmatrix}\n1&0&0&1&0&1\\\\\n0&1&0&1&1&1\\\\\n0&0&1&0&1&1\n\\end{pmatrix},\n\\end{equation}\ncorresponding respectively to\n\\[\nx,\n\\quad p,\n\\quad q,\n\\quad x+p,\n\\quad p+q,\n\\quad x+p+q.\n\\]","0&0&1&0&1&1\n\\end{pmatrix},\n\\end{equation}\ncorresponding respectively to\n\\[\nx,\n\\quad p,\n\\quad q,\n\\quad x+p,\n\\quad p+q,\n\\quad x+p+q.\n\\]","\\begin{proposition}[Regular-matroid core]\\label{g:prop:regular-matroid}\nEvery square minor of $B$ belongs to $\\{0,\\pm1\\}$, so $B$ is totally unimodular.  Hence the column-dependence matroid is the same over every field.  Under the edge labeling\n\\[\n\\begin{array}{c|cccccc}\n\\text{state}&p&x&x+p&p+q&q&x+p+q","\\begin{proposition}[Regular-matroid core]\\label{g:prop:regular-matroid}\nEvery square minor of $B$ belongs to $\\{0,\\pm1\\}$, so $B$ is totally unimodular.  Hence the column-dependence matroid is the same over every field.  Under the edge labeling\n\\[\n\\begin{array}{c|cccccc}\n\\text{state}&p&x&x+p&p+q&q&x+p+q\\\\\n\\hline\nK_4\\text{ edge}&12&13&23&14&24&34,\n\\end{array}\n\\]\nit is the graphic matroid $M(K_4)$.\n\\end{proposition}","\\hline\nK_4\\text{ edge}&12&13&23&14&24&34,\n\\end{array}\n\\]\nit is the graphic matroid $M(K_4)$.\n\\end{proposition}","\\begin{proof}\nA direct calculation of all square minors of $B$ gives only $0$, $1$, and $-1$.  Thus every subdeterminant has characteristic-independent vanishing, and a column set is independent over one field exactly when it is independent over every field.  The four minimal dependent triples are\n\\[\n\\{x,p,x+p\\},\n\\quad\n\\{p,q,p+q\\},\n\\quad\n\\{x,p+q,x+p+q\\},\n\\quad\n\\{q,x+p,x+p+q\\}.\n\\]\nUnder the displayed edge labeling, these are precisely the four triangular cycles of $K_4$.  They determine the rank-three cycle matroid $M(K_4)$, which is regular because it has a totally unimodular representation; see, for example, \\cite{g:Oxley2011}.\n\\end{proof}","only $0$, $1$, and $-1$.  Thus every subdeterminant has characteristic-independent vanishing, and a column set is independent over one field exactly when it is independent over every field.  The four minimal dependent triples are\n\\[\n\\{x,p,x+p\\},\n\\quad\n\\{p,q,p+q\\},\n\\quad\n\\{x,p+q,x+p+q\\},\n\\quad\n\\{q,x+p,x+p+q\\}.\n\\]\nUnder the displayed edge labeling, these are precisely the four triangular cycles of $K_4$.  They determine the rank-three cycle matroid $M(K_4)$, which is regular because it has a totally unimodular representation; see, for example, \\cite{g:Oxley2011}.\n\\end{proof}","has characteristic-independent vanishing, and a column set is independent over one field exactly when it is independent over every field.  The four minimal dependent triples are\n\\[\n\\{x,p,x+p\\},\n\\quad\n\\{p,q,p+q\\},\n\\quad\n\\{x,p+q,x+p+q\\},\n\\quad\n\\{q,x+p,x+p+q\\}.\n\\]\nUnder the displayed edge labeling, these are precisely the four triangular cycles of $K_4$.  They determine the rank-three cycle matroid $M(K_4)$, which is regular because it has a totally unimodular representation; see, for example, \\cite{g:Oxley2011}.\n\\end{proof}","\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.72\\linewidth]{assets/g_regular_matroid_k4.pdf}\n\\caption{The six carrier states as the edges of $K_4$.  Each triangular cycle is one of the four three-element dependencies in the Pasch configuration.  The resulting regular matroid is representable over every field.}\n\\label{g:fig:k4-matroid}\n\\end{figure}","Let $G$ be a finite group and $S=S^{-1}$ a symmetric generating set.  On $\\ell^2(G)$, let $\\lambda$ be the left regular representation,\n\\[\n\\lambda(s)\\delta_h=\\delta_{sh},\n\\]\nand define\n\\[\nA_S=\\sum_{s\\in S}\\lambda(s).\n\\]\nFor $g\\in G$, define the endpoint Green series\n\\[\n\\mathcal G_{S,g}(z)\n=\n\\left\\langle\n\\delta_g,\n(I-zA_S)^{-1}\\delta_e\n\\right\\rangle.\n\\]","\\begin{proposition}[First-arrival identity]\\label{g:prop:first-arrival}\nThe coefficient of $z^k$ in $\\mathcal G_{S,g}(z)$ is the number of length-$k$ words in $S$ with product $g$.  Hence\n\\[\nd_S(e,g)=\\ord_{z=0}\\mathcal G_{S,g}(z).\n\\]\n\\end{proposition}","\\begin{proof}\nFormally,\n\\[\n(I-zA_S)^{-1}=\\sum_{k\\ge0}z^kA_S^k.\n\\]\nThe matrix coefficient $\\langle\\delta_g,A_S^k\\delta_e\\rangle$ counts $k$-step walks from $e$ to $g$, which are exactly length-$k$ words with product $g$.  The first nonzero coefficient occurs at the word distance.\n\\end{proof}","If\n\\[\nA_S=\\sum_\\lambda \\lambda\\Pi_\\lambda\n\\]\nis the spectral resolution, then\n\\[\n\\mathcal G_{S,g}(z)\n=\n\\sum_\\lambda\n\\frac{\\langle\\delta_g,\\Pi_\\lambda\\delta_e\\rangle}{1-\\lambda z}.\n\\]\nThus eigenvalues alone do not determine first arrival.  The endpoint matrix coefficients of the spectral projectors are essential.","For the canonical target $C$, exact word counts begin\n\\begin{align*}\n\\mathcal G_{\\downarrow,C}(z)\n&=\n543z^7+25080z^8+510703z^9+10824940z^{10}+\\cdots,","For the canonical target $C$, exact word counts begin\n\\begin{align*}\n\\mathcal G_{\\downarrow,C}(z)\n&=\n543z^7+25080z^8+510703z^9+10824940z^{10}+\\cdots,\\\\\n\\mathcal G_{*,C}(z)\n&=\n2z^6+776z^7+39454z^8+\\cdots.\n\\end{align*}\nThe metric defect is the valuation drop\n\\[\n\\ord_0\\mathcal G_{\\downarrow,C}\n-\n\\ord_0\\mathcal G_{*,C}\n=1.\n\\]","\\mathcal G_{*,C}(z)\n&=\n2z^6+776z^7+39454z^8+\\cdots.\n\\end{align*}\nThe metric defect is the valuation drop\n\\[\n\\ord_0\\mathcal G_{\\downarrow,C}\n-\n\\ord_0\\mathcal G_{*,C}\n=1.\n\\]","packet&$[z^7]$&$[z^8]$&$[z^9]$&$[z^{10}]$&$[z^6]$&$[z^7]$&$[z^8]$","From this point through the rank and tensor arguments, the coefficient domain is a field $\\F$.  The six-row atom is the first defect, but it is not an isolated low-dimensional accident.  This section gives a field-independent family of connected targets in every dimension at least seven.  The lower bound is certified by a multiscale rank potential adapted to the ordered flag.","\\begin{definition}[Interval tree]\nAn \\emph{interval tree} on the ordered coordinates $0,1,\\ldots,n-1$ is a rooted binary tree whose leaves occur in that order and whose every node is a contiguous interval.  If an internal node $v$ splits its interval into consecutive left and right children $L_v$ and $R_v$, define\n\\begin{equation}\\label{g:eq:tree-potential}\n\\Phi_T(A)\n=\n\\sum_{v\\in\\operatorname{Int}(T)}\n\\rank A[R_v,L_v],\n\\qquad A\\in\\U_n(\\F).\n\\end{equation}\n\\end{definition}","\\begin{lemma}[One-cut locality]\\label{g:lem:one-cut-locality}\nLet $X\\in\\U_n(\\F)$ and let $X'$ be obtained from $X$ by one lower-directed transvection.  Then\n\\[\n\\bigl|\\Phi_T(X')-\\Phi_T(X)\\bigr|\\le1.\n\\]\n\\end{lemma}","\\begin{proof}\nSuppose the gate is $s\\to t$ with $s<t$.  Let $v_0$ be the lowest node of $T$ whose two children separate $s$ and $t$.  The order of the leaves forces $s\\in L_{v_0}$ and $t\\in R_{v_0}$.  At this one node, a single row of the cross block $X[R_{v_0},L_{v_0}]$ changes, so its rank changes by at most one.","At every ancestor of $v_0$, the two rows lie in the same child.  If that child is the right child, the gate is an internal row operation on the relevant cross block and preserves rank; if it is the left child, the cross block is unchanged.  At a descendant containing $t$ but not $s$, every column of the descendant interval lies strictly to the right of $s$.  Since $X$ remains lower triangular, row $s$ vanishes on those columns, so the descendant block is unchanged.  All other descendants are also unchanged.  Hence only the term at $v_0$ can vary.\n\\end{proof}","every ancestor of $v_0$, the two rows lie in the same child.  If that child is the right child, the gate is an internal row operation on the relevant cross block and preserves rank; if it is the left child, the cross block is unchanged.  At a descendant containing $t$ but not $s$, every column of the descendant interval lies strictly to the right of $s$.  Since $X$ remains lower triangular, row $s$ vanishes on those columns, so the descendant block is unchanged.  All other descendants are also unchanged.  Hence only the term at $v_0$ can vary.\n\\end{proof}","n the relevant cross block and preserves rank; if it is the left child, the cross block is unchanged.  At a descendant containing $t$ but not $s$, every column of the descendant interval lies strictly to the right of $s$.  Since $X$ remains lower triangular, row $s$ vanishes on those columns, so the descendant block is unchanged.  All other descendants are also unchanged.  Hence only the term at $v_0$ can vary.\n\\end{proof}","\\begin{corollary}[Interval-tree lower bound]\\label{g:cor:tree-lower-bound}\nFor every $A\\in\\U_n(\\F)$ and every interval tree $T$,\n\\[\n\\ellDown(A)\\ge\\Phi_T(A).\n\\]\n\\end{corollary}","\\begin{proof}\nThe identity has potential zero.  Apply \\cref{g:lem:one-cut-locality} along any lower-directed word from the identity to $A$.\n\\end{proof}","\\begin{definition}[Tree-tight target]\\label{g:def:tree-tight}\nA lower-unitriangular target $A$ is \\emph{tree tight} if some interval tree $T$ satisfies\n\\[\n\\Phi_T(A)=\\ellDown(A).\n\\]\n\\end{definition}","Fix integers\n\\[\nu\\ge1,\n\\qquad\n1\\le v\\le u+1,\n\\]\nand choose distinct phases\n\\[\n0\\le s_1<s_2<\\cdots<s_v\\le u.\n\\]\nOrder the coordinates as\n\\[\nx_1,\\ldots,x_u,\\ p,\\ q,\\ t_1,\\ldots,t_v.\n\\]\nFor $0\\le s\\le u$, put\n\\begin{equation}\\label{g:eq:prefix-state}\nw_s=e_p+\\sum_{i=1}^{s}e_{x_i}.\n\\end{equation}\nDefine\n\\begin{equation}\\label{g:eq:snapshot-target}\nA(u;s_1,\\ldots,s_v)\\in\\U_{u+v+2}(\\F)\n\\end{equation}\nby\n\\[\nR_{x_i}=e_{x_i},\n\\qquad\nR_p=w_u,\n\\qquad\nR_q=e_q,\n\\]\nand\n\\begin{equation}\\label{g:eq:snapshot-output}\nR_{t_j}=e_{t_j}+e_q+w_{s_j}\n\\qquad(1\\le j\\le v).\n\\end{equation}\nThe output rows record selected intermediate versions of the carrier $p$.","integers\n\\[\nu\\ge1,\n\\qquad\n1\\le v\\le u+1,\n\\]\nand choose distinct phases\n\\[\n0\\le s_1<s_2<\\cdots<s_v\\le u.\n\\]\nOrder the coordinates as\n\\[\nx_1,\\ldots,x_u,\\ p,\\ q,\\ t_1,\\ldots,t_v.\n\\]\nFor $0\\le s\\le u$, put\n\\begin{equation}\\label{g:eq:prefix-state}\nw_s=e_p+\\sum_{i=1}^{s}e_{x_i}.\n\\end{equation}\nDefine\n\\begin{equation}\\label{g:eq:snapshot-target}\nA(u;s_1,\\ldots,s_v)\\in\\U_{u+v+2}(\\F)\n\\end{equation}\nby\n\\[\nR_{x_i}=e_{x_i},\n\\qquad\nR_p=w_u,\n\\qquad\nR_q=e_q,\n\\]\nand\n\\begin{equation}\\label{g:eq:snapshot-output}\nR_{t_j}=e_{t_j}+e_q+w_{s_j}\n\\qquad(1\\le j\\le v).\n\\end{equation}\nThe output rows record selected intermediate versions of the carrier $p$.","[\n0\\le s_1<s_2<\\cdots<s_v\\le u.\n\\]\nOrder the coordinates as\n\\[\nx_1,\\ldots,x_u,\\ p,\\ q,\\ t_1,\\ldots,t_v.\n\\]\nFor $0\\le s\\le u$, put\n\\begin{equation}\\label{g:eq:prefix-state}\nw_s=e_p+\\sum_{i=1}^{s}e_{x_i}.\n\\end{equation}\nDefine\n\\begin{equation}\\label{g:eq:snapshot-target}\nA(u;s_1,\\ldots,s_v)\\in\\U_{u+v+2}(\\F)\n\\end{equation}\nby\n\\[\nR_{x_i}=e_{x_i},\n\\qquad\nR_p=w_u,\n\\qquad\nR_q=e_q,\n\\]\nand\n\\begin{equation}\\label{g:eq:snapshot-output}\nR_{t_j}=e_{t_j}+e_q+w_{s_j}\n\\qquad(1\\le j\\le v).\n\\end{equation}\nThe output rows record selected intermediate versions of the carrier $p$.","x-state}\nw_s=e_p+\\sum_{i=1}^{s}e_{x_i}.\n\\end{equation}\nDefine\n\\begin{equation}\\label{g:eq:snapshot-target}\nA(u;s_1,\\ldots,s_v)\\in\\U_{u+v+2}(\\F)\n\\end{equation}\nby\n\\[\nR_{x_i}=e_{x_i},\n\\qquad\nR_p=w_u,\n\\qquad\nR_q=e_q,\n\\]\nand\n\\begin{equation}\\label{g:eq:snapshot-output}\nR_{t_j}=e_{t_j}+e_q+w_{s_j}\n\\qquad(1\\le j\\le v).\n\\end{equation}\nThe output rows record selected intermediate versions of the carrier $p$.","\\begin{proof}[Proof of \\cref{g:thm:snapshot-family}]\nFor the ambient construction, first apply\n\\[\nq\\to p.\n\\]\nSweep through the updates\n\\[\nx_1\\to p,\\ldots,x_u\\to p.\n\\]\nWhenever phase $s_j$ is reached, apply\n\\[\np\\to t_j.\n\\]\nFinally apply $q\\to p$ with coefficient $-1$.  At phase $s$, the carrier is $e_q+w_s$, so every output has the required value, and the final gate removes $e_q$ from row $p$.  The word has length\n\\[\n1+u+v+1=u+v+2.\n\\]\nThus\n\\[\n\\ellAll\\bigl(A(u;s_1,\\ldots,s_v)\\bigr)\\le u+v+2.\n\\]","of \\cref{g:thm:snapshot-family}]\nFor the ambient construction, first apply\n\\[\nq\\to p.\n\\]\nSweep through the updates\n\\[\nx_1\\to p,\\ldots,x_u\\to p.\n\\]\nWhenever phase $s_j$ is reached, apply\n\\[\np\\to t_j.\n\\]\nFinally apply $q\\to p$ with coefficient $-1$.  At phase $s$, the carrier is $e_q+w_s$, so every output has the required value, and the final gate removes $e_q$ from row $p$.  The word has length\n\\[\n1+u+v+1=u+v+2.\n\\]\nThus\n\\[\n\\ellAll\\bigl(A(u;s_1,\\ldots,s_v)\\bigr)\\le u+v+2.\n\\]","amily}]\nFor the ambient construction, first apply\n\\[\nq\\to p.\n\\]\nSweep through the updates\n\\[\nx_1\\to p,\\ldots,x_u\\to p.\n\\]\nWhenever phase $s_j$ is reached, apply\n\\[\np\\to t_j.\n\\]\nFinally apply $q\\to p$ with coefficient $-1$.  At phase $s$, the carrier is $e_q+w_s$, so every output has the required value, and the final gate removes $e_q$ from row $p$.  The word has length\n\\[\n1+u+v+1=u+v+2.\n\\]\nThus\n\\[\n\\ellAll\\bigl(A(u;s_1,\\ldots,s_v)\\bigr)\\le u+v+2.\n\\]","For a lower word, perform the same $u$ updates without opening the catalyst.  At each selected phase, use the two gates\n\\[\np\\to t_j,\n\\qquad\nq\\to t_j.\n\\]\nThis gives length $u+2v$.","It remains to prove optimality in the lower model.  At the root, split\n\\[\n\\{x_1,\\ldots,x_u,p\\}\n\\quad\\big|\\quad\n\\{q,t_1,\\ldots,t_v\\}.\n\\]\nThe nonzero rows of the root cross block are $w_{s_1},\\ldots,w_{s_v}$.  Distinct prefix vectors are linearly independent: in a linear relation, inspect successively a coordinate occurring in $w_{s_j}$ but in none of the earlier prefixes, starting from the largest phase.  Hence the root contributes $v$.","e lower model.  At the root, split\n\\[\n\\{x_1,\\ldots,x_u,p\\}\n\\quad\\big|\\quad\n\\{q,t_1,\\ldots,t_v\\}.\n\\]\nThe nonzero rows of the root cross block are $w_{s_1},\\ldots,w_{s_v}$.  Distinct prefix vectors are linearly independent: in a linear relation, inspect successively a coordinate occurring in $w_{s_j}$ but in none of the earlier prefixes, starting from the largest phase.  Hence the root contributes $v$.","Inside the left child, use the comb that splits off $x_1$, then $x_2$, and so on while retaining $p$ in the remaining interval.  Every one of the $u$ cuts has rank one because the final row $p$ contains every $x_i$.  Inside the right child, repeatedly split off the last output while retaining $q$ in the left interval.  Each of the $v$ cuts has rank one because every output contains $q$.  Therefore\n\\[\n\\Phi_T\\bigl(A(u;s_1,\\ldots,s_v)\\bigr)\n=v+u+v=u+2v.\n\\]\nThe interval-tree lower bound matches the displayed lower circuit.\n\\end{proof}","ide the left child, use the comb that splits off $x_1$, then $x_2$, and so on while retaining $p$ in the remaining interval.  Every one of the $u$ cuts has rank one because the final row $p$ contains every $x_i$.  Inside the right child, repeatedly split off the last output while retaining $q$ in the left interval.  Each of the $v$ cuts has rank one because every output contains $q$.  Therefore\n\\[\n\\Phi_T\\bigl(A(u;s_1,\\ldots,s_v)\\bigr)\n=v+u+v=u+2v.\n\\]\nThe interval-tree lower bound matches the displayed lower circuit.\n\\end{proof}","\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.90\\linewidth]{assets/g_snapshot_family.pdf}\n\\caption{The connected snapshot family.  One temporary upper dependency lets the ambient circuit export $v$ selected prefix states.  A lower-only circuit must insert the $q$ component separately into every output.}\n\\label{g:fig:snapshot-family}\n\\end{figure}","\\begin{remark}[Support connectivity]\nFor $u,v\\ge1$, the undirected support graph of $A(u;s_1,\\ldots,s_v)-I$ is connected.  Row $p$ joins all update coordinates $x_i$, while every output joins both $p$ and $q$.  The separation below therefore does not arise from a direct sum of independent six-row gadgets.\n\\end{remark}","\\begin{theorem}[Exact binary cases in dimensions seven through nine]\\label{g:thm:higher-exact}\nOver $\\F_2$, let\n\\[\nH_7=A(2;0,1,2),\n\\qquad\nH_8=A(3;0,1,2),\n\\qquad\nH_9=A(3;0,1,2,3).\n\\]\nThen\n\\[\n(\\ellDown(H_7),\\ellAll(H_7))=(8,7),\n\\]\n\\[\n(\\ellDown(H_8),\\ellAll(H_8))=(9,8),\n\\]\nand\n\\[\n(\\ellDown(H_9),\\ellAll(H_9))=(11,9).\n\\]\n\\end{theorem}","\\begin{proof}\nThe restricted values follow from \\cref{g:thm:snapshot-family}.  The ambient words have lengths $7$, $8$, and $9$.  To exclude shorter words, the supplement computes exact ambient balls and checks that the two relevant balls do not intersect.  The certificate sizes are\n\\begin{center}\n\\begin{tabular}{@{}ccll@{}}\n\\toprule\n$n$&tested split&left ball&right ball","\\midrule\n$7$&$3+3$&$23276$&$23276$","$8$&$3+4$&$52277$&$1174097$","$9$&$4+4$&$2994949$&$2994949$","\\bottomrule\n\\end{tabular}\n\\end{center}\nEvery intersection is empty.  Since the generating sets are symmetric, an empty radius-$r$ versus radius-$s$ intersection rules out every word of length at most $r+s$.\n\\end{proof}","These are exact higher-dimensional examples, not merely padded copies of $C$.","\\begin{proof}[Proof of \\cref{g:cor:connected-gap}]\nFor $n\\ge7$, put\n\\[\nv_n=\\left\\lfloor\\frac{n-1}{2}\\right\\rfloor,\n\\qquad\nu_n=n-v_n-2,\n\\]\nand take the phases\n\\[\n0,1,\\ldots,v_n-1.\n\\]\nThe inequality $v_n\\le u_n+1$ holds, so the snapshot target is defined.  By \\cref{g:thm:snapshot-family},\n\\[\n\\ellDown(A_n)=u_n+2v_n=n+v_n-2,\n\\]\nwhile\n\\[\n\\ellAll(A_n)\\le u_n+v_n+2=n.\n\\]\nHence\n\\[\n\\ellDown(A_n)-\\ellAll(A_n)\n\\ge v_n-2\n=\n\\left\\lfloor\\frac{n-5}{2}\\right\\rfloor.\n\\]\nConnectivity follows from the preceding remark.\n\\end{proof}","\\begin{table}[t]\n\\centering\n\\caption{Successive connected cases from the optimized snapshot family.  The restricted value is exact; the ambient column is a certified construction and is exact for $n=7,8,9$ over $\\F_2$.}\n\\label{g:tab:higher-cases}\n\\begin{tabular}{@{}rrrrr@{}}\n\\toprule\n$n$&$u$&$v$&$\\ellDown$&ambient upper bound / certified gap","\\midrule\n$7$&$2$&$3$&$8$&$7\\;/\\;1$","$8$&$3$&$3$&$9$&$8\\;/\\;1$","$9$&$3$&$4$&$11$&$9\\;/\\;2$","$10$&$4$&$4$&$12$&$10\\;/\\;2$","$11$&$4$&$5$&$14$&$11\\;/\\;3$","$12$&$5$&$5$&$15$&$12\\;/\\;3$","$13$&$5$&$6$&$17$&$13\\;/\\;4$","$15$&$6$&$7$&$20$&$15\\;/\\;5$","$17$&$7$&$8$&$23$&$17\\;/\\;6$","$19$&$8$&$9$&$26$&$19\\;/\\;7$","$21$&$9$&$10$&$29$&$21\\;/\\;8$","The ratio also increases beyond the block-sum value $7/6$.","\\begin{corollary}[Asymptotic multiplicative distortion]\\label{g:cor:three-halves}\nFor every field,\n\\[\n\\limsup_{n\\to\\infty}\n\\sup_{A\\in\\U_n(\\F),\\,A\\ne I}\n\\frac{\\ellDown(A)}{\\ellAll(A)}\n\\ge\\frac32.\n\\]\n\\end{corollary}","\\begin{proof}\nAlong the optimized connected family,\n\\[\n\\frac{\\ellDown(A_n)}{\\ellAll(A_n)}\n\\ge\n\\frac{n+\\lfloor(n-1)/2\\rfloor-2}{n},\n\\]\nwhich tends to $3/2$.\n\\end{proof}","The connected snapshot family uses a single matched excursion: one upper-directed gate opens the dirty carrier and one upper-directed gate closes it.  This permits a sharper statement than the unrestricted inequality $\\ellAll<\\ellDown$.","For a circuit word $w$, let $u(w)$ denote its number of upper-directed gates.","\\begin{definition}[Upper-gate budget]\nFor $q\\ge0$ and $A\\in\\U_n(\\F)$, define\n\\[\n\\ellBudget{q}(A)\n=\n\\min\\{|w|:[w]=A,\\ u(w)\\le q\\}.\n\\]\nThus\n\\[\n\\ellBudget{0}(A)=\\ellDown(A),\n\\qquad\n\\ellBudget{q+1}(A)\\le\\ellBudget{q}(A),\n\\qquad\n\\ellAll(A)\\le\\ellBudget{q}(A).\n\\]\n\\end{definition}","\\begin{proposition}[Two upper gates give a linear advantage]\\label{g:prop:two-upper-linear}\nFor every snapshot target $A=A(u;s_1,\\ldots,s_v)$,\n\\[\n\\ellBudget{0}(A)=u+2v,\n\\qquad\n\\ellBudget{2}(A)\\le u+v+2.\n\\]\nConsequently,\n\\[\n\\ellBudget{0}(A)-\\ellBudget{2}(A)\\ge v-2.\n\\]\nFor the optimized connected target $A_n$ of \\cref{g:cor:connected-gap},\n\\[\n\\ellBudget{0}(A_n)-\\ellBudget{2}(A_n)\n\\ge\n\\left\\lfloor\\frac{n-5}{2}\\right\\rfloor.\n\\]\n\\end{proposition}","\\begin{proof}\nThe exact restricted value is \\cref{g:thm:snapshot-family}.  Its ambient construction has precisely two upper-directed gates, namely the opening and closing operations on the carrier row; all $u$ updates and all $v$ reads are lower directed.  The optimized choice has $v=\\lfloor(n-1)/2\\rfloor$.\n\\end{proof}","\\begin{corollary}[No budget-only additive simulation bound]\\label{g:cor:no-budget-only}\nThere is no finite constant $K$ such that\n\\[\n\\ellDown(A)\\le\\ellBudget{2}(A)+K\n\\]\nfor all dimensions, fields, and lower-unitriangular targets $A$.\n\\end{corollary}","\\begin{proof}\nThe left-minus-right difference in \\cref{g:prop:two-upper-linear} tends to infinity with $n$.\n\\end{proof}","For $X\\in\\SL_n(\\F)$, let $X^+$ be the matrix obtained by retaining only the strict upper entries of $X$.  Define\n\\[\nr^+(X)=\\rank X^+\n\\]\nand\n\\[\nb^+(X)=\\max\\{j-i:X_{ij}\\ne0,\\ i<j\\},\n\\]\nwith $b^+(X)=0$ when $X^+=0$.  Let $\\ellExc{r}{b}(A)$ be the minimum length of a transvection circuit from $I$ to $A$ such that every prefix matrix $X$ satisfies\n\\[\nr^+(X)\\le r,\n\\qquad\nb^+(X)\\le b.\n\\]","\\begin{lemma}\\label{g:lem:zero-excursion-equals-lower}\nFor $A\\in\\U_n(\\F)$,\n\\[\n\\ellExc{0}{0}(A)=\\ellDown(A).\n\\]\n\\end{lemma}","\\begin{proof}\nA lower-directed gate preserves lower triangularity.  Conversely, if $X$ is lower unitriangular and an upper-directed gate $R_t\\leftarrow R_t+aR_s$ with $t<s$ is applied, then the new $(t,s)$ entry equals\n\\[\nX_{t,s}+aX_{s,s}=a\\ne0.\n\\]\nThe next state is therefore not lower triangular.  A circuit whose every prefix has zero strict upper part uses only lower-directed gates.\n\\end{proof}","\\begin{theorem}[Microscopic escape, macroscopic saving]\\label{g:thm:microscopic-escape}\nFor every field $\\F$ and every $n\\ge7$, there is a connected target $A_n\\in\\U_n(\\F)$ such that\n\\[\n\\ellExc{0}{0}(A_n)-\\ellExc{1}{1}(A_n)\n\\ge\n\\left\\lfloor\\frac{n-5}{2}\\right\\rfloor.\n\\]\nThe length-$n$ circuit witnessing the second term uses exactly two upper-directed gates.\n\\end{theorem}","\\begin{proof}\nUse the connected family from \\cref{g:cor:connected-gap}, with the carrier coordinates $p<q$ adjacent.  Between the opening and closing gates, the only strict upper entry is the coefficient at position $(p,q)$.  Hence every prefix has upper rank at most one and upper bandwidth at most one.  The displayed ambient circuit has length $n$, whereas\n\\[\n\\ellExc{0}{0}(A_n)=\\ellDown(A_n)\n=n+\\left\\lfloor\\frac{n-1}{2}\\right\\rfloor-2.\n\\]\nSubtracting proves the claim.\n\\end{proof}","A hardware or fault model may penalize an upper-directed interaction more heavily than a lower-directed one.  Fix $\\lambda>0$.  If $w$ has $d(w)$ lower-directed gates and $u(w)$ upper-directed gates, put\n\\[\n\\operatorname{cost}_{\\lambda}(w)=d(w)+\\lambda u(w),\n\\]\nand define\n\\[\n\\ellWeight{\\lambda}(A)\n=\n\\inf\\{\\operatorname{cost}_{\\lambda}(w):[w]=A\\}.\n\\]\nAt $\\lambda=1$ this is the ordinary ambient length, while large $\\lambda$ increasingly penalizes flag excursions.","lt model may penalize an upper-directed interaction more heavily than a lower-directed one.  Fix $\\lambda>0$.  If $w$ has $d(w)$ lower-directed gates and $u(w)$ upper-directed gates, put\n\\[\n\\operatorname{cost}_{\\lambda}(w)=d(w)+\\lambda u(w),\n\\]\nand define\n\\[\n\\ellWeight{\\lambda}(A)\n=\n\\inf\\{\\operatorname{cost}_{\\lambda}(w):[w]=A\\}.\n\\]\nAt $\\lambda=1$ this is the ordinary ambient length, while large $\\lambda$ increasingly penalizes flag excursions.","\\begin{proposition}[Fixed penalties do not suppress catalysis]\\label{g:prop:weighted-catalysis}\nFor every snapshot target $A=A(u;s_1,\\ldots,s_v)$ and every $\\lambda>0$,\n\\[\n\\ellWeight{\\lambda}(A)\n\\le\nu+v+2\\lambda.\n\\]\nConsequently,\n\\[\n\\ellDown(A)-\\ellWeight{\\lambda}(A)\n\\ge\nv-2\\lambda.\n\\]\nFor the optimized connected family,\n\\[\n\\ellDown(A_n)-\\ellWeight{\\lambda}(A_n)\n\\ge\n\\left\\lfloor\\frac{n-1}{2}\\right\\rfloor-2\\lambda.\n\\]\nIn particular, for every fixed finite $\\lambda$, the weighted advantage grows linearly with $n$.\n\\end{proposition}","s]\\label{g:prop:weighted-catalysis}\nFor every snapshot target $A=A(u;s_1,\\ldots,s_v)$ and every $\\lambda>0$,\n\\[\n\\ellWeight{\\lambda}(A)\n\\le\nu+v+2\\lambda.\n\\]\nConsequently,\n\\[\n\\ellDown(A)-\\ellWeight{\\lambda}(A)\n\\ge\nv-2\\lambda.\n\\]\nFor the optimized connected family,\n\\[\n\\ellDown(A_n)-\\ellWeight{\\lambda}(A_n)\n\\ge\n\\left\\lfloor\\frac{n-1}{2}\\right\\rfloor-2\\lambda.\n\\]\nIn particular, for every fixed finite $\\lambda$, the weighted advantage grows linearly with $n$.\n\\end{proposition}","target $A=A(u;s_1,\\ldots,s_v)$ and every $\\lambda>0$,\n\\[\n\\ellWeight{\\lambda}(A)\n\\le\nu+v+2\\lambda.\n\\]\nConsequently,\n\\[\n\\ellDown(A)-\\ellWeight{\\lambda}(A)\n\\ge\nv-2\\lambda.\n\\]\nFor the optimized connected family,\n\\[\n\\ellDown(A_n)-\\ellWeight{\\lambda}(A_n)\n\\ge\n\\left\\lfloor\\frac{n-1}{2}\\right\\rfloor-2\\lambda.\n\\]\nIn particular, for every fixed finite $\\lambda$, the weighted advantage grows linearly with $n$.\n\\end{proposition}","\\begin{proof}\nThe displayed catalytic word contains $u+v$ lower-directed gates and two upper-directed gates, so its weighted cost is $u+v+2\\lambda$.  Subtract this from the exact restricted value $u+2v$.  The optimized family again has $v=\\lfloor(n-1)/2\\rfloor$.\n\\end{proof}","\\begin{remark}[Penalty threshold]\nFor the displayed realization of a snapshot target, catalysis beats the optimal lower-only cost whenever\n\\[\n\\lambda<\\frac v2.\n\\]\nThe critical penalty therefore scales with the number of consumers, not with the dimension of the dirty workspace.\n\\end{remark}","The preceding sections construct highly structured targets with large defects.  This section proves the complementary statement that catalytic defects are nongeneric at every fixed dimension.  Fix a field $K$.  Algebraic dimensions and closures are taken after base change to an algebraic closure of $K$.  Put\n\\[\nN=\\dim \\U_n=\\binom n2,\n\\qquad\nM=n(n-1).\n\\]","For a direction skeleton\n\\[\n\\sigma=((t_1,s_1),\\ldots,(t_L,s_L)),\n\\qquad t_i\\ne s_i,\n\\]\ndefine the polynomial word map\n\\[\n\\Phi_\\sigma:\\Aff^L\\longrightarrow\\SL_n,\n\\qquad\n(a_1,\\ldots,a_L)\n\\longmapsto\n(I+a_LE_{t_L,s_L})\\cdots(I+a_1E_{t_1,s_1}).\n\\]\nThe coefficients are allowed to vanish, so a word of length at most $L$ can be padded to length $L$.  Let\n\\[\n\\mathcal R^*_{n,\\le L}\n=\n\\U_n\\cap\n\\bigcup_{\\sigma\\in\\{(t,s):t\\ne s\\}^L}\n\\operatorname{im}\\Phi_\\sigma\n\\]\nbe the ambient reachability locus, and define $\\mathcal R^\\downarrow_{n,\\le L}$ analogously using only lower directions.","mial word map\n\\[\n\\Phi_\\sigma:\\Aff^L\\longrightarrow\\SL_n,\n\\qquad\n(a_1,\\ldots,a_L)\n\\longmapsto\n(I+a_LE_{t_L,s_L})\\cdots(I+a_1E_{t_1,s_1}).\n\\]\nThe coefficients are allowed to vanish, so a word of length at most $L$ can be padded to length $L$.  Let\n\\[\n\\mathcal R^*_{n,\\le L}\n=\n\\U_n\\cap\n\\bigcup_{\\sigma\\in\\{(t,s):t\\ne s\\}^L}\n\\operatorname{im}\\Phi_\\sigma\n\\]\nbe the ambient reachability locus, and define $\\mathcal R^\\downarrow_{n,\\le L}$ analogously using only lower directions.","Equivalently, put\n\\[\nX_*\n=\n\\bigcup_{t\\ne s}\\{I+aE_{t,s}:a\\in K\\},\n\\qquad\nX_\\downarrow\n=\n\\bigcup_{s<t}\\{I+aE_{t,s}:a\\in K\\}.\n\\]\nThen $\\ellAll(A)$ and $\\ellDown(A)$ are the corresponding transvection product lengths: the least $L$ for which $A\\in X_*^L$ or $A\\in X_\\downarrow^L$.","\\begin{proposition}[Dimension of bounded-length loci]\\label{g:prop:reachability-dimension}\nFor every $L\\ge0$, the sets $\\mathcal R^*_{n,\\le L}$ and $\\mathcal R^\\downarrow_{n,\\le L}$ are constructible and\n\\[\n\\dim\\overline{\\mathcal R^*_{n,\\le L}}\n\\le L,\n\\qquad\n\\dim\\overline{\\mathcal R^\\downarrow_{n,\\le L}}\n\\le L.\n\\]\n\\end{proposition}","\\begin{proof}\nThere are finitely many direction skeletons of length $L$.  For each skeleton, $\\Phi_\\sigma$ is a morphism from an $L$-dimensional affine space.  Chevalley's theorem makes its image constructible, and the closure of the image has dimension at most $L$ \\cite{g:Hartshorne1977}.  Finite unions preserve constructibility and the maximum dimension bound.  Intersecting with the closed subvariety $\\U_n$ cannot increase dimension.\n\\end{proof}","proof}\nThere are finitely many direction skeletons of length $L$.  For each skeleton, $\\Phi_\\sigma$ is a morphism from an $L$-dimensional affine space.  Chevalley's theorem makes its image constructible, and the closure of the image has dimension at most $L$ \\cite{g:Hartshorne1977}.  Finite unions preserve constructibility and the maximum dimension bound.  Intersecting with the closed subvariety $\\U_n$ cannot increase dimension.\n\\end{proof}","\\begin{proposition}[Dimension per gate]\\label{g:prop:dimension-per-gate}\nLet $G$ be an algebraic group, let $V\\subseteq G$ be an irreducible subvariety of dimension $D$, and let $X\\subseteq G$ be a finite union of locally closed subvarieties, each of dimension at most $r$, with the identity in $X$.  Write\n\\[\nX^{(L)}\n=\n\\{x_L\\cdots x_1:x_1,\\ldots,x_L\\in X\\}.\n\\]\nThen\n\\[\n\\dim\\overline{V\\cap X^{(L)}}\\le rL.\n\\]\nIn particular, if $rL<D$, a Zariski-generic point of $V$ has $X$-product length greater than $L$.\n\\end{proposition}","n per gate]\\label{g:prop:dimension-per-gate}\nLet $G$ be an algebraic group, let $V\\subseteq G$ be an irreducible subvariety of dimension $D$, and let $X\\subseteq G$ be a finite union of locally closed subvarieties, each of dimension at most $r$, with the identity in $X$.  Write\n\\[\nX^{(L)}\n=\n\\{x_L\\cdots x_1:x_1,\\ldots,x_L\\in X\\}.\n\\]\nThen\n\\[\n\\dim\\overline{V\\cap X^{(L)}}\\le rL.\n\\]\nIn particular, if $rL<D$, a Zariski-generic point of $V$ has $X$-product length greater than $L$.\n\\end{proposition}","per-gate}\nLet $G$ be an algebraic group, let $V\\subseteq G$ be an irreducible subvariety of dimension $D$, and let $X\\subseteq G$ be a finite union of locally closed subvarieties, each of dimension at most $r$, with the identity in $X$.  Write\n\\[\nX^{(L)}\n=\n\\{x_L\\cdots x_1:x_1,\\ldots,x_L\\in X\\}.\n\\]\nThen\n\\[\n\\dim\\overline{V\\cap X^{(L)}}\\le rL.\n\\]\nIn particular, if $rL<D$, a Zariski-generic point of $V$ has $X$-product length greater than $L$.\n\\end{proposition}","}\nLet $G$ be an algebraic group, let $V\\subseteq G$ be an irreducible subvariety of dimension $D$, and let $X\\subseteq G$ be a finite union of locally closed subvarieties, each of dimension at most $r$, with the identity in $X$.  Write\n\\[\nX^{(L)}\n=\n\\{x_L\\cdots x_1:x_1,\\ldots,x_L\\in X\\}.\n\\]\nThen\n\\[\n\\dim\\overline{V\\cap X^{(L)}}\\le rL.\n\\]\nIn particular, if $rL<D$, a Zariski-generic point of $V$ has $X$-product length greater than $L$.\n\\end{proposition}","let $V\\subseteq G$ be an irreducible subvariety of dimension $D$, and let $X\\subseteq G$ be a finite union of locally closed subvarieties, each of dimension at most $r$, with the identity in $X$.  Write\n\\[\nX^{(L)}\n=\n\\{x_L\\cdots x_1:x_1,\\ldots,x_L\\in X\\}.\n\\]\nThen\n\\[\n\\dim\\overline{V\\cap X^{(L)}}\\le rL.\n\\]\nIn particular, if $rL<D$, a Zariski-generic point of $V$ has $X$-product length greater than $L$.\n\\end{proposition}","\\begin{proof}\nEvery irreducible component of the Cartesian power $X^{\\times L}$ has dimension at most $rL$.  Multiplication $G^L\\to G$ is a morphism, so the closure of the image of each component has dimension at most $rL$.  There are finitely many components, and intersecting the resulting finite union with $V$ cannot increase dimension.  Since the identity belongs to $X$, products of length at most $L$ are already contained in $X^{(L)}$.\n\\end{proof}","n{proof}\nEvery irreducible component of the Cartesian power $X^{\\times L}$ has dimension at most $rL$.  Multiplication $G^L\\to G$ is a morphism, so the closure of the image of each component has dimension at most $rL$.  There are finitely many components, and intersecting the resulting finite union with $V$ cannot increase dimension.  Since the identity belongs to $X$, products of length at most $L$ are already contained in $X^{(L)}$.\n\\end{proof}","mponent of the Cartesian power $X^{\\times L}$ has dimension at most $rL$.  Multiplication $G^L\\to G$ is a morphism, so the closure of the image of each component has dimension at most $rL$.  There are finitely many components, and intersecting the resulting finite union with $V$ cannot increase dimension.  Since the identity belongs to $X$, products of length at most $L$ are already contained in $X^{(L)}$.\n\\end{proof}","For transvections, the discrete choice of direction contributes finitely many components and each component carries one scalar parameter.  Thus $r=1$.  Every element of $\\U_n(K)$ has lower length at most $N$: process the strict lower entries in any root order compatible with elimination.  The dimension-per-gate bound shows that this worst-case value is generic.","\\begin{theorem}[Generic rigidity]\\label{g:thm:generic-rigidity}\nLet $K$ be an infinite field.  There is a nonempty Zariski-open subvariety $\\Omega_n\\subseteq\\U_n$, defined over $K$, such that $\\Omega_n(K)$ is nonempty and\n\\[\n\\ellAll(A)=\\ellDown(A)=N\n\\qquad(A\\in\\Omega_n(K)).\n\\]\nThe same statement holds with exact product length replaced by border product length.\n\\end{theorem}","\\begin{proof}\nAfter base change to an algebraic closure $\\overline K$, \\cref{g:prop:reachability-dimension} gives\n\\[\n\\dim\\overline{\\mathcal R^*_{n,\\le N-1}}\\le N-1<N=\\dim\\U_n.\n\\]\nBecause the word maps are defined over $K$, this proper closed set descends to a proper closed subset of $\\U_n$ over $K$.  Its complement is a nonempty Zariski-open set $\\Omega_n$.  Since $K$ is infinite and $\\U_n\\cong\\Aff^N$, the set $\\Omega_n(K)$ is nonempty and Zariski dense.  For any $A\\in\\Omega_n(K)$, one has $\\ellAll(A)\\ge N$.  The inequalities\n\\[\n\\ellAll(A)\\le\\ellDown(A)\\le N\n\\]\nforce equality.  Replacing each image by its Zariski closure gives the border-length statement by the same argument.\n\\end{proof}","fter base change to an algebraic closure $\\overline K$, \\cref{g:prop:reachability-dimension} gives\n\\[\n\\dim\\overline{\\mathcal R^*_{n,\\le N-1}}\\le N-1<N=\\dim\\U_n.\n\\]\nBecause the word maps are defined over $K$, this proper closed set descends to a proper closed subset of $\\U_n$ over $K$.  Its complement is a nonempty Zariski-open set $\\Omega_n$.  Since $K$ is infinite and $\\U_n\\cong\\Aff^N$, the set $\\Omega_n(K)$ is nonempty and Zariski dense.  For any $A\\in\\Omega_n(K)$, one has $\\ellAll(A)\\ge N$.  The inequalities\n\\[\n\\ellAll(A)\\le\\ellDown(A)\\le N\n\\]\nforce equality.  Replacing each image by its Zariski closure gives the border-length statement by the same argument.\n\\end{proof}",", \\cref{g:prop:reachability-dimension} gives\n\\[\n\\dim\\overline{\\mathcal R^*_{n,\\le N-1}}\\le N-1<N=\\dim\\U_n.\n\\]\nBecause the word maps are defined over $K$, this proper closed set descends to a proper closed subset of $\\U_n$ over $K$.  Its complement is a nonempty Zariski-open set $\\Omega_n$.  Since $K$ is infinite and $\\U_n\\cong\\Aff^N$, the set $\\Omega_n(K)$ is nonempty and Zariski dense.  For any $A\\in\\Omega_n(K)$, one has $\\ellAll(A)\\ge N$.  The inequalities\n\\[\n\\ellAll(A)\\le\\ellDown(A)\\le N\n\\]\nforce equality.  Replacing each image by its Zariski closure gives the border-length statement by the same argument.\n\\end{proof}","-dimension} gives\n\\[\n\\dim\\overline{\\mathcal R^*_{n,\\le N-1}}\\le N-1<N=\\dim\\U_n.\n\\]\nBecause the word maps are defined over $K$, this proper closed set descends to a proper closed subset of $\\U_n$ over $K$.  Its complement is a nonempty Zariski-open set $\\Omega_n$.  Since $K$ is infinite and $\\U_n\\cong\\Aff^N$, the set $\\Omega_n(K)$ is nonempty and Zariski dense.  For any $A\\in\\Omega_n(K)$, one has $\\ellAll(A)\\ge N$.  The inequalities\n\\[\n\\ellAll(A)\\le\\ellDown(A)\\le N\n\\]\nforce equality.  Replacing each image by its Zariski closure gives the border-length statement by the same argument.\n\\end{proof}","thcal R^*_{n,\\le N-1}}\\le N-1<N=\\dim\\U_n.\n\\]\nBecause the word maps are defined over $K$, this proper closed set descends to a proper closed subset of $\\U_n$ over $K$.  Its complement is a nonempty Zariski-open set $\\Omega_n$.  Since $K$ is infinite and $\\U_n\\cong\\Aff^N$, the set $\\Omega_n(K)$ is nonempty and Zariski dense.  For any $A\\in\\Omega_n(K)$, one has $\\ellAll(A)\\ge N$.  The inequalities\n\\[\n\\ellAll(A)\\le\\ellDown(A)\\le N\n\\]\nforce equality.  Replacing each image by its Zariski closure gives the border-length statement by the same argument.\n\\end{proof}","For an algebraically closed field $K$ and $d\\ge1$, define\n\\[\n\\mathcal D_{n,\\ge d}\n=\n\\{A\\in\\U_n(K):\\DeltaDef(A)\\ge d\\}.\n\\]","\\begin{theorem}[Defect--codimension bound]\\label{g:thm:defect-codimension}\nLet $K$ be algebraically closed.  For every $1\\le d\\le N$,\n\\[\n\\dim\\overline{\\mathcal D_{n,\\ge d}}\\le N-d.\n\\]\nEquivalently,\n\\[\n\\codim_{\\U_n}\\overline{\\mathcal D_{n,\\ge d}}\\ge d.\n\\]\nMore finely, on the stratum where $\\ellDown(A)=L$ and $\\DeltaDef(A)\\ge d$, the closure has dimension at most $L-d$.\n\\end{theorem}","\\begin{proof}\nIf $\\DeltaDef(A)\\ge d$, then\n\\[\n\\ellAll(A)\n\\le\n\\ellDown(A)-d\n\\le\nN-d.\n\\]\nThus $\\mathcal D_{n,\\ge d}\\subseteq\\mathcal R^*_{n,\\le N-d}$, and \\cref{g:prop:reachability-dimension} gives the first assertion.  On the intrinsic-length-$L$ stratum one instead has $\\ellAll(A)\\le L-d$, proving the refinement.\n\\end{proof}","\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.78\\linewidth]{assets/g_defect_strata.pdf}\n\\caption{Bounded ambient product loci form nested algebraic strata inside $\\U_n$.  A defect of size at least $d$ lies inside the ambient reachability locus of length $N-d$, whose closure has codimension at least $d$.}\n\\label{g:fig:defect-strata}\n\\end{figure}","The dimension argument has an elementary counting counterpart.  Allowing zero coefficients, a padded word of length $L$ is specified by $L$ choices among $M$ directed positions and $L$ coefficients in $\\F_q$.  Consequently\n\\[\n|\\mathcal R^*_{n,\\le L}(\\F_q)|\\le(Mq)^L.\n\\]","\\begin{corollary}[Finite-field defect density]\\label{g:cor:finite-field-density}\nFor $1\\le d\\le N$,\n\\[\n\\Prob_{A\\in\\U_n(\\F_q)}[\\DeltaDef(A)\\ge d]\n\\le\n\\min\\{1,M^{N-d}q^{-d}\\}.\n\\]\nIn particular, for fixed $n$,\n\\[\n\\Prob[\\DeltaDef(A)>0]=O_n(q^{-1}),\n\\qquad\n\\Exp[\\DeltaDef(A)]=O_n(q^{-1})\n\\]\nas $q\\to\\infty$.\n\\end{corollary}","\\begin{proof}\nThe defect condition implies ambient length at most $N-d$, so the number of such targets is at most $(Mq)^{N-d}$.  Divide by\n\\[\n|\\U_n(\\F_q)|=q^N.\n\\]\nThe expectation bound follows from\n\\[\n\\Exp[\\DeltaDef]=\\sum_{d=1}^{N}\\Prob[\\DeltaDef\\ge d].\n\\]\n\\end{proof}","The same counting argument also gives\n\\[\n\\Prob_{A\\in\\U_n(\\F_q)}[\\ellAll(A)=\\ellDown(A)=N]\n\\ge\n1-M^{N-1}q^{-1},\n\\]\nwhenever the right side is positive.  Thus maximal ambient and intrinsic length is not only Zariski generic over infinite fields but asymptotically typical over growing finite fields at fixed dimension.","For an algebraic generating set $X$, define the border product length\n\\[\n\\underline\\ell_X(A)\n=\n\\min\\{L:A\\in\\overline{X^L}\\}.\n\\]\nThis is analogous to the passage from tensor rank to border rank \\cite{g:Landsberg2012}.  The generic value is still $N$ by \\cref{g:thm:generic-rigidity}, but exact and border defects can differ on exceptional strata.  It is presently open whether the canonical atom $C$ belongs to the closure of the six-factor lower transvection locus.  A negative answer would show that its one-gate defect is stable under algebraic degeneration; a positive answer would exhibit a rank-versus-border-rank phenomenon inside reversible linear synthesis.","\\begin{remark}[Rare but amplifiable]\\label{g:rem:rare-amplifiable}\nThe codimension theorem and the tensor construction point in opposite directions without contradiction.  At each fixed $n$, a defect of size $d$ lies on a codimension-at-least-$d$ subset.  Carefully selected points on these thin strata can nevertheless be tensorized into families with $\\Omega(n\\log n)$ additive defect.  Catalytic compression is therefore algebraically rare but asymptotically amplifiable.\n\\end{remark}","\\begin{proposition}[Block additivity]\\label{g:prop:block-additivity}\nLet $A_i\\in\\U_{n_i}(\\F)$ for $1\\le i\\le k$.  Then\n\\[\n\\ellDown\\bigl(\\operatorname{diag}(A_1,\\ldots,A_k)\\bigr)\n=\n\\sum_{i=1}^k\\ellDown(A_i).\n\\]\n\\end{proposition}","For the lower bound, view an element of $\\U_{n_1+\\cdots+n_k}(\\F)$ as a lower block-triangular matrix and project to its diagonal blocks:\n\\[\n\\pi(X)=(X_{11},\\ldots,X_{kk}).\n\\]\nBecause diagonal blocks multiply independently in a lower block-triangular product, $\\pi$ is a homomorphism.  A lower transvection internal to one block projects to one generator in that factor, while a crossing transvection projects to the identity in every factor.  If $m_i$ gates survive in factor $i$, then the total word length is at least\n\\[\n\\sum_i m_i\n\\ge\n\\sum_i\\ellDown(A_i).\n\\]\n\\end{proof}","_1+\\cdots+n_k}(\\F)$ as a lower block-triangular matrix and project to its diagonal blocks:\n\\[\n\\pi(X)=(X_{11},\\ldots,X_{kk}).\n\\]\nBecause diagonal blocks multiply independently in a lower block-triangular product, $\\pi$ is a homomorphism.  A lower transvection internal to one block projects to one generator in that factor, while a crossing transvection projects to the identity in every factor.  If $m_i$ gates survive in factor $i$, then the total word length is at least\n\\[\n\\sum_i m_i\n\\ge\n\\sum_i\\ellDown(A_i).\n\\]\n\\end{proof}","block-triangular matrix and project to its diagonal blocks:\n\\[\n\\pi(X)=(X_{11},\\ldots,X_{kk}).\n\\]\nBecause diagonal blocks multiply independently in a lower block-triangular product, $\\pi$ is a homomorphism.  A lower transvection internal to one block projects to one generator in that factor, while a crossing transvection projects to the identity in every factor.  If $m_i$ gates survive in factor $i$, then the total word length is at least\n\\[\n\\sum_i m_i\n\\ge\n\\sum_i\\ellDown(A_i).\n\\]\n\\end{proof}","iagonal blocks:\n\\[\n\\pi(X)=(X_{11},\\ldots,X_{kk}).\n\\]\nBecause diagonal blocks multiply independently in a lower block-triangular product, $\\pi$ is a homomorphism.  A lower transvection internal to one block projects to one generator in that factor, while a crossing transvection projects to the identity in every factor.  If $m_i$ gates survive in factor $i$, then the total word length is at least\n\\[\n\\sum_i m_i\n\\ge\n\\sum_i\\ellDown(A_i).\n\\]\n\\end{proof}","The ambient metric is subadditive:\n\\[\n\\ellAll\\bigl(\\operatorname{diag}(A_1,\\ldots,A_k)\\bigr)\n\\le\n\\sum_i\\ellAll(A_i).\n\\]\nCross-block ambient workspace can only improve this upper bound.","\\begin{corollary}[Block-sum amplification]\\label{g:cor:linear-gap}\nFor every field and every $n\\ge6$, there is an $A_n\\in\\U_n(\\F)$ with\n\\[\n\\ellDown(A_n)-\\ellAll(A_n)\n\\ge\n\\left\\lfloor\\frac n6\\right\\rfloor.\n\\]\nFor $n=6k$, the target $C^{\\oplus k}$ satisfies\n\\[\n\\ellDown(C^{\\oplus k})=7k,\n\\qquad\n\\ellAll(C^{\\oplus k})\\le6k.\n\\]\n\\end{corollary}","\\begin{proof}\nFor $n=6k+r$, take $C^{\\oplus k}\\oplus I_r$ and apply block additivity together with $k$ copies of the ambient six-gate word.\n\\end{proof}","\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.68\\linewidth]{assets/g_direct_sum_amplification.pdf}\n\\caption{Block-sum amplification.  The restricted length is exactly $7k$, while the displayed ambient construction has length $6k$.}\n\\label{g:fig:direct-sum}\n\\end{figure}","For $A\\in\\U_m(\\F)$, define\n\\begin{equation}\\label{g:eq:zeta-lift}\n\\mathcal Z(A)\n=\n\\begin{pmatrix}\nA&0","For $A\\in\\U_m(\\F)$, define\n\\begin{equation}\\label{g:eq:zeta-lift}\n\\mathcal Z(A)\n=\n\\begin{pmatrix}\nA&0\\\\\nA&A\n\\end{pmatrix}\n\\in\\U_{2m}(\\F).\n\\end{equation}","\\begin{proposition}[Exact restricted lift]\\label{g:prop:zeta-lift}\nFor every $A\\in\\U_m(\\F)$,\n\\[\n\\ellDown(\\mathcal Z(A))=2\\ellDown(A)+m,\n\\]\nand\n\\[\n\\ellAll(\\mathcal Z(A))\\le2\\ellAll(A)+m.\n\\]\nConsequently,\n\\[\n\\DeltaDef(\\mathcal Z(A))\\ge2\\DeltaDef(A).\n\\]\n\\end{proposition}","\\begin{proof}\nThe construction first realizes $\\operatorname{diag}(A,A)$ and then applies the $m$ root-cross gates that add top row $i$ into bottom row $i$.","For the lower bound, write a circuit state in $m\\times m$ blocks as\n\\[\nX=\n\\begin{pmatrix}\nP&0","For the lower bound, write a circuit state in $m\\times m$ blocks as\n\\[\nX=\n\\begin{pmatrix}\nP&0\\\\\nQ&R\n\\end{pmatrix}.\n\\]\nProjection to the two diagonal blocks forces at least $2\\ellDown(A)$ internal gates.  A top-internal gate leaves $Q$ unchanged.  A bottom-internal gate performs an invertible row operation on $Q$ and preserves its rank.  A root-cross gate changes $Q$ by one rank-one row contribution, so it can increase $\\rank Q$ by at most one.  The final cross block is $A$, of rank $m$, hence at least $m$ root-cross gates are necessary.\n\\end{proof}","Q&R\n\\end{pmatrix}.\n\\]\nProjection to the two diagonal blocks forces at least $2\\ellDown(A)$ internal gates.  A top-internal gate leaves $Q$ unchanged.  A bottom-internal gate performs an invertible row operation on $Q$ and preserves its rank.  A root-cross gate changes $Q$ by one rank-one row contribution, so it can increase $\\rank Q$ by at most one.  The final cross block is $A$, of rank $m$, hence at least $m$ root-cross gates are necessary.\n\\end{proof}","trix}.\n\\]\nProjection to the two diagonal blocks forces at least $2\\ellDown(A)$ internal gates.  A top-internal gate leaves $Q$ unchanged.  A bottom-internal gate performs an invertible row operation on $Q$ and preserves its rank.  A root-cross gate changes $Q$ by one rank-one row contribution, so it can increase $\\rank Q$ by at most one.  The final cross block is $A$, of rank $m$, hence at least $m$ root-cross gates are necessary.\n\\end{proof}","Let $A^{[0]}=A$ and $A^{[r+1]}=\\mathcal Z(A^{[r]})$.  If $A$ has size $m$, restricted length $a$, and ambient length $b$, then\n\\begin{equation}\\label{g:eq:zeta-iterate}\n\\ellDown(A^{[r]})\n=\n2^r a+r m2^{r-1},\n\\end{equation}\nand\n\\begin{equation}\\label{g:eq:zeta-iterate-ambient}\n\\ellAll(A^{[r]})\n\\le\n2^r b+r m2^{r-1}.\n\\end{equation}\nThus\n\\[\n\\DeltaDef(A^{[r]})\\ge2^r\\DeltaDef(A).\n\\]\nFor the canonical atom $C$, the first values are\n\\begin{center}\n\\begin{tabular}{@{}rrrrr@{}}\n\\toprule\n$r$&dimension&restricted exact&ambient upper bound&certified gap","$A^{[0]}=A$ and $A^{[r+1]}=\\mathcal Z(A^{[r]})$.  If $A$ has size $m$, restricted length $a$, and ambient length $b$, then\n\\begin{equation}\\label{g:eq:zeta-iterate}\n\\ellDown(A^{[r]})\n=\n2^r a+r m2^{r-1},\n\\end{equation}\nand\n\\begin{equation}\\label{g:eq:zeta-iterate-ambient}\n\\ellAll(A^{[r]})\n\\le\n2^r b+r m2^{r-1}.\n\\end{equation}\nThus\n\\[\n\\DeltaDef(A^{[r]})\\ge2^r\\DeltaDef(A).\n\\]\nFor the canonical atom $C$, the first values are\n\\begin{center}\n\\begin{tabular}{@{}rrrrr@{}}\n\\toprule\n$r$&dimension&restricted exact&ambient upper bound&certified gap","ted length $a$, and ambient length $b$, then\n\\begin{equation}\\label{g:eq:zeta-iterate}\n\\ellDown(A^{[r]})\n=\n2^r a+r m2^{r-1},\n\\end{equation}\nand\n\\begin{equation}\\label{g:eq:zeta-iterate-ambient}\n\\ellAll(A^{[r]})\n\\le\n2^r b+r m2^{r-1}.\n\\end{equation}\nThus\n\\[\n\\DeltaDef(A^{[r]})\\ge2^r\\DeltaDef(A).\n\\]\nFor the canonical atom $C$, the first values are\n\\begin{center}\n\\begin{tabular}{@{}rrrrr@{}}\n\\toprule\n$r$&dimension&restricted exact&ambient upper bound&certified gap","\\midrule\n$0$&$6$&$7$&$6$&$1$","$1$&$12$&$20$&$18$&$2$","$2$&$24$&$52$&$48$&$4$","$3$&$48$&$128$&$120$&$8$","$4$&$96$&$304$&$288$&$16$","\\begin{theorem}[Exact restricted tensor rule]\\label{g:thm:tensor-rule}\nLet $A\\in\\U_m(\\F)$ and $B\\in\\U_n(\\F)$ be tree tight.  Then $A\\otimes B$ is tree tight and\n\\begin{equation}\\label{g:eq:tensor-lower}\n\\ellDown(A\\otimes B)\n=\nn\\ellDown(A)+m\\ellDown(B).\n\\end{equation}\nMoreover,\n\\begin{equation}\\label{g:eq:tensor-ambient}\n\\ellAll(A\\otimes B)\n\\le\nn\\ellAll(A)+m\\ellAll(B),\n\\end{equation}\nso\n\\begin{equation}\\label{g:eq:tensor-defect}\n\\DeltaDef(A\\otimes B)\n\\ge\nn\\DeltaDef(A)+m\\DeltaDef(B).\n\\end{equation}\n\\end{theorem}","n{theorem}[Exact restricted tensor rule]\\label{g:thm:tensor-rule}\nLet $A\\in\\U_m(\\F)$ and $B\\in\\U_n(\\F)$ be tree tight.  Then $A\\otimes B$ is tree tight and\n\\begin{equation}\\label{g:eq:tensor-lower}\n\\ellDown(A\\otimes B)\n=\nn\\ellDown(A)+m\\ellDown(B).\n\\end{equation}\nMoreover,\n\\begin{equation}\\label{g:eq:tensor-ambient}\n\\ellAll(A\\otimes B)\n\\le\nn\\ellAll(A)+m\\ellAll(B),\n\\end{equation}\nso\n\\begin{equation}\\label{g:eq:tensor-defect}\n\\DeltaDef(A\\otimes B)\n\\ge\nn\\DeltaDef(A)+m\\DeltaDef(B).\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nUse lexicographic order on the $mn$ coordinates.  Substitute a copy of a certificate tree for $B$ at every leaf of a certificate tree for $A$.  At an outer node, the cross block is\n\\[\nA[R,L]\\otimes B,\n\\]\nwhose rank is $n\\rank A[R,L]$ because $B$ is invertible.  The outer nodes therefore contribute $n\\ellDown(A)$.  The $m$ diagonal copies of the inner tree contribute $m\\ellDown(B)$.  This proves the lower bound.","For the matching lower construction, first realize $I_m\\otimes B$ using $m$ copies of a circuit for $B$.  Then realize $A\\otimes I_n$.  Each gate of a circuit for $A$ lifts to $n$ coordinatewise gates.  Since\n\\[\n(A\\otimes I_n)(I_m\\otimes B)=A\\otimes B,\n\\]\nthe cost is the right side of \\cref{g:eq:tensor-lower}.  The same construction with ambient circuits proves \\cref{g:eq:tensor-ambient}.\n\\end{proof}","If $A$ is tree tight of dimension $m$, with restricted length $a$ and an ambient construction of length $b$, then\n\\begin{equation}\\label{g:eq:tensor-power-lower}\n\\ellDown(A^{\\otimes r})\n=\nr m^{r-1}a,\n\\end{equation}\nand\n\\begin{equation}\\label{g:eq:tensor-power-upper}\n\\ellAll(A^{\\otimes r})\n\\le\nr m^{r-1}b.\n\\end{equation}\nA positive one-copy defect therefore becomes an order-$N\\log N$ defect at dimension $N=m^r$.","is tree tight of dimension $m$, with restricted length $a$ and an ambient construction of length $b$, then\n\\begin{equation}\\label{g:eq:tensor-power-lower}\n\\ellDown(A^{\\otimes r})\n=\nr m^{r-1}a,\n\\end{equation}\nand\n\\begin{equation}\\label{g:eq:tensor-power-upper}\n\\ellAll(A^{\\otimes r})\n\\le\nr m^{r-1}b.\n\\end{equation}\nA positive one-copy defect therefore becomes an order-$N\\log N$ defect at dimension $N=m^r$.","\\begin{proof}[Proof of \\cref{g:thm:tensor-main}]\nApply \\cref{g:thm:tensor-rule} repeatedly to the tree-tight snapshot target $A=A(u;s_1,\\ldots,s_v)$, for which\n\\[\na=u+2v,\n\\qquad\nb=u+v+2=m.\n\\]\nThen\n\\[\n\\ellDown(A^{\\otimes r})\n-\n\\ellAll(A^{\\otimes r})\n\\ge\nr m^{r-1}(v-2).\n\\]\nSince $N=m^r$ and $r=\\log_2N/\\log_2m$, this is\n\\[\n\\frac{v-2}{m\\log_2m}N\\log_2N.\n\\]\n\\end{proof}","Within the snapshot family, the best coefficient is obtained by the twenty-one-dimensional seed\n\\[\nA_{21}=A(9;0,1,\\ldots,9),\n\\]\nfor which\n\\[\n\\ellDown(A_{21})=29,\n\\qquad\n\\ellAll(A_{21})\\le21,\n\\qquad\n\\DeltaDef(A_{21})\\ge8.\n\\]","\\begin{proposition}[Optimal snapshot seed for tensor rate]\\label{g:prop:optimal-snapshot-seed}\nAmong all snapshot targets, the coefficient\n\\[\n\\frac{v-2}{m\\log_2m},\n\\qquad m=u+v+2,\n\\]\nis maximized at\n\\[\n(m,u,v)=(21,9,10).\n\\]\n\\end{proposition}","\\begin{proof}\nFor fixed $m$, maximize $v$ subject to $v\\le u+1$ and $u+v+2=m$.  Thus\n\\[\nv=\\left\\lfloor\\frac{m-1}{2}\\right\\rfloor.\n\\]\nIf $m$ is even and at least eight, its coefficient is strictly smaller than that of $m-1$, because the numerator is the same while\n\\[\n(m-1)\\log(m-1)<m\\log m.\n\\]\nIt therefore suffices to consider odd $m$.  Apart from the constant change of logarithm base, the odd-dimensional envelope is\n\\[\nf(x)=\\frac{x-5}{2x\\log x}.\n\\]\nIts derivative has the sign of\n\\[\n5(1+\\log x)-x.\n\\]\nFor $x>5$ this expression crosses zero exactly once, between $19$ and $21$.  Direct comparison gives $f(21)>f(19)$, after which the envelope decreases.  Hence the unique integer maximizer is $m=21$.\n\\end{proof}","d $m$, maximize $v$ subject to $v\\le u+1$ and $u+v+2=m$.  Thus\n\\[\nv=\\left\\lfloor\\frac{m-1}{2}\\right\\rfloor.\n\\]\nIf $m$ is even and at least eight, its coefficient is strictly smaller than that of $m-1$, because the numerator is the same while\n\\[\n(m-1)\\log(m-1)<m\\log m.\n\\]\nIt therefore suffices to consider odd $m$.  Apart from the constant change of logarithm base, the odd-dimensional envelope is\n\\[\nf(x)=\\frac{x-5}{2x\\log x}.\n\\]\nIts derivative has the sign of\n\\[\n5(1+\\log x)-x.\n\\]\nFor $x>5$ this expression crosses zero exactly once, between $19$ and $21$.  Direct comparison gives $f(21)>f(19)$, after which the envelope decreases.  Hence the unique integer maximizer is $m=21$.\n\\end{proof}","[\nv=\\left\\lfloor\\frac{m-1}{2}\\right\\rfloor.\n\\]\nIf $m$ is even and at least eight, its coefficient is strictly smaller than that of $m-1$, because the numerator is the same while\n\\[\n(m-1)\\log(m-1)<m\\log m.\n\\]\nIt therefore suffices to consider odd $m$.  Apart from the constant change of logarithm base, the odd-dimensional envelope is\n\\[\nf(x)=\\frac{x-5}{2x\\log x}.\n\\]\nIts derivative has the sign of\n\\[\n5(1+\\log x)-x.\n\\]\nFor $x>5$ this expression crosses zero exactly once, between $19$ and $21$.  Direct comparison gives $f(21)>f(19)$, after which the envelope decreases.  Hence the unique integer maximizer is $m=21$.\n\\end{proof}","1}{2}\\right\\rfloor.\n\\]\nIf $m$ is even and at least eight, its coefficient is strictly smaller than that of $m-1$, because the numerator is the same while\n\\[\n(m-1)\\log(m-1)<m\\log m.\n\\]\nIt therefore suffices to consider odd $m$.  Apart from the constant change of logarithm base, the odd-dimensional envelope is\n\\[\nf(x)=\\frac{x-5}{2x\\log x}.\n\\]\nIts derivative has the sign of\n\\[\n5(1+\\log x)-x.\n\\]\nFor $x>5$ this expression crosses zero exactly once, between $19$ and $21$.  Direct comparison gives $f(21)>f(19)$, after which the envelope decreases.  Hence the unique integer maximizer is $m=21$.\n\\end{proof}","its coefficient is strictly smaller than that of $m-1$, because the numerator is the same while\n\\[\n(m-1)\\log(m-1)<m\\log m.\n\\]\nIt therefore suffices to consider odd $m$.  Apart from the constant change of logarithm base, the odd-dimensional envelope is\n\\[\nf(x)=\\frac{x-5}{2x\\log x}.\n\\]\nIts derivative has the sign of\n\\[\n5(1+\\log x)-x.\n\\]\nFor $x>5$ this expression crosses zero exactly once, between $19$ and $21$.  Direct comparison gives $f(21)>f(19)$, after which the envelope decreases.  Hence the unique integer maximizer is $m=21$.\n\\end{proof}","ficient is strictly smaller than that of $m-1$, because the numerator is the same while\n\\[\n(m-1)\\log(m-1)<m\\log m.\n\\]\nIt therefore suffices to consider odd $m$.  Apart from the constant change of logarithm base, the odd-dimensional envelope is\n\\[\nf(x)=\\frac{x-5}{2x\\log x}.\n\\]\nIts derivative has the sign of\n\\[\n5(1+\\log x)-x.\n\\]\nFor $x>5$ this expression crosses zero exactly once, between $19$ and $21$.  Direct comparison gives $f(21)>f(19)$, after which the envelope decreases.  Hence the unique integer maximizer is $m=21$.\n\\end{proof}","n that of $m-1$, because the numerator is the same while\n\\[\n(m-1)\\log(m-1)<m\\log m.\n\\]\nIt therefore suffices to consider odd $m$.  Apart from the constant change of logarithm base, the odd-dimensional envelope is\n\\[\nf(x)=\\frac{x-5}{2x\\log x}.\n\\]\nIts derivative has the sign of\n\\[\n5(1+\\log x)-x.\n\\]\nFor $x>5$ this expression crosses zero exactly once, between $19$ and $21$.  Direct comparison gives $f(21)>f(19)$, after which the envelope decreases.  Hence the unique integer maximizer is $m=21$.\n\\end{proof}","therefore suffices to consider odd $m$.  Apart from the constant change of logarithm base, the odd-dimensional envelope is\n\\[\nf(x)=\\frac{x-5}{2x\\log x}.\n\\]\nIts derivative has the sign of\n\\[\n5(1+\\log x)-x.\n\\]\nFor $x>5$ this expression crosses zero exactly once, between $19$ and $21$.  Direct comparison gives $f(21)>f(19)$, after which the envelope decreases.  Hence the unique integer maximizer is $m=21$.\n\\end{proof}","The first tensor powers are\n\\begin{center}\n\\begin{tabular}{@{}rrrrr@{}}\n\\toprule\n$r$&dimension $21^r$&restricted exact&ambient upper bound&certified gap","\\midrule\n$1$&$21$&$29$&$21$&$8$","$2$&$441$&$1218$&$882$&$336$","$3$&$9261$&$38367$&$27783$&$10584$","$4$&$194481$&$1074276$&$777924$&$296352$","\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.74\\linewidth]{assets/g_tensor_amplification.pdf}\n\\caption{Tensor powers of the twenty-one-dimensional snapshot seed.  Both displayed circuit sizes are of order $N\\log N$, and their difference has a positive $N\\log N$ coefficient.}\n\\label{g:fig:tensor-amplification}\n\\end{figure}","The tensor inequality also defines a regularized ambient invariant.  Let $A\\in\\U_m(\\F)$ and put\n\\[\nb_r(A)\n=\n\\frac{\\ellAll(A^{\\otimes r})}{m^{r-1}}.\n\\]\nApplying \\cref{g:eq:tensor-ambient} to $A^{\\otimes r}\\otimes A^{\\otimes s}$ gives\n\\[\nb_{r+s}(A)\\le b_r(A)+b_s(A).\n\\]","\\begin{proposition}[Stable tensor length]\\label{g:prop:stable-tensor-length}\nThe limit\n\\[\n\\tau_*(A)\n=\n\\lim_{r\\to\\infty}\n\\frac{\\ellAll(A^{\\otimes r})}{r m^{r-1}}\n=\n\\inf_{r\\ge1}\n\\frac{\\ellAll(A^{\\otimes r})}{r m^{r-1}}\n\\]\nexists.  If $A$ is tree tight, then the stable catalytic capacity\n\\[\n\\kappa_\\otimes(A)\n=\n\\ellDown(A)-\\tau_*(A)\n\\]\nsatisfies\n\\[\n\\kappa_\\otimes(A)\\ge\\DeltaDef(A).\n\\]\n\\end{proposition}","\\begin{proof}\nThe sequence $b_r(A)$ is subadditive, so Fekete's lemma gives the displayed limit.  Since the one-copy lifted construction implies $\\tau_*(A)\\le\\ellAll(A)$, one has\n\\[\n\\kappa_\\otimes(A)\n\\ge\n\\ellDown(A)-\\ellAll(A)\n=\n\\DeltaDef(A).\n\\]\nFor a tree-tight target, \\cref{g:eq:tensor-power-lower} identifies $\\ellDown(A)$ with its exact normalized restricted tensor cost.\n\\end{proof}","For the twenty-one-dimensional seed $A_{21}$,\n\\[\n\\kappa_\\otimes(A_{21})\\ge8.\n\\]\nThis regularization introduces a new possibility: a target may have $\\DeltaDef(A)=0$ but $\\kappa_\\otimes(A)>0$.  In that case one copy is rigid while many tensor copies admit a collective shortcut.  We call this \\emph{tensor activation}.","Define\n\\[\nD_n(\\F)\n=\n\\sup_{A\\in\\U_n(\\F)}\n\\bigl(\\ellDown(A)-\\ellAll(A)\\bigr).\n\\]\nFor finite fields the supremum is a maximum.  The tensor family gives the following explicit all-dimension bound.","\\begin{proof}[Proof of \\cref{g:cor:nlogn-gap}]\nLet $m=21$ and write\n\\[\nn=\\sum_{j=0}^{r}d_jm^j,\n\\qquad\n0\\le d_j<m.\n\\]\nTake the block sum of $d_j$ copies of $A_{21}^{\\otimes j}$ for every $j$, with the convention $A_{21}^{\\otimes0}=I_1$.  Block additivity and the tensor estimate give\n\\[\nD_n(\\F)\n\\ge\n\\frac{8}{21}\n\\sum_{j=0}^{r}j d_j21^j.\n\\]\nPut $S=\\sum_j j d_j21^j$.  Since $r=\\lfloor\\log_{21}n\\rfloor$,\n\\[\nr n-S\n=\n\\sum_{j<r}(r-j)d_j21^j\n\\le\n20\\sum_{h\\ge1}h21^{r-h}\n\\le\n\\frac{21^{r+1}}{20}\n\\le\n\\frac{21}{20}n.\n\\]\nAlso $r\\ge\\log_{21}n-1$.  Therefore\n\\[\nS\\ge n\\log_{21}n-\\frac{41}{20}n.\n\\]\nMultiplying by $8/21$ yields\n\\[\nD_n(\\F)\n\\ge\n\\frac{8}{21\\log_2 21}n\\log_2n\n-\n\\frac{82}{105}n.\n\\]\n\\end{proof}","in{proof}[Proof of \\cref{g:cor:nlogn-gap}]\nLet $m=21$ and write\n\\[\nn=\\sum_{j=0}^{r}d_jm^j,\n\\qquad\n0\\le d_j<m.\n\\]\nTake the block sum of $d_j$ copies of $A_{21}^{\\otimes j}$ for every $j$, with the convention $A_{21}^{\\otimes0}=I_1$.  Block additivity and the tensor estimate give\n\\[\nD_n(\\F)\n\\ge\n\\frac{8}{21}\n\\sum_{j=0}^{r}j d_j21^j.\n\\]\nPut $S=\\sum_j j d_j21^j$.  Since $r=\\lfloor\\log_{21}n\\rfloor$,\n\\[\nr n-S\n=\n\\sum_{j<r}(r-j)d_j21^j\n\\le\n20\\sum_{h\\ge1}h21^{r-h}\n\\le\n\\frac{21^{r+1}}{20}\n\\le\n\\frac{21}{20}n.\n\\]\nAlso $r\\ge\\log_{21}n-1$.  Therefore\n\\[\nS\\ge n\\log_{21}n-\\frac{41}{20}n.\n\\]\nMultiplying by $8/21$ yields\n\\[\nD_n(\\F)\n\\ge\n\\frac{8}{21\\log_2 21}n\\log_2n\n-\n\\frac{82}{105}n.\n\\]\n\\end{proof}","Take the block sum of $d_j$ copies of $A_{21}^{\\otimes j}$ for every $j$, with the convention $A_{21}^{\\otimes0}=I_1$.  Block additivity and the tensor estimate give\n\\[\nD_n(\\F)\n\\ge\n\\frac{8}{21}\n\\sum_{j=0}^{r}j d_j21^j.\n\\]\nPut $S=\\sum_j j d_j21^j$.  Since $r=\\lfloor\\log_{21}n\\rfloor$,\n\\[\nr n-S\n=\n\\sum_{j<r}(r-j)d_j21^j\n\\le\n20\\sum_{h\\ge1}h21^{r-h}\n\\le\n\\frac{21^{r+1}}{20}\n\\le\n\\frac{21}{20}n.\n\\]\nAlso $r\\ge\\log_{21}n-1$.  Therefore\n\\[\nS\\ge n\\log_{21}n-\\frac{41}{20}n.\n\\]\nMultiplying by $8/21$ yields\n\\[\nD_n(\\F)\n\\ge\n\\frac{8}{21\\log_2 21}n\\log_2n\n-\n\\frac{82}{105}n.\n\\]\n\\end{proof}","es of $A_{21}^{\\otimes j}$ for every $j$, with the convention $A_{21}^{\\otimes0}=I_1$.  Block additivity and the tensor estimate give\n\\[\nD_n(\\F)\n\\ge\n\\frac{8}{21}\n\\sum_{j=0}^{r}j d_j21^j.\n\\]\nPut $S=\\sum_j j d_j21^j$.  Since $r=\\lfloor\\log_{21}n\\rfloor$,\n\\[\nr n-S\n=\n\\sum_{j<r}(r-j)d_j21^j\n\\le\n20\\sum_{h\\ge1}h21^{r-h}\n\\le\n\\frac{21^{r+1}}{20}\n\\le\n\\frac{21}{20}n.\n\\]\nAlso $r\\ge\\log_{21}n-1$.  Therefore\n\\[\nS\\ge n\\log_{21}n-\\frac{41}{20}n.\n\\]\nMultiplying by $8/21$ yields\n\\[\nD_n(\\F)\n\\ge\n\\frac{8}{21\\log_2 21}n\\log_2n\n-\n\\frac{82}{105}n.\n\\]\n\\end{proof}","nvention $A_{21}^{\\otimes0}=I_1$.  Block additivity and the tensor estimate give\n\\[\nD_n(\\F)\n\\ge\n\\frac{8}{21}\n\\sum_{j=0}^{r}j d_j21^j.\n\\]\nPut $S=\\sum_j j d_j21^j$.  Since $r=\\lfloor\\log_{21}n\\rfloor$,\n\\[\nr n-S\n=\n\\sum_{j<r}(r-j)d_j21^j\n\\le\n20\\sum_{h\\ge1}h21^{r-h}\n\\le\n\\frac{21^{r+1}}{20}\n\\le\n\\frac{21}{20}n.\n\\]\nAlso $r\\ge\\log_{21}n-1$.  Therefore\n\\[\nS\\ge n\\log_{21}n-\\frac{41}{20}n.\n\\]\nMultiplying by $8/21$ yields\n\\[\nD_n(\\F)\n\\ge\n\\frac{8}{21\\log_2 21}n\\log_2n\n-\n\\frac{82}{105}n.\n\\]\n\\end{proof}","}^{\\otimes0}=I_1$.  Block additivity and the tensor estimate give\n\\[\nD_n(\\F)\n\\ge\n\\frac{8}{21}\n\\sum_{j=0}^{r}j d_j21^j.\n\\]\nPut $S=\\sum_j j d_j21^j$.  Since $r=\\lfloor\\log_{21}n\\rfloor$,\n\\[\nr n-S\n=\n\\sum_{j<r}(r-j)d_j21^j\n\\le\n20\\sum_{h\\ge1}h21^{r-h}\n\\le\n\\frac{21^{r+1}}{20}\n\\le\n\\frac{21}{20}n.\n\\]\nAlso $r\\ge\\log_{21}n-1$.  Therefore\n\\[\nS\\ge n\\log_{21}n-\\frac{41}{20}n.\n\\]\nMultiplying by $8/21$ yields\n\\[\nD_n(\\F)\n\\ge\n\\frac{8}{21\\log_2 21}n\\log_2n\n-\n\\frac{82}{105}n.\n\\]\n\\end{proof}","Numerically,\n\\[\n\\frac{8}{21\\log_2 21}\n=0.0867315\\ldots.\n\\]\nThe elementary bottom-to-top construction gives\n\\[\nD_n(\\F)\\le\\frac{n(n-1)}2.\n\\]\nThus the current general range is\n\\begin{equation}\\label{g:eq:defect-range}\n0.0867315\\,n\\log_2n-O(n)\n\\le\nD_n(\\F)\n\\le\n\\frac{n(n-1)}2.\n\\end{equation}","The six-row atom proves that an optimal circuit for a lower-unitriangular target need not remain in the lower-unitriangular subgroup.  The exact binary classification shows that this first failure is rigid: it occurs for exactly twenty targets, every defect is $(7,6)$, and every target is generated by the same three-cross, one-phase mechanism.","he phenomenon.  The connected snapshot family shows that exactly two upper-directed gates can support a defect linear in dimension, and that the advantage persists under every fixed finite penalty on those gates.  The recursive lift shows that a defective leaf survives arbitrarily many identical full-rank dyadic levels.  Tensor products then turn any tree-tight positive-defect atom into an order-$N\\log N$ additive separation.  Thus the defect is not a finite anomaly and not merely linear in the worst case.  At the same time, generic rigidity and the defect--codimension theorem show that large defects require correspondingly special algebraic conditions.  The central synthesis is therefore not ubiquity but amplification: thin exceptional strata can control large structured families.","Exact CNOT synthesis usually minimizes $\\ellAll$ on all of $\\GL_n(\\F_2)$.  The present problem fixes a structured target class and compares the ambient optimum with a structure-preserving optimum.  Recent exact methods explore the full Cayley graph using symmetries, bidirectional search, A* methods, SAT, and related techniques \\cite{g:ChristensenEtAl2025,g:WebsterEtAl2025,g:ShaikVanDePol2025}.  The right-coset computation in this paper is tailored to a subgroup comparison, while the higher-dimensional exact audits use direct bidirectional balls for selected targets.","\\begin{question}[Universal six-dimensional threshold]\nIs six the first defective dimension over every field, or even over every nonzero commutative ring?  The atom gives a universal defective target in dimension six.  Exact no-defect searches below six are proved here for $\\F_2$ and $\\F_3$.\n\\end{question}","\\begin{question}[Geometry of the defect locus]\nFor fixed $n$ and $d$, determine the irreducible components and exact dimension of $\\overline{\\mathcal D_{n,\\ge d}}$.  When is the lower bound $\\codim\\ge d$ sharp?\n\\end{question}","\\begin{question}[Correct growth of the maximum defect]\nIs $D_n(\\F)$ of order $n\\log n$, or can interacting catalysts produce a larger power of $n$?  The current bounds are\n\\[\n\\Omega(n\\log n)\n\\le\nD_n(\\F)\n\\le\nO(n^2).\n\\]\n\\end{question}","\\begin{question}[Multiplicative distortion]\nIs\n\\[\n\\sup_{n,A\\in\\U_n(\\F)}\n\\frac{\\ellDown(A)}{\\ellAll(A)}\n\\]\nbounded?  The connected snapshot family gives an asymptotic lower bound of $3/2$.  Exceeding $3/2$ appears to require a mechanism beyond one clean carrier exporting independent snapshots.\n\\end{question}","\\begin{question}[Exact ambient complexity of snapshot targets]\nFor which phase sets is the ambient construction of length $u+v+2$ optimal?  It is exact for the displayed binary targets in dimensions seven, eight, and nine, but the general ambient optimum remains open.\n\\end{question}","\\begin{question}[Tensor synergy and activation]\nCan $A\\otimes B$ admit an ambient circuit shorter than the lifted bound\n\\[\nn\\ellAll(A)+m\\ellAll(B)?\n\\]\nSuch synergy would only increase the defect, but understanding it is necessary for exact tensor complexity.  Does there exist a tree-tight target with $\\DeltaDef(A)=0$ but $\\kappa_\\otimes(A)>0$?\n\\end{question}","\\begin{question}[Two-gate multiplicative ceiling]\nAmong targets realizable with at most two upper-directed gates, is the asymptotic multiplicative distortion at most $3/2$?  The snapshot family reaches this value from below, but the present paper proves no matching upper bound.\n\\end{question}","\\begin{question}[Matroidal classification]\nWhich regular matroids admit a flagged realization whose shortest ambient state tour is strictly shorter than every flag-compatible lift?  The first atom is the graphic matroid $M(K_4)$ with designated source, carrier, and output roles.\n\\end{question}","\\begin{question}[Weighted phase transition]\nFor a fixed target family, determine the exact critical penalty $\\lambda_c$ at which a flag excursion ceases to improve the optimal weighted cost.  The snapshot construction proves only the lower bound $\\lambda_c\\ge v/2$ for its displayed carrier word.\n\\end{question}","\\begin{question}[Other root systems]\nLet $G$ be a split reductive group with unipotent radical $U$.  Compare word length in positive root subgroups with ambient word length using all root subgroups.  Which root systems admit a catalytic defect, and what is the first rank?\n\\end{question}","A lower-unitriangular binary matrix is encoded by reading the strict lower entries row by row:\n\\[\n(1,0),\n(2,0),(2,1),\n\\ldots,\n(5,0),\\ldots,(5,4).\n\\]\nThe resulting fifteen-bit integer is denoted by $\\operatorname{code}(A)$.  A row is displayed as the integer whose binary expansion is its support vector.  \\Cref{g:tab:twenty} lists all defects.  Every row has $\\ellDown=7$ and $\\ellAll=6$.","\\begin{longtable}{@{}rllllll@{}}\n\\caption{The twenty defective matrices in $\\U_6(\\F_2)$.}\\label{g:tab:twenty}","\\toprule\ncode&$R_0$&$R_1$&$R_2$&$R_3$&$R_4$&$R_5$","\\midrule\n\\endfirsthead\n\\toprule\ncode&$R_0$&$R_1$&$R_2$&$R_3$&$R_4$&$R_5$","For example, the canonical target has row encoding\n\\[\n(1,3,4,14,23,39)\n\\]\nand code $7601$.","For $\\F_2$, each lower-unitriangular matrix is stored as a bit string.  Standard breadth-first search gives exact distance because every edge has unit weight.  The algorithm terminates after visiting all states.","\\begin{algorithm}[Restricted breadth-first search]\nInitialize $d[I]=0$ and a queue containing $I$.  Repeatedly remove $X$ from the queue.  For every lower transvection $s\\to t$, form $Y=\\tau_{t,s}(1)X$.  If $Y$ has not been visited, set $d[Y]=d[X]+1$ and append $Y$.  Continue until the queue is empty.\n\\end{algorithm}","For an invertible binary matrix $A$, let $c_j$ be column $j$.  Starting at $j=n-1$ and moving left, reduce $c_j$ by the already canonicalized columns $c_{j+1},\\ldots,c_{n-1}$, using their pivot coordinates.  The resulting ordered reduced columns form the key $\\kappa(A)$.","\\begin{lemma}\nFor $A,B\\in\\GL_n(\\F_2)$,\n\\[\n\\kappa(A)=\\kappa(B)\n\\quad\\Longleftrightarrow\\quad\nAU_n=BU_n.\n\\]\n\\end{lemma}","\\begin{proof}\nRight multiplication by $U\\in\\U_n$ replaces column $j$ by itself plus a combination of columns with index greater than $j$.  Right-to-left reduction is therefore unchanged on a right coset.  Conversely, equal reduced columns imply that the original column frames differ by a unique lower-unitriangular column transformation, so $A=BU$ for some $U\\in\\U_n$.\n\\end{proof}","\\begin{algorithm}[Radius-four right-coset meet in the middle]\nCompute the exact radius-four ball $B_4$ in $\\GL_6(\\F_2)$.  Bucket its elements by $\\kappa$.  For each bucket and every ordered pair $(b,a)$ in that bucket, compute $u=b^{-1}a\\in\\U_6$ and update\n\\[\nd_*(u)\\leftarrow\n\\min\\bigl(d_*(u),d(b)+d(a)\\bigr).\n\\]\n\\end{algorithm}","Correctness is \\cref{g:lem:mitm-complete}.  The computation also checks that every generated pair lies in $\\U_6$ and that every one of the $32768$ targets receives a finite distance.","For the targets $H_7,H_8,H_9$ of \\cref{g:thm:higher-exact}, the program \\texttt{small\\_snapshot\\_exact.cpp} constructs exact ambient balls from the identity and from the target.  If radius-$r$ and radius-$s$ balls are disjoint, symmetry of the generating set rules out every word of length at most $r+s$.  The tested splits are $3+3$, $3+4$, and $4+4$, respectively.  The source stores matrices in one 128-bit word and uses exact row-addition transitions; no probabilistic or floating-point step occurs.","The program \\texttt{higher\\_family\\_tests.py} generates the snapshot targets over $\\F_2$, $\\F_3$, and $\\F_5$, evaluates both displayed circuits, computes the interval-tree potentials, audits a tensor certificate for the seven-dimensional seed, verifies the recursive cross rank, and reproduces the twenty-one-dimensional rate optimization.","The program \\texttt{further\\_generalization\\_tests.py} verifies the universal circuits over $\\mathbb Z/4\\mathbb Z$, $\\mathbb Z/6\\mathbb Z$, and $\\mathbb Z/8\\mathbb Z$, audits the rank-one bandwidth-one excursion, checks the finite-field counting bounds symbolically, and verifies the stable tensor subadditivity identities.","\\subsection{From transvections to quantum gates}\nLet $d\\ge2$.  On $n$ qudits with computational basis labeled by $\\Z_d^n$, define the scaled SUM gate\n\\[\n\\SUM^{(a)}_{s\\to t}\n\\ket{x_0,\\ldots,x_{n-1}}\n=\n\\ket{x_0,\\ldots,x_t+a x_s,\\ldots,x_{n-1}},\n\\qquad a\\in\\Z_d.\n\\]\nIt is a permutation unitary.  Its action on basis labels is the row transvection\n\\[\nI+aE_{t,s}.\n\\]\nFor $d=2$ and $a=1$, it is CNOT.  For prime $d$, these gates lie in the standard qudit Clifford group.  For composite $d$, they remain exact reversible modular-addition gates.","tion{From transvections to quantum gates}\nLet $d\\ge2$.  On $n$ qudits with computational basis labeled by $\\Z_d^n$, define the scaled SUM gate\n\\[\n\\SUM^{(a)}_{s\\to t}\n\\ket{x_0,\\ldots,x_{n-1}}\n=\n\\ket{x_0,\\ldots,x_t+a x_s,\\ldots,x_{n-1}},\n\\qquad a\\in\\Z_d.\n\\]\nIt is a permutation unitary.  Its action on basis labels is the row transvection\n\\[\nI+aE_{t,s}.\n\\]\nFor $d=2$ and $a=1$, it is CNOT.  For prime $d$, these gates lie in the standard qudit Clifford group.  For composite $d$, they remain exact reversible modular-addition gates.","computational basis labeled by $\\Z_d^n$, define the scaled SUM gate\n\\[\n\\SUM^{(a)}_{s\\to t}\n\\ket{x_0,\\ldots,x_{n-1}}\n=\n\\ket{x_0,\\ldots,x_t+a x_s,\\ldots,x_{n-1}},\n\\qquad a\\in\\Z_d.\n\\]\nIt is a permutation unitary.  Its action on basis labels is the row transvection\n\\[\nI+aE_{t,s}.\n\\]\nFor $d=2$ and $a=1$, it is CNOT.  For prime $d$, these gates lie in the standard qudit Clifford group.  For composite $d$, they remain exact reversible modular-addition gates.","\\begin{provedbox}\nThe universal six-row atom in the core chapter is valid over every nonzero commutative ring with identity.  Taking $R=\\Z_d$ gives a six-qudit target with restricted directed-SUM length seven and ambient all-direction SUM length six for every $d\\ge2$.\n\\end{provedbox}","\\subsection{The universal six-qudit circuit}\nUse the coordinate names\n\\[\nx,p,q,t_0,t_1,t_2.\n\\]\nThe target acts on basis labels by\n\\begin{align*}\np&\\mapsto p+x,","\\subsection{The universal six-qudit circuit}\nUse the coordinate names\n\\[\nx,p,q,t_0,t_1,t_2.\n\\]\nThe target acts on basis labels by\n\\begin{align*}\np&\\mapsto p+x,\\\\\nt_0&\\mapsto t_0+p+q,\\\\\nt_1&\\mapsto t_1+x+p+q,\\\\\nt_2&\\mapsto t_2+x+p+q,\n\\end{align*}\nwith $x$ and $q$ unchanged.  An ambient word is\n\\[\n\\SUM_{q\\to p},\n\\quad\n\\SUM_{p\\to t_0},\n\\quad\n\\SUM_{x\\to p},\n\\quad\n\\SUM_{p\\to t_1},\n\\quad\n\\SUM_{p\\to t_2},\n\\quad\n\\SUM^{(-1)}_{q\\to p}.\n\\]\nThe first gate adds the dirty $q$ component to $p$.  The middle gates export two different carrier states.  The final inverse gate restores $p$.","t_2&\\mapsto t_2+x+p+q,\n\\end{align*}\nwith $x$ and $q$ unchanged.  An ambient word is\n\\[\n\\SUM_{q\\to p},\n\\quad\n\\SUM_{p\\to t_0},\n\\quad\n\\SUM_{x\\to p},\n\\quad\n\\SUM_{p\\to t_1},\n\\quad\n\\SUM_{p\\to t_2},\n\\quad\n\\SUM^{(-1)}_{q\\to p}.\n\\]\nThe first gate adds the dirty $q$ component to $p$.  The middle gates export two different carrier states.  The final inverse gate restores $p$.","\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=.96\\textwidth]{assets/g_qudit_sum_circuit.pdf}\n\\caption{The universal six-qudit catalytic circuit.  Coefficients are interpreted modulo $d$.}\n\\label{g:fig:qudit-sum}\n\\end{figure}","\\subsection{Connected higher-dimensional qudit families}\nThe snapshot target $A(u;s_1,\\ldots,s_v)$ from the core chapter is defined over any coefficient ring for which the displayed additions make sense.  Over $\\Z_d$, it becomes a modular SUM circuit.  Its explicit ambient word uses exactly two reverse-directed SUM gates and has length\n\\[\nu+v+2.\n\\]\nThe interval-tree proof of exact restricted length is most transparent over fields, but the displayed upper-bound and resource-amortization mechanism remain valid over every $\\Z_d$.","fined over any coefficient ring for which the displayed additions make sense.  Over $\\Z_d$, it becomes a modular SUM circuit.  Its explicit ambient word uses exactly two reverse-directed SUM gates and has length\n\\[\nu+v+2.\n\\]\nThe interval-tree proof of exact restricted length is most transparent over fields, but the displayed upper-bound and resource-amortization mechanism remain valid over every $\\Z_d$.","For a field $\\F_q$, the exact formula is\n\\[\n\\ellDown(A)=u+2v,\n\\qquad\n\\ellAll(A)\\le u+v+2,\n\\]\nso\n\\[\n\\Delta(A)\\ge v-2.\n\\]\nThe connected family has asymptotic ratio approaching $3/2$.","\\subsection{Weighted physical costs}\nSuppose forward SUM gates cost one and reverse-directed SUM gates cost $\\lambda$.  The displayed word has weighted cost\n\\[\nu+v+2\\lambda,\n\\]\nwhile the restricted construction costs $u+2v$.  Hence the gain is at least\n\\[\nv-2\\lambda.\n\\]\nThis formula can be combined with hardware calibration.  If reverse orientation requires echo pulses or routing, the catalyst remains useful when the number of exported snapshots exceeds twice the effective penalty.","\\subsection{Pauli action for prime qudits}\nFor odd prime $d$, let $X$ and $Z$ denote generalized Pauli operators.  The SUM gate acts by\n\\begin{align*}\nX_s&\\mapsto X_sX_t^a,\n&\nX_t&\\mapsto X_t,","\\subsection{Pauli action for prime qudits}\nFor odd prime $d$, let $X$ and $Z$ denote generalized Pauli operators.  The SUM gate acts by\n\\begin{align*}\nX_s&\\mapsto X_sX_t^a,\n&\nX_t&\\mapsto X_t,\\\\\nZ_s&\\mapsto Z_s,\n&\nZ_t&\\mapsto Z_s^{-a}Z_t.\n\\end{align*}\nThus the modular linear action embeds into a symplectic transformation.  The catalytic row excursion has a dual Pauli interpretation: a temporary reverse dependency changes both $X$ propagation and the contragredient $Z$ propagation, then restores them.","Z_s&\\mapsto Z_s,\n&\nZ_t&\\mapsto Z_s^{-a}Z_t.\n\\end{align*}\nThus the modular linear action embeds into a symplectic transformation.  The catalytic row excursion has a dual Pauli interpretation: a temporary reverse dependency changes both $X$ propagation and the contragredient $Z$ propagation, then restores them.","\\subsection{Continuous-variable shear analogue}\nOn continuous variables, a linear coordinate map $A$ embeds symplectically as\n\\[\nA\\longmapsto\n\\begin{pmatrix}\nA&0","\\subsection{Continuous-variable shear analogue}\nOn continuous variables, a linear coordinate map $A$ embeds symplectically as\n\\[\nA\\longmapsto\n\\begin{pmatrix}\nA&0\\\\\n0&A^{-T}\n\\end{pmatrix}.\n\\]\nA row shear therefore gives a Gaussian SUM transformation.  The six-row word identity remains a matrix identity in the corresponding shear subgroup.","\\subsection{Compiler primitive}\nA compiler can search for a carrier row $p$, a temporary source $q$, and output rows requiring several successive states of $p$.  The symbolic pattern is:\n\\begin{enumerate}\n\\item add $q$ into $p$ in the disallowed direction,\n\\item update $p$ through a sequence of allowed additions,\n\\item copy selected intermediate states of $p$ into outputs,\n\\item subtract $q$ from $p$,\n\\item verify that the outputs contain the intended modular combinations.\n\\end{enumerate}\nThe Python function \\texttt{snapshot\\_target} constructs the target and \\texttt{snapshot\\_ambient\\_word} emits the word.","subsection{Compiler primitive}\nA compiler can search for a carrier row $p$, a temporary source $q$, and output rows requiring several successive states of $p$.  The symbolic pattern is:\n\\begin{enumerate}\n\\item add $q$ into $p$ in the disallowed direction,\n\\item update $p$ through a sequence of allowed additions,\n\\item copy selected intermediate states of $p$ into outputs,\n\\item subtract $q$ from $p$,\n\\item verify that the outputs contain the intended modular combinations.\n\\end{enumerate}\nThe Python function \\texttt{snapshot\\_target} constructs the target and \\texttt{snapshot\\_ambient\\_word} emits the word.","\\subsection{Open qudit questions}\n\\begin{question}\nDoes the first defective dimension remain six for every finite field and every modular ring $\\Z_d$ under the directed-SUM metric?\n\\end{question}\n\\begin{question}\nCan qudit Clifford gates outside the SUM subgroup produce a larger defect for a SUM target under total, entangling, or depth cost?\n\\end{question}\n\\begin{question}\nDo composite dimensions admit characteristic-dependent catalytic motifs unavailable over fields?\n\\end{question}","\\subsection{Binary symplectic representation}\nModulo global phase and Pauli translations, an $n$-qubit Clifford operation acts on Pauli exponent vectors\n\\[\nv=(x_0,\\ldots,x_{n-1}\\mid z_0,\\ldots,z_{n-1})^T\\in\\F_2^{2n}\n\\]\nthrough a symplectic matrix\n\\[\nM\\in\\Sp(2n,2).\n\\]\nThe elementary gates used here act as follows:\n\\begin{align*}\nH_i &: (x_i,z_i)\\mapsto(z_i,x_i),","\\subsection{Binary symplectic representation}\nModulo global phase and Pauli translations, an $n$-qubit Clifford operation acts on Pauli exponent vectors\n\\[\nv=(x_0,\\ldots,x_{n-1}\\mid z_0,\\ldots,z_{n-1})^T\\in\\F_2^{2n}\n\\]\nthrough a symplectic matrix\n\\[\nM\\in\\Sp(2n,2).\n\\]\nThe elementary gates used here act as follows:\n\\begin{align*}\nH_i &: (x_i,z_i)\\mapsto(z_i,x_i),\\\\\nS_i &: z_i\\mapsto z_i+x_i,\\\\\n\\CX_{i\\to j} &: x_j\\mapsto x_j+x_i,\n\\quad z_i\\mapsto z_i+z_j,\\\\\n\\CZ_{ij} &: z_i\\mapsto z_i+x_j,\n\\quad z_j\\mapsto z_j+x_i.\n\\end{align*}","Let $P_n\\le\\Sp(2n,2)$ be the parabolic subgroup stabilizing the $Z$ Lagrangian\n\\[\nL_Z=\\{(0\\mid z):z\\in\\F_2^n\\}.\n\\]\nThe gates $S$, $\\CZ$, and $\\CX$ preserve $L_Z$.  Hadamard gates do not.","\\begin{definition}[Parabolic and ambient Clifford metrics]\nThe restricted alphabet is\n\\[\n\\mathcal S_P\n=\n\\{S_i,\\CZ_{ij},\\CX_{i\\to j}\\},\n\\]\nand the ambient alphabet is\n\\[\n\\mathcal S_C\n=\n\\mathcal S_P\\cup\\{H_i\\}.\n\\]\nTargets lie in $P_n$.  We study both total primitive count and entangling count, where $H_i$ and $S_i$ have cost zero while $\\CX$ and $\\CZ$ have cost one.\n\\end{definition}","\\subsection{Exact group sizes}\nThe finite state spaces are\n\\[\n|P_n|\n=\n2^{n(n+1)/2}|\\GL(n,2)|,\n\\]\nand\n\\[\n|\\Sp(2n,2)|\n=\n2^{n^2}\\prod_{j=1}^n(2^{2j}-1).\n\\]\nFor $n=1,2,3$ this gives:\n\\begin{center}\n\\begin{tabular}{@{}rcc@{}}\n\\toprule\n$n$&$|P_n|$&$|\\Sp(2n,2)|$","\\subsection{Total primitive count}\n\\begin{computedbox}\nWith every gate in $\\mathcal S_P$ and $\\mathcal S_C$ assigned unit cost and with SWAP excluded as a primitive, exhaustive search gives:\n\\begin{center}\n\\begin{tabular}{@{}rcccc@{}}\n\\toprule\n$n$&restricted diameter&ambient diameter&defective targets&maximum defect","\\bottomrule\n\\end{tabular}\n\\end{center}\nAt $n=3$, six targets have distance pair $(7,6)$ and eighteen have pair $(8,7)$.\n\\end{computedbox}","\\subsubsection{A three-qubit witness}\nOne defective target has symplectic matrix\n\\[\nM_3=\n\\begin{pmatrix}\n0&1&0&0&0&0\\\\\n0&0&1&0&0&0\\\\\n1&0&0&0&0&0\\\\\n1&0&1&0&1&0\\\\\n0&1&0&0&0&1\\\\\n0&1&0&1&0&0\n\\end{pmatrix}.\n\\]\nA restricted word of length seven is\n\\[\n\\CX_{0\\to2},\n\\CZ_{12},\n\\CX_{2\\to0},\n\\CX_{0\\to2},\n\\CX_{0\\to1},\n\\CX_{1\\to0},\n\\CX_{0\\to1}.\n\\]\nAn ambient word of length six is\n\\[\nH_1,\n\\CX_{1\\to2},\n\\CX_{2\\to1},\n\\CX_{2\\to0},\n\\CX_{0\\to2},\nH_0.\n\\]","0&1&0&1&0&0\n\\end{pmatrix}.\n\\]\nA restricted word of length seven is\n\\[\n\\CX_{0\\to2},\n\\CZ_{12},\n\\CX_{2\\to0},\n\\CX_{0\\to2},\n\\CX_{0\\to1},\n\\CX_{1\\to0},\n\\CX_{0\\to1}.\n\\]\nAn ambient word of length six is\n\\[\nH_1,\n\\CX_{1\\to2},\n\\CX_{2\\to1},\n\\CX_{2\\to0},\n\\CX_{0\\to2},\nH_0.\n\\]","The Hadamards do not form a matched pair on the same wire.  They open and close the Lagrangian excursion in different coordinates.  This is a useful warning against assuming that every catalyst has the literal form $g\\cdots g^{-1}$.","\\subsection{SWAP sensitivity}\nAdd $\\SWAP_{ij}$ to both alphabets as a unit-cost primitive.  The target groups remain unchanged, but the metric changes.","\\begin{computedbox}\nFor $n=3$ with unit-cost SWAP gates, the restricted diameter drops from eight to seven, the ambient diameter drops from twelve to eleven, and every one of the $10,752$ parabolic targets has zero total-count defect.\n\\end{computedbox}","\\subsection{Entangling-count defects}\nNow assign cost zero to $H_i$ and $S_i$ and cost one to $\\CX$ and $\\CZ$.","\\begin{computedbox}\nThe exact censuses are:\n\\begin{center}\n\\begin{tabular}{@{}rccc@{}}\n\\toprule\n$n$&defective targets&distance pairs&maximum defect","2&4&$(3,2)$ with multiplicity $4$&1","3&1,624&\n$\\begin{array}{c}\n(3,2):24,\\ (4,3):432,\\ (5,4):1056,\\\\\n(6,4):48,\\ (6,5):64\n\\end{array}$\n&2","\\subsubsection{A two-qubit witness}\nA canonical target is\n\\[\nM_2=\n\\begin{pmatrix}\n0&1&0&0\\\\\n1&0&0&0\\\\\n1&1&0&1\\\\\n1&1&1&0\n\\end{pmatrix}.\n\\]\nA restricted realization is\n\\[\n\\CX_{0\\to1},\nS_1,\n\\CX_{1\\to0},\n\\CX_{0\\to1},\n\\]\nwith three entangling gates.  An ambient realization is\n\\[\nH_1,\n\\CX_{1\\to0},\n\\CX_{0\\to1},\nH_0,\nS_1,\nS_0,\n\\]\nwith two entangling gates.","1&1&1&0\n\\end{pmatrix}.\n\\]\nA restricted realization is\n\\[\n\\CX_{0\\to1},\nS_1,\n\\CX_{1\\to0},\n\\CX_{0\\to1},\n\\]\nwith three entangling gates.  An ambient realization is\n\\[\nH_1,\n\\CX_{1\\to0},\n\\CX_{0\\to1},\nH_0,\nS_1,\nS_0,\n\\]\nwith two entangling gates.","The four exceptional matrices admit one common lower-bound proof.  Work in the declared symplectic quotient and entangling metric, with $H_i$ and $S_i$ free and SWAP excluded as a primitive.  Put\n\\[\nJ_2=\\begin{pmatrix}0&1","The four exceptional matrices admit one common lower-bound proof.  Work in the declared symplectic quotient and entangling metric, with $H_i$ and $S_i$ free and SWAP excluded as a primitive.  Put\n\\[\nJ_2=\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix},\n\\qquad D=\\operatorname{diag}(d_0,d_1),\n\\qquad d_0,d_1\\in\\F_2,\n\\]\nand define\n\\[\nM_D=\\begin{pmatrix}J_2&0","1&0\\end{pmatrix},\n\\qquad D=\\operatorname{diag}(d_0,d_1),\n\\qquad d_0,d_1\\in\\F_2,\n\\]\nand define\n\\[\nM_D=\\begin{pmatrix}J_2&0\\\\I_2+DJ_2&J_2\\end{pmatrix}.\n\\]","\\begin{theorem}[Two-qubit orbit and exact costs]\\label{g:thm:clifford-two-analytic}\nEach of the four distinct matrices $M_D$ lies in $P_2$ and has restricted entangling cost three and ambient entangling cost two.  They form one orbit under left multiplication by the free restricted subgroup generated by $S_0,S_1$.\n\\end{theorem}\n\\begin{proof}\nThe ambient sequence\n\\[\nH_1,\\ \\CX_{1\\to0},\\ \\CX_{0\\to1},\\ H_0\n\\]\nhas matrix $M_0=\\left(\\begin{smallmatrix}J_2&0","ubit orbit and exact costs]\\label{g:thm:clifford-two-analytic}\nEach of the four distinct matrices $M_D$ lies in $P_2$ and has restricted entangling cost three and ambient entangling cost two.  They form one orbit under left multiplication by the free restricted subgroup generated by $S_0,S_1$.\n\\end{theorem}\n\\begin{proof}\nThe ambient sequence\n\\[\nH_1,\\ \\CX_{1\\to0},\\ \\CX_{0\\to1},\\ H_0\n\\]\nhas matrix $M_0=\\left(\\begin{smallmatrix}J_2&0\\\\I_2&J_2\\end{smallmatrix}\\right)$.  Applying $S_0^{d_0}S_1^{d_1}$ afterwards left multiplies by $\\left(\\begin{smallmatrix}I_2&0","I_2&J_2\\end{smallmatrix}\\right)$.  Applying $S_0^{d_0}S_1^{d_1}$ afterwards left multiplies by $\\left(\\begin{smallmatrix}I_2&0","I_2&J_2\\end{smallmatrix}\\right)$.  Applying $S_0^{d_0}S_1^{d_1}$ afterwards left multiplies by $\\left(\\begin{smallmatrix}I_2&0\\\\D&I_2\\end{smallmatrix}\\right)$ and gives $M_D$.  These matrices are distinct because $J_2$ is invertible.  This also proves their orbit statement and the ambient upper bound two.  The displayed restricted sequence for $M_2$ above gives $M_{I_2}$ using three CNOT gates and one free $S_1$.  Free final phase gates then give every $M_D$, proving the restricted upper bound three.","D&I_2\\end{smallmatrix}\\right)$ and gives $M_D$.  These matrices are distinct because $J_2$ is invertible.  This also proves their orbit statement and the ambient upper bound two.  The displayed restricted sequence for $M_2$ above gives $M_{I_2}$ using three CNOT gates and one free $S_1$.  Free final phase gates then give every $M_D$, proving the restricted upper bound three.","For the restricted lower bound, the upper-left block of a parabolic product is the product of its upper-left blocks.  Each $S$ or CZ contributes $I_2$, while the two directed CNOTs contribute the elementary matrices\n\\[\nL=\\begin{pmatrix}1&0","For the restricted lower bound, the upper-left block of a parabolic product is the product of its upper-left blocks.  Each $S$ or CZ contributes $I_2$, while the two directed CNOTs contribute the elementary matrices\n\\[\nL=\\begin{pmatrix}1&0\\\\1&1\\end{pmatrix},\n\\qquad U=\\begin{pmatrix}1&1\\\\0&1\\end{pmatrix}.\n\\]\nProducts using at most two CNOTs belong to $\\{I_2,L,U,LU,UL\\}$, which does not contain $J_2$.  Thus any restricted circuit for $M_D$ requires at least three charged gates.","0&1\\end{pmatrix}.\n\\]\nProducts using at most two CNOTs belong to $\\{I_2,L,U,LU,UL\\}$, which does not contain $J_2$.  Thus any restricted circuit for $M_D$ requires at least three charged gates.","For the ambient lower bound, reorder the coordinates as $(x_0,z_0\\mid x_1,z_1)$.  A local Clifford acts by an invertible block-diagonal matrix in this order.  In $M_D$, each of the two diagonal $2\\times2$ blocks is $\\left(\\begin{smallmatrix}0&0","For the ambient lower bound, reorder the coordinates as $(x_0,z_0\\mid x_1,z_1)$.  A local Clifford acts by an invertible block-diagonal matrix in this order.  In $M_D$, each of the two diagonal $2\\times2$ blocks is $\\left(\\begin{smallmatrix}0&0\\\\1&0\\end{smallmatrix}\\right)$ and has rank one.  In either directed CNOT or CZ, both diagonal blocks are $I_2$ and have rank two.  Left and right multiplication by local Cliffords preserve these diagonal-block ranks.  A circuit with one charged gate has the form $K_1 E K_2$ with $K_1,K_2$ local and $E$ a CNOT or CZ, so it cannot equal $M_D$.  A circuit with no charged gate is local and also has rank-two diagonal blocks.  The ambient cost is therefore at least two.\n\\end{proof}","1&0\\end{smallmatrix}\\right)$ and has rank one.  In either directed CNOT or CZ, both diagonal blocks are $I_2$ and have rank two.  Left and right multiplication by local Cliffords preserve these diagonal-block ranks.  A circuit with one charged gate has the form $K_1 E K_2$ with $K_1,K_2$ local and $E$ a CNOT or CZ, so it cannot equal $M_D$.  A circuit with no charged gate is local and also has rank-two diagonal blocks.  The ambient cost is therefore at least two.\n\\end{proof}","\\subsection{Search completeness}\nThe search state is the exact $2n\\times2n$ binary symplectic matrix.  Every generator acts by a fixed row update.  Ordinary breadth-first search gives exact unit-cost distances.  Zero-one breadth-first search gives exact entangling costs.  Since every state of $P_n$ and $\\Sp(2n,2)$ is reached, comparing the two distance maps is a complete census.","The optimized C++ verifier represents a $6\\times6$ matrix by a $36$-bit integer.  It enumerates the full three-qubit group in a few seconds in the present environment.  The Python implementation is slower but independently expresses the same state model.","\\subsection{Interpretation as symplectic holonomy}\nThe parabolic subgroup is a fiber over the Lagrangian Grassmannian.  A Hadamard moves one coordinate out of the chosen Lagrangian.  The ambient path traverses other Lagrangian frames and returns to a target stabilizing $L_Z$.","This is the symplectic analogue of the unitriangular Schreier loop.  The catalyst is a temporary exchange of $X$ and $Z$ roles.  The consumers are entangling gates that become cheaper in the changed frame.","\\subsection{CSS-preserving extensions}\nThe parabolic model fixes one Lagrangian, but CSS circuit classes can be defined more narrowly or more broadly.  Possible restricted alphabets include:\n\\begin{enumerate}\n\\item CNOT and Pauli gates only,\n\\item CNOT, $S$, and diagonal controlled phases,\n\\item block-diagonal symplectic transformations preserving separate $X$ and $Z$ codes,\n\\item logical Clifford gates preserving a CSS stabilizer decomposition,\n\\item architecture-constrained versions of the preceding models.\n\\end{enumerate}\nEach produces a different subgroup and metric.","\\subsection{Depth and parallelism}\nGate-count defects do not automatically imply depth defects.  Given a gate sequence, define its dependency graph and schedule commuting or disjoint gates in parallel.  The appropriate quantities are\n\\[\n\\Delta_{\\mathrm{depth}}(U),\n\\qquad\n\\Delta_{\\mathrm{2q-depth}}(U),\n\\qquad\n\\Delta_{\\mathrm{space-time}}(U).\n\\]\nA basis change applied transversally to many qubits may have small depth even when its total count is large.  Conversely, a two-gate count saving can increase depth under locality constraints.","\\subsection{Open Clifford problems}\n\\begin{question}\nWhat are the defect orbits of the twenty-four three-qubit total-count targets under qubit permutations and parabolic conjugation?\n\\end{question}\n\\begin{question}\nDoes the maximum parabolic-versus-Clifford defect grow linearly, as $n\\log n$, or faster under total primitive count?\n\\end{question}\n\\begin{question}\nCan one prove a symbolic family with entangling-count defect greater than two, rather than relying on finite census data?\n\\end{question}\n\\begin{question}\nWhich finite symplectic parabolics are geodesically convex after SWAP is included as a primitive?\n\\end{question}\n\\begin{conjecturebox}\nThe observed three-qubit defects should admit a small collection of symplectic root motifs analogous to the three-cross, one-phase classification in the unitriangular case.  The present monograph records the exact census and witnesses but does not claim that classification.\n\\end{conjecturebox}","\\subsection{The question}\nLet\n\\[\n\\operatorname{AGL}(n,2)\n=\n\\{x\\mapsto Ax+b:A\\in\\GL(n,2),\\ b\\in\\F_2^n\\}\n\\]\nbe the affine reversible group.  It is generated by NOT and CNOT gates.  The ambient gate set adds Toffoli.  On three bits it generates the full reversible group.  On $n\\ge4$ bits with no ancillas, each NOT, CNOT, or Toffoli gate is an even permutation, so this alphabet does not generate the full symmetric group.","\\subsection{Algebraic normal form}\nEvery Boolean coordinate function has a unique algebraic normal form\n\\[\nf_i(x)\n=\n\\bigoplus_{S\\subseteq[n]}a_{i,S}\\prod_{j\\in S}x_j.\n\\]\nAffine maps have degree at most one.  A Toffoli gate targeting $t$ with controls $a,b$ performs\n\\[\nx_t\\longmapsto x_t+x_ax_b,\n\\]\nwhich can raise the degree of later coordinates.  The final affine target requires every higher-degree coefficient to cancel.","A candidate lower-bound invariant must do more than track maximum degree.  One Toffoli creates degree two and a second can remove it.  Useful refinements include:\n\\begin{itemize}\n\\item the dimension of the quadratic monomial span,\n\\item directional derivatives $D_u f(x)=f(x+u)+f(x)$,\n\\item the rank of the bilinear polarization,\n\\item the number of outputs consuming a nonlinear workspace term,\n\\item restoration obligations for every nonlinear carrier.\n\\end{itemize}","\\subsection{Exact three-bit census}\nOn three bits,\n\\[\n|\\operatorname{AGL}(3,2)|\n=2^3|\\GL(3,2)|\n=1344,\n\\]\nwhile the group generated by NOT, CNOT, and Toffoli is the full permutation group\n\\[\nS_8,\n\\qquad |S_8|=40320.\n\\]\nBoth spaces can be enumerated exactly.","omputedbox}\nFor three bits, every affine target has the same minimum cost in the affine gate set and in the Toffoli-assisted gate set under each of the following metrics:\n\\begin{enumerate}\n\\item total primitive gate count, where NOT, CNOT, and Toffoli cost one,\n\\item entangling count, where NOT is free and CNOT and Toffoli cost one.\n\\end{enumerate}\nThere are no defective affine targets among all $1344$ targets.\n\\end{computedbox}","The restricted total-count diameter is seven and the ambient diameter on $S_8$ is eight.  Under entangling count, both relevant diameters are six.  Equality on affine targets is stronger than a diameter comparison because every target is checked individually.","\\subsection{A formal nonlinear catalyst}\nA clean one-workspace template is\n\\[\nT\\;h_m\\cdots h_1\\;T^{-1},\n\\]\nwhere $T$ is Toffoli and the middle gates use the quadratic term created in the workspace.  Since Toffoli is self-inverse, the displayed circuit uses two nonlinear gates.","\\subsection{Conjugation formulas}\nLet $T_{ab\\to t}$ denote Toffoli.  Conjugating an affine gate by $T$ can produce nonlinear permutations.  For example, if $X_t$ flips the target,\n\\[\nT_{ab\\to t}X_tT_{ab\\to t}=X_t,\n\\]\nso no gain occurs.  If a CNOT changes a control or exports the target, the conjugated map can contain several quadratic terms.  The exact formula is easiest in algebraic normal form: substitute the Toffoli-updated coordinate into the affine map, then apply the inverse substitution.","A systematic search should classify pairs $(T,h)$ for which\n\\[\nThT^{-1}\\in\\operatorname{AGL}(n,2).\n\\]\nFor one Toffoli and an affine middle word, this is a normalizer-intersection problem.  A useful next bounded test is to examine simple two-Toffoli sandwiches with an affine middle word of restricted length at most five.  The present archive contains no reproducible implementation or receipt for this four-bit test, so no negative result from it is claimed here.","\\subsection{Potential lower-bound invariants}\n\\subsubsection{Quadratic support debt}\nLet $Q(f)$ be the span of quadratic monomials appearing across the coordinate functions.  A Toffoli changes $Q(f)$ by at most a controlled rank-one contribution before subsequent substitutions.  If several outputs consume the same quadratic term, a dynamic debt invariant may charge its creation, distribution, and cancellation.","\\subsubsection{Derivative rank}\nFor an affine map, every derivative $D_u f$ is constant.  During a nonlinear excursion, the derivative map\n\\[\nu\\mapsto D_u f(x)\n\\]\ncan depend on $x$.  One can track the dimension of the nonconstant derivative space and require it to return to zero.","\\subsubsection{Higher-order Möbius coefficients}\nThe ANF coefficients are the Möbius transform of the truth table.  Projectors onto degree layers give an exact packet decomposition\n\\[\nf=f^{(0)}+f^{(1)}+f^{(2)}+\\cdots.\n\\]\nA restricted affine circuit remains in degrees zero and one.  The ambient circuit can populate higher packets temporarily.  A successful lower bound must charge the useful flow through those packets, not merely their instantaneous dimension.","\\subsection{Four-bit search architecture}\nThe full permutation group on four-bit states has size $16!$.  The declared no-ancilla gate alphabet lies in its alternating subgroup: NOT, CNOT, and Toffoli are respectively products of eight, four, and two disjoint transpositions.  Even this reduced ambient search is large enough to require a more structured approach.  Several exact approaches remain available.\n\\begin{enumerate}\n\\item Meet in the middle for a fixed candidate target and length.\n\\item SAT or SMT encoding of a length-$L$ circuit with affine endpoint constraints.\n\\item Bidirectional search restricted to states within a bounded nonlinear-degree or Toffoli-count budget.\n\\item Enumeration of two- or four-Toffoli normal forms modulo affine conjugation.\n\\item Mixed-integer or algebraic sol","te alphabet lies in its alternating subgroup: NOT, CNOT, and Toffoli are respectively products of eight, four, and two disjoint transpositions.  Even this reduced ambient search is large enough to require a more structured approach.  Several exact approaches remain available.\n\\begin{enumerate}\n\\item Meet in the middle for a fixed candidate target and length.\n\\item SAT or SMT encoding of a length-$L$ circuit with affine endpoint constraints.\n\\item Bidirectional search restricted to states within a bounded nonlinear-degree or Toffoli-count budget.\n\\item Enumeration of two- or four-Toffoli normal forms modulo affine conjugation.\n\\item Mixed-integer or algebraic solving of ANF coefficient equations.\n\\end{enumerate}","\\begin{protocol}[First four-bit threshold search]\n\\begin{enumerate}\n\\item Compute exact affine distances for all $322{,}560$ affine targets.\n\\item Enumerate ambient words with two Toffoli gates modulo affine gates on the left and right.\n\\item Retain words whose endpoint is affine.\n\\item Compare with the exact affine distance.\n\\item If no defect appears, increase the Toffoli budget and produce an unsatisfiability certificate for every shorter length.\n\\end{enumerate}\n\\end{protocol}\nThis protocol is a research plan.  The packaged executable census stops at three bits, and neither the four-bit normal-form search nor its lower-bound certificates is supplied.","\\subsection{Resource variants}\nTemporary nonlinearity may be useful under one metric and useless under another.  Natural costs include:\n\\[\n\\begin{gathered}\n\\text{Toffoli count},\\quad \\text{CNOT count},\\quad\nT\\text{-count after decomposition},\\quad T\\text{-depth},\\\\\n\\text{ancilla count},\\quad \\text{total depth}.\n\\end{gathered}\n\\]\nA Toffoli-assisted affine shortcut in total Boolean gates might disappear after fault-tolerant decomposition.  Conversely, a circuit with more Boolean primitives can reduce $T$-depth.","\\text{ancilla count},\\quad \\text{total depth}.\n\\end{gathered}\n\\]\nA Toffoli-assisted affine shortcut in total Boolean gates might disappear after fault-tolerant decomposition.  Conversely, a circuit with more Boolean primitives can reduce $T$-depth.","\\subsection{Related nonlinear gate families}\nThe same program applies to:\n\\begin{itemize}\n\\item CCZ-assisted diagonal Boolean phase maps,\n\\item relative-phase Toffoli gates,\n\\item multi-controlled $X$ and $Z$ gates,\n\\item Reed--Muller degree hierarchies,\n\\item reversible arithmetic over $\\Z_d$,\n\\item polynomial automorphisms over finite rings.\n\\end{itemize}","\\subsection{Reproducible code}\nThe implementation represents every reversible map as an exact permutation of $2^n$ basis states.  Breadth-first search enumerates both the affine group and the full three-bit reversible group.","\\section{Phase-polynomial and Clifford+\\texorpdfstring{$T$}{T} catalysis}\\label{g:ch:phase-cliffordt}\n\\index{Clifford+T circuit}\n\\index{phase polynomial}\n\\index{T-count}\n\\index{T-depth}\n\\index{Pareto defect}","\\subsection{Why the phase setting is different}\nFault-tolerant quantum compilation rarely assigns equal cost to every gate.  Clifford operations are often treated as inexpensive relative to non-Clifford gates, while connectivity and parity computation can dominate the remaining cost.  Consequently there is no single Clifford+$T$ defect.  One must specify at least\n\\[\n\\mathbf c(U)=\n\\bigl(\nT\\text{-count},\nT\\text{-depth},\n\\text{two-qubit count},\n\\text{two-qubit depth},\n\\text{ancillas}\n\\bigr).\n\\]\nA catalytic excursion may improve one component and worsen another.  The natural comparison is therefore Pareto-valued rather than scalar unless a hardware-specific weight vector is fixed.","Let $\\omega=e^{i\\pi/4}$.  A phase-polynomial circuit on $n$ qubits has the form \\cite{g:AmyMosca2019,g:Selinger2013}\n\\begin{equation}\\label{g:eq:phase-poly-normal}\nU\\ket{x}=\\omega^{P(x)}\\ket{Ax+b},\n\\end{equation}\nwhere $A\\in\\GL(n,2)$, $b\\in\\F_2^n$, and $P:\\F_2^n\\to\\Z_8$ is represented as a sum of binary linear forms with coefficients in $\\Z_8$.  The diagonal case has $A=I$ and $b=0$.","The restricted model may be chosen in several ways:\n\\begin{enumerate}\n\\item diagonal phase gadgets only;\n\\item CNOT-dihedral circuits generated by $X$, CNOT, and $T$;\n\\item parity networks in which every intermediate transformation remains affine on computational basis labels;\n\\item a prescribed family of commuting Pauli rotations.\n\\end{enumerate}\nThe ambient model may add Hadamards, arbitrary Clifford conjugations, measurements, or ancillas.  These choices define different problems.","\\subsection{Pauli-rotation language}\nFor a Pauli operator $P$ and angle $\\theta$, write\n\\[\nR_P(\\theta)=e^{-i\\theta P/2}.\n\\]\nEvery Clifford $C$ satisfies\n\\begin{equation}\\label{g:eq:pauli-conjugation}\nC R_P(\\theta) C^\\dagger=R_{CPC^\\dagger}(\\theta).\n\\end{equation}\nThus a short ambient basis change can rotate a large family of phase gadgets into a different commuting or nearly commuting family, apply them there, and rotate back.","The simplest symbolic sandwich is\n\\[\nC\\left(\\prod_{j=1}^m R_{P_j}(\\theta_j)\\right)C^\\dagger\n=\n\\prod_{j=1}^m R_{CP_jC^\\dagger}(\\theta_j).\n\\]\nIf the $P_j$ commute, their conjugates commute.  A catalyst can therefore change the geometry of the support without changing the algebra of the angles.","\\subsection{Phase polynomials and parity carriers}\nA CNOT network computes linear parities\n\\[\n\\lambda(x)=a\\cdot x\\in\\F_2.\n\\]\nApplying $T^k$ to a wire carrying $\\lambda(x)$ contributes\n\\[\n\\omega^{k\\lambda(x)}\n\\]\nto the phase polynomial.  A parity network must create each required parity, expose it to a phase gate, and restore the linear map required at the endpoint.","This makes phase-polynomial synthesis a close relative of the snapshot families in the linear core.  A parity row may evolve through\n\\[\n\\lambda_0,\\lambda_1,\\ldots,\\lambda_m,\n\\]\nwith selected states consumed by phase gates.  The difference is that consumption need not copy the parity to another row; it can deposit phase while leaving the carrier unchanged.","\\begin{skeletonbox}\nA candidate phase catalyst consists of:\n\\begin{enumerate}\n\\item a Clifford $C$ that opens a non-diagonal frame;\n\\item an evolving Pauli or parity carrier in that frame;\n\\item several phase rotations consuming intermediate carrier states;\n\\item $C^\\dagger$ restoring the original diagonal or CNOT-dihedral model;\n\\item a certificate that no restricted parity network attains the same resource vector.\n\\end{enumerate}\n\\end{skeletonbox}","\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=.94\\textwidth]{assets/g_phase_polynomial_skeleton.pdf}\n\\caption{Symbolic Clifford+$T$ catalyst: a basis excursion transports a family of phase obligations and is removed at the endpoint.}\n\\label{g:fig:phase-polynomial-skeleton}\n\\end{figure}","\\subsection{\\texorpdfstring{$T$}{T}-count as a coding problem}\nFor diagonal Clifford+$T$ synthesis without ancillas, equivalent phase-polynomial descriptions can differ by identities whose coefficients vanish modulo eight on Boolean inputs.  Minimizing the number of odd coefficients is naturally expressed as a coset-decoding problem in a punctured Reed--Muller-type code.","Let $a\\in\\Z_8^{2^n-1}$ index nonzero linear forms and let $\\operatorname{odd}(a)$ denote the binary vector recording which coefficients are odd.  An abstract restricted $T$-count is\n\\[\n\\ell_T(P)=\n\\min_{a:\\,P_a=P}\n\\operatorname{wt}(\\operatorname{odd}(a)).\n\\]\nAn ambient Clifford excursion changes not only the coefficient vector but also the Pauli frame in which the rotations are represented.  A lower bound that survives ambient Clifford conjugation must therefore be invariant under the relevant symplectic action or must explicitly charge the frame change.","\\subsection{\\texorpdfstring{$T$}{T}-depth and commuting layers}\nTwo phase gadgets may share a $T$ layer when a suitable Clifford network maps their parities to distinct wires or when ancilla resources permit parallel extraction.  Define the conflict graph $\\Gamma_P$ whose vertices are odd phase terms and whose edges encode inability to realize two terms in one layer under the chosen ancilla and connectivity rules.  Then\n\\[\nT\\text{-depth}(P)\\ge\\chi(\\Gamma_P)\n\\]\nfor any conflict graph whose edge relation is sound.  Catalysis can alter $\\Gamma_P$ by changing the Pauli frame.  The right question is not merely whether $C$ preserves the number of terms, but whether it reduces the chromatic or scheduling obstruction enough to r","\\subsection{\\texorpdfstring{$T$}{T}-depth and commuting layers}\nTwo phase gadgets may share a $T$ layer when a suitable Clifford network maps their parities to distinct wires or when ancilla resources permit parallel extraction.  Define the conflict graph $\\Gamma_P$ whose vertices are odd phase terms and whose edges encode inability to realize two terms in one layer under the chosen ancilla and connectivity rules.  Then\n\\[\nT\\text{-depth}(P)\\ge\\chi(\\Gamma_P)\n\\]\nfor any conflict graph whose edge relation is sound.  Catalysis can alter $\\Gamma_P$ by changing the Pauli frame.  The right question is not merely whether $C$ preserves the number of terms, but whether it reduces the chromatic or scheduling obstruction enough to repay its own depth.","g{$T$}{T}-depth and commuting layers}\nTwo phase gadgets may share a $T$ layer when a suitable Clifford network maps their parities to distinct wires or when ancilla resources permit parallel extraction.  Define the conflict graph $\\Gamma_P$ whose vertices are odd phase terms and whose edges encode inability to realize two terms in one layer under the chosen ancilla and connectivity rules.  Then\n\\[\nT\\text{-depth}(P)\\ge\\chi(\\Gamma_P)\n\\]\nfor any conflict graph whose edge relation is sound.  Catalysis can alter $\\Gamma_P$ by changing the Pauli frame.  The right question is not merely whether $C$ preserves the number of terms, but whether it reduces the chromatic or scheduling obstruction enough to repay its own depth.","s their parities to distinct wires or when ancilla resources permit parallel extraction.  Define the conflict graph $\\Gamma_P$ whose vertices are odd phase terms and whose edges encode inability to realize two terms in one layer under the chosen ancilla and connectivity rules.  Then\n\\[\nT\\text{-depth}(P)\\ge\\chi(\\Gamma_P)\n\\]\nfor any conflict graph whose edge relation is sound.  Catalysis can alter $\\Gamma_P$ by changing the Pauli frame.  The right question is not merely whether $C$ preserves the number of terms, but whether it reduces the chromatic or scheduling obstruction enough to repay its own depth.","ction.  Define the conflict graph $\\Gamma_P$ whose vertices are odd phase terms and whose edges encode inability to realize two terms in one layer under the chosen ancilla and connectivity rules.  Then\n\\[\nT\\text{-depth}(P)\\ge\\chi(\\Gamma_P)\n\\]\nfor any conflict graph whose edge relation is sound.  Catalysis can alter $\\Gamma_P$ by changing the Pauli frame.  The right question is not merely whether $C$ preserves the number of terms, but whether it reduces the chromatic or scheduling obstruction enough to repay its own depth.","\\subsection{A vector-valued defect}\nFor a target $U$ in a restricted phase model, define\n\\[\n\\mathcal C_{\\mathrm{res}}(U)\n=\n\\{\\mathbf c(\\mathcal W):\\mathcal W\\text{ is a restricted circuit for }U\\}\n\\]\nand similarly $\\mathcal C_{\\mathrm{amb}}(U)$.  Their Pareto frontiers are denoted $\\operatorname{Pareto}_{\\mathrm{res}}(U)$ and $\\operatorname{Pareto}_{\\mathrm{amb}}(U)$.","Scalarization by positive weights $w$ gives\n\\[\n\\Delta_w(U)=\n\\min_{c\\in\\mathcal C_{\\mathrm{res}}(U)}w\\cdot c\n-\n\\min_{c\\in\\mathcal C_{\\mathrm{amb}}(U)}w\\cdot c.\n\\]\nDifferent $w$ can change the magnitude of the defect and whether it is zero or positive.  It remains nonnegative because every restricted realization is also an ambient realization with the same resource vector.","\\subsection{Exact finite-search model}\nA finite exact atlas can be built \\cite{g:KliuchnikovEtAl2013,g:AmyEtAl2013} for small $n$ by representing Clifford+$T$ operators in one of three ways.\n\\begin{enumerate}\n\\item Exact matrices over the cyclotomic ring $\\Z[1/\\sqrt2,i]$ modulo global phase.\n\\item Normal forms combining a binary symplectic Clifford with a phase-polynomial record.\n\\item Quotients by left and right Clifford actions when only $T$-count is charged.\n\\end{enumerate}\nThe third model can be dramatically smaller but requires care: quotienting is sound only when the free Clifford actions in the cost model agree with the quotient relation.","te exact atlas can be built \\cite{g:KliuchnikovEtAl2013,g:AmyEtAl2013} for small $n$ by representing Clifford+$T$ operators in one of three ways.\n\\begin{enumerate}\n\\item Exact matrices over the cyclotomic ring $\\Z[1/\\sqrt2,i]$ modulo global phase.\n\\item Normal forms combining a binary symplectic Clifford with a phase-polynomial record.\n\\item Quotients by left and right Clifford actions when only $T$-count is charged.\n\\end{enumerate}\nThe third model can be dramatically smaller but requires care: quotienting is sound only when the free Clifford actions in the cost model agree with the quotient relation.","\\begin{protocol}[Small-width Clifford+$T$ threshold atlas]\n\\begin{enumerate}\n\\item Fix exact versus approximate synthesis, ancilla rules, and the resource vector.\n\\item Enumerate the restricted target subgroup or a complete set of normal forms.\n\\item Compute restricted Pareto frontiers by multiobjective shortest path.\n\\item Enumerate ambient states or use meet in the middle for candidate targets.\n\\item Store an optimal witness and a lower-bound certificate for every defect.\n\\item Classify defects under qubit permutation and free Clifford equivalence.\n\\end{enumerate}\n\\end{protocol}","\\subsubsection{Pauli support geometry}\nRepresent a commuting Pauli family by an isotropic subspace of $\\F_2^{2n}$.  Restricted diagonal circuits remain in the $Z$ Lagrangian.  Ambient Clifford circuits move through the symplectic Grassmannian.  Distance from the $Z$ Lagrangian, together with the number of deposited rotations, is the phase analogue of microscopic flag escape.","\\subsubsection{Moment and derivative packets}\nFor a phase function $P$, finite differences\n\\[\nD_uP(x)=P(x+u)-P(x)\\pmod8\n\\]\nseparate affine, quadratic, and higher interactions.  A hierarchy of derivative tensors can record temporary flow into nonlinear phase packets.","\\subsection{Tensor and repeated-module questions}\nFor targets $U$ and $V$, independent synthesis gives\n\\[\n\\ell_T(U\\otimes V)\n\\le\n\\ell_T(U)+\\ell_T(V)\n\\]\nwhen the systems are disjoint and local Clifford cost is ignored.  Joint synthesis may be better because phase identities can couple terms across modules.  This motivates a stable regularization\n\\[\n\\tau_T(U)=\n\\liminf_{r\\to\\infty}\n\\frac{\\ell_T(U^{\\otimes r})}{r}.\n\\]\nA one-copy rigid phase target could still display tensor activation.","For $n$ bosonic modes, a Gaussian unitary acts on quadratures\n\\[\n\\mathbf r=(q_1,\\ldots,q_n,p_1,\\ldots,p_n)^T\n\\]\nby an affine symplectic transformation\n\\[\n\\mathbf r\\longmapsto S\\mathbf r+d,\n\\qquad\nS\\in\\Sp(2n,\\R).\n\\]\nFor $n$ fermionic modes, a fermionic Gaussian unitary acts on $2n$ Majorana operators by\n\\[\n\\gamma\\longmapsto O\\gamma,\n\\qquad O\\in SO(2n).\n\\]\nNearest-neighbor matchgates give a circuit presentation of a fermionic Gaussian subgroup on qubits.","\\subsection{An inherited symplectic shear theorem}\nEvery invertible linear transformation $A\\in\\GL(n,R)$ has a canonical symplectic embedding\n\\begin{equation}\\label{g:eq:symplectic-embedding}\n\\iota(A)=\n\\begin{pmatrix}\nA&0","\\subsection{An inherited symplectic shear theorem}\nEvery invertible linear transformation $A\\in\\GL(n,R)$ has a canonical symplectic embedding\n\\begin{equation}\\label{g:eq:symplectic-embedding}\n\\iota(A)=\n\\begin{pmatrix}\nA&0\\\\\n0&A^{-T}\n\\end{pmatrix}.\n\\end{equation}\nIndeed, with\n\\[\nJ=\n\\begin{pmatrix}0&I","0&A^{-T}\n\\end{pmatrix}.\n\\end{equation}\nIndeed, with\n\\[\nJ=\n\\begin{pmatrix}0&I\\\\-I&0\\end{pmatrix},\n\\]\none checks\n\\[\n\\iota(A)^T J\\iota(A)=J.\n\\]\nThus every row-transvection identity in the established linear theory induces an exact identity between symplectic shear circuits.","-I&0\\end{pmatrix},\n\\]\none checks\n\\[\n\\iota(A)^T J\\iota(A)=J.\n\\]\nThus every row-transvection identity in the established linear theory induces an exact identity between symplectic shear circuits.","\\begin{provedbox}\nUnder a Gaussian gate alphabet in which each embedded elementary transvection $\\iota(I+aE_{ij})$ is one primitive and direction is the only restriction, the universal six-row catalytic atom embeds into a twelve-quadrature Gaussian atom with restricted length seven and ambient length six.\n\\end{provedbox}","\\subsection{Bosonic gate hierarchies}\nNatural nested models include\n\\[\n\\text{triangular shears}\n\\subset\n\\text{all Gaussian unitaries}\n\\subset\n\\text{Gaussian plus a non-Gaussian primitive}\n\\subset\nU(\\mathcal H).\n\\]\nTypical primitive resources are:\n\\begin{itemize}\n\\item SUM or controlled displacement gates,\n\\item beam splitters,\n\\item single-mode and two-mode squeezers,\n\\item Fourier transforms,\n\\item cubic phase gates,\n\\item photon counting or non-Gaussian state injection.\n\\end{itemize}\nEach inclusion supports a different defect.  A target Gaussian transformation might be shortened by leaving a triangular decomposition but remaining Gaussian.  A more radical question asks whether temporary non-Gaussianity can shorten a final Gaussian target.","\\subsection{Bloch--Messiah versus catalytic synthesis}\nA bosonic Gaussian unitary admits a Bloch--Messiah decomposition\n\\[\nS=K_1 D K_2,\n\\]\nwhere $K_1,K_2$ are passive symplectic transformations and $D$ is a diagonal squeezing layer.  This normal form optimizes conceptual structure, not necessarily a weighted gate count on a specific architecture.","\\begin{skeletonbox}\nA Gaussian catalytic theorem requires:\n\\begin{enumerate}\n\\item a declared primitive library and cost vector;\n\\item a target subgroup, such as triangular, number conserving, or local;\n\\item an explicit sandwich $KHK^{-1}$ returning to the target subgroup;\n\\item a lower bound based on symplectic rank, singular values, graph cuts, or mode support;\n\\item treatment of approximation error when continuous parameters are synthesized numerically.\n\\end{enumerate}\n\\end{skeletonbox}","\\subsection{Fermionic Gaussian and matchgate circuits}\nLet $c_j,c_j^\\dagger$ be fermionic creation and annihilation operators.  Quadratic Hamiltonians generate Gaussian unitaries, and their action is determined by an orthogonal transformation on Majoranas.  Number-conserving Gaussian operations correspond to a unitary action on modes, while general Gaussian operations allow particle-hole mixing.","This yields the hierarchy\n\\[\n\\begin{gathered}\nU(n)\\text{ mode transformations}\n\\subset SO(2n)\\text{ fermionic Gaussian transformations}\\\\\n\\subset \\text{parity-preserving unitaries}.\n\\end{gathered}\n\\]\nA temporary Bogoliubov transformation can violate number conservation while preserving parity.  The target may nevertheless return to the number-conserving subgroup.","\\begin{definition}[Bogoliubov catalytic defect]\nFor a number-conserving Gaussian target $U$, compare the minimum cost using number-conserving primitives with the minimum cost using general Gaussian primitives, requiring the endpoint to conserve number exactly.\n\\end{definition}","\\subsection{Covariance-matrix lower bounds}\nFor a Gaussian state, the covariance matrix $\\Gamma$ transforms as\n\\[\n\\Gamma\\longmapsto S\\Gamma S^T\n\\]\nin the bosonic case and\n\\[\n\\Gamma\\longmapsto O\\Gamma O^T\n\\]\nin the fermionic case.  Possible restricted lower bounds include:\n\\begin{itemize}\n\\item rank change of off-diagonal covariance blocks,\n\\item symplectic singular-value obligations,\n\\item growth of mode support,\n\\item graph bandwidth and cut rank,\n\\item number of nontrivial canonical angles between invariant subspaces.\n\\end{itemize}\nOne elementary gate changes only a controlled portion of these data.  A useful invariant must also charge preparation of a carrier that makes progress on many cuts simultaneously.","\\subsection{Temporary non-Gaussianity}\nSuppose a non-Gaussian primitive $g$ is introduced and later removed:\n\\[\nU=g h_m\\cdots h_1 g^{-1},\n\\qquad U\\in G_{\\mathrm{Gauss}}.\n\\]\nIf $g h_i g^{-1}$ generates several useful Gaussian effects, the excursion may be catalytic.  However, many standard non-Gaussianity monotones vanish only on states rather than unitaries, and cancellation at the endpoint complicates direct lower bounds.","\\subsection{Architecture and routing}\nOn a line or grid, Gaussian and matchgate circuits may be dominated by routing.  Let $G_{\\mathrm{conn}}$ be the hardware graph.  A mode permutation $\\pi$ can be treated as a catalyst if it is opened, many interactions are executed in the reordered frame, and $\\pi^{-1}$ restores the logical labeling.  The benefit is\n\\[\n\\text{routing saved across consumers}\n-\n2\\,\\text{cost}(\\pi).\n\\]\nThis is the same amortization law as the linear snapshot family, now applied to mode geometry.","\\subsection{Beyond subgroup word metrics}\nSome of the most important quantum resource enlargements are not inclusions of groups.  Measurement introduces branches, reset is irreversible, and ancillas enlarge the Hilbert space.  The appropriate abstraction is a nested pair of realization categories\n\\[\n\\mathcal C_{\\mathrm{res}}\n\\subseteq\n\\mathcal C_{\\mathrm{amb}},\n\\]\nwith a cost assigned to generating morphisms.","For a target channel $\\Phi$ in the restricted category, define\n\\[\nc_{\\mathrm{res}}(\\Phi)\n=\n\\inf\\{\\cost(W):W\\text{ realizes }\\Phi\\text{ in }\\mathcal C_{\\mathrm{res}}\\}\n\\]\nand similarly $c_{\\mathrm{amb}}(\\Phi)$.  The value $+\\infty$ is allowed.  A finite positive difference is a metric defect; an infinite difference is an existence defect.","\\subsection{Symmetry-preserving circuits}\nLet $Q$ be a conserved charge and let\n\\[\nH_Q=\\{U:[U,Q]=0\\}.\n\\]\nRestricted circuits use gates commuting with $Q$.  Ambient circuits may temporarily leave $H_Q$ but must return to a charge-preserving endpoint.","The Hilbert space decomposes as\n\\[\n\\mathcal H=\\bigoplus_q\\mathcal H_q.\n\\]\nA restricted circuit is block diagonal.  An ambient catalyst may couple sectors, use the enlarged route, and decouple them at the endpoint.","\\subsection{Sector-flow invariants}\nLet $P_q$ be the projector onto sector $q$.  For an intermediate unitary $U_t$, define transfer blocks\n\\[\nT_{q,r}(t)=P_q U_t P_r.\n\\]\nA restricted path has $T_{q,r}=0$ for $q\\ne r$.  Ambient paths may populate off-diagonal blocks.  Possible excursion measures include\n\\[\n\\sum_{q\\ne r}\\rank T_{q,r},\n\\qquad\n\\max_{q\\ne r}\\|T_{q,r}\\|,\n\\qquad\n\\text{bandwidth in }|q-r|.\n\\]\nThese are analogues of strict-upper rank and bandwidth in the unitriangular theory.","\\subsection{Clean and dirty ancillas}\nA clean ancilla starts in a specified state, usually $\\ket0$, and must be returned to that state.  A dirty ancilla starts in an unknown state and must be restored for every input.","Let $U$ act on the data register.  A clean-ancilla realization is a unitary $W$ satisfying\n\\[\nW(\\ket\\psi\\otimes\\ket0)\n=(U\\ket\\psi)\\otimes\\ket0\n\\quad\\text{for every }\\ket\\psi.\n\\]\nA dirty-ancilla realization requires\n\\[\nW(\\ket\\psi\\otimes\\ket\\phi)\n=(U\\ket\\psi)\\otimes\\ket\\phi\n\\quad\\text{for every }\\ket\\psi,\\ket\\phi.\n\\]\nThe second condition is strictly stronger.","Ancillas define a family of defects\n\\[\n\\Delta_{a,\\mathrm{clean}}(U),\n\\qquad\n\\Delta_{a,\\mathrm{dirty}}(U),\n\\]\nwhere $a$ is the number of ancillary qubits or modes.  Stable ancilla capacity asks whether a fixed number of reusable ancillas can yield an unbounded saving as the data size grows.","\\subsection{Measurement and feedforward}\nA measurement-assisted realization is described by an instrument with classical outcomes $m$, conditional channels $\\Phi_m$, and feedforward operations.  Exact deterministic implementation of a unitary target requires every branch, after correction, to induce the same channel.","The cost vector may include\n\\[\n(\nT\\text{-count},\n\\text{two-qubit count},\n\\text{measurement rounds},\n\\text{classical latency},\n\\text{ancilla qubits}\n).\n\\]\nA measurement can create an existence defect by realizing a channel unavailable unitarily under the same locality or depth constraints, or a metric defect by reducing one resource while increasing another.","\\subsection{Teleportation as a canonical ambient excursion}\nGate teleportation replaces direct implementation by:\n\\[\n\\text{prepare resource state}\n\\to\n\\text{entangle and measure}\n\\to\n\\text{classical correction}.\n\\]\nThe ambient category contains resources not present in the target channel: entanglement, measurement outcomes, and feedforward.  The endpoint can nevertheless be a deterministic unitary.","\\subsection{Architecture-constrained gates}\nLet $\\Gamma=(V,E)$ be a hardware graph.  A restricted two-qubit gate may act only on edges of $E$.  The ambient model may permit a small number of nonlocal interactions, temporary logical relabelings, teleportation edges, or bus-mediated gates.","For a target $U$, define\n\\[\n\\ell_\\Gamma(U)\n=\n\\text{minimum hardware-respecting cost},\n\\]\nand compare it with an enlarged graph $\\Gamma'$.  A matched routing excursion has the form\n\\[\n\\pi\n\\;W\\;\n\\pi^{-1},\n\\]\nwhere $\\pi$ is a logical permutation.  The excursion is profitable when many gates in $W$ become local in the permuted frame.","\\subsection{Depth, count, and space-time volume}\nLet $g_t$ be the set of gates scheduled in layer $t$.  Common scalar costs are\n\\[\n\\sum_t |g_t|,\n\\qquad\n\\#\\{t:g_t\\ne\\varnothing\\},\n\\qquad\n\\sum_t \\#\\{\\text{active qubits in layer }t\\}.\n\\]\nA catalyst may reduce count but increase depth, or vice versa.  Space-time volume is often the more stable hardware quantity.","For stochastic error rates $p_g$, a first-order weighted cost is\n\\[\nC_{\\mathrm{err}}(W)=\\sum_{g\\in W}-\\log(1-p_g).\n\\]\nThis additive approximation can be inserted directly into Dijkstra search.  It does not capture coherent or correlated errors, but it produces a declared and reproducible metric.","\\subsection{Approximate synthesis}\nFor continuous gate sets, define\n\\[\n\\ell_{\\mathrm{res},\\varepsilon}(U)\n=\n\\min\\{\\cost(W):\\|W-U\\|\\le\\varepsilon\\}.\n\\]\nThere are two distinct border phenomena:\n\\begin{enumerate}\n\\item algebraic border length, where a target lies in the closure of a bounded exact word locus;\n\\item analytic approximation length, where an error tolerance is explicitly permitted.\n\\end{enumerate}\nThey should not be conflated.","A state monotone $M(\\rho)$ alone may fail because the catalyst acts at the channel level.  Useful channel-level quantities include:\n\\begin{itemize}\n\\item robustness or magic capacity,\n\\item entangling power,\n\\item operator Schmidt rank,\n\\item Choi-state resource measures,\n\\item symmetry asymmetry of the channel,\n\\item amortized resource generation.\n\\end{itemize}\nA lower bound must relate the cumulative or peak resource to the declared gate cost and account for cancellation.","\\subsection{A categorical amplification principle}\nSuppose a monoidal category admits a tensor product and a cost satisfying\n\\[\nc_{\\mathrm{amb}}(f\\otimes g)\n\\le\nc_{\\mathrm{amb}}(f)+c_{\\mathrm{amb}}(g).\n\\]\nIf restricted cost is additive on a target family, then any one-copy defect amplifies linearly under independent tensor powers.  Superlinear amplification requires a normalization in which dimensions multiply, as in the matrix tensor rule of the core theory, or a joint-synthesis synergy across copies.","This chapter develops the hierarchy\n\\begin{equation}\\label{g:eq:packet-hierarchy}\n\\text{spherical packet}\n\\prec\n\\text{embedded type packet}\n\\prec\n\\text{holonomy packet}\n\\prec\n\\text{realization fiber}.\n\\end{equation}\nThe relation $\\prec$ means ``retains strictly more potentially relevant information,'' not that every refinement is complete.","\\subsection{Spherical packets}\nLet $T_1,\\ldots,T_m$ be commuting normal operators on a finite-dimensional Hilbert space.  Their unique coarsest joint spectral resolution is\n\\[\nT_j=\\sum_{\\lambda}t_j(\\lambda)B_\\lambda,\n\\qquad\n\\sum_\\lambda B_\\lambda=I.\n\\]\nThe tuple\n\\[\n\\lambda=(t_1(\\lambda),\\ldots,t_m(\\lambda))\n\\]\nis the spherical label.  The abstract spectral packet records these labels and the ranks of $B_\\lambda$.","ection{Spherical packets}\nLet $T_1,\\ldots,T_m$ be commuting normal operators on a finite-dimensional Hilbert space.  Their unique coarsest joint spectral resolution is\n\\[\nT_j=\\sum_{\\lambda}t_j(\\lambda)B_\\lambda,\n\\qquad\n\\sum_\\lambda B_\\lambda=I.\n\\]\nThe tuple\n\\[\n\\lambda=(t_1(\\lambda),\\ldots,t_m(\\lambda))\n\\]\nis the spherical label.  The abstract spectral packet records these labels and the ranks of $B_\\lambda$.","Two different embedded systems can have the same abstract packet.  The actual matrices $B_\\lambda$, endpoint coefficients, and transition operators may still differ.","\\subsection{The Latin-square association algebra}\nA Latin square $L$ of order $n$ has $n^2$ cells.  Let $R,C,S$ be adjacency matrices for equality of row, column, and symbol.  The algebra generated by $I,J,R,C,S$ is commutative and its abstract multiplication table depends only on $n$, not on the paratopy class of $L$.","Define\n\\[\nQ_R=\\frac{I+R}{n},\n\\qquad\nQ_C=\\frac{I+C}{n},\n\\qquad\nQ_S=\\frac{I+S}{n},\n\\qquad\nB_0=\\frac{J}{n^2}.\n\\]\nThen the five joint projectors are\n\\[\nB_0,\n\\quad Q_R-B_0,\n\\quad Q_C-B_0,\n\\quad Q_S-B_0,\n\\quad I-Q_R-Q_C-Q_S+2B_0.\n\\]\nTheir multiplicities are\n\\[\n1,\n\\quad n-1,\n\\quad n-1,\n\\quad n-1,\n\\quad (n-1)(n-2).\n\\]\nThe Latin-square graph $R+C+S$ has universal spectrum\n\\[\n3(n-1)^1,\n\\qquad\n(n-3)^{3(n-1)},\n\\qquad\n(-3)^{(n-1)(n-2)}.\n\\]\nAt $n=6$ this becomes\n\\[\n15^1,\\qquad3^{15},\\qquad(-3)^{20}.\n\\]\nThus the graph spectrum cannot distinguish the twelve main classes of order six.","Q_C=\\frac{I+C}{n},\n\\qquad\nQ_S=\\frac{I+S}{n},\n\\qquad\nB_0=\\frac{J}{n^2}.\n\\]\nThen the five joint projectors are\n\\[\nB_0,\n\\quad Q_R-B_0,\n\\quad Q_C-B_0,\n\\quad Q_S-B_0,\n\\quad I-Q_R-Q_C-Q_S+2B_0.\n\\]\nTheir multiplicities are\n\\[\n1,\n\\quad n-1,\n\\quad n-1,\n\\quad n-1,\n\\quad (n-1)(n-2).\n\\]\nThe Latin-square graph $R+C+S$ has universal spectrum\n\\[\n3(n-1)^1,\n\\qquad\n(n-3)^{3(n-1)},\n\\qquad\n(-3)^{(n-1)(n-2)}.\n\\]\nAt $n=6$ this becomes\n\\[\n15^1,\\qquad3^{15},\\qquad(-3)^{20}.\n\\]\nThus the graph spectrum cannot distinguish the twelve main classes of order six.","\\qquad\nQ_S=\\frac{I+S}{n},\n\\qquad\nB_0=\\frac{J}{n^2}.\n\\]\nThen the five joint projectors are\n\\[\nB_0,\n\\quad Q_R-B_0,\n\\quad Q_C-B_0,\n\\quad Q_S-B_0,\n\\quad I-Q_R-Q_C-Q_S+2B_0.\n\\]\nTheir multiplicities are\n\\[\n1,\n\\quad n-1,\n\\quad n-1,\n\\quad n-1,\n\\quad (n-1)(n-2).\n\\]\nThe Latin-square graph $R+C+S$ has universal spectrum\n\\[\n3(n-1)^1,\n\\qquad\n(n-3)^{3(n-1)},\n\\qquad\n(-3)^{(n-1)(n-2)}.\n\\]\nAt $n=6$ this becomes\n\\[\n15^1,\\qquad3^{15},\\qquad(-3)^{20}.\n\\]\nThus the graph spectrum cannot distinguish the twelve main classes of order six.","\\begin{provedbox}\nThe displayed projector formulas depend only on the Latin-square axioms and the order.  They therefore prove that the abstract $R,C,S$ joint eigenvalue packets cannot distinguish squares of the same order.  This conclusion does not require an enumeration of the order-six main classes.\n\\end{provedbox}","\\subsection{Transition fusion also collapses}\nLet $\\rho_i$ be the row permutations of a Latin square.  The symmetrized aggregate of all pair transitions satisfies\n\\begin{equation}\\label{g:eq:transition-fusion}\n\\sum_{i<j}\n\\left(\nP_{\\rho_i\\rho_j^{-1}}\n+\nP_{\\rho_j\\rho_i^{-1}}\n\\right)\n=n(J-I).\n\\end{equation}\nHence first-order fusion destroys the class information as well.","\\subsection{Holonomy packets}\nLet a family of local charts or rows be represented by permutations $\\rho_i$.  Define normalized transitions\n\\[\nh_{ij}=\\rho_i\\rho_j^{-1}\n\\]\nand the transition group\n\\[\nH_r(L)=\\langle h_{ij}:i,j\\rangle.\n\\]\nA holonomy packet may include:\n\\begin{itemize}\n\\item the isomorphism type or order of $H_r(L)$,\n\\item the cycle-type inventory of the distinguished $h_{ij}$,\n\\item block systems and orbit partitions,\n\\item characters or matrix coefficients of words in the $h_{ij}$,\n\\item noncommutative moments\n\\[\n\\operatorname{tr}(h_{i_1j_1}\\cdots h_{i_kj_k}),\n\\]\n\\item the corresponding packets for row, column, and symbol roles.\n\\end{itemize}","tions $\\rho_i$.  Define normalized transitions\n\\[\nh_{ij}=\\rho_i\\rho_j^{-1}\n\\]\nand the transition group\n\\[\nH_r(L)=\\langle h_{ij}:i,j\\rangle.\n\\]\nA holonomy packet may include:\n\\begin{itemize}\n\\item the isomorphism type or order of $H_r(L)$,\n\\item the cycle-type inventory of the distinguished $h_{ij}$,\n\\item block systems and orbit partitions,\n\\item characters or matrix coefficients of words in the $h_{ij}$,\n\\item noncommutative moments\n\\[\n\\operatorname{tr}(h_{i_1j_1}\\cdots h_{i_kj_k}),\n\\]\n\\item the corresponding packets for row, column, and symbol roles.\n\\end{itemize}","itions\n\\[\nh_{ij}=\\rho_i\\rho_j^{-1}\n\\]\nand the transition group\n\\[\nH_r(L)=\\langle h_{ij}:i,j\\rangle.\n\\]\nA holonomy packet may include:\n\\begin{itemize}\n\\item the isomorphism type or order of $H_r(L)$,\n\\item the cycle-type inventory of the distinguished $h_{ij}$,\n\\item block systems and orbit partitions,\n\\item characters or matrix coefficients of words in the $h_{ij}$,\n\\item noncommutative moments\n\\[\n\\operatorname{tr}(h_{i_1j_1}\\cdots h_{i_kj_k}),\n\\]\n\\item the corresponding packets for row, column, and symbol roles.\n\\end{itemize}","For a Latin square of order $2m$, normalize one row to the identity permutation on the symbols.  An $m\\times m$ Latin subsquare determines a partition of the symbols into its $m$ symbols and their complement.  Rows meeting the subsquare preserve the two corresponding column blocks, and the remaining rows exchange them.  Thus normalized row transitions preserve this two-block system.  Conversely, if all normalized row transitions preserve a partition into two blocks of size $m$, exactly $m$ rows send a fixed column block into a fixed symbol block, producing an $m\\times m$ Latin subsquare.  This is a structural criterion.  Counting the order-six classes with this property requires separate input representatives.","on the symbols.  An $m\\times m$ Latin subsquare determines a partition of the symbols into its $m$ symbols and their complement.  Rows meeting the subsquare preserve the two corresponding column blocks, and the remaining rows exchange them.  Thus normalized row transitions preserve this two-block system.  Conversely, if all normalized row transitions preserve a partition into two blocks of size $m$, exactly $m$ rows send a fixed column block into a fixed symbol block, producing an $m\\times m$ Latin subsquare.  This is a structural criterion.  Counting the order-six classes with this property requires separate input representatives.","ls.  An $m\\times m$ Latin subsquare determines a partition of the symbols into its $m$ symbols and their complement.  Rows meeting the subsquare preserve the two corresponding column blocks, and the remaining rows exchange them.  Thus normalized row transitions preserve this two-block system.  Conversely, if all normalized row transitions preserve a partition into two blocks of size $m$, exactly $m$ rows send a fixed column block into a fixed symbol block, producing an $m\\times m$ Latin subsquare.  This is a structural criterion.  Counting the order-six classes with this property requires separate input representatives.","nto its $m$ symbols and their complement.  Rows meeting the subsquare preserve the two corresponding column blocks, and the remaining rows exchange them.  Thus normalized row transitions preserve this two-block system.  Conversely, if all normalized row transitions preserve a partition into two blocks of size $m$, exactly $m$ rows send a fixed column block into a fixed symbol block, producing an $m\\times m$ Latin subsquare.  This is a structural criterion.  Counting the order-six classes with this property requires separate input representatives.","\\subsection{Realization fibers}\nLet $P$ denote packet data and let\n\\[\n\\operatorname{Real}_{\\mathrm{res}}(P)\n\\]\nbe the space of restricted realizations compatible with $P$.  The packet may completely classify the discrete object while the realization fiber remains empty, positive dimensional, disconnected, or singular.","\\subsection{Quantum Latin squares as an existence defect}\nA quantum Latin square is an array \\cite{s:MustoVicary2016} of vectors whose rows and columns are orthonormal bases.  Ball and Simoens prove that a pair of non-entangled mutually orthogonal quantum Latin squares of order six does not exist, while earlier work provides entangled order-six solutions and hence AME$(4,6)$ states \\cite{s:BallSimoens2026,s:RatherEtAl2022,g:RatherEtAl2023}.","This is naturally expressed as\n\\[\n\\operatorname{Real}_{\\mathrm{product}}(P_6)=\\varnothing,\n\\qquad\n\\operatorname{Real}_{\\mathrm{entangled}}(P_6)\\ne\\varnothing.\n\\]\nIt is an existence defect, not a gate-count defect.","\\subsection{First-arrival packets}\nFor a finite generator set $S$, the endpoint Green kernel\n\\[\n\\mathcal G_{S,g}(z)\n=\n\\langle\\delta_g,(I-zA_S)^{-1}\\delta_e\\rangle\n\\]\ncontains more information than the spectrum of $A_S$.  Its first nonzero degree is the distance to $g$.","If\n\\[\nA_S=\\sum_\\lambda \\lambda B_\\lambda,\n\\]\nthen\n\\[\n\\mathcal G_{S,g}(z)\n=\n\\sum_\\lambda\n\\frac{\\langle\\delta_g,B_\\lambda\\delta_e\\rangle}{1-\\lambda z}.\n\\]\nThe endpoint matrix coefficients of the embedded projectors determine cancellation between spectral components.  This is the metric analogue of retaining a type rather than only an unramified eigencharacter.","\\subsection{Typed Green kernels}\nLet $H\\le G$ and let $\\mu_k(h)$ count ambient words of length $k$ with holonomy $h\\in H$.  For an irreducible representation $\\tau$ of $H$, define\n\\[\n\\widehat\\mu_k(\\tau)\n=\n\\sum_{h\\in H}\\mu_k(h)\\tau(h)\n\\]\nand\n\\[\n\\widehat{\\mathcal G}_\\tau(z)\n=\n\\sum_{k\\ge0}\\widehat\\mu_k(\\tau)z^k.\n\\]\nFourier inversion recovers every $\\mu_k(h)$.  The trivial representation records only total return to the subgroup; nontrivial types resolve the holonomy.","\\begin{provedbox}\nFor finite $H$, the full family of typed Green kernels over $\\widehat H$ determines every endpoint Green series and therefore every ambient word distance from the identity to an element of $H$.\n\\end{provedbox}","\\subsection{Operational Langlands analogy}\nThe hierarchy resembles the local organization of representation theory:\n\\[\n\\text{spherical character}\n\\to\n\\text{type}\n\\to\n\\text{ramification or monodromy}\n\\to\n\\text{realization and descent}.\n\\]\nThe analogy is useful because it prevents coarse spectral data from being mistaken for a complete correspondence.  It does not imply that global Langlands reciprocity predicts either the circuit threshold or the quantum-Latin result.","The first threshold may then be defined as\n\\[\n\\tau=\n\\min\\{n:\\exists P\\text{ with }c_{\\mathrm{res}}(P)>c_{\\mathrm{amb}}(P)\\}.\n\\]\nThe cost can be finite or infinite.","\\subsection{State models}\n\\subsubsection{Binary matrices}\nAn $m\\times m$ matrix over $\\F_2$ is stored as\n\\[\n(r_0,\\ldots,r_{m-1}),\n\\]\nwhere bit $j$ of $r_i$ is the $(i,j)$ entry.  Matrix multiplication is exact XOR accumulation.","\\subsubsection{Symplectic Clifford states}\nAn $n$-qubit Clifford modulo global phase and Pauli translation is represented by its action on Pauli exponent vectors\n\\[\n(x\\mid z)\\in\\F_2^{2n}.\n\\]\nThe state space is $\\Sp(2n,2)$.  The restricted parabolic group is generated by $S$, CZ, and CNOT; the ambient group additionally permits $H$.","\\subsubsection{Classical reversible states}\nA reversible map on $n$ bits is stored as a permutation of $\\{0,\\ldots,2^n-1\\}$.  Composition is exact.  The affine subgroup is generated by $X$ and CNOT, and the ambient three-bit group adds Toffoli.","\\subsection{Shortest-path engines}\nThe package implements:\n\\begin{itemize}\n\\item breadth-first search for unit costs;\n\\item $0$--$1$ breadth-first search when gates are free or unit cost;\n\\item Dijkstra search for general nonnegative integer weights;\n\\item parent maps for witness reconstruction;\n\\item defect distributions on a restricted target set.\n\\end{itemize}","\\begin{algorithm}[Breadth-first defect census]\nGiven restricted generators $S_R$, ambient generators $S_A$, and identity $e$:\n\\begin{enumerate}\n\\item compute $d_R$ on $\\langle S_R\\rangle$;\n\\item compute $d_A$ on $\\langle S_A\\rangle$;\n\\item for every restricted state $x$, compare $d_R(x)$ and $d_A(x)$;\n\\item record the pair $(d_R(x),d_A(x))$ and reconstruct witnesses for strict inequalities.\n\\end{enumerate}\n\\end{algorithm}","\\subsection{Zero-cost gates}\nWhen local Clifford gates are free and entangling gates cost one, ordinary BFS is invalid.  The package uses a deque-based $0$--$1$ BFS.  A zero-cost edge is pushed to the front and a unit-cost edge to the back.  The resulting labels are exact shortest entangling costs.","A common error is to quotient by free gates without proving that the quotient preserves the target subset and all left/right actions.  The atlas avoids this issue by running $0$--$1$ BFS directly on the finite group for the reported small cases.","\\subsection{The optimized Clifford verifier}\nThe three-qubit symplectic group has\n\\[\n|\\Sp(6,2)|=1{,}451{,}520\n\\]\nstates.  A compact C++ implementation enumerates the full group and the parabolic subgroup exactly.  It supports:\n\\begin{itemize}\n\\item total primitive count;\n\\item total count with unit-cost SWAP;\n\\item entangling count with $H$ and $S$ free;\n\\item exact distance distributions and JSON output.\n\\end{itemize}\nThe C++ program and the independent Python implementation use the same gate matrices but different traversal code.  Witness reconstruction is supplied by the Python implementation, not by the C++ census executable.","\\subsection{Computed Clifford census}\n\\begin{longtable}{@{}ccccccc@{}}\n\\toprule\n$n$ & Metric & SWAP & $|P_n|$ & $|\\Sp(2n,2)|$ & Defects & Max defect","The detailed defect distributions are:\n\\[\n\\begin{array}{c|l}\n\\text{model}&\\text{restricted}\\to\\text{ambient pairs}\\\\\n\\hline\nn=2\\text{ entangling}&3\\to2:4\\\\\nn=3\\text{ total}&7\\to6:6,\\quad8\\to7:18\\\\\nn=3\\text{ entangling}&3\\to2:24,\\ 4\\to3:432,\\ 5\\to4:1056,\\ 6\\to4:48,\\ 6\\to5:64.\n\\end{array}\n\\]","\\subsection{Computed affine-versus-Toffoli census}\nFor three bits:\n\\[\n|\\operatorname{AGL}(3,2)|=1344,\n\\qquad\n|S_8|=40320.\n\\]\nThe complete census finds no affine target with a strict defect under total primitive count or under entangling count with NOT free.","\\subsection{Linear symbolic checks}\nThe Python module verifies that the displayed six-row ambient and restricted words produce the same canonical target modulo\n\\[\n2,3,5,8.\n\\]\nThe lower bounds and ring universality are proved in the core chapter; these modular checks are regression tests rather than the logical basis of the theorem.","\\subsection{Witness format}\nA witness record contains:\n\\begin{itemize}\n\\item the target matrix or permutation;\n\\item restricted and ambient costs;\n\\item a named restricted gate path;\n\\item a named ambient gate path;\n\\item the metric and gate alphabet;\n\\item the state-model convention.\n\\end{itemize}\nFor example, the two-qubit entangling Clifford witness has restricted path\n\\[\n\\CX_{0\\to1},\\ S_1,\\ \\CX_{1\\to0},\\ \\CX_{0\\to1}\n\\]\nand ambient path\n\\[\nH_1,\\ \\CX_{1\\to0},\\ \\CX_{0\\to1},\\ H_0,\\ S_1,\\ S_0.\n\\]\nOnly two gates in the ambient path are charged by the entangling metric.","ontains:\n\\begin{itemize}\n\\item the target matrix or permutation;\n\\item restricted and ambient costs;\n\\item a named restricted gate path;\n\\item a named ambient gate path;\n\\item the metric and gate alphabet;\n\\item the state-model convention.\n\\end{itemize}\nFor example, the two-qubit entangling Clifford witness has restricted path\n\\[\n\\CX_{0\\to1},\\ S_1,\\ \\CX_{1\\to0},\\ \\CX_{0\\to1}\n\\]\nand ambient path\n\\[\nH_1,\\ \\CX_{1\\to0},\\ \\CX_{0\\to1},\\ H_0,\\ S_1,\\ S_0.\n\\]\nOnly two gates in the ambient path are charged by the entangling metric.","\\subsection{Multiobjective search}\nLet each gate have a vector cost in $\\N^r$.  At each state, retain only nondominated labels.  The algorithm terminates for finite state spaces with a bounded search region, but the frontier can grow exponentially.","\\subsection{Scaling limits}\nFull group enumeration grows rapidly:\n\\[\n|\\Sp(2n,2)|\n=2^{n^2}\\prod_{j=1}^n(4^j-1).\n\\]\nThe four-qubit symplectic group is far beyond direct Python BFS.  Future work should exploit:\n\\begin{itemize}\n\\item double-coset normal forms,\n\\item Bruhat or parabolic decompositions,\n\\item admissible heuristics for A* search,\n\\item SAT encodings,\n\\item orbit reduction,\n\\item tensor-network representations,\n\\item distributed meet in the middle.\n\\end{itemize}","\\subsection{Template I: catalytic conjugation}\nLet $H\\le G$ have restricted and ambient costs with $\\ellAmb(g^{-1})=\\ellAmb(g)$.  Choose $g\\in G$ and $h_1,\\ldots,h_m\\in H$ such that\n\\[\nU=g h_m\\cdots h_1 g^{-1}\\in H.\n\\]\nThen\n\\[\n\\ellAmb(U)\n\\le\n2\\ellAmb(g)+\\sum_i\\ellAmb(h_i).\n\\]","\\begin{theorem}[Catalytic comparison template]\nAssume a restricted lower bound of the form\n\\[\n\\ellRes(U)\n\\ge\n\\sum_i r_i-\\beta,\n\\]\nwhere $r_i$ is the restricted cost of the $i$th transported obligation and $\\beta$ is the maximum certified sharing.  Then\n\\[\n\\Delta(U)\n\\ge\n\\sum_i r_i-\\beta\n-\n2\\ellAmb(g)-\\sum_i\\ellAmb(h_i).\n\\]\n\\end{theorem}","The proof is immediate.  The research content lies in bounding $\\beta$.","\\subsection{Template II: amortized consumers}\nSuppose a catalyst costs $c_{\\mathrm{open}}+c_{\\mathrm{close}}$ and each of $m$ consumers saves at least $s_i$ restricted cost.  Then\n\\[\n\\Delta(U)\n\\ge\n\\sum_{i=1}^m s_i\n-c_{\\mathrm{open}}-c_{\\mathrm{close}}-r,\n\\]\nwhere $r$ is repair or interference cost.  A threshold occurs when the right side first becomes positive.","\\subsection{Template III: first-arrival Green valuation}\nLet $S$ be a finite symmetric generating set and\n\\[\nA_S=\\sum_{s\\in S}\\rho(s).\n\\]\nThen\n\\[\n\\mathcal G_{S,g}(z)\n=\\sum_{k\\ge0}\\langle\\delta_g,A_S^k\\delta_e\\rangle z^k.\n\\]","\\begin{theorem}[First-arrival template]\nIf word multiplicities are nonnegative, then\n\\[\nd_S(e,g)=\\ord_{z=0}\\mathcal G_{S,g}(z).\n\\]\nFor restricted and ambient sets,\n\\[\n\\Delta(g)=\\ord_0\\mathcal G_{S_R,g}-\\ord_0\\mathcal G_{S_A,g}.\n\\]\n\\end{theorem}","\\subsection{Template IV: one-Lipschitz potentials}\nLet $F$ be a function on circuit states satisfying\n\\[\nF(I)=0,\n\\qquad\n|F(gs)-F(g)|\\le c(s)\n\\]\nfor every allowed restricted gate $s$.  Then\n\\[\n\\ellRes(U)\\ge F(U).\n\\]","\\subsection{Template V: dimension per gate}\nLet $X$ be an algebraic family of elementary gates whose irreducible components have dimension at most $r$.  Let $V$ be an irreducible target variety of dimension $N$.","\\begin{theorem}[Dimension-per-gate template]\nThe closure of the locus reachable in $L$ gates has dimension at most $rL$.  If $rL<N$, a generic point of $V$ is not reachable in $L$ gates.\n\\end{theorem}","If every target in $V$ has a restricted construction of length at most $N/r$, then generic restricted and ambient lengths coincide at that value, provided the same parameter bound applies to the ambient alphabet.","\\subsection{Template VI: defect codimension}\nAssume every target has restricted cost at most $M$.  A defect of size at least $d$ implies ambient cost at most $M-d$.  If the $(M-d)$-gate ambient locus has dimension at most $N-d$, then\n\\[\n\\codim\\overline{\\mathcal D_{\\ge d}}\\ge d.\n\\]\nThis is the general mechanism behind algebraic rarity.","\\subsection{Template VII: finite-field sparsity}\nSuppose there are at most $B$ gate directions and at most $q^r$ parameter choices per gate over $\\F_q$.  Then the number of words of length at most $L$ is at most\n\\[\n\\sum_{j=0}^L(Bq^r)^j.\n\\]\nIf the target family has $q^N$ points and defect $d$ forces $L\\le (N-d)/r$, this yields a density bound of order $q^{-d}$ up to a fixed combinatorial factor.","\\subsection{Template VIII: block retraction}\nLet a large restricted group contain a smaller one through a homomorphic embedding $\\iota$, and suppose a homomorphic retraction $\\pi$ satisfies $\\pi\\iota=\\operatorname{id}$.  Require that every large restricted generator maps to a small restricted word of no greater cost.  Then\n\\[\n\\ell_{\\mathrm{large,res}}(\\iota U)\n\\ge\n\\ell_{\\mathrm{small,res}}(U).\n\\]\nIf the ambient embedding preserves gate cost,\n\\[\n\\Delta_{\\mathrm{large}}(\\iota U)\n\\ge\n\\Delta_{\\mathrm{small}}(U).\n\\]\nThis transfers a catalytic atom to architectures containing it as a retract.","a large restricted group contain a smaller one through a homomorphic embedding $\\iota$, and suppose a homomorphic retraction $\\pi$ satisfies $\\pi\\iota=\\operatorname{id}$.  Require that every large restricted generator maps to a small restricted word of no greater cost.  Then\n\\[\n\\ell_{\\mathrm{large,res}}(\\iota U)\n\\ge\n\\ell_{\\mathrm{small,res}}(U).\n\\]\nIf the ambient embedding preserves gate cost,\n\\[\n\\Delta_{\\mathrm{large}}(\\iota U)\n\\ge\n\\Delta_{\\mathrm{small}}(U).\n\\]\nThis transfers a catalytic atom to architectures containing it as a retract.","\\subsection{Template IX: exact block coupling}\nFor block-lower matrices\n\\[\nM=\n\\begin{pmatrix}\nA&0\\\\\nXA&D\n\\end{pmatrix},\n\\]\nprojection to diagonal blocks forces the costs of $A$ and $D$, while rank of the cross block forces at least $\\rank X$ cross transmissions.  If $X$ is a partial monomial matrix, the lower bound is attained by direct construction.","XA&D\n\\end{pmatrix},\n\\]\nprojection to diagonal blocks forces the costs of $A$ and $D$, while rank of the cross block forces at least $\\rank X$ cross transmissions.  If $X$ is a partial monomial matrix, the lower bound is attained by direct construction.","\\subsection{Template X: tensor Leibniz rule}\nSuppose a restricted complexity is tree tight and satisfies\n\\[\n\\ellRes(A\\otimes B)\n=n\\ellRes(A)+m\\ellRes(B)\n\\]\nfor $A$ of size $m$ and $B$ of size $n$.  If ambient complexity satisfies the corresponding upper bound, then\n\\[\n\\Delta(A\\otimes B)\n\\ge\nn\\Delta(A)+m\\Delta(B).\n\\]\nFor tensor powers,\n\\[\n\\Delta(A^{\\otimes r})\n\\ge\nr m^{r-1}\\Delta(A).\n\\]\nThis is the standard route from a finite atom to an $N\\log N$ defect.","e}\nSuppose a restricted complexity is tree tight and satisfies\n\\[\n\\ellRes(A\\otimes B)\n=n\\ellRes(A)+m\\ellRes(B)\n\\]\nfor $A$ of size $m$ and $B$ of size $n$.  If ambient complexity satisfies the corresponding upper bound, then\n\\[\n\\Delta(A\\otimes B)\n\\ge\nn\\Delta(A)+m\\Delta(B).\n\\]\nFor tensor powers,\n\\[\n\\Delta(A^{\\otimes r})\n\\ge\nr m^{r-1}\\Delta(A).\n\\]\nThis is the standard route from a finite atom to an $N\\log N$ defect.","\\subsection{Template XI: stable capacity}\nIf\n\\[\na_{r+s}\\le m^s a_r+m^r a_s,\n\\]\nthen\n\\[\nb_r=\\frac{a_r}{m^{r-1}}\n\\]\nis subadditive and\n\\[\n\\lim_{r\\to\\infty}\\frac{a_r}{r m^{r-1}}\n\\]\nexists by Fekete's lemma.  This defines a regularized ambient cost and exposes possible tensor activation.","\\subsection{Template XII: vector costs}\nLet gates carry costs in $\\R_+^k$.  A lower bound may be a vector inequality\n\\[\nF(U)\\preceq\\sum_{g\\in W}c(g)\n\\]\nor a family of scalar inequalities indexed by dual weights $w\\in\\R_+^k$:\n\\[\nw\\cdot F(U)\\le w\\cdot c(W).\n\\]\nA complete Pareto proof may require enough supporting weights to separate the restricted and ambient convex lower hulls.","\\subsection{Template XIII: ancilla restoration}\nLet $W$ act on data $D$ and ancilla $A$.  To certify a dirty catalyst, prove the operator identity\n\\[\nW=U_D\\otimes I_A\n\\]\non the full joint space, not only on one ancilla input.  For a clean catalyst, prove\n\\[\nW(\\ket\\psi\\otimes\\ket0)\n=(U\\ket\\psi)\\otimes\\ket0\n\\]\nfor all $\\ket\\psi$.  The distinction changes both the target equations and lower bounds.","\\subsection{Template XIV: symmetry-sector escape}\nGiven projectors $P_q$, define off-sector transfer blocks\n\\[\nT_{qr}=P_qUP_r.\n\\]\nA restricted gate preserves all diagonal sectors.  An ambient gate changes a controlled collection of blocks.  A candidate potential is a weighted sum of ranks or norms of obligations relative to the final target.  The proof must charge both creation and useful consumption of off-sector coherence.","\\subsection{Template XV: exhaustive threshold classification}\nA finite first-threshold theorem should contain:\n\\begin{enumerate}\n\\item no defects below size $n_0$;\n\\item at least one defect at $n_0$;\n\\item exact classification of every defect at $n_0$ modulo declared equivalence;\n\\item analytic generation of the exceptional motifs;\n\\item equality between the analytic set and computational census;\n\\item witness and completeness certificates.\n\\end{enumerate}\nThis is the standard achieved by the six-wire linear core and is the target standard for future gate logics.","\\subsection{The central research problem}\nFor a nested pair of gate models, determine the distortion function\n\\[\nD_n=\n\\sup_{U\\in\\mathcal T_n}\n\\bigl(c_{\\mathrm{res}}(U)-c_{\\mathrm{amb}}(U)\\bigr),\n\\]\nwhere $\\mathcal T_n$ is the target class.  The basic questions are:\n\\begin{enumerate}\n\\item Does any defect exist?\n\\item What is the first size threshold?\n\\item What is the minimal catalytic motif?\n\\item Is the effect generic, sparse, or confined to special strata?\n\\item How does it scale under direct sum, recursion, and tensor products?\n\\item Which resource vector reveals the first defect?\n\\item Can the effect survive realistic architecture and fault-tolerant costs?\n\\end{enumerate}","{itemize}\n\\item a universal six-row atom over nonzero commutative rings;\n\\item exact binary and ternary first-threshold evidence;\n\\item exact classification of twenty binary minimal defects;\n\\item connected families with two reverse gates and linear saving;\n\\item microscopic rank-one, bandwidth-one escape;\n\\item generic rigidity and defect codimension;\n\\item recursive and tensor amplification to $\\Omega(n\\log n)$.\n\\end{itemize}\nThe small Clifford atlas adds a second gate logic with genuine exact defects and shows that changing the primitive library by adding SWAP can remove them.","\\subsubsection{Longer-term questions}\n\\begin{conjecture}\nThe parabolic subgroup of $\\Sp(2n,2)$ has unbounded additive distortion in the full symplectic group for at least one natural Clifford cost model.\n\\end{conjecture}\n\\begin{question}\nDoes the maximum total-count defect grow as $n$, $n\\log n$, or faster?\n\\end{question}\n\\begin{question}\nIs the disappearance of three-qubit defects after adding unit-cost SWAP a small-width accident or a structural geodesic-convexity phenomenon?\n\\end{question}","\\subsection{Program B: temporary nonlinearity}\nThe three-bit affine-versus-Toffoli census is rigid.  The next target is a complete four-bit normal-form search without enumerating $16!$ permutations.","\\subsubsection{Algorithmic plan}\n\\begin{enumerate}\n\\item Compute exact affine distances for all $|\\operatorname{AGL}(4,2)|=322{,}560$ targets.\n\\item Enumerate Toffoli sandwiches modulo affine left and right actions.\n\\item Use ANF equations to retain only affine endpoints.\n\\item Compare total, Toffoli, CNOT, and fault-tolerant costs.\n\\item Produce SAT certificates below every claimed optimum.\n\\end{enumerate}","Priority metrics:\n\\[\nT\\text{-depth},\n\\quad\n\\text{CNOT count at fixed }T\\text{-count},\n\\quad\n\\text{two-qubit depth},\n\\quad\n\\text{architecture-weighted cost}.\n\\]\nA first paper should choose one metric and avoid a diffuse comparison across all resources.","\\begin{question}\nCan a restored Hadamard or Clifford frame reduce $T$-depth of a diagonal target without additional ancillas?\n\\end{question}\n\\begin{question}\nDoes tensor activation occur for phase-polynomial $T$-count?\n\\end{question}","\\subsection{Program E: symmetry and measurement}\nThis program should begin with a narrowly declared resource theory, such as:\n\\begin{itemize}\n\\item excitation-preserving two-qubit gates versus arbitrary two-qubit gates;\n\\item unitary Clifford+$T$ versus measurement-assisted Clifford+$T$;\n\\item clean versus dirty ancillas for a fixed arithmetic family;\n\\item local architecture versus one reusable nonlocal bus gate.\n\\end{itemize}\nThe first objective is not maximal generality but one theorem with exact restoration and cost accounting.","\\subsection{Program F: root-system catalysis}\nFor a split reductive group with allowed roots $\\Psi\\subset\\Phi$, classify configurations in which a forbidden root acts as a reusable catalyst through Chevalley commutator relations.","Suggested sequence:\n\\[\nA_2\\to B_2=C_2\\to G_2\\to A_r.\n\\]\nLow-rank nonsimply laced systems are attractive because longer root strings may create more than one useful effect per consumer.","\\begin{question}\nIs there a root-system analogue of the three-cross, one-phase threshold?\n\\end{question}\n\\begin{question}\nCan a $G_2$ root string produce a multiplicative distortion exceeding the clean type-$A$ value $3/2$?\n\\end{question}","\\begin{conjecture}\nIn a broad class of one-parameter elementary gate varieties, defect at least $d$ is generically a codimension-at-least-$d$ condition, with equality on explicit families of components.\n\\end{conjecture}","VII & Root-system catalytic complexity & Chevalley commutators and thresholds beyond type $A$.","\\subsection{Software roadmap}\nVersion $0.1$ of the atlas supports exact small finite groups.  A mature package should add:\n\\begin{enumerate}\n\\item generic multiobjective Dijkstra search;\n\\item disk-backed and parallel BFS;\n\\item SAT and SMT export;\n\\item group action and orbit classification;\n\\item symbolic word-map elimination;\n\\item circuit diagrams in Quantikz and SVG;\n\\item architecture graphs and routing costs;\n\\item certificate verification independent of the search engine;\n\\item a machine-readable theorem-status ledger.\n\\end{enumerate}","is bounded.\n\\item Determine the lower border length of the canonical six-row atom.\n\\item Classify tree-tight catalytic matroids.\n\\item Find a one-copy rigid target with tensor activation.\n\\item Extend the two-qubit analytic orbit certificate to larger Clifford systems.\n\\item Analytically classify the three-qubit total-count Clifford defects.\n\\item Determine whether SWAP restores rigidity for all $n$ in the chosen primitive model.\n\\item Find the first affine target shortened by temporary nonlinearity, or prove none exists.\n\\item Find the first diagonal target improved by non-diagonal Clifford catalysis.\n\\item Determine a phase-polynomial Pareto defect.\n\\item Find a number-conserving Gaussian target shortened by Bogoliubov escape.\n\\item Find a matchgate routing catalyst with a certified architecture advantage.\n\\item Establish a symmetry-sector microscopic-escape theorem.\n\\item Classify when a fixed number of dirty ancillas gives unbounded saving.\n\\item Develop typed Green-kernel lower bounds that avoid full enumeration.\n\\item Determine finite moment bounds sufficient for transition-hol…","ate routing catalyst with a certified architecture advantage.\n\\item Establish a symmetry-sector microscopic-escape theorem.\n\\item Classify when a fixed number of dirty ancillas gives unbounded saving.\n\\item Develop typed Green-kernel lower bounds that avoid full enumeration.\n\\item Determine finite moment bounds sufficient for transition-holonomy classification.\n\\item Classify catalytic motifs in $B_2$, $C_2$, and $G_2$.\n\\item Prove a general theorem connecting algebraic codimension and catalytic defect.\n\\end{enumerate}","ting catalyst with a certified architecture advantage.\n\\item Establish a symmetry-sector microscopic-escape theorem.\n\\item Classify when a fixed number of dirty ancillas gives unbounded saving.\n\\item Develop typed Green-kernel lower bounds that avoid full enumeration.\n\\item Determine finite moment bounds sufficient for transition-holonomy classification.\n\\item Classify catalytic motifs in $B_2$, $C_2$, and $G_2$.\n\\item Prove a general theorem connecting algebraic codimension and catalytic defect.\n\\end{enumerate}","st with a certified architecture advantage.\n\\item Establish a symmetry-sector microscopic-escape theorem.\n\\item Classify when a fixed number of dirty ancillas gives unbounded saving.\n\\item Develop typed Green-kernel lower bounds that avoid full enumeration.\n\\item Determine finite moment bounds sufficient for transition-holonomy classification.\n\\item Classify catalytic motifs in $B_2$, $C_2$, and $G_2$.\n\\item Prove a general theorem connecting algebraic codimension and catalytic defect.\n\\end{enumerate}","Coherence block & $P_\\alpha\\rho P_\\beta$ with possibly distinct left and right packet labels.","Packet transfer graph & Directed graph recording nonzero blocks $P_\\alpha G P_\\beta$.","\\section{The typed mathematical target}\nThe complex matrices act on vectors indexed by \\(\\operatorname{Fin}(12)\\).\nThe Fourier convention is\n\\[\n F_{nk}=\\frac{1}{\\sqrt{12}}\\exp\\!\\left(\\frac{2\\pi i nk}{12}\\right),\n \\qquad n,k\\in\\{0,\\ldots,11\\}.\n\\]\nThe adjoint \\(F^*\\) therefore carries the negative exponential sign. The\nselector \\(D\\) is diagonal with value one precisely at the residues\n\\(U=\\{1,5,7,11\\}\\). The transported operator is \\(P=FDF^*\\).","t}\nThe complex matrices act on vectors indexed by \\(\\operatorname{Fin}(12)\\).\nThe Fourier convention is\n\\[\n F_{nk}=\\frac{1}{\\sqrt{12}}\\exp\\!\\left(\\frac{2\\pi i nk}{12}\\right),\n \\qquad n,k\\in\\{0,\\ldots,11\\}.\n\\]\nThe adjoint \\(F^*\\) therefore carries the negative exponential sign. The\nselector \\(D\\) is diagonal with value one precisely at the residues\n\\(U=\\{1,5,7,11\\}\\). The transported operator is \\(P=FDF^*\\).","Write\n\\[\n c=(4,0,2,0,-2,0,-4,0,-2,0,2,0),\\qquad\n \\psi=\\frac{c}{\\sqrt{48}},\\qquad \\chi=\\frac{|c|}{\\sqrt{48}}.\n\\]\nBoth vectors have norm one. Their initial probability histograms are identical,\nbut their inner product is zero. The vector \\(u\\) has entry \\(1/2\\) on \\(U\\)\nand zero elsewhere. Lean proves the actual complex identities \\(Fu=\\psi\\) and\n\\(F^*\\psi=u\\), as well as transformed coprime probability one for \\(\\psi\\)\nand zero for \\(\\chi\\).","0,2,0,-2,0,-4,0,-2,0,2,0),\\qquad\n \\psi=\\frac{c}{\\sqrt{48}},\\qquad \\chi=\\frac{|c|}{\\sqrt{48}}.\n\\]\nBoth vectors have norm one. Their initial probability histograms are identical,\nbut their inner product is zero. The vector \\(u\\) has entry \\(1/2\\) on \\(U\\)\nand zero elsewhere. Lean proves the actual complex identities \\(Fu=\\psi\\) and\n\\(F^*\\psi=u\\), as well as transformed coprime probability one for \\(\\psi\\)\nand zero for \\(\\chi\\).",",0),\\qquad\n \\psi=\\frac{c}{\\sqrt{48}},\\qquad \\chi=\\frac{|c|}{\\sqrt{48}}.\n\\]\nBoth vectors have norm one. Their initial probability histograms are identical,\nbut their inner product is zero. The vector \\(u\\) has entry \\(1/2\\) on \\(U\\)\nand zero elsewhere. Lean proves the actual complex identities \\(Fu=\\psi\\) and\n\\(F^*\\psi=u\\), as well as transformed coprime probability one for \\(\\psi\\)\nand zero for \\(\\chi\\).","The original 22 finite-table statements are retained inside both checkpoints.\nThey are not counted a second time. Likewise, \\(59+96\\) is not a meaningful\ntheorem total: these are overlapping revisions of one formal development.","Exact coordinates & Algebra in the basis \\(1,\\sqrt3,i,i\\sqrt3\\).","Finite matrices & All entries indexed by \\(\\operatorname{Fin}(12)\\), with exact finite sums.","\\section{The rational signal and its filter}\nThe first layer uses literal rational lists. The chosen perturbation is\n\\(r_n=3\\cos(\\pi n/2)+2\\), whose displayed rational entries can be certified\nwithout numerical trigonometric evaluation. Its sum with \\(c\\) is the test signal.","The cyclic matrix has entry \\(c_{(n-m)\\bmod12}/12\\). The original layer proves\nits symmetry, idempotence, preservation of \\(c\\), rejection of the chosen\nperturbation, and exact recovery from the displayed contaminated signal.\nThe exact energy contract is \\(150=48+102\\).","The list operations use finite lists, including zero defaults and zip-based dot\nproducts. These facts are therefore claims about the supplied twelve-entry\nobjects. They should not be silently interpreted as universal matrix theorems\nfor arbitrary ragged lists. The later \\(\\operatorname{Fin}(12)\\) layer supplies\nthe typed matrix identities used in the publication.","\\section{The exact phase arithmetic}\nThe structure \\texttt{Cyclo12} represents\n\\[\n (a,b,c,d)\\longmapsto a+b\\sqrt3+i(c+d\\sqrt3),\\qquad a,b,c,d\\in\\mathbb Q.\n\\]\nIts multiplication explicitly incorporates \\((\\sqrt3)^2=3\\) and \\(i^2=-1\\).\nIt is a concrete arithmetic representation. The development does not need or\ninstall a global field instance to certify the finite example.","The phase \\(\\zeta=(0,1/2,1/2,0)\\) has complex meaning\n\\(\\sqrt3/2+i/2\\). Exact finite certificates establish its twelfth power, inverse,\nand absence of a smaller positive power below twelve. Later the convention\nmodule identifies it with \\(\\exp(\\pi i/6)\\).","The Ramanujan vector is obtained by summing powers of this phase over\n\\(1,5,7,11\\). The negative-sign unnormalized DFT has value 12 at these four\nfrequencies and zero elsewhere. The absolute-value vector has Fourier support\non the even frequencies instead. These are exact phase calculations over\nrational coordinates.","\\section{Finite indices and normalized Fourier certificates}\nThe finite layer uses \\texttt{Index := Fin 12}. Its normalized Fourier scale is\nrepresented exactly as \\(\\sqrt3/6=1/\\sqrt{12}\\), and its state scale as\n\\(\\sqrt3/12=1/\\sqrt{48}\\). The use of a finite index type makes every row and\ncolumn well defined without list-length side conditions.","The two Fourier identities above quantify over all \\(12\\times12\\) entries.\nKernel reduction proves these concrete universal statements over a finite type.\nThe selected-kernel equality and state actions are also complete finite\ncertificates, rather than floating-point matrix comparisons.","\\section{Matrix transport and actual unitarity}\nThe semantics layer proves that coordinate matrix multiplication, adjoint,\nidentity, and matrix-vector action have their ordinary complex meanings.\nThe matrices \\(F,P,D\\) and the normalized states are then defined directly\nin that complex representation.","This is the key strengthening over the earlier checkpoint. The rational matrix,\nthe exact coordinate kernel, and the complex matrix \\(FDF^*\\) are explicitly\nconnected. State preparation and recovery are equalities of actual complex\nvectors. The theorem name \\texttt{state\\_preparation} describes a matrix identity,\nnot an efficient hardware preparation algorithm.","Its normalization lemmas prove \\(\\sqrt{12}=2\\sqrt3\\) and\n\\(\\sqrt{48}=4\\sqrt3\\), then identify the exact scales with their ordinary\nreciprocals. The matrix entry formula and state formula below are therefore\nliteral equalities in \\(\\mathbb C\\).","The convention is positive sign for \\(F\\) and negative sign for \\(F^*\\). The\nearlier unnormalized \\texttt{dft12} uses the negative sign. The normalization\nand adjoint are the precise link between those notations.","\\section{Hermitian projection and phase-sensitive states}\nThe selector is Hermitian and idempotent. Transport through a unitary matrix\nthen proves the corresponding facts for \\(P\\). They are ordinary matrix\nidentities, not claims that every rational list operation is an orthogonal\nprojection.","The vectors \\(\\psi\\) and \\(\\chi\\) are separately defined states. One is the\nsigned arithmetic amplitude vector and the other its entrywise absolute value.\nThey have the same squared amplitudes at each initial index but retain different\nrelative phase. The development neither claims nor requires a universal quantum\noperation implementing the nonlinear map \\(v\\mapsto |v|\\).","The transformed probability statement is exact. The measurement event is the\ncoprime-frequency subset. After applying \\(F^*\\), its probability is one for\n\\(\\psi\\) and zero for \\(\\chi\\). This basis change reveals information that their\nshared initial histogram does not contain.","\\section{Two exact information-loss statements}\nThe formal development isolates distinct obstructions. First, the initial\nhistogram cannot be inverted on both states by any reconstruction function.\nSecond, the projector itself is noninjective because it annihilates the nonzero\nnormalized state \\(\\chi\\).","Its proof is algebraic. The identity \\(Pv+(I-P)v=v\\) itself does not require\nidempotence. Hermitian idempotence supplies the stronger interpretation as\northogonal retained and discarded components. This separates the reconstruction\nidentity from the additional geometric structure.","The manuscript's QA18 additionally includes filtering and energy, normalization\nof \\(u\\), the projector actions on both states, and noninjectivity. Those clauses\nuse separately exported declarations, including\n\\texttt{projector12\\_recovers\\_test\\_signal}, \\texttt{energy\\_decomposition},\n\\texttt{unitState\\_normalized}, \\texttt{projector\\_fixes\\_psi},\n\\texttt{projector\\_kills\\_chi}, and \\texttt{projector\\_not\\_injective}.\nThe complete machine concordance lists these supporting results. The manuscript\nbundle is not identified with a literal statement that omits its extra clauses.","QA19 & Relative-phase family & Prose extension \\(\\sin^2(\\theta/2)\\); not an inherited Lean theorem.","QA20 & Dephased probability & Prose density-matrix calculation \\(1/3\\); not an inherited Lean theorem.","The later source explicitly supplies \\(FDF^\\dagger=P\\), both normalized\nstate-action identities, the conventional exponential formula for \\(F\\),\nidentical initial probabilities, orthogonality, exact selected\nprobability one versus zero, histogram nonreconstruction, and projector\nnoninjectivity. It therefore closes the representation-transport\nobligations that the earlier package leaves outside its formal boundary.","\\texttt{projector12\\_idempotent} & Exact rational multiplication gives\n\\(P^2=P\\)","\\texttt{projector12\\_fixes\\_ramanujan} & The matrix fixes the signal\n\\(c\\)","\\texttt{projector12\\_kills\\_noise} & The matrix sends the displayed\nperturbation \\(b\\) to zero","\\texttt{projector12\\_recovers\\_test\\_signal} & Applying it to \\(x\\)\nreturns \\(c\\)","\\texttt{ramanujan12\\_energy} & \\(\\|c\\|^2=48\\)","\\texttt{noise12\\_energy} & \\(\\|b\\|^2=102\\)","\\texttt{testSignal12\\_energy} & \\(\\|x\\|^2=150\\)","\\texttt{energy\\_decomposition} & The exact energy balance is\n\\(150=48+102\\)","\\texttt{ramanujan\\_dft\\_table} & The negative-sign DFT gives twelve\nexactly at \\(1,5,7,11\\)","\\texttt{recovered\\_mode\\_probability} & The scalar normalization gives\n\\(12^2/(48\\cdot12)=1/4\\)","\\[\nP=FDF^\\dagger.\n\\]","Here \\(D\\) is a diagonal zero-or-one matrix, \\(F\\) is unitary, and\n\\(\\dagger\\) denotes conjugate transpose. Consequently \\(P\\) is an\northogonal projector. The transformation \\(F\\) is reversible. The\nselection \\(P\\) need not be. A complete decomposition consists of both\n\\(Px\\) and \\((I-P)x\\), whose sum recovers \\(x\\).","We work over \\(\\mathbb C\\) for Fourier transformations and use\n\\(\\mathbb Q\\) for the rational kernels. All index sets in the main\nfinite construction are specified, and no limit argument is needed. A\npacket in this chapter means a component together with its stated\nrealization map. This definition does not turn every kind of component\ninto the same object. Character-order packets and coefficient-residue\nlanes will be distinguished explicitly in the polyphase section.","\\[\n\\begin{aligned}\nc&=(4,0,2,0,-2,0,-4,0,-2,0,2,0),\\\\\nb&=(5,2,-1,2,5,2,-1,2,5,2,-1,2),\\\\\nx&=(9,2,1,2,3,2,-5,2,3,2,1,2).\n\\end{aligned}\n\\]","These are the literal coefficients of the formal source, and \\(x=c+b\\).\nKeeping one example fixed makes every arrow independently checkable.","Let \\(q\\) be a positive integer. Write \\(I_q=\\{0,1,\\ldots,q-1\\}\\),\ninterpreting indices modulo \\(q\\), and put","\\[\n\\zeta_q=e^{2\\pi i/q}.\n\\]","A signal is a vector \\(x:I_q\\to\\mathbb C\\). We write \\(\\operatorname{Id}_q\\) for the identity matrix of dimension \\(q\\), and \\(I\\) for an identity matrix when its dimension is clear. Our inner product is\nconjugate-linear in the first argument:","\\[\n\\langle x,y\\rangle=\\sum_{n=0}^{q-1}\\overline{x(n)}y(n),\n\\qquad \\|x\\|^2=\\langle x,x\\rangle.\n\\]","\\[\n\\widehat{x}(k)=\\sum_{n=0}^{q-1}x(n)\\zeta_q^{-kn}.\n\\]","\\[\n(F_q)_{na}=q^{-1/2}\\zeta_q^{na}.\n\\]","Thus \\(F_q\\) synthesizes a signal from normalized frequency coordinates,\nwhile \\(F_q^\\dagger x=\\widehat{x}/\\sqrt q\\) analyzes it. This convention\nexplains all signs and all square-root factors below. The negative-sign\ntransform and the positive-sign synthesis are adjoints of one another.","For \\(a\\in I_q\\), the multiplicative order of \\(\\zeta_q^a\\) is","\\[\n\\frac q{\\gcd(a,q)}.\n\\]","\\textbf{Proof.} The equation \\((\\zeta_q^a)^t=1\\) holds exactly when\n\\(q\\) divides \\(at\\). Write \\(g=\\gcd(a,q)\\), \\(q=gq'\\), and \\(a=ga'\\),\nwith \\(\\gcd(a',q')=1\\). The divisibility condition is \\(q'\\mid a't\\),\nwhich is equivalent to \\(q'\\mid t\\). Its smallest positive solution is\n\\(t=q'\\). This includes \\(a=0\\): then \\(g=q\\) and the order is one.\n\\(\\square\\)","For any integer \\(h\\),","\\[\n\\sum_{n=0}^{q-1}\\zeta_q^{hn}\n=\\begin{cases}q,&q\\mid h,\\\\0,&q\\nmid h.\\end{cases}\n\\]","\\textbf{Proof.} If \\(q\\mid h\\), every summand is one. Otherwise\n\\(r=\\zeta_q^h\\ne1\\) and \\(r^q=1\\). Multiplying the sum by \\(1-r\\) gives\n\\(1-r^q=0\\), so the sum is zero. This argument also covers \\(q=1\\),\nwhere only the first case occurs. \\(\\square\\)","For every positive \\(q\\),","\\[\nF_q^\\dagger F_q=\\operatorname{Id}_q,\n\\qquad F_qF_q^\\dagger=\\operatorname{Id}_q.\n\\]","\\[\nx(n)=\\frac1q\\sum_{k=0}^{q-1}\\widehat{x}(k)\\zeta_q^{kn},\n\\qquad\n\\sum_n|x(n)|^2=\\frac1q\\sum_k|\\widehat{x}(k)|^2.\n\\]","\\textbf{Proof.} The \\((a,b)\\) entry of the first matrix product is","\\[\n\\frac1q\\sum_{n=0}^{q-1}\\zeta_q^{n(b-a)}.\n\\]","By QA02 it is one when \\(a=b\\) and zero otherwise. The \\((n,m)\\) entry\nof the second product is the same orthogonality sum with indices\ninterchanged. Since \\(F_q^\\dagger x=\\widehat{x}/\\sqrt q\\), the identity\n\\(F_qF_q^\\dagger x=x\\) is the displayed inverse formula. Finally, a\nunitary matrix preserves inner products:","\\[\n\\|F_q^\\dagger x\\|^2\n=x^\\dagger F_qF_q^\\dagger x\n=x^\\dagger x.\n\\]","Replacing \\(F_q^\\dagger x\\) by \\(\\widehat{x}/\\sqrt q\\) proves Parseval.\n\\(\\square\\)","These proofs use finite sums, ordinary complex multiplication, and the\npositive real square root of \\(q\\). No asymptotic approximation enters.\nAt \\(q=1\\) the Fourier matrix is the one-by-one identity, and all\nidentities have their literal scalar meanings.","\\[\nU_q=\\{a\\in I_q:\\gcd(a,q)=1\\},\n\\qquad \\varphi(q)=|U_q|.\n\\]","\\[\nc_q(n)=\\sum_{a\\in U_q}\\zeta_q^{an},\n\\qquad n\\in\\mathbb Z.\n\\]","It adds the primitive \\(q\\)th-root phases selected by QA01. Its\ndefinition is complex, but its value is an integer. That integrality is\nthe step that makes the sample-coordinate projector rational. Ramanujan\nsums and their divisor formulas are classical arithmetic constructions\n\\cite{q:Ramanujan1918}.","Let \\(\\mu\\) be the Möbius function, with \\(\\mu(1)=1\\). For every\npositive \\(q\\) and every integer \\(n\\),","\\[\n\\boxed{c_q(n)=\\sum_{d\\mid\\gcd(q,n)}d\\,\\mu(q/d)\\in\\mathbb Z.}\n\\]","In particular \\(c_q\\) has period \\(q\\), is real-valued, satisfies\n\\(c_q(-n)=c_q(n)\\), and has \\(c_q(0)=\\varphi(q)\\).","\\textbf{Proof.} For a positive integer \\(g\\),","\\[\n\\sum_{d\\mid g}\\mu(d)\n=\\begin{cases}1,&g=1,\\\\0,&g>1.\\end{cases}\n\\]","To see this, only squarefree divisors contribute. If \\(g\\) has \\(r\\)\ndistinct prime factors, the sum is \\((1-1)^r\\), with value one when\n\\(r=0\\). Therefore","\\[\n\\mathbf1_{\\gcd(a,q)=1}=\\sum_{d\\mid\\gcd(a,q)}\\mu(d).\n\\]","Substitute this into the phase sum over all \\(a\\in I_q\\), exchange the\nfinite sums, and write \\(a=db\\) in the terms where \\(d\\mid a\\):","\\[\n\\begin{aligned}\nc_q(n)\n&=\\sum_{d\\mid q}\\mu(d)\n  \\sum_{b=0}^{q/d-1}e^{2\\pi ibn/(q/d)}\\\\\n&=\\sum_{\\substack{d\\mid q\\\\(q/d)\\mid n}}\\mu(d)\\frac qd.\n\\end{aligned}\n\\]","The second equality is QA02. Replacing \\(q/d\\) by \\(e\\) gives the\ndisplayed divisor sum. It is an integer. Periodicity follows already\nfrom the phase definition. The divisor formula is unchanged by\n\\(n\\mapsto -n\\), so \\(c_q(-n)=c_q(n)\\). At \\(n=0\\), every selected phase\nis one, giving \\(c_q(0)=|U_q|\\). \\(\\square\\)","At \\(q=1\\), \\(U_1=\\{0\\}\\) because \\(\\gcd(0,1)=1\\). The phase sum and the\ndivisor formula both give \\(c_1(n)=1\\). This boundary case is included\nin the proofs even though the inherited arithmetic regression range\nbegins at modulus two.","\\[\n\\boxed{\\widehat{c_q}(k)=q\\,\\mathbf1_{k\\in U_q},\\qquad k\\in I_q.}\n\\]","\\[\n\\sum_{n=0}^{q-1}|c_q(n)|^2=q\\varphi(q).\n\\]","\\[\n\\widehat{c_q}(k)\n=\\sum_{a\\in U_q}\\sum_{n=0}^{q-1}\\zeta_q^{(a-k)n}.\n\\]","QA02 makes the inner sum \\(q\\) exactly when \\(a=k\\), and zero otherwise.\nThe energy identity follows from QA03:","\\[\n\\|c_q\\|^2\n=\\frac1q\\sum_k|\\widehat{c_q}(k)|^2\n=\\frac1q\\varphi(q)q^2.\n\\] \\(\\square\\)","It is helpful to distinguish the size of an ambient signal from the\norder of a character. Fix a positive ambient length \\(N\\). For each\ndivisor \\(d\\) of \\(N\\), define","\\[\nS_d=\\{a\\in I_N:N/\\gcd(a,N)=d\\}.\n\\]","The set \\(S_d\\) contains the labels of characters of exact order \\(d\\).\nPut \\(D_d=\\operatorname{diag}(\\mathbf1_{a\\in S_d})\\) and","\\[\nP_d^{(N)}=F_ND_dF_N^\\dagger.\n\\]","The superscript records the ambient dimension. In particular, the kernel\nnormalization below is \\(1/N\\), even when \\(d<N\\).","For every \\(d\\mid N\\),","\\[\n\\boxed{(P_d^{(N)})_{nm}=\\frac1N c_d(n-m).}\n\\]","\\[\n\\begin{aligned}\n(P_d^{(N)})^\\dagger&=P_d^{(N)},\\\\\nP_d^{(N)}P_e^{(N)}&=\\delta_{de}P_d^{(N)},\\\\\n\\sum_{d\\mid N}P_d^{(N)}&=\\operatorname{Id}_N,\\\\\n\\operatorname{rank}P_d^{(N)}&=\\operatorname{tr}P_d^{(N)}=\\varphi(d).\n\\end{aligned}\n\\]","\\textbf{Proof.} A label in \\(S_d\\) has the unique form \\(a=(N/d)b\\),\nwhere \\(0\\le b<d\\) and \\(\\gcd(b,d)=1\\). Indeed,","\\[\n\\gcd((N/d)b,N)=(N/d)\\gcd(b,d).\n\\]","\\[\n\\begin{aligned}\n(F_ND_dF_N^\\dagger)_{nm}\n&=\\frac1N\\sum_{a\\in S_d}\\zeta_N^{a(n-m)}\\\\\n&=\\frac1N\\sum_{b\\in U_d}\\zeta_d^{b(n-m)}\n=\\frac1N c_d(n-m).\n\\end{aligned}\n\\]","QA04 proves rationality. Each \\(D_d\\) is a self-adjoint diagonal\nprojector. The sets \\(S_d\\) partition \\(I_N\\), because every character\nhas exactly one order dividing \\(N\\). Hence \\(D_dD_e=\\delta_{de}D_d\\)\nand \\(\\sum_{d\\mid N}D_d=\\operatorname{Id}_N\\). Conjugating these identities by the\nunitary \\(F_N\\) proves the projector identities. The rank is unchanged\nunder multiplication by invertible matrices, so it equals the number of\nselected diagonal entries, \\(|S_d|=\\varphi(d)\\). The diagonal kernel\nentries are all \\(c_d(0)/N=\\varphi(d)/N\\), whose sum is \\(\\varphi(d)\\).\n\\(\\square\\)","nal\nprojector. The sets \\(S_d\\) partition \\(I_N\\), because every character\nhas exactly one order dividing \\(N\\). Hence \\(D_dD_e=\\delta_{de}D_d\\)\nand \\(\\sum_{d\\mid N}D_d=\\operatorname{Id}_N\\). Conjugating these identities by the\nunitary \\(F_N\\) proves the projector identities. The rank is unchanged\nunder multiplication by invertible matrices, so it equals the number of\nselected diagonal entries, \\(|S_d|=\\varphi(d)\\). The diagonal kernel\nentries are all \\(c_d(0)/N=\\varphi(d)/N\\), whose sum is \\(\\varphi(d)\\).\n\\(\\square\\)","every character\nhas exactly one order dividing \\(N\\). Hence \\(D_dD_e=\\delta_{de}D_d\\)\nand \\(\\sum_{d\\mid N}D_d=\\operatorname{Id}_N\\). Conjugating these identities by the\nunitary \\(F_N\\) proves the projector identities. The rank is unchanged\nunder multiplication by invertible matrices, so it equals the number of\nselected diagonal entries, \\(|S_d|=\\varphi(d)\\). The diagonal kernel\nentries are all \\(c_d(0)/N=\\varphi(d)/N\\), whose sum is \\(\\varphi(d)\\).\n\\(\\square\\)","ividing \\(N\\). Hence \\(D_dD_e=\\delta_{de}D_d\\)\nand \\(\\sum_{d\\mid N}D_d=\\operatorname{Id}_N\\). Conjugating these identities by the\nunitary \\(F_N\\) proves the projector identities. The rank is unchanged\nunder multiplication by invertible matrices, so it equals the number of\nselected diagonal entries, \\(|S_d|=\\varphi(d)\\). The diagonal kernel\nentries are all \\(c_d(0)/N=\\varphi(d)/N\\), whose sum is \\(\\varphi(d)\\).\n\\(\\square\\)","ence \\(D_dD_e=\\delta_{de}D_d\\)\nand \\(\\sum_{d\\mid N}D_d=\\operatorname{Id}_N\\). Conjugating these identities by the\nunitary \\(F_N\\) proves the projector identities. The rank is unchanged\nunder multiplication by invertible matrices, so it equals the number of\nselected diagonal entries, \\(|S_d|=\\varphi(d)\\). The diagonal kernel\nentries are all \\(c_d(0)/N=\\varphi(d)/N\\), whose sum is \\(\\varphi(d)\\).\n\\(\\square\\)","This orthogonal divisor-shell decomposition has established antecedents\nin Ramanujan subspaces and filter banks \\cite{q:Vaidyanathan2014FIR,q:Vaidyanathan2014I,q:Vaidyanathan2014II}. The general theorem above is\nproved independently in this text. The formal companion certifies the\nconcrete \\(N=d=12\\) projector and its transport identity, rather than an\narbitrary-divisor Lean theorem.","Let \\(A\\) be the cyclic shift \\((Ax)(n)=x(n-1)\\). A Fourier column\n\\(f_a(n)=N^{-1/2}\\zeta_N^{na}\\) satisfies \\(Af_a=\\zeta_N^{-a}f_a\\).\nTherefore","\\[\nE_a=\\frac1N\\sum_{\\ell=0}^{N-1}\\zeta_N^{a\\ell}A^\\ell\n\\]","acts as the identity on \\(f_a\\) and as zero on every other column, by\nQA02. Summing \\(E_a\\) over \\(S_d\\) gives","\\[\nP_d^{(N)}=\\frac1N\\sum_{\\ell=0}^{N-1}c_d(\\ell)A^\\ell.\n\\]","This formula also follows entrywise because \\((A^\\ell)_{nm}\\) is one\nexactly when \\(n-m=\\ell\\) modulo \\(N\\). A group-algebra convention with\nthe opposite shift sign writes individual idempotents with the opposite\nphase sign. The divisor shells are unchanged under \\(a\\mapsto -a\\), and\n\\(c_d(-\\ell)=c_d(\\ell)\\), so the rational shell formula remains the\nsame.","If \\(\\mathcal D\\) is any set of divisors of \\(N\\), then","\\[\nP_{\\mathcal D}=\\sum_{d\\in\\mathcal D}P_d^{(N)}\n\\]","is a rational orthogonal projector of rank\n\\(\\sum_{d\\in\\mathcal D}\\varphi(d)\\). Its complement is the sum over the\nremaining divisors. Individual split Fourier projectors can have\nnonrational entries. Summing complete order classes is a sufficient\nmechanism for rationality. No converse classification of every rational\nselector is needed for the present bridge.","For any set \\(S\\subseteq I_N\\), let \\(D_S\\) be its diagonal indicator\nand \\(P_S=F_ND_SF_N^\\dagger\\). This construction does not require \\(S\\)\nto be an order class. Rationality is special, but orthogonal projection\nis general.","For every \\(x\\in\\mathbb C^N\\),","\\[\n\\begin{aligned}\nx&=P_Sx+(I-P_S)x,\\\\\n\\langle P_Sx,(I-P_S)x\\rangle&=0,\\\\\n\\|x\\|^2&=\\|P_Sx\\|^2+\\|(I-P_S)x\\|^2.\n\\end{aligned}\n\\]","Furthermore, \\(P_S\\) is injective exactly when \\(S=I_N\\). If \\(S\\) is\nproper, its kernel has dimension \\(N-|S|\\).","\\textbf{Proof.} The first equality is distributivity. Since\n\\(P_S^\\dagger=P_S\\) and \\(P_S^2=P_S\\),","\\[\n\\langle P_Sx,(I-P_S)x\\rangle\n=x^\\dagger P_S(I-P_S)x=0.\n\\]","Expanding the norm square of the orthogonal sum proves the energy\nidentity. In Fourier coordinates, \\(P_S\\) is \\(D_S\\), whose kernel\nconsists exactly of the vectors supported outside \\(S\\). Its dimension\nis \\(N-|S|\\). This kernel is zero exactly when every label is selected.\n\\(\\square\\)","This gives a precise reconstruction contract. The pair of packets\n\\((P_Sx,(I-P_S)x)\\) retains the complete vector, subject to the\ncompatibility condition that the first lies in the range and the second\nin the kernel. Keeping only the first packet is an exact projection but\nan incomplete representation of an arbitrary input.","\\[\nx=\\sum_{d\\mid N}P_d^{(N)}x,\n\\qquad\n\\|x\\|^2=\\sum_{d\\mid N}\\|P_d^{(N)}x\\|^2.\n\\]","Let \\(L\\ge1\\) and let \\(f(z)=\\sum_{n\\ge0}a_nz^n\\) be a formal power\nseries over a field containing a primitive \\(L\\)th root \\(\\omega\\) and\n\\(1/L\\). For \\(0\\le r<L\\), define","\\[\nE_rf(t)=\\sum_{m\\ge0}a_{r+Lm}t^m,\n\\qquad\nR_rf(z)=z^rE_rf(z^L).\n\\]","\\[\n\\begin{aligned}\nR_rf(z)&=\\frac1L\\sum_{j=0}^{L-1}\\omega^{-rj}f(\\omega^jz),\\\\\nR_rR_s&=\\delta_{rs}R_r,\\\\\n\\sum_{r=0}^{L-1}R_rf&=f.\n\\end{aligned}\n\\]","\\textbf{Proof.} The coefficient of \\(z^n\\) in the first right-hand side\nis","\\[\n\\frac{a_n}{L}\\sum_{j=0}^{L-1}\\omega^{j(n-r)}.\n\\]","Character orthogonality makes it \\(a_n\\) when \\(n\\equiv r\\pmod L\\) and\nzero otherwise. This is precisely the coefficient of \\(R_rf\\). Each\ncoefficient position belongs to a unique lane, so the projector and\ncompleteness identities follow coefficientwise. The equality is formal,\nwith no analytic convergence hypothesis. \\(\\square\\)","Now suppose \\(N=LM\\) with positive integers \\(L\\) and \\(M\\). For a\nvector \\(x\\in\\mathbb C^N\\), define its \\(L\\) lanes by","\\[\nx_r(m)=x(r+Lm),\n\\qquad 0\\le r<L,\\quad 0\\le m<M.\n\\]","The addressing map \\(T:x\\mapsto(x_0,\\ldots,x_{L-1})\\) is bijective.\nInterleaving the lanes reconstructs \\(x\\), and","\\[\n\\sum_{n=0}^{N-1}|x(n)|^2\n=\\sum_{r=0}^{L-1}\\sum_{m=0}^{M-1}|x_r(m)|^2.\n\\]","Write \\(\\widehat{x_r}\\) for the unscaled \\(M\\)-point transform of lane\n\\(r\\). Then for \\(k\\in I_N\\),","\\[\n\\boxed{\\widehat{x}(k)\n=\\sum_{r=0}^{L-1}\\zeta_N^{-kr}\\widehat{x_r}(k\\bmod M).}\n\\]","Conversely, for \\(\\ell\\in I_M\\),","\\[\n\\boxed{\\widehat{x_r}(\\ell)\n=\\frac1L\\sum_{j=0}^{L-1}\n\\zeta_N^{r(\\ell+jM)}\\widehat{x}(\\ell+jM).}\n\\]","\\textbf{Proof.} Partition the \\(N\\) samples by their unique\nrepresentation \\(n=r+Lm\\):","\\[\n\\begin{aligned}\n\\widehat{x}(k)\n&=\\sum_{r=0}^{L-1}\\sum_{m=0}^{M-1}\nx_r(m)\\zeta_N^{-k(r+Lm)}\\\\\n&=\\sum_r\\zeta_N^{-kr}\\sum_mx_r(m)\\zeta_M^{-km}.\n\\end{aligned}\n\\]","The inner sum is \\(\\widehat{x_r}(k\\bmod M)\\), proving the first\nidentity. For the second, insert Fourier inversion into the lane\ntransform:","\\[\n\\begin{aligned}\n\\widehat{x_r}(\\ell)\n&=\\sum_{m=0}^{M-1}x(r+Lm)\\zeta_M^{-\\ell m}\\\\\n&=\\frac1N\\sum_{k=0}^{N-1}\\widehat{x}(k)\\zeta_N^{kr}\n\\sum_{m=0}^{M-1}\\zeta_M^{(k-\\ell)m}.\n\\end{aligned}\n\\]","QA02 makes the inner sum \\(M\\) exactly when \\(k=\\ell+jM\\) for some\n\\(0\\le j<L\\). Since \\(M/N=1/L\\), this is the second displayed identity.\n\\(\\square\\)","Every lane transform combines an alias class of \\(L\\) full-spectrum\ncoefficients. One decimated lane generally cannot distinguish those\ncoefficients. All \\(L\\) offsets together retain the original signal. For\na fixed alias class \\(\\ell\\), the equations across \\(r\\) are an\ninvertible Fourier mixing system after the nonzero phases\n\\(\\zeta_N^{r\\ell}\\) are removed. The apparent collapse is resolved by\npreserving the complete collection of offsets.","A coefficient-residue selector keeps samples such as\n\\(x(0),x(3),x(6),x(9)\\). A character-order projector keeps spectral\nmodes such as \\(1,5,7,11\\). The former is diagonal in sample\ncoordinates; the latter is diagonal in frequency coordinates. Under a\nFourier basis change, a sample-residue mask generally produces a\ndifferent operator, with its own alias structure.","We now fix \\(q=12\\). The coprime labels are","\\[\nU_{12}=\\{1,5,7,11\\},\n\\qquad \\varphi(12)=4,\n\\qquad \\zeta=e^{\\pi i/6}=\\frac{\\sqrt3}{2}+\\frac i2.\n\\]","\\[\nc(n)=\\zeta^n+\\zeta^{5n}+\\zeta^{7n}+\\zeta^{11n}\n=2\\cos(\\pi n/6)+2\\cos(5\\pi n/6).\n\\]","\\[\nc=(4,0,2,0,-2,0,-4,0,-2,0,2,0),\n\\]","\\[\n\\widehat c=(0,12,0,0,0,12,0,12,0,0,0,12).\n\\]","\\textbf{Proof.} For \\(q=12\\), QA04 becomes","\\[\nc(n)=2\\mathbf1_{2\\mid n}-4\\mathbf1_{4\\mid n}\n-6\\mathbf1_{6\\mid n}+12\\mathbf1_{12\\mid n}.\n\\]","Indeed, the only divisors \\(d\\) of twelve for which \\(\\mu(12/d)\\ne0\\)\nare \\(2,4,6,12\\), with coefficients \\(2,-4,-6,12\\). Evaluating the\ndivisibility indicators at \\(0,\\ldots,11\\) gives the sample table. QA05\ngives value twelve at the four coprime labels and zero elsewhere.\n\\(\\square\\)","Put \\(P=P_{12}^{(12)}\\). Its entries are","\\[\nP_{nm}=\\frac{c(n-m)}{12}.\n\\]","\\[\n\\left(\\frac13,0,\\frac16,0,-\\frac16,0,-\\frac13,0,-\\frac16,0,\\frac16,0\\right),\n\\]","With \\(D=\\operatorname{diag}(\\mathbf1_{a\\in U_{12}})\\) and \\(F=F_{12}\\),","\\[\nP=FDF^\\dagger,\n\\qquad P^2=P,\n\\qquad P^\\dagger=P,\n\\qquad \\operatorname{rank}P=\\operatorname{tr}P=4.\n\\]","The six divisor shells of length twelve have orders \\(1,2,3,4,6,12\\) and\nranks \\(1,1,2,2,2,4\\). Their ranks add to twelve, and their orthogonal\nsum is the identity by QA06.","For \\(L=3\\) and \\(M=4\\), the coefficient-residue lanes are","\\[\n\\begin{aligned}\nc_0&=(4,0,-4,0),\\\\\nc_1&=(0,-2,0,2),\\\\\nc_2&=(2,0,-2,0).\n\\end{aligned}\n\\]","\\[\n\\widehat c_0=(0,8,0,8),\n\\quad \\widehat c_1=(0,4i,0,-4i),\n\\quad \\widehat c_2=(0,4,0,4).\n\\]","For example, a full coefficient at \\(k=1\\) is reconstructed as","\\[\n\\widehat c(1)=8+4i\\zeta^{-1}+4\\zeta^{-2}=12.\n\\]","At \\(k=9\\), the same lane-frequency index \\(1\\) gives","\\[\n\\widehat c(9)=8+4i\\zeta^{-9}+4\\zeta^{-18}=8-4-4=0.\n\\]","\\[\nb(n)=3\\cos(\\pi n/2)+2.\n\\]","The constant occupies frequency zero, and the cosine occupies labels\nthree and nine. These labels all lie outside \\(U_{12}\\). The\nperturbation has the literal rational vector","\\[\nb=(5,2,-1,2,5,2,-1,2,5,2,-1,2).\n\\]","Set \\(x=c+b\\), giving","\\[\nx=(9,2,1,2,3,2,-5,2,3,2,1,2).\n\\]","\\[\nPc=c,\n\\qquad Pb=0,\n\\qquad Px=c,\n\\qquad (I-P)x=b,\n\\]","\\[\n\\langle c,b\\rangle=0,\n\\qquad \\|c\\|^2=48,\n\\qquad \\|b\\|^2=102,\n\\qquad \\|x\\|^2=150=48+102.\n\\]","\\textbf{Proof.} QA05 gives the support of \\(\\widehat c\\). Write the\ncosine as","\\[\n3\\cos(\\pi n/2)=\\frac32(\\zeta^{3n}+\\zeta^{9n}).\n\\]","Thus \\(\\widehat b\\) equals twenty-four at zero, eighteen at three and\nnine, and zero elsewhere. Multiplication by \\(D\\) fixes the spectrum of\n\\(c\\) and kills the spectrum of \\(b\\), proving the filter identities.\nTheir supports are disjoint, so Parseval for inner products gives\n\\(\\langle c,b\\rangle=0\\). QA05 gives \\(\\|c\\|^2=12\\cdot4=48\\), and","\\[\n\\|b\\|^2=\\frac{24^2+18^2+18^2}{12}=102.\n\\]","QA07 then gives \\(\\|x\\|^2=48+102=150\\). The same numbers follow by\ndirect rational squaring of the displayed vectors. \\(\\square\\)","The discarded component is part of a complete explanation of the input.\nIt is called a perturbation because the example declares \\(c\\) to be the\ntarget. It is a perfectly exact signal component rather than an error\nintroduced by the representation. If the task were to recover all of\n\\(x\\), deleting \\(b\\) would lose valid information.","In a twelve-dimensional complex Hilbert space, let\n\\(|0\\rangle,\\ldots,|11\\rangle\\) be an orthonormal basis. Define","\\[\n|u\\rangle=\\frac{|1\\rangle+|5\\rangle+|7\\rangle+|11\\rangle}{2}.\n\\]","Its squared norm is \\(4(1/2)^2=1\\). Define the amplitude states","\\[\n|\\psi\\rangle=\\frac1{\\sqrt{48}}\\sum_{n=0}^{11}c(n)|n\\rangle,\n\\qquad\n|\\chi\\rangle=\\frac1{\\sqrt{48}}\\sum_{n=0}^{11}|c(n)|\\,|n\\rangle.\n\\]","Both have norm one because \\(\\sum_nc(n)^2=48\\). The second state is a\nseparately specified state, constructed from the absolute-value vector.\nNo universal quantum operation implementing the nonlinear rule\n\\(z\\mapsto|z|\\) is being asserted.","\\[\n\\boxed{F|u\\rangle=|\\psi\\rangle,\n\\qquad F^\\dagger|\\psi\\rangle=|u\\rangle.}\n\\]","\\textbf{Proof.} At coordinate \\(n\\), the first left-hand side is","\\[\n\\frac1{2\\sqrt{12}}\\sum_{a\\in U_{12}}\\zeta^{na}\n=\\frac{c(n)}{\\sqrt{48}}.\n\\]","This is the defining coordinate of \\(\\psi\\). Applying \\(F^\\dagger\\) and\nusing \\(F^\\dagger F=I\\) proves the second equality. \\(\\square\\)","The normalization relation is \\(\\sqrt{48}=2\\sqrt{12}\\). In the exact\ncoordinate implementation the Fourier scale is \\(\\sqrt3/6=1/\\sqrt{12}\\),\nand the state scale is \\(\\sqrt3/12=1/\\sqrt{48}\\). The later formal\nsource proves those scale identities and the exponential entry formula,\nrather than leaving them as informal interpretations of a rational\ntuple.","This is a mathematical quantum-state construction. An actual realization\nwould additionally require state preparation, a twelve-dimensional\nunitary implementation, coherence control, and sampling. A\ntwelve-dimensional Fourier matrix is not the ordinary\nsixteen-dimensional Fourier transformation on four qubits. The unitary\ndirect sum \\(F_{12}\\oplus I_4\\) supplies a mathematical\nsixteen-dimensional embedding, but it supplies no gate synthesis or\nresource bound by itself.","For a normalized state \\(|v\\rangle=\\sum_nv(n)|n\\rangle\\), the\nstandard-basis probabilities are \\(|v(n)|^2\\). In the basis analyzed by\n\\(F^\\dagger\\), the probabilities are \\(|(F^\\dagger v)(k)|^2\\).","The states \\(\\psi\\) and \\(\\chi\\) have identical probabilities in the\noriginal basis and are orthogonal:","\\[\n|\\psi(n)|^2=|\\chi(n)|^2\\quad\\text{for all }n,\n\\qquad \\langle\\psi,\\chi\\rangle=0.\n\\]","\\[\n\\langle\\psi,\\chi\\rangle\n=\\frac1{48}\\sum_{n=0}^{11}c(n)|c(n)|.\n\\]","The positive entries contribute \\(16+4+4=24\\), and the negative entries\ncontribute \\(-4-16-4=-24\\). Their sum is zero. \\(\\square\\)","After applying \\(F^\\dagger\\),","\\[\n\\Pr_{F^\\dagger\\psi}(k\\in U_{12})=1,\n\\qquad\n\\Pr_{F^\\dagger\\chi}(k\\in U_{12})=0.\n\\]","\\[\n(F^\\dagger\\psi)(k)\n=\\begin{cases}1/2,&k\\in\\{1,5,7,11\\},\\\\0,&\\text{otherwise},\\end{cases}\n\\]","\\[\n(F^\\dagger\\chi)(k)\n=\\begin{cases}\n2/3,&k\\in\\{0,6\\},\\\\\n1/6,&k\\in\\{2,4,8,10\\},\\\\\n0,&\\text{otherwise}.\n\\end{cases}\n\\]","\\textbf{Proof.} QA12 gives the first amplitude table. For the second,\nset \\(a(n)=|c(n)|\\). Its vector is","\\[\na=(4,0,2,0,2,0,4,0,2,0,2,0).\n\\]","It has period six. Pairing \\(n\\) and \\(n+6\\) in its twelve-point\ntransform produces a factor \\(1+(-1)^k\\), so every odd transform\ncoefficient vanishes. For \\(k=2\\ell\\), its transform is twice the\nsix-point transform of \\((4,0,2,0,2,0)\\). The doubled transform is\nsixteen when \\(\\ell\\) is divisible by three and four otherwise, as one\nsees from","\\[\n2\\left(4+2e^{-2\\pi i\\ell/3}+2e^{-4\\pi i\\ell/3}\\right).\n\\]","When \\(\\ell\\equiv0\\pmod3\\), the bracket is eight. Otherwise the two\ncubic phases sum to minus one and the bracket is two. Thus","\\[\n\\widehat a=(16,0,4,0,4,0,16,0,4,0,4,0).\n\\]","State normalization and the unitary inverse transform divide this table\nby \\(\\sqrt{48}\\sqrt{12}=24\\), producing the second amplitude table. Its\nsupport is even, while every label in \\(U_{12}\\) is odd. The selected\nprobability is therefore zero. The signed state has four probabilities\nof \\(1/4\\), with total one. \\(\\square\\)",">{\\centering\\arraybackslash}p{\\dimexpr 0.2500\\columnwidth - 1.5000\\tabcolsep\\relax}\n  >{\\centering\\arraybackslash}p{\\dimexpr 0.2500\\columnwidth - 1.5000\\tabcolsep\\relax}@{}}\n\\toprule\\noalign{}\n\\begin{minipage}[b]{\\linewidth}\\centering\nLabel\n\\end{minipage} & \\begin{minipage}[b]{\\linewidth}\\centering\nInitial probability, both states\n\\end{minipage} & \\begin{minipage}[b]{\\linewidth}\\centering\nAfter \\(F^\\dagger\\), signed state\n\\end{minipage} & \\begin{minipage}[b]{\\linewidth}\\centering\nAfter \\(F^\\dagger\\), unsigned state\n\\end{minipage}","ering\\arraybackslash}p{\\dimexpr 0.2500\\columnwidth - 1.5000\\tabcolsep\\relax}@{}}\n\\toprule\\noalign{}\n\\begin{minipage}[b]{\\linewidth}\\centering\nLabel\n\\end{minipage} & \\begin{minipage}[b]{\\linewidth}\\centering\nInitial probability, both states\n\\end{minipage} & \\begin{minipage}[b]{\\linewidth}\\centering\nAfter \\(F^\\dagger\\), signed state\n\\end{minipage} & \\begin{minipage}[b]{\\linewidth}\\centering\nAfter \\(F^\\dagger\\), unsigned state\n\\end{minipage}","\\midrule\\noalign{}\n\\endhead\n\\bottomrule\\noalign{}\n\\endlastfoot\n0 & \\(1/3\\) & \\(0\\) & \\(4/9\\)","1 & \\(0\\) & \\(1/4\\) & \\(0\\)","2 & \\(1/12\\) & \\(0\\) & \\(1/36\\)","3 & \\(0\\) & \\(0\\) & \\(0\\)","4 & \\(1/12\\) & \\(0\\) & \\(1/36\\)","5 & \\(0\\) & \\(1/4\\) & \\(0\\)","6 & \\(1/3\\) & \\(0\\) & \\(4/9\\)","7 & \\(0\\) & \\(1/4\\) & \\(0\\)","8 & \\(1/12\\) & \\(0\\) & \\(1/36\\)","9 & \\(0\\) & \\(0\\) & \\(0\\)","10 & \\(1/12\\) & \\(0\\) & \\(1/36\\)","11 & \\(0\\) & \\(1/4\\) & \\(0\\)","There is no function from original-basis probability histograms to state\nvectors that reconstructs both \\(\\psi\\) and \\(\\chi\\). There is also no\nreconstruction of both states up to overall phase from that histogram\nalone.","\\textbf{Proof.} The two histograms are equal by QA13. A function assigns\nthe same output to equal inputs, so it cannot output the two distinct\nvectors. For the phase-insensitive formulation, QA13 shows that the\nstates are orthogonal and normalized. If \\(\\chi=\\lambda\\psi\\) for any\ncomplex scalar of modulus one, then\n\\(\\langle\\psi,\\chi\\rangle=\\lambda\\ne0\\), a contradiction. \\(\\square\\)","The pair \\(\\{P,I-P\\}\\) is a projective measurement because its two\nmembers are self-adjoint, idempotent, orthogonal, and sum to the\nidentity. For the conditional update we choose the ideal Lüders\ninstrument, whose unnormalized branches send a density matrix \\(\\rho\\)\nto \\(P\\rho P\\) and \\((I-P)\\rho(I-P)\\). The projector effects determine\nthe outcome probabilities; this chosen instrument determines the state\nupdate. The probability rule and this conditional update can be stated\nindependently of the arithmetic example.","Let \\(P\\) be an orthogonal projector on \\(\\mathbb C^N\\) and let \\(v\\) be\na normalized state. For the ideal Lüders instrument just specified, the\ntwo probabilities are","\\[\np=\\|Pv\\|^2=\\langle v,Pv\\rangle,\n\\qquad 1-p=\\|(I-P)v\\|^2.\n\\]","When \\(p>0\\), the normalized selected state is \\(Pv/\\sqrt p\\). When\n\\(1-p>0\\), the normalized complementary state is \\((I-P)v/\\sqrt{1-p}\\).\nThe probabilities belong to \\([0,1]\\) and sum to one.","\\textbf{Proof.} Since \\(P^\\dagger P=P\\), the quadratic form equals\n\\(\\|Pv\\|^2\\) and is nonnegative. QA07 applied to a unit vector gives the\nprobability sum and the upper bound. Division by the positive square\nroot produces a state of squared norm one in each nonzero branch.\n\\(\\square\\)","For the explicit normalized test signal \\(v=x/\\sqrt{150}\\), QA11 gives","\\[\np=\\frac{48}{150}=\\frac8{25},\n\\qquad 1-p=\\frac{102}{150}=\\frac{17}{25}.\n\\]","The selected conditional state is \\(c/\\sqrt{48}=\\psi\\), and the\ncomplementary conditional state is \\(b/\\sqrt{102}\\). An exact classical\nfilter applied to the full vector returns \\(c\\). A projective\nmeasurement on the normalized input instead returns a branch with these\nprobabilities. These are compatible uses of the same operator with\ndifferent observation rules.","The vector \\(\\chi\\) makes noninjectivity concrete: \\(P\\chi=0\\) although\n\\(\\|\\chi\\|=1\\). A reversible Fourier change of basis cannot be\nsubstituted for this loss of information. Conversely, retaining both\ncomplementary packets reconstructs every vector, as QA07 and the\npublication module explicitly state.","The twelfth root contains \\(\\sqrt3\\) and \\(i\\). Its relevant exact\narithmetic can be represented without floating-point trigonometric\nevaluation. Let a tuple","\\[\n(a,b,c,d)\\in\\mathbb Q^4\n\\]","\\[\n\\iota(a,b,c,d)=a+b\\sqrt3+i(c+d\\sqrt3).\n\\]","The coordinate root is \\((0,1/2,1/2,0)\\), whose interpretation is\n\\(\\sqrt3/2+i/2\\). Coordinate conjugation sends \\((a,b,c,d)\\) to\n\\((a,b,-c,-d)\\). Addition is entrywise, and multiplication is","\\[\n\\begin{aligned}\n(a,b,c,d)(A,B,C,D)=\\big(&aA+3bB-cC-3dD,\\\\\n&aB+bA-cD-dC,\\\\\n&aC+3bD+cA+3dB,\\\\\n&aD+bC+cB+dA\\big).\n\\end{aligned}\n\\]","The map \\(\\iota\\) preserves zero, one, rational scalars, addition,\nmultiplication, conjugation, natural powers, and finite sums. Therefore\nan equality obtained with these exact coordinate operations yields the\ncorresponding equality in \\(\\mathbb C\\).","\\textbf{Proof.} The claims for zero, one, rational scalars, addition,\nand conjugation follow directly from the definition. For multiplication,\nwrite the two interpreted values as \\((a+b\\sqrt3)+i(c+d\\sqrt3)\\) and\n\\((A+B\\sqrt3)+i(C+D\\sqrt3)\\). Their product has real part","\\[\n(a+b\\sqrt3)(A+B\\sqrt3)-(c+d\\sqrt3)(C+D\\sqrt3)\n\\]","\\[\n(a+b\\sqrt3)(C+D\\sqrt3)+(c+d\\sqrt3)(A+B\\sqrt3).\n\\]","Expanding and using \\((\\sqrt3)^2=3\\) gives exactly the displayed tuple\nproduct. Power preservation follows by induction from multiplication and\none. Finite-sum preservation follows by induction from addition and\nzero. Applying a function to equal coordinate tuples gives equal complex\nnumbers. \\(\\square\\)","Only sound transport of equalities is needed here. The formal source\ndoes not install the tuple type as a global field instance, and it does\nnot need an injectivity theorem for \\(\\iota\\). It proves preservation of\nthe actual operations and explicitly connects the scales and root to\ntheir conventional complex formulas.","Let \\(F\\), \\(D\\), \\(P\\), \\(c\\), \\(b\\), \\(x\\), \\(u\\), \\(\\psi\\), and\n\\(\\chi\\) have the explicit modulus-twelve definitions above. Then:","\\item\n  \\(F^\\dagger F=FF^\\dagger=I\\).\n\\item\n  \\(P=FDF^\\dagger\\), \\(P^2=P\\), and \\(P^\\dagger=P\\).\n\\item\n  \\(Px=c\\) and \\((I-P)x=b\\), with \\(\\|x\\|^2=48+102=150\\).\n\\item\n  \\(Fu=\\psi\\) and \\(F^\\dagger\\psi=u\\).\n\\item\n  \\(u\\), \\(\\psi\\), and \\(\\chi\\) are normalized, and \\(\\psi\\) is\n  orthogonal to \\(\\chi\\).\n\\item\n  \\(\\psi\\) and \\(\\chi\\) have identical original-basis probabilities.\n\\item\n  Their inverse-Fourier coprime probabilities are respectively one and\n  zero.\n\\item\n  \\(P\\psi=\\psi\\), \\(P\\chi=0\\), and \\(P\\) is not injective.\n\\end{enumerate}","\\), and \\(P^\\dagger=P\\).\n\\item\n  \\(Px=c\\) and \\((I-P)x=b\\), with \\(\\|x\\|^2=48+102=150\\).\n\\item\n  \\(Fu=\\psi\\) and \\(F^\\dagger\\psi=u\\).\n\\item\n  \\(u\\), \\(\\psi\\), and \\(\\chi\\) are normalized, and \\(\\psi\\) is\n  orthogonal to \\(\\chi\\).\n\\item\n  \\(\\psi\\) and \\(\\chi\\) have identical original-basis probabilities.\n\\item\n  Their inverse-Fourier coprime probabilities are respectively one and\n  zero.\n\\item\n  \\(P\\psi=\\psi\\), \\(P\\chi=0\\), and \\(P\\) is not injective.\n\\end{enumerate}","ger=P\\).\n\\item\n  \\(Px=c\\) and \\((I-P)x=b\\), with \\(\\|x\\|^2=48+102=150\\).\n\\item\n  \\(Fu=\\psi\\) and \\(F^\\dagger\\psi=u\\).\n\\item\n  \\(u\\), \\(\\psi\\), and \\(\\chi\\) are normalized, and \\(\\psi\\) is\n  orthogonal to \\(\\chi\\).\n\\item\n  \\(\\psi\\) and \\(\\chi\\) have identical original-basis probabilities.\n\\item\n  Their inverse-Fourier coprime probabilities are respectively one and\n  zero.\n\\item\n  \\(P\\psi=\\psi\\), \\(P\\chi=0\\), and \\(P\\) is not injective.\n\\end{enumerate}","\\(Px=c\\) and \\((I-P)x=b\\), with \\(\\|x\\|^2=48+102=150\\).\n\\item\n  \\(Fu=\\psi\\) and \\(F^\\dagger\\psi=u\\).\n\\item\n  \\(u\\), \\(\\psi\\), and \\(\\chi\\) are normalized, and \\(\\psi\\) is\n  orthogonal to \\(\\chi\\).\n\\item\n  \\(\\psi\\) and \\(\\chi\\) have identical original-basis probabilities.\n\\item\n  Their inverse-Fourier coprime probabilities are respectively one and\n  zero.\n\\item\n  \\(P\\psi=\\psi\\), \\(P\\chi=0\\), and \\(P\\) is not injective.\n\\end{enumerate}","\\textbf{Proof.} QA03 gives the unitarity identities. QA06 gives\nprojector transport and the two projector identities. QA11 gives the\nfilter and energy statements. QA12 gives preparation and recovery; the\ndisplayed normalizations and QA13 give statement five and statement six.\nQA14 gives statement seven. In Fourier coordinates \\(\\psi\\) is supported\non \\(U_{12}\\) while \\(\\chi\\) is supported outside it, so \\(P\\psi=\\psi\\)\nand \\(P\\chi=0\\). The normalized nonzero vector \\(\\chi\\) is therefore in\nthe kernel, proving noninjectivity. \\(\\square\\)","roof.} QA03 gives the unitarity identities. QA06 gives\nprojector transport and the two projector identities. QA11 gives the\nfilter and energy statements. QA12 gives preparation and recovery; the\ndisplayed normalizations and QA13 give statement five and statement six.\nQA14 gives statement seven. In Fourier coordinates \\(\\psi\\) is supported\non \\(U_{12}\\) while \\(\\chi\\) is supported outside it, so \\(P\\psi=\\psi\\)\nand \\(P\\chi=0\\). The normalized nonzero vector \\(\\chi\\) is therefore in\nthe kernel, proving noninjectivity. \\(\\square\\)","he unitarity identities. QA06 gives\nprojector transport and the two projector identities. QA11 gives the\nfilter and energy statements. QA12 gives preparation and recovery; the\ndisplayed normalizations and QA13 give statement five and statement six.\nQA14 gives statement seven. In Fourier coordinates \\(\\psi\\) is supported\non \\(U_{12}\\) while \\(\\chi\\) is supported outside it, so \\(P\\psi=\\psi\\)\nand \\(P\\chi=0\\). The normalized nonzero vector \\(\\chi\\) is therefore in\nthe kernel, proving noninjectivity. \\(\\square\\)","or transport and the two projector identities. QA11 gives the\nfilter and energy statements. QA12 gives preparation and recovery; the\ndisplayed normalizations and QA13 give statement five and statement six.\nQA14 gives statement seven. In Fourier coordinates \\(\\psi\\) is supported\non \\(U_{12}\\) while \\(\\chi\\) is supported outside it, so \\(P\\psi=\\psi\\)\nand \\(P\\chi=0\\). The normalized nonzero vector \\(\\chi\\) is therefore in\nthe kernel, proving noninjectivity. \\(\\square\\)","filter and energy statements. QA12 gives preparation and recovery; the\ndisplayed normalizations and QA13 give statement five and statement six.\nQA14 gives statement seven. In Fourier coordinates \\(\\psi\\) is supported\non \\(U_{12}\\) while \\(\\chi\\) is supported outside it, so \\(P\\psi=\\psi\\)\nand \\(P\\chi=0\\). The normalized nonzero vector \\(\\chi\\) is therefore in\nthe kernel, proving noninjectivity. \\(\\square\\)","The two endpoint states are especially simple, but the mechanism is\ncontinuous. Define the nonnegative vectors \\(v_+\\) and \\(v_-\\) by\nplacing \\(|c(n)|/\\sqrt{48}\\) respectively on the positive and negative\nentries of \\(c\\), with zero elsewhere. Thus","\\[\n\\chi=v_++v_-,\n\\qquad \\psi=v_+-v_-.\n\\]","Their supports are disjoint, and each has squared norm \\(1/2\\). For real\n\\(\\theta\\), define","\\[\n\\chi_\\theta=v_++e^{i\\theta}v_-.\n\\]","It is a normalized state with the same original-basis probabilities for\nevery \\(\\theta\\). The endpoints are \\(\\chi_0=\\chi\\) and\n\\(\\chi_\\pi=\\psi\\).","\\[\n\\boxed{\\Pr_{F^\\dagger\\chi_\\theta}(k\\in U_{12})\n=\\|P\\chi_\\theta\\|^2=\\sin^2(\\theta/2).}\n\\]","\\textbf{Proof.} Disjoint support gives\n\\(\\|\\chi_\\theta\\|^2=\\|v_+\\|^2+\\|v_-\\|^2=1\\) and unchanged entrywise\nsquared magnitudes. Solving for \\(v_+\\) and \\(v_-\\) in terms of \\(\\chi\\)\nand \\(\\psi\\) gives","\\[\n\\chi_\\theta\n=\\frac{1+e^{i\\theta}}2\\chi\n+\\frac{1-e^{i\\theta}}2\\psi.\n\\]","Since \\(P\\chi=0\\) and \\(P\\psi=\\psi\\),","\\[\nP\\chi_\\theta=\\frac{1-e^{i\\theta}}2\\psi.\n\\]","\\[\n\\frac{|1-e^{i\\theta}|^2}{4}\n=\\frac{1-\\cos\\theta}{2}\n=\\sin^2(\\theta/2).\n\\]","Finally, \\(P=FDF^\\dagger\\) and unitarity imply\n\\(\\|P\\chi_\\theta\\|^2=\\|DF^\\dagger\\chi_\\theta\\|^2\\), exactly the sum of\nthe selected Fourier-basis probabilities. \\(\\square\\)","For a normalized vector \\(v\\), its density matrix is \\(vv^\\dagger\\).\nComplete dephasing in the sample basis keeps only its diagonal entries.\nIn the present example, dephasing either \\(\\psi\\) or \\(\\chi\\) gives the\nsame diagonal matrix","\\[\n\\rho_{\\mathrm{diag}}=\\operatorname{diag}\\big(c(n)^2/48\\big)_{n=0}^{11}.\n\\]","\\[\n\\operatorname{tr}(P\\rho_{\\mathrm{diag}})=\\frac13.\n\\]","\\textbf{Proof.} Every diagonal entry of \\(P\\) is \\(c(0)/12=4/12=1/3\\).\nMultiplication by a diagonal matrix and taking the trace gives","\\[\n\\operatorname{tr}(P\\rho_{\\mathrm{diag}})\n=\\sum_n P_{nn}\\frac{c(n)^2}{48}\n=\\frac13\\sum_n\\frac{c(n)^2}{48}\n=\\frac13.\n\\] \\(\\square\\)","twelve will become an integer signal, an exact rational filter, and a normalized quantum state. Every bridge will be written down.}\nThe guide follows one object through three coordinate systems. The starting condition is coprimality. The signal realization adds its selected waves. The quantum realization normalizes the amplitude vector and distinguishes coherent transformation from measurement.\n\\[\n U_{12}=\\{1,5,7,11\\},\\qquad\n c_{12}(n)=\\sum_{a\\in U_{12}}e^{2\\pi ian/12}.\n\\]\nWe will build a projector that keeps exactly those modes, apply it to a signal with a complementary component, and account for all retained and discarded energy. Then we will compare two orthogonal states with identical initial measurement probabilities.\n\\begin{center}\n\\begin{tabular}{lll}\\toprule\nArithmetic & Signal & Quantum model","\\clearpage\\section*{Coprimality selects primitive roots}\\addcontentsline{toc}{section}{Coprimality selects primitive roots}\n\\lead{A selected residue is a label for a wave whose phase has exact order twelve.}\nSet $\\zeta=e^{2\\pi i/12}$. The phase $\\zeta^a$ first returns to $1$ after $12/\\gcd(a,12)$ powers. Thus the coprime labels $1,5,7,11$ select exactly the primitive twelfth roots.\n\\fig{01_primitive_roots}{0.91}{The circle is the complex unit circle. Teal points are the four primitive roots; pale points are the other eight roots. Labels are residues $a$, not measured outcomes.}\n\\[\n \\gcd(a,12)=1\\quad\\Longleftrightarrow\\quad\\operatorname{ord}(\\zeta^a)=12.\n\\]\nArithmetic has supplied a precise selector. Nothing probabilistic has happened at this stage.","el for a wave whose phase has exact order twelve.}\nSet $\\zeta=e^{2\\pi i/12}$. The phase $\\zeta^a$ first returns to $1$ after $12/\\gcd(a,12)$ powers. Thus the coprime labels $1,5,7,11$ select exactly the primitive twelfth roots.\n\\fig{01_primitive_roots}{0.91}{The circle is the complex unit circle. Teal points are the four primitive roots; pale points are the other eight roots. Labels are residues $a$, not measured outcomes.}\n\\[\n \\gcd(a,12)=1\\quad\\Longleftrightarrow\\quad\\operatorname{ord}(\\zeta^a)=12.\n\\]\nArithmetic has supplied a precise selector. Nothing probabilistic has happened at this stage.","exact order twelve.}\nSet $\\zeta=e^{2\\pi i/12}$. The phase $\\zeta^a$ first returns to $1$ after $12/\\gcd(a,12)$ powers. Thus the coprime labels $1,5,7,11$ select exactly the primitive twelfth roots.\n\\fig{01_primitive_roots}{0.91}{The circle is the complex unit circle. Teal points are the four primitive roots; pale points are the other eight roots. Labels are residues $a$, not measured outcomes.}\n\\[\n \\gcd(a,12)=1\\quad\\Longleftrightarrow\\quad\\operatorname{ord}(\\zeta^a)=12.\n\\]\nArithmetic has supplied a precise selector. Nothing probabilistic has happened at this stage.","\\clearpage\\section*{The selected phases add and cancel}\\addcontentsline{toc}{section}{The selected phases add and cancel}\n\\lead{The same four waves can reinforce, cancel, or reinforce with the opposite sign.}\nAt each sample $n$, add the four numbers $\\zeta^{an}$ with $a\\in U_{12}$. Each has magnitude one. Their sum need not have magnitude one.\n\\fig{02_phase_addition}{1.0}{At $n=0$ the sum is $4$. At $n=1$ the four phases cancel. At $n=6$ the sum is $-4$. Tiny parallel drawing offsets expose coincident arrows at $n=0$ and $n=6$; each arrow's displacement is the actual complex phase.}\n\\[\n c_{12}(0)=4,\\qquad c_{12}(1)=0,\\qquad c_{12}(6)=-4.\n\\]\nNegative samples encode relative phase. They are part of the information that subsequent transforms use.","arpage\\section*{The selected phases add and cancel}\\addcontentsline{toc}{section}{The selected phases add and cancel}\n\\lead{The same four waves can reinforce, cancel, or reinforce with the opposite sign.}\nAt each sample $n$, add the four numbers $\\zeta^{an}$ with $a\\in U_{12}$. Each has magnitude one. Their sum need not have magnitude one.\n\\fig{02_phase_addition}{1.0}{At $n=0$ the sum is $4$. At $n=1$ the four phases cancel. At $n=6$ the sum is $-4$. Tiny parallel drawing offsets expose coincident arrows at $n=0$ and $n=6$; each arrow's displacement is the actual complex phase.}\n\\[\n c_{12}(0)=4,\\qquad c_{12}(1)=0,\\qquad c_{12}(6)=-4.\n\\]\nNegative samples encode relative phase. They are part of the information that subsequent transforms use.","es add and cancel}\\addcontentsline{toc}{section}{The selected phases add and cancel}\n\\lead{The same four waves can reinforce, cancel, or reinforce with the opposite sign.}\nAt each sample $n$, add the four numbers $\\zeta^{an}$ with $a\\in U_{12}$. Each has magnitude one. Their sum need not have magnitude one.\n\\fig{02_phase_addition}{1.0}{At $n=0$ the sum is $4$. At $n=1$ the four phases cancel. At $n=6$ the sum is $-4$. Tiny parallel drawing offsets expose coincident arrows at $n=0$ and $n=6$; each arrow's displacement is the actual complex phase.}\n\\[\n c_{12}(0)=4,\\qquad c_{12}(1)=0,\\qquad c_{12}(6)=-4.\n\\]\nNegative samples encode relative phase. They are part of the information that subsequent transforms use.","}\\addcontentsline{toc}{section}{The selected phases add and cancel}\n\\lead{The same four waves can reinforce, cancel, or reinforce with the opposite sign.}\nAt each sample $n$, add the four numbers $\\zeta^{an}$ with $a\\in U_{12}$. Each has magnitude one. Their sum need not have magnitude one.\n\\fig{02_phase_addition}{1.0}{At $n=0$ the sum is $4$. At $n=1$ the four phases cancel. At $n=6$ the sum is $-4$. Tiny parallel drawing offsets expose coincident arrows at $n=0$ and $n=6$; each arrow's displacement is the actual complex phase.}\n\\[\n c_{12}(0)=4,\\qquad c_{12}(1)=0,\\qquad c_{12}(6)=-4.\n\\]\nNegative samples encode relative phase. They are part of the information that subsequent transforms use.","el}\n\\lead{The same four waves can reinforce, cancel, or reinforce with the opposite sign.}\nAt each sample $n$, add the four numbers $\\zeta^{an}$ with $a\\in U_{12}$. Each has magnitude one. Their sum need not have magnitude one.\n\\fig{02_phase_addition}{1.0}{At $n=0$ the sum is $4$. At $n=1$ the four phases cancel. At $n=6$ the sum is $-4$. Tiny parallel drawing offsets expose coincident arrows at $n=0$ and $n=6$; each arrow's displacement is the actual complex phase.}\n\\[\n c_{12}(0)=4,\\qquad c_{12}(1)=0,\\qquad c_{12}(6)=-4.\n\\]\nNegative samples encode relative phase. They are part of the information that subsequent transforms use.","The same four waves can reinforce, cancel, or reinforce with the opposite sign.}\nAt each sample $n$, add the four numbers $\\zeta^{an}$ with $a\\in U_{12}$. Each has magnitude one. Their sum need not have magnitude one.\n\\fig{02_phase_addition}{1.0}{At $n=0$ the sum is $4$. At $n=1$ the four phases cancel. At $n=6$ the sum is $-4$. Tiny parallel drawing offsets expose coincident arrows at $n=0$ and $n=6$; each arrow's displacement is the actual complex phase.}\n\\[\n c_{12}(0)=4,\\qquad c_{12}(1)=0,\\qquad c_{12}(6)=-4.\n\\]\nNegative samples encode relative phase. They are part of the information that subsequent transforms use.","opposite sign.}\nAt each sample $n$, add the four numbers $\\zeta^{an}$ with $a\\in U_{12}$. Each has magnitude one. Their sum need not have magnitude one.\n\\fig{02_phase_addition}{1.0}{At $n=0$ the sum is $4$. At $n=1$ the four phases cancel. At $n=6$ the sum is $-4$. Tiny parallel drawing offsets expose coincident arrows at $n=0$ and $n=6$; each arrow's displacement is the actual complex phase.}\n\\[\n c_{12}(0)=4,\\qquad c_{12}(1)=0,\\qquad c_{12}(6)=-4.\n\\]\nNegative samples encode relative phase. They are part of the information that subsequent transforms use.","\\clearpage\\section*{Interference yields an integer signal}\\addcontentsline{toc}{section}{Interference yields an integer signal}\n\\lead{The samples are exact integers, even though the individual waves have complex amplitudes.}\nPairing conjugates gives\n\\[\n c_{12}(n)=2\\cos(\\pi n/6)+2\\cos(5\\pi n/6).\n\\]\n\\fig{03_ramanujan_signal}{1.0}{One complete period of the Ramanujan signal. The dots are discrete samples, not samples from a continuous probability distribution.}\n\\[\n (c_{12}(0),\\ldots,c_{12}(11))=(4,0,2,0,-2,0,-4,0,-2,0,2,0).\n\\]\nThe antisymmetry $c_{12}(n+6)=-c_{12}(n)$ will later expose why erasing the signs changes the recovered arithmetic.","tentsline{toc}{section}{Interference yields an integer signal}\n\\lead{The samples are exact integers, even though the individual waves have complex amplitudes.}\nPairing conjugates gives\n\\[\n c_{12}(n)=2\\cos(\\pi n/6)+2\\cos(5\\pi n/6).\n\\]\n\\fig{03_ramanujan_signal}{1.0}{One complete period of the Ramanujan signal. The dots are discrete samples, not samples from a continuous probability distribution.}\n\\[\n (c_{12}(0),\\ldots,c_{12}(11))=(4,0,2,0,-2,0,-4,0,-2,0,2,0).\n\\]\nThe antisymmetry $c_{12}(n+6)=-c_{12}(n)$ will later expose why erasing the signs changes the recovered arithmetic.","e exact integers, even though the individual waves have complex amplitudes.}\nPairing conjugates gives\n\\[\n c_{12}(n)=2\\cos(\\pi n/6)+2\\cos(5\\pi n/6).\n\\]\n\\fig{03_ramanujan_signal}{1.0}{One complete period of the Ramanujan signal. The dots are discrete samples, not samples from a continuous probability distribution.}\n\\[\n (c_{12}(0),\\ldots,c_{12}(11))=(4,0,2,0,-2,0,-4,0,-2,0,2,0).\n\\]\nThe antisymmetry $c_{12}(n+6)=-c_{12}(n)$ will later expose why erasing the signs changes the recovered arithmetic.","\\clearpage\\section*{Why the phase sum is an integer}\\addcontentsline{toc}{section}{Why the phase sum is an integer}\n\\lead{A divisor identity certifies integrality without rounding trigonometric values.}\nFor $q\\geq1$, insert the M\\\"obius coprimality identity into the finite phase sum:\n\\[\n \\mathbf1_{\\gcd(a,q)=1}=\\sum_{d\\mid\\gcd(a,q)}\\mu(d).\n\\]\nWrite $a=db$ after interchanging the finite sums. Character orthogonality evaluates the inner geometric series:\n\\[\n \\begin{aligned}\n c_q(n)&=\\sum_{d\\mid q}\\mu(d)\\sum_{b=0}^{q/d-1}e^{2\\pi ibn/(q/d)}","ntentsline{toc}{section}{Why the phase sum is an integer}\n\\lead{A divisor identity certifies integrality without rounding trigonometric values.}\nFor $q\\geq1$, insert the M\\\"obius coprimality identity into the finite phase sum:\n\\[\n \\mathbf1_{\\gcd(a,q)=1}=\\sum_{d\\mid\\gcd(a,q)}\\mu(d).\n\\]\nWrite $a=db$ after interchanging the finite sums. Character orthogonality evaluates the inner geometric series:\n\\[\n \\begin{aligned}\n c_q(n)&=\\sum_{d\\mid q}\\mu(d)\\sum_{b=0}^{q/d-1}e^{2\\pi ibn/(q/d)}\\\\\n &=\\sum_{\\substack{d\\mid q\\\\q/d\\mid n}}\\mu(d)\\frac qd\n =\\sum_{e\\mid\\gcd(q,n)}e\\mu(q/e)\\in\\mathbb Z.\n \\end{aligned}\n\\]\n\\begin{center}\n\\begin{tabular}{lll}\\toprule\nSample & Relevant divisor contribution & Exact result","midrule\n$n=0$ & $2\\mu(6)+4\\mu(3)+6\\mu(2)+12\\mu(1)$ & $4$","$n=2$ & $\\mu(12)+2\\mu(6)$ & $2$","$n=4$ & $\\mu(12)+2\\mu(6)+4\\mu(3)$ & $-2$","$n=6$ & $\\mu(12)+2\\mu(6)+3\\mu(4)+6\\mu(2)$ & $-4$","bottomrule\n\\end{tabular}\n\\end{center}\n\\note{The finite verifier also evaluates the twelve phase sums in the exact ring $\\mathbb Z[z]/(z^4-z^2+1)$. The plots use floating coordinates for rendering, while the sample identities use exact arithmetic.}","\\clearpage\\section*{The spectrum recovers the selector}\\addcontentsline{toc}{section}{The spectrum recovers the selector}\n\\lead{The Fourier transform returns exactly the four labels selected by coprimality.}\nUse the negative-sign, unnormalized discrete Fourier transform\n\\[\n \\widehat x(k)=\\sum_{n=0}^{11}x(n)e^{-2\\pi ikn/12}.\n\\]\n\\fig{04_exact_spectrum}{1.0}{The coefficient is exactly $12$ at $k=1,5,7,11$ and exactly zero elsewhere. This is an algebraic identity, not a spectral thresholding decision.}\n\\[\n \\widehat{c_{12}}(k)=\\sum_{a\\in U_{12}}\\sum_{n=0}^{11}\\zeta^{(a-k)n}\n =12\\,\\mathbf1_{k\\in U_{12}}.\n\\]\nThe inner geometric sum is twelve when $a=k$ modulo twelve and zero otherwise. The selector and the Fourier support are now the same exact data.","s the selector}\n\\lead{The Fourier transform returns exactly the four labels selected by coprimality.}\nUse the negative-sign, unnormalized discrete Fourier transform\n\\[\n \\widehat x(k)=\\sum_{n=0}^{11}x(n)e^{-2\\pi ikn/12}.\n\\]\n\\fig{04_exact_spectrum}{1.0}{The coefficient is exactly $12$ at $k=1,5,7,11$ and exactly zero elsewhere. This is an algebraic identity, not a spectral thresholding decision.}\n\\[\n \\widehat{c_{12}}(k)=\\sum_{a\\in U_{12}}\\sum_{n=0}^{11}\\zeta^{(a-k)n}\n =12\\,\\mathbf1_{k\\in U_{12}}.\n\\]\nThe inner geometric sum is twelve when $a=k$ modulo twelve and zero otherwise. The selector and the Fourier support are now the same exact data.","ete Fourier transform\n\\[\n \\widehat x(k)=\\sum_{n=0}^{11}x(n)e^{-2\\pi ikn/12}.\n\\]\n\\fig{04_exact_spectrum}{1.0}{The coefficient is exactly $12$ at $k=1,5,7,11$ and exactly zero elsewhere. This is an algebraic identity, not a spectral thresholding decision.}\n\\[\n \\widehat{c_{12}}(k)=\\sum_{a\\in U_{12}}\\sum_{n=0}^{11}\\zeta^{(a-k)n}\n =12\\,\\mathbf1_{k\\in U_{12}}.\n\\]\nThe inner geometric sum is twelve when $a=k$ modulo twelve and zero otherwise. The selector and the Fourier support are now the same exact data.","\\clearpage\\section*{A diagonal selector becomes a rational filter}\\addcontentsline{toc}{section}{A diagonal selector becomes a rational filter}\n\\lead{The filter is diagonal in one basis and a rational convolution in the other.}\nLet $F_{na}=\\zeta^{na}/\\sqrt{12}$ and $D_{aa}=\\mathbf1_{a\\in U_{12}}$. With the positive-sign unitary matrix $F$, the negative-sign DFT satisfies $F^\\dagger x=\\widehat x/\\sqrt{12}$.\n\\fig{05_transported_projector}{1.0}{Left: the frequency selector $D$. Right: its transported sample-basis matrix $P$. The right colorbar displays its five possible rational entries. The two panels use different scales.}\n\\[\n P=FDF^\\dagger,\\qquad P_{nm}=\\frac{c_{12}(n-m)}{12}.\n\\]\nBecause $D^2=D$ and $F^\\dagger F=I$, we obtain $P^2=P$. Because $D^\\dagger=D$, we also obtain $P^\\dagger=P$. This is an orthogonal projector of rank four.","tsline{toc}{section}{A diagonal selector becomes a rational filter}\n\\lead{The filter is diagonal in one basis and a rational convolution in the other.}\nLet $F_{na}=\\zeta^{na}/\\sqrt{12}$ and $D_{aa}=\\mathbf1_{a\\in U_{12}}$. With the positive-sign unitary matrix $F$, the negative-sign DFT satisfies $F^\\dagger x=\\widehat x/\\sqrt{12}$.\n\\fig{05_transported_projector}{1.0}{Left: the frequency selector $D$. Right: its transported sample-basis matrix $P$. The right colorbar displays its five possible rational entries. The two panels use different scales.}\n\\[\n P=FDF^\\dagger,\\qquad P_{nm}=\\frac{c_{12}(n-m)}{12}.\n\\]\nBecause $D^2=D$ and $F^\\dagger F=I$, we obtain $P^2=P$. Because $D^\\dagger=D$, we also obtain $P^\\dagger=P$. This is an orthogonal projector of rank four.","a rational filter}\n\\lead{The filter is diagonal in one basis and a rational convolution in the other.}\nLet $F_{na}=\\zeta^{na}/\\sqrt{12}$ and $D_{aa}=\\mathbf1_{a\\in U_{12}}$. With the positive-sign unitary matrix $F$, the negative-sign DFT satisfies $F^\\dagger x=\\widehat x/\\sqrt{12}$.\n\\fig{05_transported_projector}{1.0}{Left: the frequency selector $D$. Right: its transported sample-basis matrix $P$. The right colorbar displays its five possible rational entries. The two panels use different scales.}\n\\[\n P=FDF^\\dagger,\\qquad P_{nm}=\\frac{c_{12}(n-m)}{12}.\n\\]\nBecause $D^2=D$ and $F^\\dagger F=I$, we obtain $P^2=P$. Because $D^\\dagger=D$, we also obtain $P^\\dagger=P$. This is an orthogonal projector of rank four.","$F_{na}=\\zeta^{na}/\\sqrt{12}$ and $D_{aa}=\\mathbf1_{a\\in U_{12}}$. With the positive-sign unitary matrix $F$, the negative-sign DFT satisfies $F^\\dagger x=\\widehat x/\\sqrt{12}$.\n\\fig{05_transported_projector}{1.0}{Left: the frequency selector $D$. Right: its transported sample-basis matrix $P$. The right colorbar displays its five possible rational entries. The two panels use different scales.}\n\\[\n P=FDF^\\dagger,\\qquad P_{nm}=\\frac{c_{12}(n-m)}{12}.\n\\]\nBecause $D^2=D$ and $F^\\dagger F=I$, we obtain $P^2=P$. Because $D^\\dagger=D$, we also obtain $P^\\dagger=P$. This is an orthogonal projector of rank four.","With the positive-sign unitary matrix $F$, the negative-sign DFT satisfies $F^\\dagger x=\\widehat x/\\sqrt{12}$.\n\\fig{05_transported_projector}{1.0}{Left: the frequency selector $D$. Right: its transported sample-basis matrix $P$. The right colorbar displays its five possible rational entries. The two panels use different scales.}\n\\[\n P=FDF^\\dagger,\\qquad P_{nm}=\\frac{c_{12}(n-m)}{12}.\n\\]\nBecause $D^2=D$ and $F^\\dagger F=I$, we obtain $P^2=P$. Because $D^\\dagger=D$, we also obtain $P^\\dagger=P$. This is an orthogonal projector of rank four.","sitive-sign unitary matrix $F$, the negative-sign DFT satisfies $F^\\dagger x=\\widehat x/\\sqrt{12}$.\n\\fig{05_transported_projector}{1.0}{Left: the frequency selector $D$. Right: its transported sample-basis matrix $P$. The right colorbar displays its five possible rational entries. The two panels use different scales.}\n\\[\n P=FDF^\\dagger,\\qquad P_{nm}=\\frac{c_{12}(n-m)}{12}.\n\\]\nBecause $D^2=D$ and $F^\\dagger F=I$, we obtain $P^2=P$. Because $D^\\dagger=D$, we also obtain $P^\\dagger=P$. This is an orthogonal projector of rank four.","$F$, the negative-sign DFT satisfies $F^\\dagger x=\\widehat x/\\sqrt{12}$.\n\\fig{05_transported_projector}{1.0}{Left: the frequency selector $D$. Right: its transported sample-basis matrix $P$. The right colorbar displays its five possible rational entries. The two panels use different scales.}\n\\[\n P=FDF^\\dagger,\\qquad P_{nm}=\\frac{c_{12}(n-m)}{12}.\n\\]\nBecause $D^2=D$ and $F^\\dagger F=I$, we obtain $P^2=P$. Because $D^\\dagger=D$, we also obtain $P^\\dagger=P$. This is an orthogonal projector of rank four.","-sign DFT satisfies $F^\\dagger x=\\widehat x/\\sqrt{12}$.\n\\fig{05_transported_projector}{1.0}{Left: the frequency selector $D$. Right: its transported sample-basis matrix $P$. The right colorbar displays its five possible rational entries. The two panels use different scales.}\n\\[\n P=FDF^\\dagger,\\qquad P_{nm}=\\frac{c_{12}(n-m)}{12}.\n\\]\nBecause $D^2=D$ and $F^\\dagger F=I$, we obtain $P^2=P$. Because $D^\\dagger=D$, we also obtain $P^\\dagger=P$. This is an orthogonal projector of rank four.","x=\\widehat x/\\sqrt{12}$.\n\\fig{05_transported_projector}{1.0}{Left: the frequency selector $D$. Right: its transported sample-basis matrix $P$. The right colorbar displays its five possible rational entries. The two panels use different scales.}\n\\[\n P=FDF^\\dagger,\\qquad P_{nm}=\\frac{c_{12}(n-m)}{12}.\n\\]\nBecause $D^2=D$ and $F^\\dagger F=I$, we obtain $P^2=P$. Because $D^\\dagger=D$, we also obtain $P^\\dagger=P$. This is an orthogonal projector of rank four.","\\clearpage\\section*{Give the projector something to remove}\\addcontentsline{toc}{section}{Give the projector something to remove}\n\\lead{The input contains an arithmetic component and a deliberately complementary component.}\nTake $x(n)=c_{12}(n)+3\\cos(\\pi n/2)+2$. Its extra cosine uses modes $3,9$ and its constant uses mode $0$. All three are outside $U_{12}$.\n\\fig{06_filter_components}{0.94}{The projector retains the primitive-order component exactly. The bottom panel shows everything it removes, so the loss remains visible.}\n\\[\n Px=c_{12},\\qquad (I-P)x=3\\cos(\\pi n/2)+2.\n\\]\nThe exact input is $(9,2,1,2,3,2,-5,2,3,2,1,2)$. Calling the complementary component ``noise'' is a modeling choice. Arbitrary noise need not lie in the kernel of $P$.","d{The input contains an arithmetic component and a deliberately complementary component.}\nTake $x(n)=c_{12}(n)+3\\cos(\\pi n/2)+2$. Its extra cosine uses modes $3,9$ and its constant uses mode $0$. All three are outside $U_{12}$.\n\\fig{06_filter_components}{0.94}{The projector retains the primitive-order component exactly. The bottom panel shows everything it removes, so the loss remains visible.}\n\\[\n Px=c_{12},\\qquad (I-P)x=3\\cos(\\pi n/2)+2.\n\\]\nThe exact input is $(9,2,1,2,3,2,-5,2,3,2,1,2)$. Calling the complementary component ``noise'' is a modeling choice. Arbitrary noise need not lie in the kernel of $P$.","plementary component.}\nTake $x(n)=c_{12}(n)+3\\cos(\\pi n/2)+2$. Its extra cosine uses modes $3,9$ and its constant uses mode $0$. All three are outside $U_{12}$.\n\\fig{06_filter_components}{0.94}{The projector retains the primitive-order component exactly. The bottom panel shows everything it removes, so the loss remains visible.}\n\\[\n Px=c_{12},\\qquad (I-P)x=3\\cos(\\pi n/2)+2.\n\\]\nThe exact input is $(9,2,1,2,3,2,-5,2,3,2,1,2)$. Calling the complementary component ``noise'' is a modeling choice. Arbitrary noise need not lie in the kernel of $P$.","utside $U_{12}$.\n\\fig{06_filter_components}{0.94}{The projector retains the primitive-order component exactly. The bottom panel shows everything it removes, so the loss remains visible.}\n\\[\n Px=c_{12},\\qquad (I-P)x=3\\cos(\\pi n/2)+2.\n\\]\nThe exact input is $(9,2,1,2,3,2,-5,2,3,2,1,2)$. Calling the complementary component ``noise'' is a modeling choice. Arbitrary noise need not lie in the kernel of $P$.","\\clearpage\\section*{Account for the entire input}\\addcontentsline{toc}{section}{Account for the entire input}\n\\lead{Exact filtering removes energy as well as modes.}\nThe selected and complementary components are orthogonal. With the unnormalized squared norm $\\|x\\|^2=\\sum_{n=0}^{11}|x(n)|^2$,\n\\[\n \\|x\\|^2=150,\\qquad\\|Px\\|^2=48,\\qquad\\|(I-P)x\\|^2=102.\n\\]\n\\fig{07_energy_and_loss}{1.0}{The energy identity is $150=48+102$. The normalized input has selected squared-norm share $48/150=8/25$ and complementary share $17/25$.}\nOn the normalized state $x/\\sqrt{150}$, the ideal L\\\"uders projective measurement $\\{P,I-P\\}$ has selected-outcome probability $8/25$. Conditional on that outcome, the resulting normalized state is $c_{12}/\\sqrt{48}$.\n\\note{This conditional measurement is not deterministic unitary denoising. The transform $F$ is reversible; the projection $P$ discards the complementary subspace.}","e\\section*{Account for the entire input}\\addcontentsline{toc}{section}{Account for the entire input}\n\\lead{Exact filtering removes energy as well as modes.}\nThe selected and complementary components are orthogonal. With the unnormalized squared norm $\\|x\\|^2=\\sum_{n=0}^{11}|x(n)|^2$,\n\\[\n \\|x\\|^2=150,\\qquad\\|Px\\|^2=48,\\qquad\\|(I-P)x\\|^2=102.\n\\]\n\\fig{07_energy_and_loss}{1.0}{The energy identity is $150=48+102$. The normalized input has selected squared-norm share $48/150=8/25$ and complementary share $17/25$.}\nOn the normalized state $x/\\sqrt{150}$, the ideal L\\\"uders projective measurement $\\{P,I-P\\}$ has selected-outcome probability $8/25$. Conditional on that outcome, the resulting normalized state is $c_{12}/\\sqrt{48}$.\n\\note{This conditional measurement is not deterministic unitary denoising. The transform $F$ is reversible; the projection $P$ discards the complementary subspace.}","on}{Account for the entire input}\n\\lead{Exact filtering removes energy as well as modes.}\nThe selected and complementary components are orthogonal. With the unnormalized squared norm $\\|x\\|^2=\\sum_{n=0}^{11}|x(n)|^2$,\n\\[\n \\|x\\|^2=150,\\qquad\\|Px\\|^2=48,\\qquad\\|(I-P)x\\|^2=102.\n\\]\n\\fig{07_energy_and_loss}{1.0}{The energy identity is $150=48+102$. The normalized input has selected squared-norm share $48/150=8/25$ and complementary share $17/25$.}\nOn the normalized state $x/\\sqrt{150}$, the ideal L\\\"uders projective measurement $\\{P,I-P\\}$ has selected-outcome probability $8/25$. Conditional on that outcome, the resulting normalized state is $c_{12}/\\sqrt{48}$.\n\\note{This conditional measurement is not deterministic unitary denoising. The transform $F$ is reversible; the projection $P$ discards the complementary subspace.}","d{Exact filtering removes energy as well as modes.}\nThe selected and complementary components are orthogonal. With the unnormalized squared norm $\\|x\\|^2=\\sum_{n=0}^{11}|x(n)|^2$,\n\\[\n \\|x\\|^2=150,\\qquad\\|Px\\|^2=48,\\qquad\\|(I-P)x\\|^2=102.\n\\]\n\\fig{07_energy_and_loss}{1.0}{The energy identity is $150=48+102$. The normalized input has selected squared-norm share $48/150=8/25$ and complementary share $17/25$.}\nOn the normalized state $x/\\sqrt{150}$, the ideal L\\\"uders projective measurement $\\{P,I-P\\}$ has selected-outcome probability $8/25$. Conditional on that outcome, the resulting normalized state is $c_{12}/\\sqrt{48}$.\n\\note{This conditional measurement is not deterministic unitary denoising. The transform $F$ is reversible; the projection $P$ discards the complementary subspace.}","s well as modes.}\nThe selected and complementary components are orthogonal. With the unnormalized squared norm $\\|x\\|^2=\\sum_{n=0}^{11}|x(n)|^2$,\n\\[\n \\|x\\|^2=150,\\qquad\\|Px\\|^2=48,\\qquad\\|(I-P)x\\|^2=102.\n\\]\n\\fig{07_energy_and_loss}{1.0}{The energy identity is $150=48+102$. The normalized input has selected squared-norm share $48/150=8/25$ and complementary share $17/25$.}\nOn the normalized state $x/\\sqrt{150}$, the ideal L\\\"uders projective measurement $\\{P,I-P\\}$ has selected-outcome probability $8/25$. Conditional on that outcome, the resulting normalized state is $c_{12}/\\sqrt{48}$.\n\\note{This conditional measurement is not deterministic unitary denoising. The transform $F$ is reversible; the projection $P$ discards the complementary subspace.}","s are orthogonal. With the unnormalized squared norm $\\|x\\|^2=\\sum_{n=0}^{11}|x(n)|^2$,\n\\[\n \\|x\\|^2=150,\\qquad\\|Px\\|^2=48,\\qquad\\|(I-P)x\\|^2=102.\n\\]\n\\fig{07_energy_and_loss}{1.0}{The energy identity is $150=48+102$. The normalized input has selected squared-norm share $48/150=8/25$ and complementary share $17/25$.}\nOn the normalized state $x/\\sqrt{150}$, the ideal L\\\"uders projective measurement $\\{P,I-P\\}$ has selected-outcome probability $8/25$. Conditional on that outcome, the resulting normalized state is $c_{12}/\\sqrt{48}$.\n\\note{This conditional measurement is not deterministic unitary denoising. The transform $F$ is reversible; the projection $P$ discards the complementary subspace.}","ed norm $\\|x\\|^2=\\sum_{n=0}^{11}|x(n)|^2$,\n\\[\n \\|x\\|^2=150,\\qquad\\|Px\\|^2=48,\\qquad\\|(I-P)x\\|^2=102.\n\\]\n\\fig{07_energy_and_loss}{1.0}{The energy identity is $150=48+102$. The normalized input has selected squared-norm share $48/150=8/25$ and complementary share $17/25$.}\nOn the normalized state $x/\\sqrt{150}$, the ideal L\\\"uders projective measurement $\\{P,I-P\\}$ has selected-outcome probability $8/25$. Conditional on that outcome, the resulting normalized state is $c_{12}/\\sqrt{48}$.\n\\note{This conditional measurement is not deterministic unitary denoising. The transform $F$ is reversible; the projection $P$ discards the complementary subspace.}","^2=48,\\qquad\\|(I-P)x\\|^2=102.\n\\]\n\\fig{07_energy_and_loss}{1.0}{The energy identity is $150=48+102$. The normalized input has selected squared-norm share $48/150=8/25$ and complementary share $17/25$.}\nOn the normalized state $x/\\sqrt{150}$, the ideal L\\\"uders projective measurement $\\{P,I-P\\}$ has selected-outcome probability $8/25$. Conditional on that outcome, the resulting normalized state is $c_{12}/\\sqrt{48}$.\n\\note{This conditional measurement is not deterministic unitary denoising. The transform $F$ is reversible; the projection $P$ discards the complementary subspace.}","lized input has selected squared-norm share $48/150=8/25$ and complementary share $17/25$.}\nOn the normalized state $x/\\sqrt{150}$, the ideal L\\\"uders projective measurement $\\{P,I-P\\}$ has selected-outcome probability $8/25$. Conditional on that outcome, the resulting normalized state is $c_{12}/\\sqrt{48}$.\n\\note{This conditional measurement is not deterministic unitary denoising. The transform $F$ is reversible; the projection $P$ discards the complementary subspace.}","orm share $48/150=8/25$ and complementary share $17/25$.}\nOn the normalized state $x/\\sqrt{150}$, the ideal L\\\"uders projective measurement $\\{P,I-P\\}$ has selected-outcome probability $8/25$. Conditional on that outcome, the resulting normalized state is $c_{12}/\\sqrt{48}$.\n\\note{This conditional measurement is not deterministic unitary denoising. The transform $F$ is reversible; the projection $P$ discards the complementary subspace.}","\\clearpage\\section*{Projection and unitary change are different operations}\\addcontentsline{toc}{section}{Projection and unitary change are different operations}\n\\lead{A basis change preserves the whole vector. Projection keeps only one orthogonal component.}\nLet $e_s=c_{12}/\\sqrt{48}$ and $e_r=(I-P)x/\\sqrt{102}$. They are orthonormal, so the normalized input has the exact two-dimensional representation\n\\[\n \\frac{x}{\\sqrt{150}}=\\sqrt{\\frac8{25}}e_s+\\sqrt{\\frac{17}{25}}e_r.\n\\]\n\\fig{20_projection_geometry}{0.97}{This plane is the actual span of the selected signal and its complementary component. The dashed segment is the discarded component. Its length is $\\sqrt{17/25}$; the retained vector has length $\\sqrt{8/25}$.}\nThe Fourier transform preserves the input norm and permits exact inversion. The projector maps every complementary vector to zero. That distinction controls what may legitimately be called reconstruction, filtering, or measurement.","e\\section*{Projection and unitary change are different operations}\\addcontentsline{toc}{section}{Projection and unitary change are different operations}\n\\lead{A basis change preserves the whole vector. Projection keeps only one orthogonal component.}\nLet $e_s=c_{12}/\\sqrt{48}$ and $e_r=(I-P)x/\\sqrt{102}$. They are orthonormal, so the normalized input has the exact two-dimensional representation\n\\[\n \\frac{x}{\\sqrt{150}}=\\sqrt{\\frac8{25}}e_s+\\sqrt{\\frac{17}{25}}e_r.\n\\]\n\\fig{20_projection_geometry}{0.97}{This plane is the actual span of the selected signal and its complementary component. The dashed segment is the discarded component. Its length is $\\sqrt{17/25}$; the retained vector has length $\\sqrt{8/25}$.}\nThe Fourier transform preserves the input norm and permits exact inversion. The projector maps every complementary vector to zero. That distinction controls what may legitimately be called reconstruction, filtering, or measurement.","$e_s=c_{12}/\\sqrt{48}$ and $e_r=(I-P)x/\\sqrt{102}$. They are orthonormal, so the normalized input has the exact two-dimensional representation\n\\[\n \\frac{x}{\\sqrt{150}}=\\sqrt{\\frac8{25}}e_s+\\sqrt{\\frac{17}{25}}e_r.\n\\]\n\\fig{20_projection_geometry}{0.97}{This plane is the actual span of the selected signal and its complementary component. The dashed segment is the discarded component. Its length is $\\sqrt{17/25}$; the retained vector has length $\\sqrt{8/25}$.}\nThe Fourier transform preserves the input norm and permits exact inversion. The projector maps every complementary vector to zero. That distinction controls what may legitimately be called reconstruction, filtering, or measurement.","2}$. They are orthonormal, so the normalized input has the exact two-dimensional representation\n\\[\n \\frac{x}{\\sqrt{150}}=\\sqrt{\\frac8{25}}e_s+\\sqrt{\\frac{17}{25}}e_r.\n\\]\n\\fig{20_projection_geometry}{0.97}{This plane is the actual span of the selected signal and its complementary component. The dashed segment is the discarded component. Its length is $\\sqrt{17/25}$; the retained vector has length $\\sqrt{8/25}$.}\nThe Fourier transform preserves the input norm and permits exact inversion. The projector maps every complementary vector to zero. That distinction controls what may legitimately be called reconstruction, filtering, or measurement.","\\clearpage\\section*{All twelve modes belong to six order classes}\\addcontentsline{toc}{section}{All twelve modes belong to six order classes}\n\\lead{The primitive-twelve projector is one packet in a complete orthogonal decomposition.}\nA frequency label $k$ has root order $d=12/\\gcd(k,12)$. The possible orders are the divisors $1,2,3,4,6,12$.\n\\fig{08_order_partition}{1.0}{The number inside each tile is its root order $d$. The horizontal label is frequency $k$. Every label belongs to one class; no labels are dropped.}\n\\[\n P_d{}_{nm}=\\frac{c_d(n-m)}{12},\\quad\n P_dP_e=\\delta_{de}P_d,\\quad\n \\sum_{d\\mid12}P_d=I.\n\\]\nThese are character-order projectors. They select Fourier modes by root order. They are different from coefficient-residue selectors, which select formal-series coefficients by their index modulo an integer.","All twelve modes belong to six order classes}\\addcontentsline{toc}{section}{All twelve modes belong to six order classes}\n\\lead{The primitive-twelve projector is one packet in a complete orthogonal decomposition.}\nA frequency label $k$ has root order $d=12/\\gcd(k,12)$. The possible orders are the divisors $1,2,3,4,6,12$.\n\\fig{08_order_partition}{1.0}{The number inside each tile is its root order $d$. The horizontal label is frequency $k$. Every label belongs to one class; no labels are dropped.}\n\\[\n P_d{}_{nm}=\\frac{c_d(n-m)}{12},\\quad\n P_dP_e=\\delta_{de}P_d,\\quad\n \\sum_{d\\mid12}P_d=I.\n\\]\nThese are character-order projectors. They select Fourier modes by root order. They are different from coefficient-residue selectors, which select formal-series coefficients by their index modulo an integer.","asses}\\addcontentsline{toc}{section}{All twelve modes belong to six order classes}\n\\lead{The primitive-twelve projector is one packet in a complete orthogonal decomposition.}\nA frequency label $k$ has root order $d=12/\\gcd(k,12)$. The possible orders are the divisors $1,2,3,4,6,12$.\n\\fig{08_order_partition}{1.0}{The number inside each tile is its root order $d$. The horizontal label is frequency $k$. Every label belongs to one class; no labels are dropped.}\n\\[\n P_d{}_{nm}=\\frac{c_d(n-m)}{12},\\quad\n P_dP_e=\\delta_{de}P_d,\\quad\n \\sum_{d\\mid12}P_d=I.\n\\]\nThese are character-order projectors. They select Fourier modes by root order. They are different from coefficient-residue selectors, which select formal-series coefficients by their index modulo an integer.","six order classes}\n\\lead{The primitive-twelve projector is one packet in a complete orthogonal decomposition.}\nA frequency label $k$ has root order $d=12/\\gcd(k,12)$. The possible orders are the divisors $1,2,3,4,6,12$.\n\\fig{08_order_partition}{1.0}{The number inside each tile is its root order $d$. The horizontal label is frequency $k$. Every label belongs to one class; no labels are dropped.}\n\\[\n P_d{}_{nm}=\\frac{c_d(n-m)}{12},\\quad\n P_dP_e=\\delta_{de}P_d,\\quad\n \\sum_{d\\mid12}P_d=I.\n\\]\nThese are character-order projectors. They select Fourier modes by root order. They are different from coefficient-residue selectors, which select formal-series coefficients by their index modulo an integer.","\\clearpage\\section*{Six exact filters in one ambient space}\\addcontentsline{toc}{section}{Six exact filters in one ambient space}\n\\lead{Every order class has an integer Ramanujan kernel and a rational projector.}\n\\fig{09_six_rational_kernels}{0.92}{Each panel displays $c_d(n)$ over the common twelve-sample ambient period. The associated matrix is $P_d{}_{nm}=c_d(n-m)/12$, so every kernel uses the same denominator twelve.}\nFor $d<12$, the kernel repeats with period dividing $d$. Its projector still acts on all twelve sample coordinates. The ranks are $1,1,2,2,2,4$, respectively.","ilters in one ambient space}\\addcontentsline{toc}{section}{Six exact filters in one ambient space}\n\\lead{Every order class has an integer Ramanujan kernel and a rational projector.}\n\\fig{09_six_rational_kernels}{0.92}{Each panel displays $c_d(n)$ over the common twelve-sample ambient period. The associated matrix is $P_d{}_{nm}=c_d(n-m)/12$, so every kernel uses the same denominator twelve.}\nFor $d<12$, the kernel repeats with period dividing $d$. Its projector still acts on all twelve sample coordinates. The ranks are $1,1,2,2,2,4$, respectively.","}{section}{Six exact filters in one ambient space}\n\\lead{Every order class has an integer Ramanujan kernel and a rational projector.}\n\\fig{09_six_rational_kernels}{0.92}{Each panel displays $c_d(n)$ over the common twelve-sample ambient period. The associated matrix is $P_d{}_{nm}=c_d(n-m)/12$, so every kernel uses the same denominator twelve.}\nFor $d<12$, the kernel repeats with period dividing $d$. Its projector still acts on all twelve sample coordinates. The ranks are $1,1,2,2,2,4$, respectively.","an integer Ramanujan kernel and a rational projector.}\n\\fig{09_six_rational_kernels}{0.92}{Each panel displays $c_d(n)$ over the common twelve-sample ambient period. The associated matrix is $P_d{}_{nm}=c_d(n-m)/12$, so every kernel uses the same denominator twelve.}\nFor $d<12$, the kernel repeats with period dividing $d$. Its projector still acts on all twelve sample coordinates. The ranks are $1,1,2,2,2,4$, respectively.","\\clearpage\\section*{A packet's capacity is not its occupied energy}\\addcontentsline{toc}{section}{A packet's capacity is not its occupied energy}\n\\lead{Ranks describe the subspaces. Energies describe a particular input.}\n\\fig{10_dimensions_and_shell_energies}{1.0}{Left: the six subspace dimensions sum to twelve. Right: the particular input $x$ has energy $48$ in order one, $54$ in order four, and $48$ in order twelve. The other packets exist but have zero component for this input.}\n\\[\n x=\\sum_{d\\mid12}P_dx,\\qquad\\|x\\|^2=\\sum_{d\\mid12}\\|P_dx\\|^2.\n\\]\nThe constant $2$ contributes $12\\cdot2^2=48$ to order one. The cosine contributes $54$ to order four. The Ramanujan component contributes $48$ to order twelve. Keeping all six packets gives exact synthesis. Keeping only one packet performs a lossy selection.","clearpage\\section*{A packet's capacity is not its occupied energy}\\addcontentsline{toc}{section}{A packet's capacity is not its occupied energy}\n\\lead{Ranks describe the subspaces. Energies describe a particular input.}\n\\fig{10_dimensions_and_shell_energies}{1.0}{Left: the six subspace dimensions sum to twelve. Right: the particular input $x$ has energy $48$ in order one, $54$ in order four, and $48$ in order twelve. The other packets exist but have zero component for this input.}\n\\[\n x=\\sum_{d\\mid12}P_dx,\\qquad\\|x\\|^2=\\sum_{d\\mid12}\\|P_dx\\|^2.\n\\]\nThe constant $2$ contributes $12\\cdot2^2=48$ to order one. The cosine contributes $54$ to order four. The Ramanujan component contributes $48$ to order twelve. Keeping all six packets gives exact synthesis. Keeping only one packet performs a lossy selection.","section}{A packet's capacity is not its occupied energy}\n\\lead{Ranks describe the subspaces. Energies describe a particular input.}\n\\fig{10_dimensions_and_shell_energies}{1.0}{Left: the six subspace dimensions sum to twelve. Right: the particular input $x$ has energy $48$ in order one, $54$ in order four, and $48$ in order twelve. The other packets exist but have zero component for this input.}\n\\[\n x=\\sum_{d\\mid12}P_dx,\\qquad\\|x\\|^2=\\sum_{d\\mid12}\\|P_dx\\|^2.\n\\]\nThe constant $2$ contributes $12\\cdot2^2=48$ to order one. The cosine contributes $54$ to order four. The Ramanujan component contributes $48$ to order twelve. Keeping all six packets gives exact synthesis. Keeping only one packet performs a lossy selection.","e subspaces. Energies describe a particular input.}\n\\fig{10_dimensions_and_shell_energies}{1.0}{Left: the six subspace dimensions sum to twelve. Right: the particular input $x$ has energy $48$ in order one, $54$ in order four, and $48$ in order twelve. The other packets exist but have zero component for this input.}\n\\[\n x=\\sum_{d\\mid12}P_dx,\\qquad\\|x\\|^2=\\sum_{d\\mid12}\\|P_dx\\|^2.\n\\]\nThe constant $2$ contributes $12\\cdot2^2=48$ to order one. The cosine contributes $54$ to order four. The Ramanujan component contributes $48$ to order twelve. Keeping all six packets gives exact synthesis. Keeping only one packet performs a lossy selection.","rgies describe a particular input.}\n\\fig{10_dimensions_and_shell_energies}{1.0}{Left: the six subspace dimensions sum to twelve. Right: the particular input $x$ has energy $48$ in order one, $54$ in order four, and $48$ in order twelve. The other packets exist but have zero component for this input.}\n\\[\n x=\\sum_{d\\mid12}P_dx,\\qquad\\|x\\|^2=\\sum_{d\\mid12}\\|P_dx\\|^2.\n\\]\nThe constant $2$ contributes $12\\cdot2^2=48$ to order one. The cosine contributes $54$ to order four. The Ramanujan component contributes $48$ to order twelve. Keeping all six packets gives exact synthesis. Keeping only one packet performs a lossy selection.","s_and_shell_energies}{1.0}{Left: the six subspace dimensions sum to twelve. Right: the particular input $x$ has energy $48$ in order one, $54$ in order four, and $48$ in order twelve. The other packets exist but have zero component for this input.}\n\\[\n x=\\sum_{d\\mid12}P_dx,\\qquad\\|x\\|^2=\\sum_{d\\mid12}\\|P_dx\\|^2.\n\\]\nThe constant $2$ contributes $12\\cdot2^2=48$ to order one. The cosine contributes $54$ to order four. The Ramanujan component contributes $48$ to order twelve. Keeping all six packets gives exact synthesis. Keeping only one packet performs a lossy selection.","ions sum to twelve. Right: the particular input $x$ has energy $48$ in order one, $54$ in order four, and $48$ in order twelve. The other packets exist but have zero component for this input.}\n\\[\n x=\\sum_{d\\mid12}P_dx,\\qquad\\|x\\|^2=\\sum_{d\\mid12}\\|P_dx\\|^2.\n\\]\nThe constant $2$ contributes $12\\cdot2^2=48$ to order one. The cosine contributes $54$ to order four. The Ramanujan component contributes $48$ to order twelve. Keeping all six packets gives exact synthesis. Keeping only one packet performs a lossy selection.","\\clearpage\\section*{The arithmetic signal is a coherent amplitude vector}\\addcontentsline{toc}{section}{The arithmetic signal is a coherent amplitude vector}\n\\lead{Normalization makes the signal a unit vector without changing its relative phases.}\nDefine $u_k=1/2$ for $k\\in U_{12}$ and zero otherwise. Then $\\|u\\|=1$, and\n\\[\n \\psi=Fu,\\qquad\\psi_n=\\frac{c_{12}(n)}{\\sqrt{48}},\\qquad F^\\dagger\\psi=u.\n\\]\n\\fig{11_same_state_two_bases}{1.0}{Separate panels preserve the different basis labels: $k$ for the frequency-basis coordinates and $n$ for the sample-basis coordinates. Heights are amplitudes, including their signs.}\nThe normalization is exact because $\\sum_n c_{12}(n)^2=48$. The matrix $F$ is unitary, so no probability mass is lost during this coherent change of basis.","c}{section}{The arithmetic signal is a coherent amplitude vector}\n\\lead{Normalization makes the signal a unit vector without changing its relative phases.}\nDefine $u_k=1/2$ for $k\\in U_{12}$ and zero otherwise. Then $\\|u\\|=1$, and\n\\[\n \\psi=Fu,\\qquad\\psi_n=\\frac{c_{12}(n)}{\\sqrt{48}},\\qquad F^\\dagger\\psi=u.\n\\]\n\\fig{11_same_state_two_bases}{1.0}{Separate panels preserve the different basis labels: $k$ for the frequency-basis coordinates and $n$ for the sample-basis coordinates. Heights are amplitudes, including their signs.}\nThe normalization is exact because $\\sum_n c_{12}(n)^2=48$. The matrix $F$ is unitary, so no probability mass is lost during this coherent change of basis.","ent amplitude vector}\n\\lead{Normalization makes the signal a unit vector without changing its relative phases.}\nDefine $u_k=1/2$ for $k\\in U_{12}$ and zero otherwise. Then $\\|u\\|=1$, and\n\\[\n \\psi=Fu,\\qquad\\psi_n=\\frac{c_{12}(n)}{\\sqrt{48}},\\qquad F^\\dagger\\psi=u.\n\\]\n\\fig{11_same_state_two_bases}{1.0}{Separate panels preserve the different basis labels: $k$ for the frequency-basis coordinates and $n$ for the sample-basis coordinates. Heights are amplitudes, including their signs.}\nThe normalization is exact because $\\sum_n c_{12}(n)^2=48$. The matrix $F$ is unitary, so no probability mass is lost during this coherent change of basis.","_k=1/2$ for $k\\in U_{12}$ and zero otherwise. Then $\\|u\\|=1$, and\n\\[\n \\psi=Fu,\\qquad\\psi_n=\\frac{c_{12}(n)}{\\sqrt{48}},\\qquad F^\\dagger\\psi=u.\n\\]\n\\fig{11_same_state_two_bases}{1.0}{Separate panels preserve the different basis labels: $k$ for the frequency-basis coordinates and $n$ for the sample-basis coordinates. Heights are amplitudes, including their signs.}\nThe normalization is exact because $\\sum_n c_{12}(n)^2=48$. The matrix $F$ is unitary, so no probability mass is lost during this coherent change of basis.","therwise. Then $\\|u\\|=1$, and\n\\[\n \\psi=Fu,\\qquad\\psi_n=\\frac{c_{12}(n)}{\\sqrt{48}},\\qquad F^\\dagger\\psi=u.\n\\]\n\\fig{11_same_state_two_bases}{1.0}{Separate panels preserve the different basis labels: $k$ for the frequency-basis coordinates and $n$ for the sample-basis coordinates. Heights are amplitudes, including their signs.}\nThe normalization is exact because $\\sum_n c_{12}(n)^2=48$. The matrix $F$ is unitary, so no probability mass is lost during this coherent change of basis.","\\clearpage\\section*{Amplitudes and probabilities tell different stories}\\addcontentsline{toc}{section}{Amplitudes and probabilities tell different stories}\n\\lead{Measurement squares magnitudes; interference depends on coherent amplitude before that step.}\n\\fig{12_alternative_measurements}{1.0}{Two alternative procedures on fresh preparations. The first measures $\\psi$ directly. The second applies $F^\\dagger$ before measurement. Neither histogram is the amplitude plot from the previous page.}\nThe first procedure gives probabilities $c_{12}(n)^2/48$. The second gives probability $1/4$ at each of $1,5,7,11$ and zero elsewhere.\nOne run returns one outcome. Measuring the sample basis first does not leave the original coherent state available for the second procedure. These distributions are alternative experimental choices, rather than successive readouts of one unchanged state.","clearpage\\section*{Amplitudes and probabilities tell different stories}\\addcontentsline{toc}{section}{Amplitudes and probabilities tell different stories}\n\\lead{Measurement squares magnitudes; interference depends on coherent amplitude before that step.}\n\\fig{12_alternative_measurements}{1.0}{Two alternative procedures on fresh preparations. The first measures $\\psi$ directly. The second applies $F^\\dagger$ before measurement. Neither histogram is the amplitude plot from the previous page.}\nThe first procedure gives probabilities $c_{12}(n)^2/48$. The second gives probability $1/4$ at each of $1,5,7,11$ and zero elsewhere.\nOne run returns one outcome. Measuring the sample basis first does not leave the original coherent state available for the second procedure. These distributions are alternative experimental choices, rather than successive readouts of one unchanged state.","ifferent stories}\n\\lead{Measurement squares magnitudes; interference depends on coherent amplitude before that step.}\n\\fig{12_alternative_measurements}{1.0}{Two alternative procedures on fresh preparations. The first measures $\\psi$ directly. The second applies $F^\\dagger$ before measurement. Neither histogram is the amplitude plot from the previous page.}\nThe first procedure gives probabilities $c_{12}(n)^2/48$. The second gives probability $1/4$ at each of $1,5,7,11$ and zero elsewhere.\nOne run returns one outcome. Measuring the sample basis first does not leave the original coherent state available for the second procedure. These distributions are alternative experimental choices, rather than successive readouts of one unchanged state.","nitudes; interference depends on coherent amplitude before that step.}\n\\fig{12_alternative_measurements}{1.0}{Two alternative procedures on fresh preparations. The first measures $\\psi$ directly. The second applies $F^\\dagger$ before measurement. Neither histogram is the amplitude plot from the previous page.}\nThe first procedure gives probabilities $c_{12}(n)^2/48$. The second gives probability $1/4$ at each of $1,5,7,11$ and zero elsewhere.\nOne run returns one outcome. Measuring the sample basis first does not leave the original coherent state available for the second procedure. These distributions are alternative experimental choices, rather than successive readouts of one unchanged state.","ence depends on coherent amplitude before that step.}\n\\fig{12_alternative_measurements}{1.0}{Two alternative procedures on fresh preparations. The first measures $\\psi$ directly. The second applies $F^\\dagger$ before measurement. Neither histogram is the amplitude plot from the previous page.}\nThe first procedure gives probabilities $c_{12}(n)^2/48$. The second gives probability $1/4$ at each of $1,5,7,11$ and zero elsewhere.\nOne run returns one outcome. Measuring the sample basis first does not leave the original coherent state available for the second procedure. These distributions are alternative experimental choices, rather than successive readouts of one unchanged state.","\\clearpage\\section*{Identical histograms can hide orthogonal states}\\addcontentsline{toc}{section}{Identical histograms can hide orthogonal states}\n\\lead{Erasing the signs leaves sample probabilities unchanged and changes the state.}\nLet $\\chi_n=|c_{12}(n)|/\\sqrt{48}$. Both states are normalized and\n\\[\n |\\psi_n|^2=|\\chi_n|^2\\quad\\text{for every }n,\n \\qquad\\langle\\psi,\\chi\\rangle=0.\n\\]\n\\fig{13_same_histogram_distinct_states}{1.0}{The top panel shows the phase information carried by negative amplitudes. The bottom panel is shared by the two states because measurement in this basis depends only on squared magnitudes.}\nThe orthogonality follows from equal positive-support and negative-support energies: each contributes $24$ before division by $48$. A sample-basis probability histogram therefore cannot reconstruct the coherent state.","\\quad\\text{for every }n,\n \\qquad\\langle\\psi,\\chi\\rangle=0.\n\\]\n\\fig{13_same_histogram_distinct_states}{1.0}{The top panel shows the phase information carried by negative amplitudes. The bottom panel is shared by the two states because measurement in this basis depends only on squared magnitudes.}\nThe orthogonality follows from equal positive-support and negative-support energies: each contributes $24$ before division by $48$. A sample-basis probability histogram therefore cannot reconstruct the coherent state.","\\qquad\\langle\\psi,\\chi\\rangle=0.\n\\]\n\\fig{13_same_histogram_distinct_states}{1.0}{The top panel shows the phase information carried by negative amplitudes. The bottom panel is shared by the two states because measurement in this basis depends only on squared magnitudes.}\nThe orthogonality follows from equal positive-support and negative-support energies: each contributes $24$ before division by $48$. A sample-basis probability histogram therefore cannot reconstruct the coherent state.","\\clearpage\\section*{A basis change makes the hidden distinction observable}\\addcontentsline{toc}{section}{A basis change makes the hidden distinction observable}\n\\lead{The signed state recovers all four coprime labels. The unsigned state recovers none.}\n\\fig{14_phase_changes_measurement}{1.0}{The pale vertical strips mark the coprime labels. After $F^\\dagger$, the signed state assigns them total probability one; the unsigned state assigns them total probability zero.}\nThe exact unnormalized DFT of the unsigned integer signal is\n\\[\n (16,0,4,0,4,0,16,0,4,0,4,0).\n\\]\nAfter state normalization and unitary transformation, its probabilities are $4/9$ at $0,6$, $1/36$ at $2,4,8,10$, and zero elsewhere. Since $|c_{12}(n)|$ has period six, its odd Fourier labels vanish.","n distinction observable}\n\\lead{The signed state recovers all four coprime labels. The unsigned state recovers none.}\n\\fig{14_phase_changes_measurement}{1.0}{The pale vertical strips mark the coprime labels. After $F^\\dagger$, the signed state assigns them total probability one; the unsigned state assigns them total probability zero.}\nThe exact unnormalized DFT of the unsigned integer signal is\n\\[\n (16,0,4,0,4,0,16,0,4,0,4,0).\n\\]\nAfter state normalization and unitary transformation, its probabilities are $4/9$ at $0,6$, $1/36$ at $2,4,8,10$, and zero elsewhere. Since $|c_{12}(n)|$ has period six, its odd Fourier labels vanish.","none.}\n\\fig{14_phase_changes_measurement}{1.0}{The pale vertical strips mark the coprime labels. After $F^\\dagger$, the signed state assigns them total probability one; the unsigned state assigns them total probability zero.}\nThe exact unnormalized DFT of the unsigned integer signal is\n\\[\n (16,0,4,0,4,0,16,0,4,0,4,0).\n\\]\nAfter state normalization and unitary transformation, its probabilities are $4/9$ at $0,6$, $1/36$ at $2,4,8,10$, and zero elsewhere. Since $|c_{12}(n)|$ has period six, its odd Fourier labels vanish.","ig{14_phase_changes_measurement}{1.0}{The pale vertical strips mark the coprime labels. After $F^\\dagger$, the signed state assigns them total probability one; the unsigned state assigns them total probability zero.}\nThe exact unnormalized DFT of the unsigned integer signal is\n\\[\n (16,0,4,0,4,0,16,0,4,0,4,0).\n\\]\nAfter state normalization and unitary transformation, its probabilities are $4/9$ at $0,6$, $1/36$ at $2,4,8,10$, and zero elsewhere. Since $|c_{12}(n)|$ has period six, its odd Fourier labels vanish.","hase_changes_measurement}{1.0}{The pale vertical strips mark the coprime labels. After $F^\\dagger$, the signed state assigns them total probability one; the unsigned state assigns them total probability zero.}\nThe exact unnormalized DFT of the unsigned integer signal is\n\\[\n (16,0,4,0,4,0,16,0,4,0,4,0).\n\\]\nAfter state normalization and unitary transformation, its probabilities are $4/9$ at $0,6$, $1/36$ at $2,4,8,10$, and zero elsewhere. Since $|c_{12}(n)|$ has period six, its odd Fourier labels vanish.","es_measurement}{1.0}{The pale vertical strips mark the coprime labels. After $F^\\dagger$, the signed state assigns them total probability one; the unsigned state assigns them total probability zero.}\nThe exact unnormalized DFT of the unsigned integer signal is\n\\[\n (16,0,4,0,4,0,16,0,4,0,4,0).\n\\]\nAfter state normalization and unitary transformation, its probabilities are $4/9$ at $0,6$, $1/36$ at $2,4,8,10$, and zero elsewhere. Since $|c_{12}(n)|$ has period six, its odd Fourier labels vanish.","strips mark the coprime labels. After $F^\\dagger$, the signed state assigns them total probability one; the unsigned state assigns them total probability zero.}\nThe exact unnormalized DFT of the unsigned integer signal is\n\\[\n (16,0,4,0,4,0,16,0,4,0,4,0).\n\\]\nAfter state normalization and unitary transformation, its probabilities are $4/9$ at $0,6$, $1/36$ at $2,4,8,10$, and zero elsewhere. Since $|c_{12}(n)|$ has period six, its odd Fourier labels vanish.","\\clearpage\\section*{Move continuously between the two states}\\addcontentsline{toc}{section}{Move continuously between the two states}\n\\lead{Changing only relative phase moves the selected probability from zero to one.}\nLet $v_+$ contain the positive entries of $c_{12}/\\sqrt{48}$ and $v_-$ contain the absolute values of its negative entries. Their supports are disjoint and each has squared norm $1/2$. Define\n\\[\n \\chi_\\theta=v_++e^{i\\theta}v_-,\\qquad\n \\chi_0=\\chi,\\quad\\chi_\\pi=\\psi.\n\\]\n\\fig{15_phase_probability}{1.0}{The selected probability is the exact formula $\\sin^2(\\theta/2)$. The continuous curve is its floating rendering. Black dots mark exactly checked quarter-turn cases $0,1/2,1,1/2,0$.}\nIndeed, $\\chi_\\theta=\\tfrac{1+e^{i\\theta}}2\\chi+\\tfrac{1-e^{i\\theta}}2\\psi$. Since $P\\chi=0$ and $P\\psi=\\psi$, we get $\\|P\\chi_\\theta\\|^2=|1-e^{i\\theta}|^2/4=\\sin^2(\\theta/2)$.\n\\note{The endpoint statements are covered by t","\\clearpage\\section*{Move continuously between the two states}\\addcontentsline{toc}{section}{Move continuously between the two states}\n\\lead{Changing only relative phase moves the selected probability from zero to one.}\nLet $v_+$ contain the positive entries of $c_{12}/\\sqrt{48}$ and $v_-$ contain the absolute values of its negative entries. Their supports are disjoint and each has squared norm $1/2$. Define\n\\[\n \\chi_\\theta=v_++e^{i\\theta}v_-,\\qquad\n \\chi_0=\\chi,\\quad\\chi_\\pi=\\psi.\n\\]\n\\fig{15_phase_probability}{1.0}{The selected probability is the exact formula $\\sin^2(\\theta/2)$. The continuous curve is its floating rendering. Black dots mark exactly checked quarter-turn cases $0,1/2,1,1/2,0$.}\nIndeed, $\\chi_\\theta=\\tfrac{1+e^{i\\theta}}2\\chi+\\tfrac{1-e^{i\\theta}}2\\psi$. Since $P\\chi=0$ and $P\\psi=\\psi$, we get $\\|P\\chi_\\theta\\|^2=|1-e^{i\\theta}|^2/4=\\sin^2(\\theta/2)$.\n\\note{The endpoint statements are covered by the inherited Lean development. This continuous fami","\\clearpage\\section*{Move continuously between the two states}\\addcontentsline{toc}{section}{Move continuously between the two states}\n\\lead{Changing only relative phase moves the selected probability from zero to one.}\nLet $v_+$ contain the positive entries of $c_{12}/\\sqrt{48}$ and $v_-$ contain the absolute values of its negative entries. Their supports are disjoint and each has squared norm $1/2$. Define\n\\[\n \\chi_\\theta=v_++e^{i\\theta}v_-,\\qquad\n \\chi_0=\\chi,\\quad\\chi_\\pi=\\psi.\n\\]\n\\fig{15_phase_probability}{1.0}{The selected probability is the exact formula $\\sin^2(\\theta/2)$. The continuous curve is its floating rendering. Black dots mark exactly checked quarter-turn cases $0,1/2,1,1/2,0$.}\nIndeed, $\\chi_\\theta=\\tfrac{1+e^{i\\theta}}2\\chi+\\tfrac{1-e^{i\\theta}}2\\psi$. Since $P\\chi=0$ and $P\\psi=\\psi$, we get $\\|P\\chi_\\theta\\|^2=|1-e^{i\\theta}|^2/4=\\sin^2(\\theta/2)$.\n\\note{The endpoint statements are covered by the inherited Lean development. This continuous family is an e","\\clearpage\\section*{Move continuously between the two states}\\addcontentsline{toc}{section}{Move continuously between the two states}\n\\lead{Changing only relative phase moves the selected probability from zero to one.}\nLet $v_+$ contain the positive entries of $c_{12}/\\sqrt{48}$ and $v_-$ contain the absolute values of its negative entries. Their supports are disjoint and each has squared norm $1/2$. Define\n\\[\n \\chi_\\theta=v_++e^{i\\theta}v_-,\\qquad\n \\chi_0=\\chi,\\quad\\chi_\\pi=\\psi.\n\\]\n\\fig{15_phase_probability}{1.0}{The selected probability is the exact formula $\\sin^2(\\theta/2)$. The continuous curve is its floating rendering. Black dots mark exactly checked quarter-turn cases $0,1/2,1,1/2,0$.}\nIndeed, $\\chi_\\theta=\\tfrac{1+e^{i\\theta}}2\\chi+\\tfrac{1-e^{i\\theta}}2\\psi$. Since $P\\chi=0$ and $P\\psi=\\psi$, we get $\\|P\\chi_\\theta\\|^2=|1-e^{i\\theta}|^2/4=\\sin^2(\\theta/2)$.\n\\note{The endpoint statements are covered by the inherited Lean development. This continuous family is an elementary prose derivation and a checked illustration, not an additional inherited Lean theorem.}","ction*{Move continuously between the two states}\\addcontentsline{toc}{section}{Move continuously between the two states}\n\\lead{Changing only relative phase moves the selected probability from zero to one.}\nLet $v_+$ contain the positive entries of $c_{12}/\\sqrt{48}$ and $v_-$ contain the absolute values of its negative entries. Their supports are disjoint and each has squared norm $1/2$. Define\n\\[\n \\chi_\\theta=v_++e^{i\\theta}v_-,\\qquad\n \\chi_0=\\chi,\\quad\\chi_\\pi=\\psi.\n\\]\n\\fig{15_phase_probability}{1.0}{The selected probability is the exact formula $\\sin^2(\\theta/2)$. The continuous curve is its floating rendering. Black dots mark exactly checked quarter-turn cases $0,1/2,1,1/2,0$.}\nIndeed, $\\chi_\\theta=\\tfrac{1+e^{i\\theta}}2\\chi+\\tfrac{1-e^{i\\theta}}2\\psi$. Since $P\\chi=0$ and $P\\psi=\\psi$, we get $\\|P\\chi_\\theta\\|^2=|1-e^{i\\theta}|^2/4=\\sin^2(\\theta/2)$.\n\\note{The endpoint statements are covered by the inherited Lean development. This continuous family is an elementary prose derivation and a checked illustration, not an additional inherited Lean theorem.}","moves the selected probability from zero to one.}\nLet $v_+$ contain the positive entries of $c_{12}/\\sqrt{48}$ and $v_-$ contain the absolute values of its negative entries. Their supports are disjoint and each has squared norm $1/2$. Define\n\\[\n \\chi_\\theta=v_++e^{i\\theta}v_-,\\qquad\n \\chi_0=\\chi,\\quad\\chi_\\pi=\\psi.\n\\]\n\\fig{15_phase_probability}{1.0}{The selected probability is the exact formula $\\sin^2(\\theta/2)$. The continuous curve is its floating rendering. Black dots mark exactly checked quarter-turn cases $0,1/2,1,1/2,0$.}\nIndeed, $\\chi_\\theta=\\tfrac{1+e^{i\\theta}}2\\chi+\\tfrac{1-e^{i\\theta}}2\\psi$. Since $P\\chi=0$ and $P\\psi=\\psi$, we get $\\|P\\chi_\\theta\\|^2=|1-e^{i\\theta}|^2/4=\\sin^2(\\theta/2)$.\n\\note{The endpoint statements are covered by the inherited Lean development. This continuous family is an elementary prose derivation and a checked illustration, not an additional inherited Lean theorem.}","-$ contain the absolute values of its negative entries. Their supports are disjoint and each has squared norm $1/2$. Define\n\\[\n \\chi_\\theta=v_++e^{i\\theta}v_-,\\qquad\n \\chi_0=\\chi,\\quad\\chi_\\pi=\\psi.\n\\]\n\\fig{15_phase_probability}{1.0}{The selected probability is the exact formula $\\sin^2(\\theta/2)$. The continuous curve is its floating rendering. Black dots mark exactly checked quarter-turn cases $0,1/2,1,1/2,0$.}\nIndeed, $\\chi_\\theta=\\tfrac{1+e^{i\\theta}}2\\chi+\\tfrac{1-e^{i\\theta}}2\\psi$. Since $P\\chi=0$ and $P\\psi=\\psi$, we get $\\|P\\chi_\\theta\\|^2=|1-e^{i\\theta}|^2/4=\\sin^2(\\theta/2)$.\n\\note{The endpoint statements are covered by the inherited Lean development. This continuous family is an elementary prose derivation and a checked illustration, not an additional inherited Lean theorem.}","lues of its negative entries. Their supports are disjoint and each has squared norm $1/2$. Define\n\\[\n \\chi_\\theta=v_++e^{i\\theta}v_-,\\qquad\n \\chi_0=\\chi,\\quad\\chi_\\pi=\\psi.\n\\]\n\\fig{15_phase_probability}{1.0}{The selected probability is the exact formula $\\sin^2(\\theta/2)$. The continuous curve is its floating rendering. Black dots mark exactly checked quarter-turn cases $0,1/2,1,1/2,0$.}\nIndeed, $\\chi_\\theta=\\tfrac{1+e^{i\\theta}}2\\chi+\\tfrac{1-e^{i\\theta}}2\\psi$. Since $P\\chi=0$ and $P\\psi=\\psi$, we get $\\|P\\chi_\\theta\\|^2=|1-e^{i\\theta}|^2/4=\\sin^2(\\theta/2)$.\n\\note{The endpoint statements are covered by the inherited Lean development. This continuous family is an elementary prose derivation and a checked illustration, not an additional inherited Lean theorem.}","red norm $1/2$. Define\n\\[\n \\chi_\\theta=v_++e^{i\\theta}v_-,\\qquad\n \\chi_0=\\chi,\\quad\\chi_\\pi=\\psi.\n\\]\n\\fig{15_phase_probability}{1.0}{The selected probability is the exact formula $\\sin^2(\\theta/2)$. The continuous curve is its floating rendering. Black dots mark exactly checked quarter-turn cases $0,1/2,1,1/2,0$.}\nIndeed, $\\chi_\\theta=\\tfrac{1+e^{i\\theta}}2\\chi+\\tfrac{1-e^{i\\theta}}2\\psi$. Since $P\\chi=0$ and $P\\psi=\\psi$, we get $\\|P\\chi_\\theta\\|^2=|1-e^{i\\theta}|^2/4=\\sin^2(\\theta/2)$.\n\\note{The endpoint statements are covered by the inherited Lean development. This continuous family is an elementary prose derivation and a checked illustration, not an additional inherited Lean theorem.}",". Define\n\\[\n \\chi_\\theta=v_++e^{i\\theta}v_-,\\qquad\n \\chi_0=\\chi,\\quad\\chi_\\pi=\\psi.\n\\]\n\\fig{15_phase_probability}{1.0}{The selected probability is the exact formula $\\sin^2(\\theta/2)$. The continuous curve is its floating rendering. Black dots mark exactly checked quarter-turn cases $0,1/2,1,1/2,0$.}\nIndeed, $\\chi_\\theta=\\tfrac{1+e^{i\\theta}}2\\chi+\\tfrac{1-e^{i\\theta}}2\\psi$. Since $P\\chi=0$ and $P\\psi=\\psi$, we get $\\|P\\chi_\\theta\\|^2=|1-e^{i\\theta}|^2/4=\\sin^2(\\theta/2)$.\n\\note{The endpoint statements are covered by the inherited Lean development. This continuous family is an elementary prose derivation and a checked illustration, not an additional inherited Lean theorem.}","eta=v_++e^{i\\theta}v_-,\\qquad\n \\chi_0=\\chi,\\quad\\chi_\\pi=\\psi.\n\\]\n\\fig{15_phase_probability}{1.0}{The selected probability is the exact formula $\\sin^2(\\theta/2)$. The continuous curve is its floating rendering. Black dots mark exactly checked quarter-turn cases $0,1/2,1,1/2,0$.}\nIndeed, $\\chi_\\theta=\\tfrac{1+e^{i\\theta}}2\\chi+\\tfrac{1-e^{i\\theta}}2\\psi$. Since $P\\chi=0$ and $P\\psi=\\psi$, we get $\\|P\\chi_\\theta\\|^2=|1-e^{i\\theta}|^2/4=\\sin^2(\\theta/2)$.\n\\note{The endpoint statements are covered by the inherited Lean development. This continuous family is an elementary prose derivation and a checked illustration, not an additional inherited Lean theorem.}","\\clearpage\\section*{A real three-dimensional view of one amplitude}\\addcontentsline{toc}{section}{A real three-dimensional view of one amplitude}\n\\lead{The plot follows an actual complex amplitude as the relative phase advances.}\nAt sample $n=6$, the phase-family amplitude is\n\\[\n \\chi_\\theta(6)=\\frac{e^{i\\theta}}{\\sqrt3}.\n\\]\n\\fig{16_complex_amplitude_trajectory}{0.84}{Coordinates are $(\\theta,\\operatorname{Re}\\chi_\\theta(6),\\operatorname{Im}\\chi_\\theta(6))$. In the real-imaginary plane the amplitude traces a radius-$1/\\sqrt3$ circle. Advancing the phase coordinate produces the three-dimensional trajectory.}\nIts magnitude is constant, so the sample-six probability stays $1/3$ throughout. Nevertheless, interference changes the probability of the selected subspace according to the preceding page. The geometry separates phase change from magnitude change.","view of one amplitude}\n\\lead{The plot follows an actual complex amplitude as the relative phase advances.}\nAt sample $n=6$, the phase-family amplitude is\n\\[\n \\chi_\\theta(6)=\\frac{e^{i\\theta}}{\\sqrt3}.\n\\]\n\\fig{16_complex_amplitude_trajectory}{0.84}{Coordinates are $(\\theta,\\operatorname{Re}\\chi_\\theta(6),\\operatorname{Im}\\chi_\\theta(6))$. In the real-imaginary plane the amplitude traces a radius-$1/\\sqrt3$ circle. Advancing the phase coordinate produces the three-dimensional trajectory.}\nIts magnitude is constant, so the sample-six probability stays $1/3$ throughout. Nevertheless, interference changes the probability of the selected subspace according to the preceding page. The geometry separates phase change from magnitude change.","\\chi_\\theta(6)=\\frac{e^{i\\theta}}{\\sqrt3}.\n\\]\n\\fig{16_complex_amplitude_trajectory}{0.84}{Coordinates are $(\\theta,\\operatorname{Re}\\chi_\\theta(6),\\operatorname{Im}\\chi_\\theta(6))$. In the real-imaginary plane the amplitude traces a radius-$1/\\sqrt3$ circle. Advancing the phase coordinate produces the three-dimensional trajectory.}\nIts magnitude is constant, so the sample-six probability stays $1/3$ throughout. Nevertheless, interference changes the probability of the selected subspace according to the preceding page. The geometry separates phase change from magnitude change.","\\clearpage\\section*{Rearrange twelve samples into three exact lanes}\\addcontentsline{toc}{section}{Rearrange twelve samples into three exact lanes}\n\\lead{Lane extraction changes addresses while preserving all twelve values.}\nFor $r=0,1,2$ and $m=0,1,2,3$, define $(E_rx)(m)=x(r+3m)$.\n\\fig{17_polyphase_lanes}{0.94}{The lane numbers and original addresses are explicit. Interleaving all three lanes reconstructs the original vector exactly. Keeping fewer lanes would be a different, lossy observation.}\nThe lanes are $(9,2,-5,2)$, $(2,3,2,1)$, and $(1,2,3,2)$. Twelve values remain twelve values. A shorter address in each lane does not by itself reduce information content or computation cost.","amples into three exact lanes}\n\\lead{Lane extraction changes addresses while preserving all twelve values.}\nFor $r=0,1,2$ and $m=0,1,2,3$, define $(E_rx)(m)=x(r+3m)$.\n\\fig{17_polyphase_lanes}{0.94}{The lane numbers and original addresses are explicit. Interleaving all three lanes reconstructs the original vector exactly. Keeping fewer lanes would be a different, lossy observation.}\nThe lanes are $(9,2,-5,2)$, $(2,3,2,1)$, and $(1,2,3,2)$. Twelve values remain twelve values. A shorter address in each lane does not by itself reduce information content or computation cost.","ree exact lanes}\n\\lead{Lane extraction changes addresses while preserving all twelve values.}\nFor $r=0,1,2$ and $m=0,1,2,3$, define $(E_rx)(m)=x(r+3m)$.\n\\fig{17_polyphase_lanes}{0.94}{The lane numbers and original addresses are explicit. Interleaving all three lanes reconstructs the original vector exactly. Keeping fewer lanes would be a different, lossy observation.}\nThe lanes are $(9,2,-5,2)$, $(2,3,2,1)$, and $(1,2,3,2)$. Twelve values remain twelve values. A shorter address in each lane does not by itself reduce information content or computation cost.","\\lead{Lane extraction changes addresses while preserving all twelve values.}\nFor $r=0,1,2$ and $m=0,1,2,3$, define $(E_rx)(m)=x(r+3m)$.\n\\fig{17_polyphase_lanes}{0.94}{The lane numbers and original addresses are explicit. Interleaving all three lanes reconstructs the original vector exactly. Keeping fewer lanes would be a different, lossy observation.}\nThe lanes are $(9,2,-5,2)$, $(2,3,2,1)$, and $(1,2,3,2)$. Twelve values remain twelve values. A shorter address in each lane does not by itself reduce information content or computation cost.","\\clearpage\\section*{Phase factors connect the lane spectra}\\addcontentsline{toc}{section}{Phase factors connect the lane spectra}\n\\lead{Each local Fourier transform contributes to every appropriate global frequency.}\nWith four-point negative-sign DFTs in the lanes,\n\\[\n \\widehat x(k)=\\sum_{r=0}^{2}e^{-2\\pi ikr/12}\n \\widehat{E_rx}(k\\bmod4).\n\\]\n\\fig{18_polyphase_alias_assembly}{1.0}{Left: magnitudes of the three local transforms. Right: the recovered global spectrum. Assembly uses the complex coefficients and phase factors; adding the displayed magnitudes would be incorrect.}\nThe identity follows by writing $n=r+3m$ in the full DFT sum. Its phase factors retain the distinction between the lanes. The diagnostic replay compares this assembly with an independent direct twelve-point DFT at a declared tolerance of $10^{-12}$.","y.}\nWith four-point negative-sign DFTs in the lanes,\n\\[\n \\widehat x(k)=\\sum_{r=0}^{2}e^{-2\\pi ikr/12}\n \\widehat{E_rx}(k\\bmod4).\n\\]\n\\fig{18_polyphase_alias_assembly}{1.0}{Left: magnitudes of the three local transforms. Right: the recovered global spectrum. Assembly uses the complex coefficients and phase factors; adding the displayed magnitudes would be incorrect.}\nThe identity follows by writing $n=r+3m$ in the full DFT sum. Its phase factors retain the distinction between the lanes. The diagnostic replay compares this assembly with an independent direct twelve-point DFT at a declared tolerance of $10^{-12}$.","transforms. Right: the recovered global spectrum. Assembly uses the complex coefficients and phase factors; adding the displayed magnitudes would be incorrect.}\nThe identity follows by writing $n=r+3m$ in the full DFT sum. Its phase factors retain the distinction between the lanes. The diagnostic replay compares this assembly with an independent direct twelve-point DFT at a declared tolerance of $10^{-12}$.","ends beyond this picture}\n\\lead{The integer waveform changes with the modulus, while the same arithmetic identity supplies it.}\n\\fig{19_other_arithmetic_signals}{1.0}{Four exact Ramanujan signals constructed by the divisor formula. Their periods and amplitudes differ. These examples illustrate the general formulas; their finite verification does not prove those formulas universally.}\nFor general $q$, character orthogonality gives\n\\[\n \\widehat{c_q}(k)=q\\mathbf1_{\\gcd(k,q)=1},\\qquad\n \\sum_{n=0}^{q-1}|c_q(n)|^2=q\\varphi(q).\n\\]\nThe associated primitive-order projector has rank $\\varphi(q)$. The general mathematical conclusions follow from the finite-sum derivations in the main paper. The source verifier separately checks its declared finite range $2\\leq q\\leq64$.","integer waveform changes with the modulus, while the same arithmetic identity supplies it.}\n\\fig{19_other_arithmetic_signals}{1.0}{Four exact Ramanujan signals constructed by the divisor formula. Their periods and amplitudes differ. These examples illustrate the general formulas; their finite verification does not prove those formulas universally.}\nFor general $q$, character orthogonality gives\n\\[\n \\widehat{c_q}(k)=q\\mathbf1_{\\gcd(k,q)=1},\\qquad\n \\sum_{n=0}^{q-1}|c_q(n)|^2=q\\varphi(q).\n\\]\nThe associated primitive-order projector has rank $\\varphi(q)$. The general mathematical conclusions follow from the finite-sum derivations in the main paper. The source verifier separately checks its declared finite range $2\\leq q\\leq64$.","anujan signals constructed by the divisor formula. Their periods and amplitudes differ. These examples illustrate the general formulas; their finite verification does not prove those formulas universally.}\nFor general $q$, character orthogonality gives\n\\[\n \\widehat{c_q}(k)=q\\mathbf1_{\\gcd(k,q)=1},\\qquad\n \\sum_{n=0}^{q-1}|c_q(n)|^2=q\\varphi(q).\n\\]\nThe associated primitive-order projector has rank $\\varphi(q)$. The general mathematical conclusions follow from the finite-sum derivations in the main paper. The source verifier separately checks its declared finite range $2\\leq q\\leq64$.","ove those formulas universally.}\nFor general $q$, character orthogonality gives\n\\[\n \\widehat{c_q}(k)=q\\mathbf1_{\\gcd(k,q)=1},\\qquad\n \\sum_{n=0}^{q-1}|c_q(n)|^2=q\\varphi(q).\n\\]\nThe associated primitive-order projector has rank $\\varphi(q)$. The general mathematical conclusions follow from the finite-sum derivations in the main paper. The source verifier separately checks its declared finite range $2\\leq q\\leq64$.","midrule\nSelector $D$ to $P=FDF^\\dagger$ & Rank, idempotence, orthogonality & A specified unitary Fourier basis","[5pt]\nVector $x$ to $F^\\dagger x$ & Full vector, inner product, norm & Retain all complex coefficients","[5pt]\nVector $x$ to $Px$ & Selected component & Complementary component removed","bottomrule\n\\end{tabular}\n\\end{center}\nThe two states $\\psi$ and $\\chi$ give a concrete obstruction: the map from coherent amplitudes to a single-basis probability histogram is not injective. Their common histogram cannot determine which arithmetic selection will appear after the Fourier transform.\n\\note{The guide uses an actual twelve-dimensional Hilbert space. A four-qubit realization would require a specified embedding and implementation. Neither a gate synthesis nor an efficiency bound is supplied here.}","\\textbf{Floating diagnostics.} Fourier matrix checks, lane assembly, phase curves, and figure coordinates use NumPy and Matplotlib. Their residuals are reported against an explicit $10^{-12}$ tolerance. Passing that tolerance is numerical evidence, not exact arithmetic or an interval certificate.","\\bibitem{g:AmyMosca2019}\nM.~Amy and M.~Mosca,\n\\emph{$T$-count optimization and Reed--Muller codes},\nIEEE Trans. Inform. Theory \\textbf{65} (2019), 4771--4784.","\\bibitem{g:DehaeneDeMoor2003}\nJ.~Dehaene and B.~De Moor,\n\\emph{Clifford group, stabilizer states, and linear and quadratic operations over $\\mathrm{GF}(2)$},\nPhys. Rev. A \\textbf{68} (2003), 042318.","\\bibitem{g:GongYu2026}\nS.~Gong and A.~Yu,\n\\emph{Explicit matrices over $\\mathbb Z_2$ with CNOT and row complexity $4n-o(n)$ and local logic gates},\narXiv:2607.28598, 2026.","\\bibitem{g:KliuchnikovEtAl2013}\nV.~Kliuchnikov, D.~Maslov, and M.~Mosca,\n\\emph{Fast and efficient exact synthesis of single-qubit unitaries generated by Clifford and $T$ gates},\nQuantum Inf. Comput. \\textbf{13} (2013), 607--630.","\\bibitem{g:Selinger2013}\nP.~Selinger,\n\\emph{Quantum circuits of $T$-depth one},\nPhys. Rev. A \\textbf{87} (2013), 042302.","\\subsection{Residue projector formula}\nLet $Q\\ket{\\psi_k}=k\\ket{\\psi_k}$. Then\n\\[\n\\frac1M\\sum_{\\ell=0}^{M-1}\n\\omega_M^{-r\\ell}e^{2\\pi i\\ell Q/M}\\ket{\\psi_k}\n=\n\\left(\n\\frac1M\\sum_{\\ell=0}^{M-1}\n\\omega_M^{(k-r)\\ell}\n\\right)\\ket{\\psi_k}.\n\\]\nThe bracket equals one when $k\\equiv r\\pmod M$ and zero otherwise. This proves \\cref{s:eq:residue-projector} and the orthogonality of the residue packets.","\\subsection{Circuit leakage telescoping}\nLet $V_j=PG_jP$. Insert and subtract one factor at a time:\n\\[\nG_m\\cdots G_1P-V_m\\cdots V_1P\n=\n\\sum_{k=1}^m\nG_m\\cdots G_{k+1}(G_k-V_k)V_{k-1}\\cdots V_1P.\n\\]\nEvery $G_j$ has norm one and every $V_j$ has norm at most one. On $P\\Hh$,\n\\[\n(G_k-V_k)P=(I-P)G_kP.\n\\]\nTaking norms gives \\cref{s:prop:sum-leakage}.","\\subsection{Hypercube quotient coupling}\nLet\n\\[\n\\ket k\n=\n\\binom nk^{-1/2}\n\\sum_{|x|=k}\\ket x.\n\\]\nEvery weight-$k$ string has $n-k$ neighbors of weight $k+1$, while every weight-$(k+1)$ string has $k+1$ neighbors of weight $k$. Counting the bipartite edges yields\n\\[\n\\bra{k+1}A\\ket k\n=\n\\frac{\\binom nk(n-k)}\n{\\sqrt{\\binom nk\\binom n{k+1}}}\n=\n\\sqrt{(k+1)(n-k)},\n\\]\nproving \\cref{s:eq:hypercube-coupling}.","\\subsection{Symmetric-subspace dimension}\nA symmetric basis state of $n$ qudits is indexed by occupation numbers\n\\[\n(n_1,\\ldots,n_d),\n\\qquad\nn_1+\\cdots+n_d=n.\n\\]\nThe stars-and-bars count is\n\\[\n\\binom{n+d-1}{d-1},\n\\]\nproving \\cref{s:eq:symmetric-qudit-dim}.","\\subsection{Bosonic and fermionic dimensions}\nBosonic occupations are weak compositions of $N$ into $m$ parts, giving $\\binom{N+m-1}{N}$. Fermionic occupations are $N$-element subsets of $m$ modes, giving $\\binom mN$.","\\subsection{Acceptance-probability stability}\nIf normalized states $\\ket\\psi,\\ket\\phi$ satisfy\n\\[\n\\|\\ket\\psi-\\ket\\phi\\|\\le\\varepsilon\n\\]\nand $0\\le M\\le I$, then\n\\[\n|\\bra\\psi M\\ket\\psi-\\bra\\phi M\\ket\\phi|\n\\le 2\\varepsilon.\n\\]\nIndeed, add and subtract $\\bra\\phi M\\ket\\psi$ and use Cauchy--Schwarz. This converts state-vector leakage bounds into measurement-error bounds.","\\subsection{Orbit quotient invariance}\nFor an automorphism group $G$, the adjacency matrix commutes with the permutation representation. The fixed subspace of the action is spanned by normalized orbit states. Therefore adjacency preserves that span. Polynomial-time simulation additionally requires direct computation of the quotient entries.","\\toprule\n$n$ & $k$ & full dim. & sector dim. & apply gain & one-shot gain & memory gain","\\midrule\n\\endfirsthead\n\\toprule\n$n$ & $k$ & full dim. & sector dim. & apply gain & one-shot gain & memory gain","\\toprule\n$n$ & full dim. & packet dim. & apply gain & one-shot gain & memory gain","\\midrule\n\\endfirsthead\n\\toprule\n$n$ & full dim. & packet dim. & apply gain & one-shot gain & memory gain","Break parity with $X$ rotation & 0.869 & 0.131","Fixed $k$ number-preserving circuits lie in $P$ & Proved corollary & $k$ fixed and local restricted gates efficiently applicable.","Hamiltonian leakage bounded by $|t|\\|[H,P]\\|$ & Proved & Bounded self-adjoint Hamiltonian.","Permutation-symmetric qubits lie in $P$ & Derived theorem & Gate matrices and measurement efficient in Dicke coordinates.","Fixed-$d$ symmetric qudits lie in $P$ & Derived theorem & Fixed local dimension and efficient occupation updates.","Polynomial-orbit quantum walks lie in $P$ & Derived theorem & Quotient data computable from succinct input.","Fixed-particle boson and fermion circuits lie in $P$ & Derived theorem & Particle number fixed and updates efficient.","Universal polynomial packetization & Open conjecture & Would imply $P=BQP$.","\\subsection{The computational mismatch}\nAn $n$-qubit pure state has $2^n$ complex amplitudes in the computational basis. That description is universal but frequently redundant. A number-preserving circuit started with two excitations never visits basis strings of weight zero, one, three, or more. A permutation-symmetric circuit started in a symmetric state never distinguishes two strings that differ only by a relabeling of qubits. A quantum walk on a highly symmetric graph may evolve entirely in the span of a polynomial number of orbit states. In each case the full Hilbert space is a valid container but not the intrinsic state space of the computation.","\\begin{definition}[Approximate symmetry collapse]\nA uniform circuit family exhibits an approximate symmetry collapse to error $\\varepsilon(n)$ when a polynomial-size classical representation reproduces its acceptance probability to additive error $\\varepsilon(n)$, with the error below the promise gap required by the decision problem.\n\\end{definition}","The first three occur in this monograph for declared subclasses. The fourth would imply $BQP\\subseteq P$ and is not established. A benchmark can support the second conclusion without proving the third. A small invariant label set can support the first conclusion while leaving exponentially large internal blocks and therefore failing to prove the third.","\\subsection{The four audit quantities}\nFor a proposed reduction, define the schematic complexity record\n\\[\n\\mathfrak C_n=\n\\bigl(\nN_{\\mathrm{packets}},\nD_{\\mathrm{internal}},\nS_{\\mathrm{transfer}},\nL_{\\mathrm{leakage}}\n\\bigr).\n\\]\nHere:","\\begin{itemize}\n\\item $N_{\\mathrm{packets}}$ is the number of reachable symmetry labels.\n\\item $D_{\\mathrm{internal}}$ is the packet dimension or the size of a compressed internal representation.\n\\item $S_{\\mathrm{transfer}}$ records the sparsity and cost of gate-induced transitions.\n\\item $L_{\\mathrm{leakage}}$ measures departure from the proposed representation.\n\\end{itemize}","\\subsection{Why symmetry is only the beginning}\nSuppose a group $G$ acts on the Hilbert space and the circuit commutes with this action. Representation theory decomposes the space into isotypic pieces. This gives a canonical block structure, but three questions remain.","First, how many pieces are reachable from the input? Second, how large is each piece or how compactly can its states be encoded? Third, can each gate and measurement be applied in those coordinates without reconstructing the full state? The monograph repeatedly returns to this three-stage distinction:\n\\[\n\\text{symmetry label}\n\\longrightarrow\n\\text{reachable representation}\n\\longrightarrow\n\\text{efficient classical algorithm}.\n\\]","\\subsection{Orthogonal packet decompositions}\nLet $\\Hh$ be a finite-dimensional Hilbert space and let $\\{P_\\alpha\\}_{\\alpha\\in A}$ be orthogonal projectors satisfying\n\\[\nP_\\alpha P_\\beta=\\delta_{\\alpha\\beta}P_\\alpha,\n\\qquad\n\\sum_{\\alpha\\in A}P_\\alpha=I.\n\\]\nEvery state and operator decompose as\n\\[\n\\ket\\psi=\\sum_\\alpha P_\\alpha\\ket\\psi,\n\\qquad\nX=\\sum_{\\alpha,\\beta}P_\\alpha X P_\\beta.\n\\]\nThe diagonal blocks act within packets. The off-diagonal blocks transfer amplitude or coherence between packets.","Define the packet-transfer graph of an operator $X$ by placing an edge $\\beta\\to\\alpha$ whenever\n\\[\nP_\\alpha X P_\\beta\\neq0.\n\\]\nAn exactly symmetry-preserving operator has only diagonal edges. A sparse transfer graph may still permit efficient simulation if the active packets and internal states remain compact.","\\subsection{Conserved observables}\nLet\n\\[\nQ=\\sum_k q_kP_k\n\\]\nbe a self-adjoint observable with spectral projectors $P_k$.","\\begin{theorem}[Invariant quantum packets]\n\\label{s:thm:invariant-packets}\nIf a unitary $U$ satisfies $[U,Q]=0$, then\n\\[\nP_iUP_j=0\\qquad(i\\neq j),\n\\]\nand\n\\[\nU=\\bigoplus_i U_i,\n\\qquad\nU_i=P_iUP_i\\big|_{P_i\\Hh}.\n\\]\nIf $P_k\\ket\\psi=\\ket\\psi$, then $P_kU\\ket\\psi=U\\ket\\psi$.\n\\end{theorem}","\\begin{proof}\nSince $U$ commutes with $Q$, it commutes with every spectral projector obtained from the functional calculus of $Q$. Thus\n\\[\nP_iUP_j=UP_iP_j=0\n\\]\nfor $i\\neq j$. Summing the diagonal blocks gives the direct-sum form. The final statement follows from\n\\[\nP_kU\\ket\\psi=UP_k\\ket\\psi=U\\ket\\psi.\n\\]\n\\end{proof}","\\subsection{Residue and character packets}\nSuppose $Q$ has integer spectrum. For a modulus $M$, define\n\\[\nB_r^{(M)}=\n\\sum_{k\\equiv r\\pmod M}P_k.\n\\]\nLet\n\\[\nV_M=\\exp\\left(\\frac{2\\pi i}{M}Q\\right),\n\\qquad\n\\omega_M=e^{2\\pi i/M}.\n\\]\nThen\n\\begin{equation}\nB_r^{(M)}\n=\n\\frac1M\\sum_{\\ell=0}^{M-1}\n\\omega_M^{-r\\ell}V_M^\\ell.\n\\label{s:eq:residue-projector}\n\\end{equation}\nOn a $Q$-eigenvector, the right side is the roots-of-unity filter. The same finite-character formula selects residue classes in signal polyphase analysis and charge sectors in quantum theory.","\\subsection{Finite-group projectors}\nLet a finite group $G$ act unitarily by $U_g$. For an irreducible character $\\chi$ of degree $d_\\chi$, the central isotypic projector is\n\\begin{equation}\nP_\\chi\n=\n\\frac{d_\\chi}{|G|}\n\\sum_{g\\in G}\n\\overline{\\chi(g)}U_g.\n\\label{s:eq:character-projector}\n\\end{equation}\nCharacter orthogonality gives\n\\[\nP_\\chi P_\\psi=\\delta_{\\chi\\psi}P_\\chi,\n\\qquad\n\\sum_\\chi P_\\chi=I\n\\]\non the represented space. If an operator $A$ commutes with every $U_g$, then\n\\[\nP_\\chi A P_\\psi=0\n\\qquad(\\chi\\neq\\psi).\n\\]","te group $G$ act unitarily by $U_g$. For an irreducible character $\\chi$ of degree $d_\\chi$, the central isotypic projector is\n\\begin{equation}\nP_\\chi\n=\n\\frac{d_\\chi}{|G|}\n\\sum_{g\\in G}\n\\overline{\\chi(g)}U_g.\n\\label{s:eq:character-projector}\n\\end{equation}\nCharacter orthogonality gives\n\\[\nP_\\chi P_\\psi=\\delta_{\\chi\\psi}P_\\chi,\n\\qquad\n\\sum_\\chi P_\\chi=I\n\\]\non the represented space. If an operator $A$ commutes with every $U_g$, then\n\\[\nP_\\chi A P_\\psi=0\n\\qquad(\\chi\\neq\\psi).\n\\]","ly by $U_g$. For an irreducible character $\\chi$ of degree $d_\\chi$, the central isotypic projector is\n\\begin{equation}\nP_\\chi\n=\n\\frac{d_\\chi}{|G|}\n\\sum_{g\\in G}\n\\overline{\\chi(g)}U_g.\n\\label{s:eq:character-projector}\n\\end{equation}\nCharacter orthogonality gives\n\\[\nP_\\chi P_\\psi=\\delta_{\\chi\\psi}P_\\chi,\n\\qquad\n\\sum_\\chi P_\\chi=I\n\\]\non the represented space. If an operator $A$ commutes with every $U_g$, then\n\\[\nP_\\chi A P_\\psi=0\n\\qquad(\\chi\\neq\\psi).\n\\]","For an irreducible character $\\chi$ of degree $d_\\chi$, the central isotypic projector is\n\\begin{equation}\nP_\\chi\n=\n\\frac{d_\\chi}{|G|}\n\\sum_{g\\in G}\n\\overline{\\chi(g)}U_g.\n\\label{s:eq:character-projector}\n\\end{equation}\nCharacter orthogonality gives\n\\[\nP_\\chi P_\\psi=\\delta_{\\chi\\psi}P_\\chi,\n\\qquad\n\\sum_\\chi P_\\chi=I\n\\]\non the represented space. If an operator $A$ commutes with every $U_g$, then\n\\[\nP_\\chi A P_\\psi=0\n\\qquad(\\chi\\neq\\psi).\n\\]","\\subsection{Nested refinements}\nIf one conserved label is too coarse, intersect commuting projector families. For mutually commuting observables $Q_1,\\ldots,Q_s$, a joint packet has the form\n\\[\nP_{\\bm\\alpha}\n=\nP^{(1)}_{\\alpha_1}\\cdots P^{(s)}_{\\alpha_s}.\n\\]\nThe dimension can decrease dramatically, but each extra label must be efficiently computable and preserved. A refinement that requires exponential preprocessing merely relocates the complexity.","\\subsection{Circuit families}\nA quantum decision procedure is a polynomial-time uniform family of circuits $\\{C_x\\}$, where the input string $x$ determines a circuit on $n=n(|x|)$ qubits with at most $\\poly(|x|)$ gates. Let\n\\[\nC_x=G_m\\cdots G_1.\n\\]\nA classical simulation theorem must describe an algorithm receiving the same succinct description of $x$ and the circuit. It is not enough to assume that an exponentially large matrix or a complete orbit list has already been supplied.","\\subsection{The packet access model}\nFor each input length, suppose there is a packet projector $P_n$ and a basis\n\\[\n\\B_n=\n\\{\\ket{b_1},\\ldots,\\ket{b_{D_n}}\\}\n\\]\nof $P_n\\Hh_n$. The simulation interface has four routines.","\\begin{enumerate}\n\\item \\textbf{Discovery and indexing.} Produce or index $\\B_n$ in polynomial time.\n\\item \\textbf{Initialization.} Express the initial state in this basis.\n\\item \\textbf{Gate update.} Apply each restricted gate $P_nG_jP_n$ to the packet coordinates in polynomial time.\n\\item \\textbf{Measurement.} Compute the final acceptance probability to the needed precision.\n\\end{enumerate}","\\subsection{State-vector cost inside a packet}\nIf the simulator stores $D$ amplitudes and a local gate can be applied in $O(nD)$ arithmetic operations, a depth-$d$ packet simulation has schematic cost\n\\[\nC_{\\mathrm{packet}}=O(dnD)+C_{\\mathrm{compile}}.\n\\]\nThe corresponding full state-vector path is\n\\[\nC_{\\mathrm{full}}=O(dn2^n).\n\\]\nThese formulas are intentionally schematic. Cache behavior, indexing, vectorization, and basis compilation determine actual runtime. The asymptotic distinction is exact when $D$ is polynomial and $2^n$ is exponential.","applied in $O(nD)$ arithmetic operations, a depth-$d$ packet simulation has schematic cost\n\\[\nC_{\\mathrm{packet}}=O(dnD)+C_{\\mathrm{compile}}.\n\\]\nThe corresponding full state-vector path is\n\\[\nC_{\\mathrm{full}}=O(dn2^n).\n\\]\nThese formulas are intentionally schematic. Cache behavior, indexing, vectorization, and basis compilation determine actual runtime. The asymptotic distinction is exact when $D$ is polynomial and $2^n$ is exponential.","ithmetic operations, a depth-$d$ packet simulation has schematic cost\n\\[\nC_{\\mathrm{packet}}=O(dnD)+C_{\\mathrm{compile}}.\n\\]\nThe corresponding full state-vector path is\n\\[\nC_{\\mathrm{full}}=O(dn2^n).\n\\]\nThese formulas are intentionally schematic. Cache behavior, indexing, vectorization, and basis compilation determine actual runtime. The asymptotic distinction is exact when $D$ is polynomial and $2^n$ is exponential.","\\begin{scopebox}\nA polynomial-dimensional invariant sector does not guarantee a practical speedup at small $n$. It guarantees that the state-vector representation within that sector has polynomial asymptotic size under the stated access assumptions.\n\\end{scopebox}","\\subsection{Gatewise invariance}\nLet $P$ be an orthogonal projector and suppose the initial state obeys $P\\ket{\\psi_0}=\\ket{\\psi_0}$. If every gate satisfies\n\\begin{equation}\nG_jP=PG_jP,\n\\label{s:eq:gatewise-invariance}\n\\end{equation}\nthen $G_j$ maps $P\\Hh$ into itself. For a general linear map, this equation asserts only invariance of the selected subspace. For the finite-dimensional unitary gates considered here, invariance also implies invariance of the orthogonal complement, so it is equivalent to commutation with $P$.","psi_0}$. If every gate satisfies\n\\begin{equation}\nG_jP=PG_jP,\n\\label{s:eq:gatewise-invariance}\n\\end{equation}\nthen $G_j$ maps $P\\Hh$ into itself. For a general linear map, this equation asserts only invariance of the selected subspace. For the finite-dimensional unitary gates considered here, invariance also implies invariance of the orthogonal complement, so it is equivalent to commutation with $P$.","\\begin{theorem}[Invariant-basis simulation]\n\\label{s:thm:invariant-basis}\nLet $U=G_m\\cdots G_1$ and assume \\cref{s:eq:gatewise-invariance} for every gate. If $P\\ket{\\psi_0}=\\ket{\\psi_0}$, then\n\\[\nU\\ket{\\psi_0}\n=\n(PG_mP)\\cdots(PG_1P)\\ket{\\psi_0}.\n\\]\nIn any basis of $P\\Hh$, the restricted circuit gives exactly the same amplitudes after embedding.\n\\end{theorem}","\\begin{proof}\nInductively, if $P\\ket{\\psi_{j-1}}=\\ket{\\psi_{j-1}}$, then\n\\[\nG_j\\ket{\\psi_{j-1}}\n=\nG_jP\\ket{\\psi_{j-1}}\n=\nPG_jP\\ket{\\psi_{j-1}},\n\\]\nso the next state remains in $P\\Hh$. Iterating proves the identity.\n\\end{proof}","\\subsection{Polynomial packet-simulation theorem}\n\\begin{theorem}[Polynomial packet-simulation theorem]\n\\label{s:thm:poly-packet}\nLet $\\{C_x\\}$ be a polynomial-time uniform family of quantum circuits. Suppose there is an efficiently computable packet representation such that:","\\begin{enumerate}\n\\item the initial state lies in the represented packet and its coordinates are computable in polynomial time;\n\\item every gate preserves the representation;\n\\item the representation size is at most $\\poly(|x|)$;\n\\item each gate update is computable to polynomial precision in polynomial time;\n\\item the final acceptance probability is computable from the representation in polynomial time.\n\\end{enumerate}","Then the language decided by the circuit family belongs to $P$.\n\\end{theorem}","\\begin{provedbox}\nThe theorem is a genuine quantum-to-classical complexity collapse for the declared subclass. It does not assume that a full $2^n$ state vector is ever constructed.\n\\end{provedbox}","\\subsection{Dimension-only corollary}\nA convenient special case uses an explicit basis of a packet of dimension $D_n$.","\\begin{corollary}[Polynomial-dimensional invariant packet]\nIf $D_n\\le\\poly(n)$, the packet basis is efficiently indexed, every gate has a polynomial-time sparse or structured action in that basis, and the measurement is efficiently evaluable, then the circuit family is classically polynomial-time simulable.\n\\end{corollary}","\\subsection{A broader compressed representation}\nThe state description need not be a vector of packet amplitudes. Let $\\mathcal R_n$ be a polynomial-size classical representation class with an embedding into the packet Hilbert space. If initialization, gate updates, and measurement stay within $\\mathcal R_n$ and are efficient, the same proof applies. This observation will later combine packet labels with stabilizer, Gaussian, or tensor-network coordinates.","\\subsection{Hamiltonian leakage}\nExact conservation is not the only useful regime. Let $P$ be an orthogonal projector and let\n\\[\nU(t)=e^{-itH}\n\\]\nfor a bounded self-adjoint Hamiltonian $H$. The leakage operator from the packet is\n\\[\nL_P(t)=(I-P)e^{-itH}P.\n\\]","\\begin{theorem}[Commutator leakage bound]\n\\label{s:thm:commutator-leakage}\nFor every real $t$,\n\\[\n\\|(I-P)e^{-itH}P\\|\n\\le\n|t|\\,\\|[H,P]\\|.\n\\]\n\\end{theorem}","\\begin{proof}\nDefine\n\\[\nF(s)=e^{-i(t-s)H}Pe^{-isH}P.\n\\]\nThen $F(0)=e^{-itH}P$ and $F(t)=Pe^{-itH}P$. Differentiation gives\n\\[\nF'(s)=ie^{-i(t-s)H}[H,P]e^{-isH}P.\n\\]\nTherefore\n\\[\n(I-P)e^{-itH}P\n=-(I-P)\\int_0^tF'(s)\\,ds.\n\\]\nThe unitary factors and projectors have norm at most one, so the integral is bounded by $|t|\\|[H,P]\\|$.\n\\end{proof}","\\subsection{Circuit leakage}\nFor a gate $G_j$, define\n\\[\n\\delta_j=\\|(I-P)G_jP\\|.\n\\]\nLet $V_j=PG_jP$. A telescoping expansion gives\n\\[\nG_m\\cdots G_1P-V_m\\cdots V_1P\n=\n\\sum_{k=1}^m\nG_m\\cdots G_{k+1}(G_k-V_k)V_{k-1}\\cdots V_1P.\n\\]\nTaking norms yields the following.","\\begin{proposition}[Sum-of-leakages bound]\n\\label{s:prop:sum-leakage}\nFor an initial state in $P\\Hh$,\n\\[\n\\|G_m\\cdots G_1\\ket{\\psi_0}\n-\nV_m\\cdots V_1\\ket{\\psi_0}\\|\n\\le\n\\sum_{j=1}^m\\delta_j.\n\\]\n\\end{proposition}","\\subsection{Decision-problem consequence}\nIf the packet representation is polynomial, the restricted updates are efficient, and\n\\[\n\\sum_j\\delta_j\\le\\frac1{\\poly(n)},\n\\]\nthen the approximate packet simulator reproduces the final acceptance probability to inverse-polynomial accuracy. After ordinary promise-gap amplification, this suffices for a deterministic polynomial-time decision algorithm.","\\subsection{Adaptive packet enlargement}\nA practical simulator need not fix one packet forever. It can maintain an active set $A_t$ of packets, estimate transfers $P_\\alpha G_tP_\\beta$, and add a packet when its predicted mass exceeds a tolerance. The proof obligation becomes a global accounting identity:\n\\[\n\\text{total error}\n\\le\n\\text{discarded transfer}\n+\n\\text{numerical error}\n+\n\\text{measurement approximation}.\n\\]\nThe key research problem is to retain polynomial active size while certifying the tail.","\\begin{definition}[Packet schedule]\nFor a circuit\n\\[\nU=G_m\\cdots G_1\n\\]\non a Hilbert space $\\Hh$, a packet schedule is a sequence of orthogonal projectors\n\\[\nP_0,P_1,\\ldots,P_m.\n\\]\nIt is exact when\n\\begin{equation}\nG_tP_{t-1}=P_tG_tP_{t-1}\n\\qquad(1\\le t\\le m).\n\\label{s:eq:dynamic-packet-condition}\n\\end{equation}\n\\end{definition}","The condition says that gate $G_t$ maps the currently active packet into the next active packet. The dimensions may vary with time.","\\subsection{Exact dynamic-packet simulation}\nChoose isometries\n\\[\nB_t:\\C^{D_t}\\longrightarrow\\Hh,\n\\qquad\nB_tB_t^*=P_t,\n\\qquad\nB_t^*B_t=I_{D_t},\n\\]\nwhere $D_t=\\rank P_t$. Define the rectangular coordinate update\n\\begin{equation}\nM_t=B_t^*G_tB_{t-1}.\n\\label{s:eq:dynamic-coordinate-update}\n\\end{equation}","\\begin{theorem}[Exact dynamic-packet theorem]\n\\label{s:thm:dynamic-packet}\nSuppose $P_0\\ket{\\psi_0}=\\ket{\\psi_0}$ and \\cref{s:eq:dynamic-packet-condition} holds for every gate. Put\n\\[\nz_0=B_0^*\\ket{\\psi_0},\n\\qquad\nz_t=M_tz_{t-1}.\n\\]\nThen\n\\[\nG_t\\cdots G_1\\ket{\\psi_0}=B_tz_t\n\\]\nfor every $t$.\n\\end{theorem}","\\begin{proof}\nThe claim is true at $t=0$. If the state after $t-1$ gates is $B_{t-1}z_{t-1}$, then\n\\[\nG_tB_{t-1}z_{t-1}\n=P_tG_tP_{t-1}B_{t-1}z_{t-1}\n=B_tB_t^*G_tB_{t-1}z_{t-1}\n=B_tM_tz_{t-1}.\n\\]\nThis is $B_tz_t$.\n\\end{proof}","\\begin{theorem}[Polynomial moving-packet collapse]\n\\label{s:thm:moving-packet-collapse}\nLet $\\{C_x\\}$ be a polynomial-time uniform circuit family. Suppose an exact packet schedule can be produced from $x$ in polynomial time such that:\n\\begin{enumerate}\n\\item $\\max_tD_t\\le\\poly(|x|)$;\n\\item the coordinate description of the initial state is computable in polynomial time;\n\\item each rectangular update $M_t$ can be applied to polynomial precision in polynomial time;\n\\item the final acceptance probability is computable from $z_m$ in polynomial time.\n\\end{enumerate}\nThen the language decided by the circuit family belongs to $P$.\n\\end{theorem}","in{theorem}[Polynomial moving-packet collapse]\n\\label{s:thm:moving-packet-collapse}\nLet $\\{C_x\\}$ be a polynomial-time uniform circuit family. Suppose an exact packet schedule can be produced from $x$ in polynomial time such that:\n\\begin{enumerate}\n\\item $\\max_tD_t\\le\\poly(|x|)$;\n\\item the coordinate description of the initial state is computable in polynomial time;\n\\item each rectangular update $M_t$ can be applied to polynomial precision in polynomial time;\n\\item the final acceptance probability is computable from $z_m$ in polynomial time.\n\\end{enumerate}\nThen the language decided by the circuit family belongs to $P$.\n\\end{theorem}","orm circuit family. Suppose an exact packet schedule can be produced from $x$ in polynomial time such that:\n\\begin{enumerate}\n\\item $\\max_tD_t\\le\\poly(|x|)$;\n\\item the coordinate description of the initial state is computable in polynomial time;\n\\item each rectangular update $M_t$ can be applied to polynomial precision in polynomial time;\n\\item the final acceptance probability is computable from $z_m$ in polynomial time.\n\\end{enumerate}\nThen the language decided by the circuit family belongs to $P$.\n\\end{theorem}","that:\n\\begin{enumerate}\n\\item $\\max_tD_t\\le\\poly(|x|)$;\n\\item the coordinate description of the initial state is computable in polynomial time;\n\\item each rectangular update $M_t$ can be applied to polynomial precision in polynomial time;\n\\item the final acceptance probability is computable from $z_m$ in polynomial time.\n\\end{enumerate}\nThen the language decided by the circuit family belongs to $P$.\n\\end{theorem}","\\subsection{Moving frames and hidden symmetry}\nThe packet schedule has a useful coordinate form. Let $P$ be one fixed projector and let $W_t$ be unitaries with\n\\[\nP_t=W_tPW_t^*.\n\\]\nIn the moving frame, the transformed gate is\n\\begin{equation}\n\\widetilde G_t\n=W_t^*G_tW_{t-1}.\n\\label{s:eq:moving-frame-gate}\n\\end{equation}\nThe schedule condition is equivalent to\n\\[\n\\widetilde G_tP=P\\widetilde G_tP.\n\\]","\\begin{corollary}[Moving-frame collapse]\n\\label{s:cor:moving-frame-collapse}\nSuppose a polynomial-time uniform circuit admits efficiently described frames $W_t$ for which the transformed gates in \\cref{s:eq:moving-frame-gate} preserve one fixed polynomial-dimensional packet, the initial state has efficiently computable coordinates in $W_0P\\Hh$, and their restricted matrices and the transformed final measurement are efficiently computable. Then the circuit family is classically simulable in polynomial time.\n\\end{corollary}","\\begin{scopebox}\nThe frame must be independently discoverable and efficiently usable. Choosing $W_t=G_t\\cdots G_1$ makes the transformed state trivial but is circular: computing that frame is the original simulation problem. The theorem is useful only when the frames have a compact classical description and their restricted actions can be computed without constructing the full state.\n\\end{scopebox}","\\subsection{Sparse packet-transfer graphs}\nLet\n\\[\nI=\\sum_{\\alpha\\in\\Lambda}P_\\alpha\n\\]\nbe an orthogonal packet decomposition. Write the gate blocks as\n\\[\nG_{t;\\beta\\alpha}=P_\\beta G_tP_\\alpha.\n\\]\nStarting from an active packet set $S_0$, define recursively\n\\begin{equation}\nS_t\n=\n\\left\\{\n\\beta:\\exists\\alpha\\in S_{t-1}\n\\text{ with }G_{t;\\beta\\alpha}\\ne0\n\\right\\}.\n\\label{s:eq:reachable-packet-set}\n\\end{equation}","\\begin{corollary}[Reachable packet-graph collapse]\n\\label{s:cor:packet-graph-collapse}\nSuppose the packet sets $S_t$ and nonzero transfer blocks are computable in polynomial time, and\n\\[\nD_t\n=\n\\sum_{\\alpha\\in S_t}\\rank P_\\alpha\n\\le\\poly(n)\n\\]\nfor every $t$. If all active block updates and the final measurement are efficiently computable, then the circuit family is classically polynomial-time simulable.\n\\end{corollary}","\\begin{proof}\nTake\n\\[\nP_t=\\sum_{\\alpha\\in S_t}P_\\alpha.\n\\]\nThe definition of $S_t$ gives $G_tP_{t-1}=P_tG_tP_{t-1}$. Apply \\cref{s:thm:moving-packet-collapse}.\n\\end{proof}","\\begin{corollary}[Bounded-excitation collapse]\n\\label{s:cor:bounded-excitation-collapse}\nConsider $n$ qubits and suppose the initial state and every circuit prefix are certified to lie in\n\\[\n\\Hh_{\\le K}\n=\n\\bigoplus_{k=0}^{K}\\Hh_{n,k}\n\\]\nfor a fixed constant $K$. If initialization, the gate actions, and the measurement are computable in polynomial time in the occupation basis, then the circuit family lies in $P$, even when individual gates change excitation number.\n\\end{corollary}","lary}[Bounded-excitation collapse]\n\\label{s:cor:bounded-excitation-collapse}\nConsider $n$ qubits and suppose the initial state and every circuit prefix are certified to lie in\n\\[\n\\Hh_{\\le K}\n=\n\\bigoplus_{k=0}^{K}\\Hh_{n,k}\n\\]\nfor a fixed constant $K$. If initialization, the gate actions, and the measurement are computable in polynomial time in the occupation basis, then the circuit family lies in $P$, even when individual gates change excitation number.\n\\end{corollary}","\\begin{proof}\nThe active dimension is\n\\[\n\\sum_{k=0}^{K}\\binom nk\n=O(n^K).\n\\]\nLet $P$ project onto $\\Hh_{\\le K}$. By the hypothesis on the actual circuit prefixes, $P\\psi_t=\\psi_t$ at every step. Consequently $PG_tP\\psi_{t-1}=G_t\\psi_{t-1}$ on the evolving state, even if the gate does not preserve all of $P\\Hh$. The restricted update therefore reproduces that trajectory exactly. The hypothesis supplies efficient initialization, updates, and measurement in these polynomially many coordinates.\n\\end{proof}","\\subsection{Approximate dynamic packets}\nDefine the one-step leakage from the scheduled packet by\n\\[\n\\delta_t\n=\n\\|(I-P_t)G_tP_{t-1}\\|.\n\\]\nLet\n\\[\n\\widehat G_t=P_tG_tP_{t-1}.\n\\]","\\begin{theorem}[Dynamic sum-of-leakages bound]\n\\label{s:thm:dynamic-leakage}\nFor an initial unit vector in $P_0\\Hh$,\n\\[\n\\left\\|\nG_m\\cdots G_1\\ket{\\psi_0}\n-\n\\widehat G_m\\cdots\\widehat G_1\\ket{\\psi_0}\n\\right\\|\n\\le\n\\sum_{t=1}^{m}\\delta_t.\n\\]\nFor any acceptance operator $0\\le M\\le I$, the corresponding acceptance probabilities differ by at most\n\\[\n2\\sum_{t=1}^{m}\\delta_t.\n\\]\n\\end{theorem}","\\begin{proof}\nA telescoping expansion inserts $G_t-\\widehat G_t$ at one time. On $P_{t-1}\\Hh$ this difference equals $(I-P_t)G_tP_{t-1}$, while all other unitary or projected factors have norm at most one. Summing the operator norms proves the state bound. If $\\psi$ and $\\phi$ have norm at most one, then\n\\[\n|\\langle\\psi,M\\psi\\rangle-\\langle\\phi,M\\phi\\rangle|\n\\le\n(\\|\\psi\\|+\\|\\phi\\|)\\|\\psi-\\phi\\|\n\\le2\\|\\psi-\\phi\\|.\n\\]\n\\end{proof}","\\begin{corollary}[Approximate moving-packet collapse]\nIf the scheduled dimensions and updates are polynomial and\n\\[\n\\sum_t\\delta_t\\le\\frac1{\\poly(n)},\n\\]\nthen, after ordinary promise-gap amplification, the circuit family remains deterministically classically simulable in polynomial time.\n\\end{corollary}","\\begin{theorem}[Adaptive discarded-state certificate]\n\\label{s:thm:adaptive-discarded-state}\nLet $G_1,\\ldots,G_m$ be unitary operators on a finite-dimensional Hilbert space, and let $\\psi_0=\\phi_0$ be a unit vector. At each step choose an orthogonal projector $P_t$, possibly as a function of the previously computed approximate states. Define\n\\[\n\\psi_t=G_t\\psi_{t-1},\\qquad\n\\phi_t=P_tG_t\\phi_{t-1},\\qquad\n\\eta_t=\\|(I-P_t)G_t\\phi_{t-1}\\|.\n\\]\nThen\n\\[\n\\|\\psi_m-\\phi_m\\|\\le\\sum_{t=1}^m\\eta_t.\n\\]\nFor an effect $0\\le M\\le I$,\n\\[\n\\left|\\langle\\psi_m,M\\psi_m\\rangle-\\langle\\phi_m,M\\phi_m\\rangle\\right|\n\\le2\\sum_{t=1}^m\\eta_t.\n\\]\nNo renormalization of $\\phi_t$ is performed. The second quadratic form is an unnormalized acceptance estimate.\n\\end{theorem}","ate]\n\\label{s:thm:adaptive-discarded-state}\nLet $G_1,\\ldots,G_m$ be unitary operators on a finite-dimensional Hilbert space, and let $\\psi_0=\\phi_0$ be a unit vector. At each step choose an orthogonal projector $P_t$, possibly as a function of the previously computed approximate states. Define\n\\[\n\\psi_t=G_t\\psi_{t-1},\\qquad\n\\phi_t=P_tG_t\\phi_{t-1},\\qquad\n\\eta_t=\\|(I-P_t)G_t\\phi_{t-1}\\|.\n\\]\nThen\n\\[\n\\|\\psi_m-\\phi_m\\|\\le\\sum_{t=1}^m\\eta_t.\n\\]\nFor an effect $0\\le M\\le I$,\n\\[\n\\left|\\langle\\psi_m,M\\psi_m\\rangle-\\langle\\phi_m,M\\phi_m\\rangle\\right|\n\\le2\\sum_{t=1}^m\\eta_t.\n\\]\nNo renormalization of $\\phi_t$ is performed. The second quadratic form is an unnormalized acceptance estimate.\n\\end{theorem}",",G_m$ be unitary operators on a finite-dimensional Hilbert space, and let $\\psi_0=\\phi_0$ be a unit vector. At each step choose an orthogonal projector $P_t$, possibly as a function of the previously computed approximate states. Define\n\\[\n\\psi_t=G_t\\psi_{t-1},\\qquad\n\\phi_t=P_tG_t\\phi_{t-1},\\qquad\n\\eta_t=\\|(I-P_t)G_t\\phi_{t-1}\\|.\n\\]\nThen\n\\[\n\\|\\psi_m-\\phi_m\\|\\le\\sum_{t=1}^m\\eta_t.\n\\]\nFor an effect $0\\le M\\le I$,\n\\[\n\\left|\\langle\\psi_m,M\\psi_m\\rangle-\\langle\\phi_m,M\\phi_m\\rangle\\right|\n\\le2\\sum_{t=1}^m\\eta_t.\n\\]\nNo renormalization of $\\phi_t$ is performed. The second quadratic form is an unnormalized acceptance estimate.\n\\end{theorem}","operators on a finite-dimensional Hilbert space, and let $\\psi_0=\\phi_0$ be a unit vector. At each step choose an orthogonal projector $P_t$, possibly as a function of the previously computed approximate states. Define\n\\[\n\\psi_t=G_t\\psi_{t-1},\\qquad\n\\phi_t=P_tG_t\\phi_{t-1},\\qquad\n\\eta_t=\\|(I-P_t)G_t\\phi_{t-1}\\|.\n\\]\nThen\n\\[\n\\|\\psi_m-\\phi_m\\|\\le\\sum_{t=1}^m\\eta_t.\n\\]\nFor an effect $0\\le M\\le I$,\n\\[\n\\left|\\langle\\psi_m,M\\psi_m\\rangle-\\langle\\phi_m,M\\phi_m\\rangle\\right|\n\\le2\\sum_{t=1}^m\\eta_t.\n\\]\nNo renormalization of $\\phi_t$ is performed. The second quadratic form is an unnormalized acceptance estimate.\n\\end{theorem}","nal projector $P_t$, possibly as a function of the previously computed approximate states. Define\n\\[\n\\psi_t=G_t\\psi_{t-1},\\qquad\n\\phi_t=P_tG_t\\phi_{t-1},\\qquad\n\\eta_t=\\|(I-P_t)G_t\\phi_{t-1}\\|.\n\\]\nThen\n\\[\n\\|\\psi_m-\\phi_m\\|\\le\\sum_{t=1}^m\\eta_t.\n\\]\nFor an effect $0\\le M\\le I$,\n\\[\n\\left|\\langle\\psi_m,M\\psi_m\\rangle-\\langle\\phi_m,M\\phi_m\\rangle\\right|\n\\le2\\sum_{t=1}^m\\eta_t.\n\\]\nNo renormalization of $\\phi_t$ is performed. The second quadratic form is an unnormalized acceptance estimate.\n\\end{theorem}","\\begin{proof}\nFix the projectors chosen during the particular run. Put $d_t=(I-P_t)G_t\\phi_{t-1}$. Subtraction gives the exact recurrence\n\\[\n\\psi_t-\\phi_t=G_t(\\psi_{t-1}-\\phi_{t-1})+d_t.\n\\]\nSince $\\psi_0-\\phi_0=0$, iteration yields\n\\[\n\\psi_m-\\phi_m=\\sum_{t=1}^mG_m\\cdots G_{t+1}d_t.\n\\]\nEvery unitary product preserves norm, so the triangle inequality proves the state estimate. Also $\\|\\psi_m\\|=1$ and $\\|\\phi_m\\|\\le1$ because each projector is contractive. Expanding the difference of the two quadratic forms and using $\\|M\\|\\le1$ gives the factor two. This proof applies to the chosen run even when the projectors were selected adaptively.\n\\end{proof}","gin{proof}\nFix the projectors chosen during the particular run. Put $d_t=(I-P_t)G_t\\phi_{t-1}$. Subtraction gives the exact recurrence\n\\[\n\\psi_t-\\phi_t=G_t(\\psi_{t-1}-\\phi_{t-1})+d_t.\n\\]\nSince $\\psi_0-\\phi_0=0$, iteration yields\n\\[\n\\psi_m-\\phi_m=\\sum_{t=1}^mG_m\\cdots G_{t+1}d_t.\n\\]\nEvery unitary product preserves norm, so the triangle inequality proves the state estimate. Also $\\|\\psi_m\\|=1$ and $\\|\\phi_m\\|\\le1$ because each projector is contractive. Expanding the difference of the two quadratic forms and using $\\|M\\|\\le1$ gives the factor two. This proof applies to the chosen run even when the projectors were selected adaptively.\n\\end{proof}","exact recurrence\n\\[\n\\psi_t-\\phi_t=G_t(\\psi_{t-1}-\\phi_{t-1})+d_t.\n\\]\nSince $\\psi_0-\\phi_0=0$, iteration yields\n\\[\n\\psi_m-\\phi_m=\\sum_{t=1}^mG_m\\cdots G_{t+1}d_t.\n\\]\nEvery unitary product preserves norm, so the triangle inequality proves the state estimate. Also $\\|\\psi_m\\|=1$ and $\\|\\phi_m\\|\\le1$ because each projector is contractive. Expanding the difference of the two quadratic forms and using $\\|M\\|\\le1$ gives the factor two. This proof applies to the chosen run even when the projectors were selected adaptively.\n\\end{proof}","\\begin{example}[A rational two-step certificate]\nTake\n\\[\nG=\\begin{pmatrix}3/5&-4/5\\\\4/5&3/5\\end{pmatrix},\n\\qquad\nP=\\begin{pmatrix}1&0\\\\0&0\\end{pmatrix},\n\\qquad\n\\psi_0=\\phi_0=\\begin{pmatrix}1\\\\0\\end{pmatrix}.\n\\]\nUsing $G_1=G_2=G$ and $P_1=P_2=P$ gives\n\\[\n\\phi_2=\\begin{pmatrix}9/25","0\\end{pmatrix}.\n\\]\nUsing $G_1=G_2=G$ and $P_1=P_2=P$ gives\n\\[\n\\phi_2=\\begin{pmatrix}9/25","0\\end{pmatrix}.\n\\]\nUsing $G_1=G_2=G$ and $P_1=P_2=P$ gives\n\\[\n\\phi_2=\\begin{pmatrix}9/25\\\\0\\end{pmatrix},\\qquad\n\\psi_2=\\begin{pmatrix}-7/25\\\\24/25\\end{pmatrix},\\qquad\n\\eta_1=4/5,\\quad\\eta_2=12/25.\n\\]\nThe squared state error is $832/625$, below $(\\eta_1+\\eta_2)^2=1024/625$. The rational validation script checks these values exactly. The deliberately large tail illustrates a valid but loose certificate, not an efficient approximation.\n\\end{example}","24/25\\end{pmatrix},\\qquad\n\\eta_1=4/5,\\quad\\eta_2=12/25.\n\\]\nThe squared state error is $832/625$, below $(\\eta_1+\\eta_2)^2=1024/625$. The rational validation script checks these values exactly. The deliberately large tail illustrates a valid but loose certificate, not an efficient approximation.\n\\end{example}","\\subsection{Excitation-number packets}\nFor $n$ qubits, define\n\\[\n\\widehat N\n=\n\\sum_{q=1}^n\\frac{I-Z_q}{2}.\n\\]\nIts eigenvalues are $k=0,1,\\ldots,n$. The projector onto Hamming weight $k$ is\n\\[\nP_{n,k}\n=\n\\sum_{|x|=k}\\ket x\\bra x,\n\\]\nand\n\\[\n\\dim P_{n,k}\\Hh\n=\n\\binom nk.\n\\]\nThe decomposition\n\\[\n(\\C^2)^{\\otimes n}\n=\n\\bigoplus_{k=0}^n\\Hh_{n,k}\n\\]\nrecovers the full dimension through\n\\[\n\\sum_{k=0}^n\\binom nk=2^n.\n\\]","\\subsection{Number-preserving gates}\nSingle-qubit $R_Z$ gates commute with $\\widehat N$. A two-qubit exchange rotation acts in the ordered basis $\\ket{00},\\ket{01},\\ket{10},\\ket{11}$ as\n\\[\nU_{XY}(\\theta,\\phi)\n=\n\\begin{pmatrix}\n1&0&0&0","\\subsection{Number-preserving gates}\nSingle-qubit $R_Z$ gates commute with $\\widehat N$. A two-qubit exchange rotation acts in the ordered basis $\\ket{00},\\ket{01},\\ket{10},\\ket{11}$ as\n\\[\nU_{XY}(\\theta,\\phi)\n=\n\\begin{pmatrix}\n1&0&0&0\\\\\n0&\\cos\\theta&-ie^{-i\\phi}\\sin\\theta&0\\\\\n0&-ie^{i\\phi}\\sin\\theta&\\cos\\theta&0\\\\\n0&0&0&1\n\\end{pmatrix}.\n\\]\nIt rotates $\\ket{01}$ and $\\ket{10}$ while fixing $\\ket{00}$ and $\\ket{11}$, so it preserves exact Hamming weight.","0&0&0&1\n\\end{pmatrix}.\n\\]\nIt rotates $\\ket{01}$ and $\\ket{10}$ while fixing $\\ket{00}$ and $\\ket{11}$, so it preserves exact Hamming weight.","\\begin{corollary}[Fixed-charge polynomial simulation]\n\\label{s:cor:fixed-charge-poly}\nFor every fixed $k$, a polynomial-time uniform family of number-preserving circuits initialized in $\\Hh_{n,k}$ and measured by an efficiently computable observable is classically simulable in polynomial time, provided the initial weight-$k$ coordinates can be computed efficiently and each restricted local gate is efficiently applied in that basis.\n\\end{corollary}","\\begin{proof}\nFor fixed $k$,\n\\[\n\\dim\\Hh_{n,k}=\\binom nk=O(n^k).\n\\]\nThe weight-$k$ basis is efficiently indexed by $k$-element subsets. A local number-preserving gate changes only basis states containing the affected occupation pattern. Apply \\cref{s:thm:poly-packet}.\n\\end{proof}","\\subsection{Sparse update structure}\nA one-qubit diagonal phase changes one amplitude at a time. An exchange gate on modes $p,q$ couples a basis state only to the state obtained by moving one excitation between those modes. Thus every column of the restricted gate has at most two nonzero entries. For fixed $k$, the total update cost is polynomial in $n$.","\\subsection{Uniformity and input encoding}\nThe fixed-charge corollary assumes $k$ is a fixed constant independent of $n$. It also applies when $k=O(1)$ is supplied by the circuit family. If $k$ is written in binary and allowed to scale as $n/2$, the sector can become exponential. The theorem is about the dimension and update cost of the actual active sector, not the name of the symmetry.","\\begin{recordbox}\nAt $n=20$, the full state has $1{,}048{,}576$ amplitudes and the exact weight-two packet has $190$. The archived repeated-application gain is $958.8$, the one-shot gain is $54.8$, and the amplitude-memory ratio is $5518.8$. The recorded relative reconstruction error is zero at the displayed precision.\n\\end{recordbox}","\\toprule\n$n$ & full dim. & packet dim. & apply gain & one-shot gain & memory gain & rel. error","\\midrule\n\\endfirsthead\n\\toprule\n$n$ & full dim. & packet dim. & apply gain & one-shot gain & memory gain & rel. error","\\subsection{What is machine independent}\nThe dimension identity\n\\[\n\\frac{2^n}{\\binom n2}\n\\]\nis independent of implementation. It predicts an exponential memory advantage for fixed $k=2$. The measured runtime ratio need not equal the dimension ratio. Packet indexing, compilation, sparse update kernels, allocation, and cache behavior determine the constant factors.","\\subsection{Repeated application versus one shot}\nLet $C_{\\rm compile}$ be the cost of constructing the packet basis and restricted gate plans, and let $C_{\\rm apply}$ be one packet application. Reusing the compiled plan $R$ times costs\n\\[\nC_{\\rm packet}(R)\n=\nC_{\\rm compile}+RC_{\\rm apply}.\n\\]\nIf the full path costs $RC_{\\rm full}$, the packet method breaks even when\n\\[\nR>\n\\frac{C_{\\rm compile}}\n{C_{\\rm full}-C_{\\rm apply}}.\n\\]\nThis is why time evolution, parameter sweeps, variational loops, and repeated likelihood evaluations are natural applications.","\\subsection{Independent reconstruction check}\nThe release code generates a random packet state, embeds it in the full Hilbert space, applies the same $R_Z$ and exchange gates through both arithmetic paths, and compares the final states. The representative checks at $(n,k)=(6,1),(8,2),(10,2)$ produce errors near machine precision.","\\subsection{Parity}\nThe even- and odd-parity packets each have dimension\n\\[\n2^{n-1}.\n\\]\nParity removes half the state vector. This can produce a useful constant factor, but it does not change exponential scaling.","A pair-creation rotation mixes $\\ket{00}$ and $\\ket{11}$ and therefore changes excitation number by two while preserving parity. The packet is exact for a parity-preserving circuit, but its internal dimension remains exponential.","\\subsection{Weight modulo four}\nA four-qubit rotation between $\\ket{0000}$ and $\\ket{1111}$ changes excitation number by four. It can mix exact-weight sectors while preserving weight modulo four. A single residue packet contains asymptotically one quarter of all basis states. Again, the reduction is a constant factor.","\\subsection{Half filling}\nAt half filling,\n\\[\n\\binom n{\\lfloor n/2\\rfloor}\n\\sim\n\\frac{2^n}{\\sqrt{\\pi n/2}}.\n\\]\nThe packet is smaller by a polynomial factor, but it remains exponential. At $n=20$, the half-filled packet contains $184{,}756$ amplitudes. The archived repeated-application gain is approximately $7.396$, while the one-shot gain is $0.6529$, meaning the packet path was slower in that single-run test after compilation overhead.","\\subsection{Entanglement inside a packet}\nA small number of symmetry labels does not imply low entanglement. The half-filled packet can support volume-law entanglement. Tensor-network bond dimension and packet dimension are independent resources. A robust simulator should therefore track both\n\\[\n(D_{\\rm packet},\\chi_{\\rm internal}),\n\\]\nwhere $\\chi_{\\rm internal}$ measures the cost of representing the state inside the packet.","\\subsection{The symmetric subspace}\nLet the symmetric group $S_n$ act by permuting the $n$ tensor factors. The fully symmetric qubit subspace is spanned by the Dicke states\n\\[\n\\ket{D_n^k}\n=\n\\binom nk^{-1/2}\n\\sum_{|x|=k}\\ket x,\n\\qquad\n0\\le k\\le n.\n\\]\nIts dimension is $n+1$.","\\begin{derivedbox}\nIf the initial state is permutation symmetric with efficiently computable Dicke coordinates, every gate preserves the fully symmetric subspace, the gate matrices in the Dicke basis are computable in polynomial time, and the final measurement is permutation invariant and efficient, then the circuit family lies in $P$.\n\\end{derivedbox}","\\begin{theorem}[Permutation-symmetric collapse]\n\\label{s:thm:permutation-collapse}\nUnder the hypotheses above, an $n$-qubit circuit can be simulated using an $(n+1)$-component classical state vector and polynomial-time updates.\n\\end{theorem}","\\begin{proof}\nThe projector onto the trivial $S_n$ representation has rank $n+1$. The Dicke basis is explicitly indexed by $k=0,\\ldots,n$. Apply \\cref{s:thm:poly-packet}.\n\\end{proof}","\\subsection{Collective spin operators}\nDefine\n\\[\nJ_a=\\frac12\\sum_{q=1}^n\\sigma_a^{(q)},\n\\qquad a\\in\\{x,y,z\\}.\n\\]\nThe operators $J_a$ preserve the symmetric subspace. We use the computational convention $\\sigma_z\\ket0=\\ket0$ and $\\sigma_z\\ket1=-\\ket1$. Thus $J_z\\ket{D_n^k}=(n/2-k)\\ket{D_n^k}$. For the following occupation formulas define $J_+=J_x-iJ_y$ and $J_-=J_x+iJ_y$, so that the subscript records increasing or decreasing excitation number rather than magnetic quantum number. Then\n\\[\nJ_+\\ket{D_n^k}\n=\n\\sqrt{(n-k)(k+1)}\\ket{D_n^{k+1}},\n\\]\n\\[\nJ_-\\ket{D_n^k}\n=\n\\sqrt{k(n-k+1)}\\ket{D_n^{k-1}}.\n\\]\nThus collective rotations and polynomial functions of collective spin admit banded or structured updates on $n+1$ amplitudes.","uad a\\in\\{x,y,z\\}.\n\\]\nThe operators $J_a$ preserve the symmetric subspace. We use the computational convention $\\sigma_z\\ket0=\\ket0$ and $\\sigma_z\\ket1=-\\ket1$. Thus $J_z\\ket{D_n^k}=(n/2-k)\\ket{D_n^k}$. For the following occupation formulas define $J_+=J_x-iJ_y$ and $J_-=J_x+iJ_y$, so that the subscript records increasing or decreasing excitation number rather than magnetic quantum number. Then\n\\[\nJ_+\\ket{D_n^k}\n=\n\\sqrt{(n-k)(k+1)}\\ket{D_n^{k+1}},\n\\]\n\\[\nJ_-\\ket{D_n^k}\n=\n\\sqrt{k(n-k+1)}\\ket{D_n^{k-1}}.\n\\]\nThus collective rotations and polynomial functions of collective spin admit banded or structured updates on $n+1$ amplitudes.","mmetric subspace. We use the computational convention $\\sigma_z\\ket0=\\ket0$ and $\\sigma_z\\ket1=-\\ket1$. Thus $J_z\\ket{D_n^k}=(n/2-k)\\ket{D_n^k}$. For the following occupation formulas define $J_+=J_x-iJ_y$ and $J_-=J_x+iJ_y$, so that the subscript records increasing or decreasing excitation number rather than magnetic quantum number. Then\n\\[\nJ_+\\ket{D_n^k}\n=\n\\sqrt{(n-k)(k+1)}\\ket{D_n^{k+1}},\n\\]\n\\[\nJ_-\\ket{D_n^k}\n=\n\\sqrt{k(n-k+1)}\\ket{D_n^{k-1}}.\n\\]\nThus collective rotations and polynomial functions of collective spin admit banded or structured updates on $n+1$ amplitudes.","g occupation formulas define $J_+=J_x-iJ_y$ and $J_-=J_x+iJ_y$, so that the subscript records increasing or decreasing excitation number rather than magnetic quantum number. Then\n\\[\nJ_+\\ket{D_n^k}\n=\n\\sqrt{(n-k)(k+1)}\\ket{D_n^{k+1}},\n\\]\n\\[\nJ_-\\ket{D_n^k}\n=\n\\sqrt{k(n-k+1)}\\ket{D_n^{k-1}}.\n\\]\nThus collective rotations and polynomial functions of collective spin admit banded or structured updates on $n+1$ amplitudes.","\\subsection{Symmetric qudits}\nFor local dimension $d$, the symmetric subspace has occupation-number basis\n\\[\n(n_1,\\ldots,n_d),\n\\qquad\nn_1+\\cdots+n_d=n,\n\\]\nand dimension\n\\begin{equation}\n\\dim\\Sym^n(\\C^d)\n=\n\\binom{n+d-1}{d-1}.\n\\label{s:eq:symmetric-qudit-dim}\n\\end{equation}\nFor fixed $d$, this is polynomial in $n$.","\\begin{corollary}[Fixed-local-dimension symmetric qudits]\nA uniform family of fully symmetric $d$-level circuits is classically polynomial-time simulable for fixed $d$, provided its occupation-basis updates and measurement are polynomial-time computable.\n\\end{corollary}","\\subsection{Decomposition}\nFor $n$ qudits of local dimension $d$, Schur--Weyl duality decomposes\n\\[\n(\\C^d)^{\\otimes n}\n\\cong\n\\bigoplus_{\\lambda\\vdash n,\\,\\ell(\\lambda)\\le d}\n\\mathcal U_\\lambda\\otimes\\mathcal V_\\lambda,\n\\]\nwhere $\\mathcal U_\\lambda$ carries an irreducible unitary-group representation and $\\mathcal V_\\lambda$ carries a symmetric-group representation. A permutation-invariant operator acts as\n\\[\nA\n=\n\\bigoplus_\\lambda\nA_\\lambda\\otimes I_{\\mathcal V_\\lambda}.\n\\]","{Decomposition}\nFor $n$ qudits of local dimension $d$, Schur--Weyl duality decomposes\n\\[\n(\\C^d)^{\\otimes n}\n\\cong\n\\bigoplus_{\\lambda\\vdash n,\\,\\ell(\\lambda)\\le d}\n\\mathcal U_\\lambda\\otimes\\mathcal V_\\lambda,\n\\]\nwhere $\\mathcal U_\\lambda$ carries an irreducible unitary-group representation and $\\mathcal V_\\lambda$ carries a symmetric-group representation. A permutation-invariant operator acts as\n\\[\nA\n=\n\\bigoplus_\\lambda\nA_\\lambda\\otimes I_{\\mathcal V_\\lambda}.\n\\]","The label $\\lambda$ can be polynomially short even when the multiplicity space $\\mathcal V_\\lambda$ is exponentially large. This is a canonical example of the difference between a small packet label and a small active representation.","\\subsection{Qubit total spin}\nFor qubits, the decomposition may be labeled by total spin $j$ and a multiplicity index. A collective $SU(2)$ operation acts nontrivially only on the $(2j+1)$-dimensional spin factor and as the identity on multiplicity. If the initial state and observable are insensitive to multiplicity, the effective dimension can be small. If arbitrary information is stored in multiplicity, total-spin labeling alone does not compress it.","\\begin{principle}[Multiplicity audit]\nA representation-theoretic packet must be recorded as\n\\[\n\\text{irrep label}\n\\quad+\n\\text{irrep carrier}\n\\quad+\n\\text{multiplicity space}.\n\\]\nDropping the multiplicity space is valid only when the dynamics, input, and measurement make it redundant.\n\\end{principle}","Let $S_n$ act on\n\\[\n\\Hh_{n,d}=(\\C^d)^{\\otimes n}\n\\]\nby permuting tensor factors. Schur--Weyl duality gives\n\\begin{equation}\n\\Hh_{n,d}\n\\cong\n\\bigoplus_{\\lambda\\vdash n,\\,\\ell(\\lambda)\\le d}\n\\mathcal U_\\lambda\\otimes\\mathcal V_\\lambda,\n\\label{s:eq:schur-decomp-commutant}\n\\end{equation}\nwhere $S_n$ acts trivially on the first tensor factor and irreducibly on $\\mathcal V_\\lambda$.","\\subsection{The permutation twirl}\nDefine the twirling channel\n\\begin{equation}\n\\mathcal T_n(X)\n=\n\\frac1{n!}\n\\sum_{\\pi\\in S_n}\nU_\\pi XU_\\pi^*.\n\\label{s:eq:permutation-twirl}\n\\end{equation}\nIts image is the commutant of the permutation action.","\\begin{lemma}[Invariant observables see only the twirl]\n\\label{s:lem:twirl-observable}\nIf $A$ commutes with every $U_\\pi$, then\n\\[\n\\Tr(A\\rho)\n=\n\\Tr\\bigl(A\\mathcal T_n(\\rho)\\bigr)\n\\]\nfor every density operator $\\rho$.\n\\end{lemma}","\\begin{proof}\nUsing cyclicity of the trace and $U_\\pi^*AU_\\pi=A$,\n\\[\n\\Tr\\bigl(A\\mathcal T_n(\\rho)\\bigr)\n=\n\\frac1{n!}\\sum_\\pi\n\\Tr(AU_\\pi\\rho U_\\pi^*)\n=\n\\frac1{n!}\\sum_\\pi\n\\Tr(U_\\pi^*AU_\\pi\\rho)\n=\n\\Tr(A\\rho).\n\\]\n\\end{proof}","If a unitary $U$ and acceptance operator $M$ are permutation invariant, then $U^*MU$ is invariant. Therefore the final probability depends on the input only through its twirl. This permits an observable-specific collapse even when the original pure input is not itself symmetric.","\\subsection{Exact commutant record}\nLet\n\\[\nu_\\lambda=\\dim\\mathcal U_\\lambda,\n\\qquad\nf_\\lambda=\\dim\\mathcal V_\\lambda.\n\\]\nSchur's lemma gives the following normal form.","\\begin{lemma}[Permutation-invariant state normal form]\n\\label{s:lem:invariant-state-normal-form}\nA density operator $\\rho$ commutes with every permutation if and only if it can be written as\n\\begin{equation}\n\\rho\n=\n\\bigoplus_\\lambda\np_\\lambda\\,\\rho_\\lambda\\otimes\n\\frac{I_{\\mathcal V_\\lambda}}{f_\\lambda},\n\\label{s:eq:commutant-state-record}\n\\end{equation}\nwhere $p_\\lambda\\ge0$, $\\sum_\\lambda p_\\lambda=1$, and each $\\rho_\\lambda$ is a density operator on $\\mathcal U_\\lambda$ whenever $p_\\lambda>0$. The weights $p_\\lambda$ and the unnormalized blocks $p_\\lambda\\rho_\\lambda$ are unique. The normalized block $\\rho_\\lambda$ is unique only when $p_\\lambda>0$ and is omitted when $p_\\lambda=0$.\n\\end{lemma}","utation-invariant state normal form]\n\\label{s:lem:invariant-state-normal-form}\nA density operator $\\rho$ commutes with every permutation if and only if it can be written as\n\\begin{equation}\n\\rho\n=\n\\bigoplus_\\lambda\np_\\lambda\\,\\rho_\\lambda\\otimes\n\\frac{I_{\\mathcal V_\\lambda}}{f_\\lambda},\n\\label{s:eq:commutant-state-record}\n\\end{equation}\nwhere $p_\\lambda\\ge0$, $\\sum_\\lambda p_\\lambda=1$, and each $\\rho_\\lambda$ is a density operator on $\\mathcal U_\\lambda$ whenever $p_\\lambda>0$. The weights $p_\\lambda$ and the unnormalized blocks $p_\\lambda\\rho_\\lambda$ are unique. The normalized block $\\rho_\\lambda$ is unique only when $p_\\lambda>0$ and is omitted when $p_\\lambda=0$.\n\\end{lemma}","bel{s:lem:invariant-state-normal-form}\nA density operator $\\rho$ commutes with every permutation if and only if it can be written as\n\\begin{equation}\n\\rho\n=\n\\bigoplus_\\lambda\np_\\lambda\\,\\rho_\\lambda\\otimes\n\\frac{I_{\\mathcal V_\\lambda}}{f_\\lambda},\n\\label{s:eq:commutant-state-record}\n\\end{equation}\nwhere $p_\\lambda\\ge0$, $\\sum_\\lambda p_\\lambda=1$, and each $\\rho_\\lambda$ is a density operator on $\\mathcal U_\\lambda$ whenever $p_\\lambda>0$. The weights $p_\\lambda$ and the unnormalized blocks $p_\\lambda\\rho_\\lambda$ are unique. The normalized block $\\rho_\\lambda$ is unique only when $p_\\lambda>0$ and is omitted when $p_\\lambda=0$.\n\\end{lemma}","al-form}\nA density operator $\\rho$ commutes with every permutation if and only if it can be written as\n\\begin{equation}\n\\rho\n=\n\\bigoplus_\\lambda\np_\\lambda\\,\\rho_\\lambda\\otimes\n\\frac{I_{\\mathcal V_\\lambda}}{f_\\lambda},\n\\label{s:eq:commutant-state-record}\n\\end{equation}\nwhere $p_\\lambda\\ge0$, $\\sum_\\lambda p_\\lambda=1$, and each $\\rho_\\lambda$ is a density operator on $\\mathcal U_\\lambda$ whenever $p_\\lambda>0$. The weights $p_\\lambda$ and the unnormalized blocks $p_\\lambda\\rho_\\lambda$ are unique. The normalized block $\\rho_\\lambda$ is unique only when $p_\\lambda>0$ and is omitted when $p_\\lambda=0$.\n\\end{lemma}","$\\rho$ commutes with every permutation if and only if it can be written as\n\\begin{equation}\n\\rho\n=\n\\bigoplus_\\lambda\np_\\lambda\\,\\rho_\\lambda\\otimes\n\\frac{I_{\\mathcal V_\\lambda}}{f_\\lambda},\n\\label{s:eq:commutant-state-record}\n\\end{equation}\nwhere $p_\\lambda\\ge0$, $\\sum_\\lambda p_\\lambda=1$, and each $\\rho_\\lambda$ is a density operator on $\\mathcal U_\\lambda$ whenever $p_\\lambda>0$. The weights $p_\\lambda$ and the unnormalized blocks $p_\\lambda\\rho_\\lambda$ are unique. The normalized block $\\rho_\\lambda$ is unique only when $p_\\lambda>0$ and is omitted when $p_\\lambda=0$.\n\\end{lemma}","if and only if it can be written as\n\\begin{equation}\n\\rho\n=\n\\bigoplus_\\lambda\np_\\lambda\\,\\rho_\\lambda\\otimes\n\\frac{I_{\\mathcal V_\\lambda}}{f_\\lambda},\n\\label{s:eq:commutant-state-record}\n\\end{equation}\nwhere $p_\\lambda\\ge0$, $\\sum_\\lambda p_\\lambda=1$, and each $\\rho_\\lambda$ is a density operator on $\\mathcal U_\\lambda$ whenever $p_\\lambda>0$. The weights $p_\\lambda$ and the unnormalized blocks $p_\\lambda\\rho_\\lambda$ are unique. The normalized block $\\rho_\\lambda$ is unique only when $p_\\lambda>0$ and is omitted when $p_\\lambda=0$.\n\\end{lemma}","=\n\\bigoplus_\\lambda\np_\\lambda\\,\\rho_\\lambda\\otimes\n\\frac{I_{\\mathcal V_\\lambda}}{f_\\lambda},\n\\label{s:eq:commutant-state-record}\n\\end{equation}\nwhere $p_\\lambda\\ge0$, $\\sum_\\lambda p_\\lambda=1$, and each $\\rho_\\lambda$ is a density operator on $\\mathcal U_\\lambda$ whenever $p_\\lambda>0$. The weights $p_\\lambda$ and the unnormalized blocks $p_\\lambda\\rho_\\lambda$ are unique. The normalized block $\\rho_\\lambda$ is unique only when $p_\\lambda>0$ and is omitted when $p_\\lambda=0$.\n\\end{lemma}","o_\\lambda\\otimes\n\\frac{I_{\\mathcal V_\\lambda}}{f_\\lambda},\n\\label{s:eq:commutant-state-record}\n\\end{equation}\nwhere $p_\\lambda\\ge0$, $\\sum_\\lambda p_\\lambda=1$, and each $\\rho_\\lambda$ is a density operator on $\\mathcal U_\\lambda$ whenever $p_\\lambda>0$. The weights $p_\\lambda$ and the unnormalized blocks $p_\\lambda\\rho_\\lambda$ are unique. The normalized block $\\rho_\\lambda$ is unique only when $p_\\lambda>0$ and is omitted when $p_\\lambda=0$.\n\\end{lemma}","V_\\lambda}}{f_\\lambda},\n\\label{s:eq:commutant-state-record}\n\\end{equation}\nwhere $p_\\lambda\\ge0$, $\\sum_\\lambda p_\\lambda=1$, and each $\\rho_\\lambda$ is a density operator on $\\mathcal U_\\lambda$ whenever $p_\\lambda>0$. The weights $p_\\lambda$ and the unnormalized blocks $p_\\lambda\\rho_\\lambda$ are unique. The normalized block $\\rho_\\lambda$ is unique only when $p_\\lambda>0$ and is omitted when $p_\\lambda=0$.\n\\end{lemma}","The complete state record is therefore\n\\[\n\\left\\{\n(p_\\lambda,\\rho_\\lambda)\n\\right\\}_\\lambda.\n\\]\nThe multiplicity dimension $f_\\lambda$ affects normalization and degeneracy, but no $f_\\lambda\\times f_\\lambda$ matrix needs to be stored.","A permutation-invariant unitary has the form\n\\begin{equation}\nU\n=\n\\bigoplus_\\lambda\nU_\\lambda\\otimes I_{\\mathcal V_\\lambda},\n\\label{s:eq:commutant-unitary-form}\n\\end{equation}\nand a permutation-invariant measurement operator has the analogous form\n\\[\nM\n=\n\\bigoplus_\\lambda\nM_\\lambda\\otimes I_{\\mathcal V_\\lambda}.\n\\]\nThe output probability is\n\\begin{equation}\n\\Tr(MU\\rho U^*)\n=\n\\sum_\\lambda\np_\\lambda\n\\Tr\\bigl(M_\\lambda U_\\lambda\\rho_\\lambda U_\\lambda^*\\bigr).\n\\label{s:eq:commutant-output}\n\\end{equation}","\\subsection{Polynomial parameter bound}\nThe number of partitions of $n$ with at most $d$ parts is bounded by the number of weak compositions:\n\\[\n\\#\\{\\lambda:\\ell(\\lambda)\\le d\\}\n\\le\n\\binom{n+d-1}{d-1}\n=O_d(n^{d-1}).\n\\]\nThe Weyl dimension formula gives\n\\begin{equation}\nu_\\lambda\n=\n\\prod_{1\\le i<j\\le d}\n\\frac{\\lambda_i-\\lambda_j+j-i}{j-i}\n=O_d\\left(n^{d(d-1)/2}\\right).\n\\label{s:eq:weyl-dimension-bound}\n\\end{equation}\nTherefore\n\\begin{equation}\n\\sum_\\lambda u_\\lambda^2\n=O_d\\left(n^{d^2-1}\\right).\n\\label{s:eq:commutant-parameter-bound}\n\\end{equation}","umber of partitions of $n$ with at most $d$ parts is bounded by the number of weak compositions:\n\\[\n\\#\\{\\lambda:\\ell(\\lambda)\\le d\\}\n\\le\n\\binom{n+d-1}{d-1}\n=O_d(n^{d-1}).\n\\]\nThe Weyl dimension formula gives\n\\begin{equation}\nu_\\lambda\n=\n\\prod_{1\\le i<j\\le d}\n\\frac{\\lambda_i-\\lambda_j+j-i}{j-i}\n=O_d\\left(n^{d(d-1)/2}\\right).\n\\label{s:eq:weyl-dimension-bound}\n\\end{equation}\nTherefore\n\\begin{equation}\n\\sum_\\lambda u_\\lambda^2\n=O_d\\left(n^{d^2-1}\\right).\n\\label{s:eq:commutant-parameter-bound}\n\\end{equation}","\\begin{theorem}[Schur-commutant collapse]\n\\label{s:thm:schur-commutant-collapse}\nFix the local dimension $d$. Consider a polynomial-time uniform family of permutation-invariant quantum circuits or channels on $n$ qudits. Suppose:\n\\begin{enumerate}\n\\item the Schur-block record of the twirled input can be constructed in polynomial time;\n\\item the induced update on the invariant operator record in \\cref{s:eq:commutant-state-record} is computable to polynomial precision in polynomial time;\n\\item the final measurement is permutation invariant and its Schur blocks are efficiently computable.\n\\end{enumerate}\nThen the associated decision problem belongs to $P$.\n\\end{theorem}","Schur-block record of the twirled input can be constructed in polynomial time;\n\\item the induced update on the invariant operator record in \\cref{s:eq:commutant-state-record} is computable to polynomial precision in polynomial time;\n\\item the final measurement is permutation invariant and its Schur blocks are efficiently computable.\n\\end{enumerate}\nThen the associated decision problem belongs to $P$.\n\\end{theorem}","\\begin{proof}\nPermutation covariance keeps the density operator inside the commutant algebra. By \\cref{s:eq:commutant-parameter-bound}, that algebra has polynomial dimension for fixed $d$. The hypotheses supply polynomial-time initialization, updates, and measurement on this record. Apply \\cref{s:thm:compressed-packet}.\n\\end{proof}","\\subsection{Qubit corollary}\nFor qubits, Schur labels are total spins $j$. The carrier dimension is\n\\[\nu_j=2j+1\\le n+1,\n\\]\nand there are at most $\\lfloor n/2\\rfloor+1$ labels. Hence\n\\begin{equation}\n\\sum_j(2j+1)^2=O(n^3).\n\\label{s:eq:qubit-commutant-size}\n\\end{equation}","\\begin{corollary}[Permutation-invariant qubit channels]\n\\label{s:cor:perm-invariant-qubit-channels}\nA uniform permutation-covariant qubit channel family with efficiently computable total-spin block updates, efficiently available twirled input blocks, and a permutation-invariant measurement is classically simulable using $O(n^3)$ density-matrix parameters, up to constant factors and numerical precision.\n\\end{corollary}","Collective Hamiltonians that are polynomial expressions in $J_x,J_y,J_z$ are a basic test class because their total-spin matrices can be generated directly without constructing the full $2^n\\times2^n$ operator.","What this theorem does not cover}\nThe theorem does not make an arbitrary local circuit easy. A gate acting on one selected pair generally fails to commute with all permutations. An arbitrary input may have Schur blocks that are difficult to compute from its succinct description. A non-invariant measurement can read information stored in multiplicity coordinates. Finally, allowing local dimension $d$ to grow with $n$ can make the commutant record superpolynomial.","does not cover}\nThe theorem does not make an arbitrary local circuit easy. A gate acting on one selected pair generally fails to commute with all permutations. An arbitrary input may have Schur blocks that are difficult to compute from its succinct description. A non-invariant measurement can read information stored in multiplicity coordinates. Finally, allowing local dimension $d$ to grow with $n$ can make the commutant record superpolynomial.","\\subsection{Orbit states}\nLet a group $G$ act by automorphisms on a finite graph with vertex set $V$. The orbits are $\\mathcal O_1,\\ldots,\\mathcal O_r$. Define normalized orbit states\n\\[\n\\ket{\\mathcal O_a}\n=\n\\frac1{\\sqrt{|\\mathcal O_a|}}\n\\sum_{x\\in\\mathcal O_a}\\ket x.\n\\]\nIf the adjacency or Hamiltonian $H$ commutes with the group action, the span of these states is invariant whenever the partition is equitable.","The quotient matrix is\n\\[\n\\widetilde H_{ab}\n=\n\\bra{\\mathcal O_a}H\\ket{\\mathcal O_b}.\n\\]","\\begin{theorem}[Orbit-quotient collapse]\n\\label{s:thm:orbit-collapse}\nSuppose a uniformly described quantum walk has a polynomial number $r(n)$ of relevant orbits, an orbit-uniform initial state, an orbit-invariant final measurement, and polynomial-time algorithms for orbit sizes, quotient entries, initial orbit coordinates, and the restricted final measurement. For continuous-time evolution, also assume a polynomial bound on $|t|\\|\\widetilde H\\|$ and efficient entry access to the precision required by the simulation. Then the walk can be simulated classically in polynomial time by the $r(n)\\times r(n)$ quotient dynamics.\n\\end{theorem}","collapse]\n\\label{s:thm:orbit-collapse}\nSuppose a uniformly described quantum walk has a polynomial number $r(n)$ of relevant orbits, an orbit-uniform initial state, an orbit-invariant final measurement, and polynomial-time algorithms for orbit sizes, quotient entries, initial orbit coordinates, and the restricted final measurement. For continuous-time evolution, also assume a polynomial bound on $|t|\\|\\widetilde H\\|$ and efficient entry access to the precision required by the simulation. Then the walk can be simulated classically in polynomial time by the $r(n)\\times r(n)$ quotient dynamics.\n\\end{theorem}",", an orbit-invariant final measurement, and polynomial-time algorithms for orbit sizes, quotient entries, initial orbit coordinates, and the restricted final measurement. For continuous-time evolution, also assume a polynomial bound on $|t|\\|\\widetilde H\\|$ and efficient entry access to the precision required by the simulation. Then the walk can be simulated classically in polynomial time by the $r(n)\\times r(n)$ quotient dynamics.\n\\end{theorem}","\\begin{proof}\nThe orbit-state span is invariant. The state is represented by $r(n)$ amplitudes. Polynomial-time computation of quotient entries and ordinary polynomial-precision linear algebra satisfy \\cref{s:thm:poly-packet}.\n\\end{proof}","\\subsection{Hypercube example}\nThe $n$-dimensional hypercube is invariant under coordinate permutations. Its vertex orbits are Hamming-weight layers $0,1,\\ldots,n$. In the normalized orbit basis, the quotient adjacency has couplings\n\\begin{equation}\n\\bra{k+1}A\\ket{k}\n=\n\\sqrt{(k+1)(n-k)}.\n\\label{s:eq:hypercube-coupling}\n\\end{equation}\nThus a walk on $2^n$ vertices initialized at $0^n$ evolves in an $(n+1)$-dimensional subspace.","ubsection{Hypercube example}\nThe $n$-dimensional hypercube is invariant under coordinate permutations. Its vertex orbits are Hamming-weight layers $0,1,\\ldots,n$. In the normalized orbit basis, the quotient adjacency has couplings\n\\begin{equation}\n\\bra{k+1}A\\ket{k}\n=\n\\sqrt{(k+1)(n-k)}.\n\\label{s:eq:hypercube-coupling}\n\\end{equation}\nThus a walk on $2^n$ vertices initialized at $0^n$ evolves in an $(n+1)$-dimensional subspace.","\\subsection{The succinct-input caveat}\nIf the full graph is supplied explicitly, a polynomial-in-$|V|$ algorithm may still be exponential in the succinct parameter $n$. The theorem requires quotient data computable directly from the succinct graph description. Discovering graph automorphisms or orbit partitions can itself be difficult. For regular families such as hypercubes, Johnson graphs, association schemes, and Cayley graphs with known character theory, the quotient is explicit.","\\subsection{Fourier packets}\nOn a cycle, translation characters diagonalize the adjacency operator. The continuous-time walk\n\\[\n\\ket{\\psi(t)}=e^{-itA}\\ket0\n\\]\nis a sum of Fourier-character packets. This is the same discrete Fourier transform that appears as a signal-processing coordinate change, a group-character decomposition, and a quantum basis transformation.","\\subsection{Bosonic occupation space}\nFor $N$ indistinguishable bosons in $m$ modes, a Fock basis state is an occupation vector\n\\[\n(n_1,\\ldots,n_m),\n\\qquad\nn_1+\\cdots+n_m=N.\n\\]\nThe dimension is\n\\begin{equation}\nD_{\\rm boson}(m,N)\n=\n\\binom{N+m-1}{N}.\n\\label{s:eq:boson-dim}\n\\end{equation}\nFor fixed $N$, this is $O(m^N)$.","\\subsection{Fermionic occupation space}\nFor $N$ fermions in $m$ modes, each mode is occupied at most once and\n\\begin{equation}\nD_{\\rm fermion}(m,N)\n=\n\\binom mN.\n\\label{s:eq:fermion-dim}\n\\end{equation}\nFor fixed $N$, this is also polynomial in $m$.","\\subsection{Gate updates}\nA number-preserving one-body unitary acts on mode creation operators by\n\\[\na_i^\\dagger\n\\longmapsto\n\\sum_j U_{ji}a_j^\\dagger.\n\\]\nA generic dense mode transformation can induce many occupation-basis transitions, but for fixed $N$ the basis dimension remains polynomial. Local hopping and phase gates are sparse. Few-body interactions remain polynomial to apply for fixed $N$, although the exponent can grow with $N$.","eserving one-body unitary acts on mode creation operators by\n\\[\na_i^\\dagger\n\\longmapsto\n\\sum_j U_{ji}a_j^\\dagger.\n\\]\nA generic dense mode transformation can induce many occupation-basis transitions, but for fixed $N$ the basis dimension remains polynomial. Local hopping and phase gates are sparse. Few-body interactions remain polynomial to apply for fixed $N$, although the exponent can grow with $N$.","\\subsection{Where the theorem stops}\nIf $N$ scales linearly with $m$, both dimensions become exponential. The fixed-number symmetry still reduces the full Fock space, but it does not prove polynomial simulation. Boson-sampling-type interference also shows that a fixed total particle number is not enough when the particle count itself scales with input size and the output asks for detailed many-body amplitudes.","\\subsection{Exact sector size is a complexity phase variable}\nThe statement ``fixed particle number is polynomial'' has a sharp extension. Let the particle number depend on the system size:\n\\[\nk=k(n).\n\\]\nThe explicit-amplitude simulation cost of one fermionic or qubit weight packet is governed by\n\\[\nD_{n,k}=\\binom nk.\n\\]\nFor $1\\le k\\le n$, the standard bounds\n\\begin{equation}\n\\left(\\frac nk\\right)^k\n\\le\n\\binom nk\n\\le\n\\left(\\frac{en}{k}\\right)^k\n\\label{s:eq:binomial-basic-bounds}\n\\end{equation}\nshow that the growth changes continuously between polynomial and exponential regimes.","Because\n\\[\n\\binom nk=\\binom n{n-k},\n\\]\nwrite\n\\[\nr(n)=\\min\\{k(n),n-k(n)\\}.\n\\]","\\begin{theorem}[Explicit fixed-charge phase diagram]\n\\label{s:thm:charge-phase-diagram}\nFor explicit amplitude storage in a weight-$k(n)$ packet:\n\\begin{enumerate}\n\\item the packet dimension is polynomial in $n$ if $r(n)=O(1)$;\n\\item the packet dimension is superpolynomial whenever $r(n)\\to\\infty$;\n\\item if $r(n)=O(\\log n)$, then\n\\[\nD_{n,k}=2^{O((\\log n)^2)},\n\\]\nso the explicit packet simulation is quasipolynomial;\n\\item if $r(n)=O(n^\\alpha)$ for a fixed $0<\\alpha<1$, then\n\\[\nD_{n,k}=2^{O(n^\\alpha\\log n)},\n\\]\nwhich is subexponential;\n\\item if $k(n)/n\\to p\\in(0,1)$, then\n\\[\n\\log_2 D_{n,k}\n=\nnH_2(k(n)/n)+O(\\log n)\n=\nnH_2(p)+o(n),\n\\]\nwhere $H_2$ is binary entropy, and the packet remains exponential.\n\\end{enumerate}\n\\end{theorem}","d-charge phase diagram]\n\\label{s:thm:charge-phase-diagram}\nFor explicit amplitude storage in a weight-$k(n)$ packet:\n\\begin{enumerate}\n\\item the packet dimension is polynomial in $n$ if $r(n)=O(1)$;\n\\item the packet dimension is superpolynomial whenever $r(n)\\to\\infty$;\n\\item if $r(n)=O(\\log n)$, then\n\\[\nD_{n,k}=2^{O((\\log n)^2)},\n\\]\nso the explicit packet simulation is quasipolynomial;\n\\item if $r(n)=O(n^\\alpha)$ for a fixed $0<\\alpha<1$, then\n\\[\nD_{n,k}=2^{O(n^\\alpha\\log n)},\n\\]\nwhich is subexponential;\n\\item if $k(n)/n\\to p\\in(0,1)$, then\n\\[\n\\log_2 D_{n,k}\n=\nnH_2(k(n)/n)+O(\\log n)\n=\nnH_2(p)+o(n),\n\\]\nwhere $H_2$ is binary entropy, and the packet remains exponential.\n\\end{enumerate}\n\\end{theorem}","s:thm:charge-phase-diagram}\nFor explicit amplitude storage in a weight-$k(n)$ packet:\n\\begin{enumerate}\n\\item the packet dimension is polynomial in $n$ if $r(n)=O(1)$;\n\\item the packet dimension is superpolynomial whenever $r(n)\\to\\infty$;\n\\item if $r(n)=O(\\log n)$, then\n\\[\nD_{n,k}=2^{O((\\log n)^2)},\n\\]\nso the explicit packet simulation is quasipolynomial;\n\\item if $r(n)=O(n^\\alpha)$ for a fixed $0<\\alpha<1$, then\n\\[\nD_{n,k}=2^{O(n^\\alpha\\log n)},\n\\]\nwhich is subexponential;\n\\item if $k(n)/n\\to p\\in(0,1)$, then\n\\[\n\\log_2 D_{n,k}\n=\nnH_2(k(n)/n)+O(\\log n)\n=\nnH_2(p)+o(n),\n\\]\nwhere $H_2$ is binary entropy, and the packet remains exponential.\n\\end{enumerate}\n\\end{theorem}","iagram}\nFor explicit amplitude storage in a weight-$k(n)$ packet:\n\\begin{enumerate}\n\\item the packet dimension is polynomial in $n$ if $r(n)=O(1)$;\n\\item the packet dimension is superpolynomial whenever $r(n)\\to\\infty$;\n\\item if $r(n)=O(\\log n)$, then\n\\[\nD_{n,k}=2^{O((\\log n)^2)},\n\\]\nso the explicit packet simulation is quasipolynomial;\n\\item if $r(n)=O(n^\\alpha)$ for a fixed $0<\\alpha<1$, then\n\\[\nD_{n,k}=2^{O(n^\\alpha\\log n)},\n\\]\nwhich is subexponential;\n\\item if $k(n)/n\\to p\\in(0,1)$, then\n\\[\n\\log_2 D_{n,k}\n=\nnH_2(k(n)/n)+O(\\log n)\n=\nnH_2(p)+o(n),\n\\]\nwhere $H_2$ is binary entropy, and the packet remains exponential.\n\\end{enumerate}\n\\end{theorem}","in{enumerate}\n\\item the packet dimension is polynomial in $n$ if $r(n)=O(1)$;\n\\item the packet dimension is superpolynomial whenever $r(n)\\to\\infty$;\n\\item if $r(n)=O(\\log n)$, then\n\\[\nD_{n,k}=2^{O((\\log n)^2)},\n\\]\nso the explicit packet simulation is quasipolynomial;\n\\item if $r(n)=O(n^\\alpha)$ for a fixed $0<\\alpha<1$, then\n\\[\nD_{n,k}=2^{O(n^\\alpha\\log n)},\n\\]\nwhich is subexponential;\n\\item if $k(n)/n\\to p\\in(0,1)$, then\n\\[\n\\log_2 D_{n,k}\n=\nnH_2(k(n)/n)+O(\\log n)\n=\nnH_2(p)+o(n),\n\\]\nwhere $H_2$ is binary entropy, and the packet remains exponential.\n\\end{enumerate}\n\\end{theorem}","t dimension is polynomial in $n$ if $r(n)=O(1)$;\n\\item the packet dimension is superpolynomial whenever $r(n)\\to\\infty$;\n\\item if $r(n)=O(\\log n)$, then\n\\[\nD_{n,k}=2^{O((\\log n)^2)},\n\\]\nso the explicit packet simulation is quasipolynomial;\n\\item if $r(n)=O(n^\\alpha)$ for a fixed $0<\\alpha<1$, then\n\\[\nD_{n,k}=2^{O(n^\\alpha\\log n)},\n\\]\nwhich is subexponential;\n\\item if $k(n)/n\\to p\\in(0,1)$, then\n\\[\n\\log_2 D_{n,k}\n=\nnH_2(k(n)/n)+O(\\log n)\n=\nnH_2(p)+o(n),\n\\]\nwhere $H_2$ is binary entropy, and the packet remains exponential.\n\\end{enumerate}\n\\end{theorem}","imension is superpolynomial whenever $r(n)\\to\\infty$;\n\\item if $r(n)=O(\\log n)$, then\n\\[\nD_{n,k}=2^{O((\\log n)^2)},\n\\]\nso the explicit packet simulation is quasipolynomial;\n\\item if $r(n)=O(n^\\alpha)$ for a fixed $0<\\alpha<1$, then\n\\[\nD_{n,k}=2^{O(n^\\alpha\\log n)},\n\\]\nwhich is subexponential;\n\\item if $k(n)/n\\to p\\in(0,1)$, then\n\\[\n\\log_2 D_{n,k}\n=\nnH_2(k(n)/n)+O(\\log n)\n=\nnH_2(p)+o(n),\n\\]\nwhere $H_2$ is binary entropy, and the packet remains exponential.\n\\end{enumerate}\n\\end{theorem}","\\begin{proof}\nThe upper bounds follow from \\cref{s:eq:binomial-basic-bounds}. If $r=O(1)$, then $\\binom nr=O(n^r)$. Conversely, suppose $r\\to\\infty$ and $r\\le n/2$. If $r\\le\\sqrt n$, then\n\\[\n\\log\\binom nr\n\\ge\nr\\log(n/r)\n\\ge\n\\frac r2\\log n,\n\\]\nwhose ratio to $\\log n$ diverges. If $r>\\sqrt n$, then\n\\[\n\\log\\binom nr\n\\ge\nr\\log2\n\\ge\n\\sqrt n\\log2,\n\\]\nwhich also dominates every constant multiple of $\\log n$. For the final assertion, $p_n=k(n)/n$ stays in a compact subinterval of $(0,1)$. Stirling's formula gives\n\\[\n\\log_2\\binom n{k(n)}\n=nH_2(p_n)-\\frac12\\log_2\\!\\bigl(2\\pi n p_n(1-p_n)\\bigr)+O(1/n).\n\\]\nThe logarithmic term is $O(\\log n)$, uniformly on that subinterval. Continuity of $H_2$ gives $nH_2(p_n)=nH_2(p)+o(n)$. Replacing $p_n$ by its limit inside an $O(\\log n)$ estimate would","\\begin{proof}\nThe upper bounds follow from \\cref{s:eq:binomial-basic-bounds}. If $r=O(1)$, then $\\binom nr=O(n^r)$. Conversely, suppose $r\\to\\infty$ and $r\\le n/2$. If $r\\le\\sqrt n$, then\n\\[\n\\log\\binom nr\n\\ge\nr\\log(n/r)\n\\ge\n\\frac r2\\log n,\n\\]\nwhose ratio to $\\log n$ diverges. If $r>\\sqrt n$, then\n\\[\n\\log\\binom nr\n\\ge\nr\\log2\n\\ge\n\\sqrt n\\log2,\n\\]\nwhich also dominates every constant multiple of $\\log n$. For the final assertion, $p_n=k(n)/n$ stays in a compact subinterval of $(0,1)$. Stirling's formula gives\n\\[\n\\log_2\\binom n{k(n)}\n=nH_2(p_n)-\\frac12\\log_2\\!\\bigl(2\\pi n p_n(1-p_n)\\bigr)+O(1/n).\n\\]\nThe logarithmic term is $O(\\log n)$, uniformly on that subinterval. Continuity of $H_2$ gives $nH_2(p_n)=nH_2(p)+o(n)$. Replacing $p_n$ by its limit inside an $O(\\log n)$ estimate would require a convergence rat","\\begin{proof}\nThe upper bounds follow from \\cref{s:eq:binomial-basic-bounds}. If $r=O(1)$, then $\\binom nr=O(n^r)$. Conversely, suppose $r\\to\\infty$ and $r\\le n/2$. If $r\\le\\sqrt n$, then\n\\[\n\\log\\binom nr\n\\ge\nr\\log(n/r)\n\\ge\n\\frac r2\\log n,\n\\]\nwhose ratio to $\\log n$ diverges. If $r>\\sqrt n$, then\n\\[\n\\log\\binom nr\n\\ge\nr\\log2\n\\ge\n\\sqrt n\\log2,\n\\]\nwhich also dominates every constant multiple of $\\log n$. For the final assertion, $p_n=k(n)/n$ stays in a compact subinterval of $(0,1)$. Stirling's formula gives\n\\[\n\\log_2\\binom n{k(n)}\n=nH_2(p_n)-\\frac12\\log_2\\!\\bigl(2\\pi n p_n(1-p_n)\\bigr)+O(1/n).\n\\]\nThe logarithmic term is $O(\\log n)$, uniformly on that subinterval. Continuity of $H_2$ gives $nH_2(p_n)=nH_2(p)+o(n)$. Replacing $p_n$ by its limit inside an $O(\\log n)$ estimate would require a convergence rate, which is not assumed here.\n\\end","\\begin{proof}\nThe upper bounds follow from \\cref{s:eq:binomial-basic-bounds}. If $r=O(1)$, then $\\binom nr=O(n^r)$. Conversely, suppose $r\\to\\infty$ and $r\\le n/2$. If $r\\le\\sqrt n$, then\n\\[\n\\log\\binom nr\n\\ge\nr\\log(n/r)\n\\ge\n\\frac r2\\log n,\n\\]\nwhose ratio to $\\log n$ diverges. If $r>\\sqrt n$, then\n\\[\n\\log\\binom nr\n\\ge\nr\\log2\n\\ge\n\\sqrt n\\log2,\n\\]\nwhich also dominates every constant multiple of $\\log n$. For the final assertion, $p_n=k(n)/n$ stays in a compact subinterval of $(0,1)$. Stirling's formula gives\n\\[\n\\log_2\\binom n{k(n)}\n=nH_2(p_n)-\\frac12\\log_2\\!\\bigl(2\\pi n p_n(1-p_n)\\bigr)+O(1/n).\n\\]\nThe logarithmic term is $O(\\log n)$, uniformly on that subinterval. Continuity of $H_2$ gives $nH_2(p_n)=nH_2(p)+o(n)$. Replacing $p_n$ by its limit inside an $O(\\log n)$ estimate would require a convergence rate, which is not assumed here.\n\\end{proof}","follow from \\cref{s:eq:binomial-basic-bounds}. If $r=O(1)$, then $\\binom nr=O(n^r)$. Conversely, suppose $r\\to\\infty$ and $r\\le n/2$. If $r\\le\\sqrt n$, then\n\\[\n\\log\\binom nr\n\\ge\nr\\log(n/r)\n\\ge\n\\frac r2\\log n,\n\\]\nwhose ratio to $\\log n$ diverges. If $r>\\sqrt n$, then\n\\[\n\\log\\binom nr\n\\ge\nr\\log2\n\\ge\n\\sqrt n\\log2,\n\\]\nwhich also dominates every constant multiple of $\\log n$. For the final assertion, $p_n=k(n)/n$ stays in a compact subinterval of $(0,1)$. Stirling's formula gives\n\\[\n\\log_2\\binom n{k(n)}\n=nH_2(p_n)-\\frac12\\log_2\\!\\bigl(2\\pi n p_n(1-p_n)\\bigr)+O(1/n).\n\\]\nThe logarithmic term is $O(\\log n)$, uniformly on that subinterval. Continuity of $H_2$ gives $nH_2(p_n)=nH_2(p)+o(n)$. Replacing $p_n$ by its limit inside an $O(\\log n)$ estimate would require a convergence rate, which is not assumed here.\n\\end{proof}","If $r=O(1)$, then $\\binom nr=O(n^r)$. Conversely, suppose $r\\to\\infty$ and $r\\le n/2$. If $r\\le\\sqrt n$, then\n\\[\n\\log\\binom nr\n\\ge\nr\\log(n/r)\n\\ge\n\\frac r2\\log n,\n\\]\nwhose ratio to $\\log n$ diverges. If $r>\\sqrt n$, then\n\\[\n\\log\\binom nr\n\\ge\nr\\log2\n\\ge\n\\sqrt n\\log2,\n\\]\nwhich also dominates every constant multiple of $\\log n$. For the final assertion, $p_n=k(n)/n$ stays in a compact subinterval of $(0,1)$. Stirling's formula gives\n\\[\n\\log_2\\binom n{k(n)}\n=nH_2(p_n)-\\frac12\\log_2\\!\\bigl(2\\pi n p_n(1-p_n)\\bigr)+O(1/n).\n\\]\nThe logarithmic term is $O(\\log n)$, uniformly on that subinterval. Continuity of $H_2$ gives $nH_2(p_n)=nH_2(p)+o(n)$. Replacing $p_n$ by its limit inside an $O(\\log n)$ estimate would require a convergence rate, which is not assumed here.\n\\end{proof}","$. Conversely, suppose $r\\to\\infty$ and $r\\le n/2$. If $r\\le\\sqrt n$, then\n\\[\n\\log\\binom nr\n\\ge\nr\\log(n/r)\n\\ge\n\\frac r2\\log n,\n\\]\nwhose ratio to $\\log n$ diverges. If $r>\\sqrt n$, then\n\\[\n\\log\\binom nr\n\\ge\nr\\log2\n\\ge\n\\sqrt n\\log2,\n\\]\nwhich also dominates every constant multiple of $\\log n$. For the final assertion, $p_n=k(n)/n$ stays in a compact subinterval of $(0,1)$. Stirling's formula gives\n\\[\n\\log_2\\binom n{k(n)}\n=nH_2(p_n)-\\frac12\\log_2\\!\\bigl(2\\pi n p_n(1-p_n)\\bigr)+O(1/n).\n\\]\nThe logarithmic term is $O(\\log n)$, uniformly on that subinterval. Continuity of $H_2$ gives $nH_2(p_n)=nH_2(p)+o(n)$. Replacing $p_n$ by its limit inside an $O(\\log n)$ estimate would require a convergence rate, which is not assumed here.\n\\end{proof}","ac r2\\log n,\n\\]\nwhose ratio to $\\log n$ diverges. If $r>\\sqrt n$, then\n\\[\n\\log\\binom nr\n\\ge\nr\\log2\n\\ge\n\\sqrt n\\log2,\n\\]\nwhich also dominates every constant multiple of $\\log n$. For the final assertion, $p_n=k(n)/n$ stays in a compact subinterval of $(0,1)$. Stirling's formula gives\n\\[\n\\log_2\\binom n{k(n)}\n=nH_2(p_n)-\\frac12\\log_2\\!\\bigl(2\\pi n p_n(1-p_n)\\bigr)+O(1/n).\n\\]\nThe logarithmic term is $O(\\log n)$, uniformly on that subinterval. Continuity of $H_2$ gives $nH_2(p_n)=nH_2(p)+o(n)$. Replacing $p_n$ by its limit inside an $O(\\log n)$ estimate would require a convergence rate, which is not assumed here.\n\\end{proof}","qrt n$, then\n\\[\n\\log\\binom nr\n\\ge\nr\\log2\n\\ge\n\\sqrt n\\log2,\n\\]\nwhich also dominates every constant multiple of $\\log n$. For the final assertion, $p_n=k(n)/n$ stays in a compact subinterval of $(0,1)$. Stirling's formula gives\n\\[\n\\log_2\\binom n{k(n)}\n=nH_2(p_n)-\\frac12\\log_2\\!\\bigl(2\\pi n p_n(1-p_n)\\bigr)+O(1/n).\n\\]\nThe logarithmic term is $O(\\log n)$, uniformly on that subinterval. Continuity of $H_2$ gives $nH_2(p_n)=nH_2(p)+o(n)$. Replacing $p_n$ by its limit inside an $O(\\log n)$ estimate would require a convergence rate, which is not assumed here.\n\\end{proof}","\\[\n\\log\\binom nr\n\\ge\nr\\log2\n\\ge\n\\sqrt n\\log2,\n\\]\nwhich also dominates every constant multiple of $\\log n$. For the final assertion, $p_n=k(n)/n$ stays in a compact subinterval of $(0,1)$. Stirling's formula gives\n\\[\n\\log_2\\binom n{k(n)}\n=nH_2(p_n)-\\frac12\\log_2\\!\\bigl(2\\pi n p_n(1-p_n)\\bigr)+O(1/n).\n\\]\nThe logarithmic term is $O(\\log n)$, uniformly on that subinterval. Continuity of $H_2$ gives $nH_2(p_n)=nH_2(p)+o(n)$. Replacing $p_n$ by its limit inside an $O(\\log n)$ estimate would require a convergence rate, which is not assumed here.\n\\end{proof}","rt n\\log2,\n\\]\nwhich also dominates every constant multiple of $\\log n$. For the final assertion, $p_n=k(n)/n$ stays in a compact subinterval of $(0,1)$. Stirling's formula gives\n\\[\n\\log_2\\binom n{k(n)}\n=nH_2(p_n)-\\frac12\\log_2\\!\\bigl(2\\pi n p_n(1-p_n)\\bigr)+O(1/n).\n\\]\nThe logarithmic term is $O(\\log n)$, uniformly on that subinterval. Continuity of $H_2$ gives $nH_2(p_n)=nH_2(p)+o(n)$. Replacing $p_n$ by its limit inside an $O(\\log n)$ estimate would require a convergence rate, which is not assumed here.\n\\end{proof}","nates every constant multiple of $\\log n$. For the final assertion, $p_n=k(n)/n$ stays in a compact subinterval of $(0,1)$. Stirling's formula gives\n\\[\n\\log_2\\binom n{k(n)}\n=nH_2(p_n)-\\frac12\\log_2\\!\\bigl(2\\pi n p_n(1-p_n)\\bigr)+O(1/n).\n\\]\nThe logarithmic term is $O(\\log n)$, uniformly on that subinterval. Continuity of $H_2$ gives $nH_2(p_n)=nH_2(p)+o(n)$. Replacing $p_n$ by its limit inside an $O(\\log n)$ estimate would require a convergence rate, which is not assumed here.\n\\end{proof}","\\begin{derivedbox}\nFor the explicit occupation-amplitude representation, ``few particles'' means a bounded number of particles or holes. A slowly growing charge can still improve on full state-vector scaling, but it no longer guarantees membership in $P$.\n\\end{derivedbox}","\\subsection{A bounded-excitation hierarchy}\nThe union of the first $K$ weight sectors has dimension\n\\[\nD_{\\le K}(n)\n=\n\\sum_{k=0}^{K}\\binom nk.\n\\]\nFor fixed $K$,\n\\[\nD_{\\le K}(n)=O(n^K).\n\\]\nIf $K=O(\\log n)$, the same union is quasipolynomial. This refines \\cref{s:cor:bounded-excitation-collapse}: bounded symmetry breaking preserves polynomial simulation, while logarithmically growing excitation gives a controlled quasipolynomial regime.","\\subsection{Bosonic and fermionic analogues}\nFor $N$ fermions in $m$ modes,\n\\[\nD_{\\mathrm F}(m,N)=\\binom mN.\n\\]\nFor $N$ bosons in $m$ modes,\n\\[\nD_{\\mathrm B}(m,N)=\\binom{m+N-1}{N}.\n\\]\nThe same binomial analysis gives:","\\begin{corollary}[Few-particle scaling]\n\\label{s:cor:few-particle-scaling}\nWith the number of modes as the input-size parameter:\n\\begin{enumerate}\n\\item fixed $N$ gives polynomial explicit-amplitude simulation;\n\\item $N=O(\\log m)$ gives quasipolynomial sector size;\n\\item $N=O(m^\\alpha)$, $0<\\alpha<1$, gives subexponential sector size;\n\\item finite-density fermionic sectors remain exponential.\n\\end{enumerate}\nFor bosons, the statements hold in the dilute regime with the same occupation-basis access assumptions.\n\\end{corollary}","\\subsection{Symmetric qudits with growing local dimension}\nThe symmetric $d$-level subspace has dimension\n\\[\nS(n,d)\n=\n\\binom{n+d-1}{d-1}.\n\\]\nFor fixed $d$, this is polynomial in $n$. If $d=d(n)$ grows, the same phase transition appears.","\\begin{proposition}[Growing-local-dimension boundary]\n\\label{s:prop:growing-d-boundary}\nIn the explicit occupation basis:\n\\begin{enumerate}\n\\item fixed $d$ gives $S(n,d)=O(n^{d-1})$;\n\\item $d=O(\\log n)$ gives $S(n,d)=2^{O((\\log n)^2)}$;\n\\item unbounded $d$ is not covered by the fixed-$d$ polynomial-collapse theorem without an additional internal compression.\n\\end{enumerate}\n\\end{proposition}","\\subsection{Complexity class versus practical runtime}\nA polynomial sector can still have a large exponent. Fixed $k=20$ gives $O(n^{20})$ explicit amplitudes, which is in $P$ but may be impractical. Conversely, a subexponential sector may be useful at moderate size without changing the formal complexity class.","A complete performance claim should therefore report both:\n\\[\n\\text{asymptotic class}\n\\qquad\\text{and}\\qquad\n\\text{actual representation size and update cost}.\n\\]\nThis is why the benchmark chapters keep wall-clock evidence separate from the dimension theorems.","\\subsection{Syndrome projectors}\nLet $S_1,\\ldots,S_m$ be commuting Pauli stabilizers with eigenvalues $\\pm1$. A syndrome $s\\in\\{0,1\\}^m$ defines\n\\[\nP_s\n=\n\\prod_{j=1}^m\n\\frac{I+(-1)^{s_j}S_j}{2}.\n\\]\nThese projectors are orthogonal and sum to the identity. Syndrome measurement selects one packet, while recovery applies a syndrome-dependent transfer toward the code packet.","\\subsection{The syndrome label is not the logical state}\nIf the code encodes $k$ logical qubits, a fixed syndrome packet can have dimension $2^k$. The syndrome has only $m$ bits, but arbitrary logical information remains inside the packet. Syndrome packetization therefore does not by itself prove efficient simulation.","\\subsection{Error-correction interpretation}\nThe packet view separates three objects:\n\\[\n\\text{syndrome label},\n\\qquad\n\\text{logical state},\n\\qquad\n\\text{recovery transfer}.\n\\]\nThis separation is useful for compressed simulation of fault-tolerant protocols, but a full complexity statement must include repeated measurements, classical control, and the size of the logical state.","on{Polynomial covariance representations}\nGaussian bosonic and fermionic states are specified by first and second moments rather than by all Fock-basis amplitudes. A bosonic covariance matrix has size polynomial in the number of modes, and a fermionic Gaussian state is represented by an antisymmetric Majorana covariance matrix. Gaussian unitaries act by symplectic or orthogonal transformations:\n\\[\n\\Gamma\\longmapsto S\\Gamma S^T\n\\quad\\text{or}\\quad\n\\Gamma\\longmapsto O\\Gamma O^T.\n\\]\nThis is a compact internal encoding even when the corresponding Hilbert-space packet is exponentially large.","\\subsection{Hybrid symmetry and Gaussianity}\nNumber conservation, parity, translation symmetry, and point-group symmetry can further block the covariance matrices. A simulator can therefore combine two reductions:\n\\[\n\\text{symmetry packetization}\n+\n\\text{Gaussian internal encoding}.\n\\]\nThe first removes unreachable sectors. The second compresses states within the remaining sectors.","\\subsection{Low-rank non-Gaussian corrections}\nOne research direction is to represent a weakly non-Gaussian state as\n\\[\n\\rho\n=\n\\sum_{a=1}^R c_a\\rho_a^{\\rm Gauss}\n\\]\nwith small Gaussian rank $R$, or as a Gaussian reference plus sparse packet corrections. A polynomial collapse requires $R$ and the update cost to remain polynomial.","\\subsection{Two packet indices}\nA pure state uses one packet label. A density operator uses two:\n\\[\n\\rho\n=\n\\sum_{\\alpha,\\beta}\nP_\\alpha\\rho P_\\beta.\n\\]\nThe diagonal block $P_\\alpha\\rho P_\\alpha$ is population inside packet $\\alpha$. The off-diagonal block $P_\\alpha\\rho P_\\beta$ is coherence between packets $\\alpha$ and $\\beta$.","\\subsection{Hilbert--Schmidt decomposition}\nOrthogonality of the left-right operator packets gives\n\\[\n\\|\\rho\\|_2^2\n=\n\\sum_{\\alpha,\\beta}\n\\|P_\\alpha\\rho P_\\beta\\|_2^2.\n\\]\nThis identity supplies an exact coherence-energy ledger. It can be used to decide which blocks are large, to audit dephasing, or to bound truncation errors in approximate packet simulations.","\\subsection{Symmetry-preserving unitary dynamics}\nIf $U$ commutes with every packet projector, then\n\\[\nP_\\alpha U\\rho U^\\dagger P_\\beta\n=\nU(P_\\alpha\\rho P_\\beta)U^\\dagger.\n\\]\nThe left and right packet labels are preserved. Populations and coherences evolve independently by their declared labels.","\\subsection{Observable sparsity}\nFor an observable $M$,\n\\[\n\\Tr(M\\rho)\n=\n\\sum_{\\alpha,\\beta}\n\\Tr(P_\\beta M P_\\alpha\\,P_\\alpha\\rho P_\\beta).\n\\]\nOnly operator packets coupled by $M$ contribute. A measurement that is diagonal in the symmetry labels depends only on populations. A phase-sensitive measurement can require off-diagonal coherences.","\\begin{derivedbox}\nFor a polynomial active set $A_n$ of packet labels, a density-matrix simulator has at most $|A_n|^2$ label pairs. Polynomial-time simulation follows when each active coherence block has a polynomial classical representation and channel updates remain sparse and efficient.\n\\end{derivedbox}","\\subsection{Four-index transfer}\nA quantum channel with Kraus operators $K_\\mu$ acts by\n\\[\n\\Phi(\\rho)\n=\n\\sum_\\mu K_\\mu\\rho K_\\mu^\\dagger.\n\\]\nRelative to operator packets\n\\[\n\\mathcal P_{\\alpha\\beta}(X)\n=\nP_\\alpha X P_\\beta,\n\\]\nthe channel has blocks\n\\[\n\\Phi_{\\gamma\\delta,\\alpha\\beta}\n=\n\\mathcal P_{\\gamma\\delta}\n\\,\\Phi\\,\n\\mathcal P_{\\alpha\\beta}.\n\\]\nThus a channel carries four packet labels: an input coherence pair and an output coherence pair.","\\subsection{Symmetry-preserving channels}\nIf every Kraus operator commutes with the symmetry action, the channel is diagonal in the isotypic labels. More generally, Kraus operators may carry definite charges, producing a sparse selection rule. Amplitude damping transfers population between number sectors. Dephasing damps selected off-diagonal blocks. Particle loss moves weight-$k$ population to weight $k-1$.","ction{Symmetry-preserving channels}\nIf every Kraus operator commutes with the symmetry action, the channel is diagonal in the isotypic labels. More generally, Kraus operators may carry definite charges, producing a sparse selection rule. Amplitude damping transfers population between number sectors. Dephasing damps selected off-diagonal blocks. Particle loss moves weight-$k$ population to weight $k-1$.","\\subsection{Lindblad dynamics}\nA Markovian open system obeys\n\\[\n\\frac{d\\rho}{dt}\n=\n\\mathcal L(\\rho)\n=\n-i[H,\\rho]\n+\n\\sum_a\n\\left(\nL_a\\rho L_a^\\dagger\n-\n\\frac12\\{L_a^\\dagger L_a,\\rho\\}\n\\right).\n\\]\nPacket selection rules of $H$ and $L_a$ determine a sparse directed graph on coherence pairs. If the reachable graph and each block representation remain polynomial, a classical integrator can evolve the reduced system.","\\subsection{Error accounting for truncated channels}\nLet $\\Pi_A$ project operator space onto an active set of coherence blocks. A reduced evolution uses\n\\[\n\\mathcal L_A\n=\n\\Pi_A\\mathcal L\\Pi_A.\n\\]\nThe discarded transfer is\n\\[\n(I-\\Pi_A)\\mathcal L\\Pi_A.\n\\]\nDuhamel's formula yields an error bound analogous to the Hamiltonian leakage theorem. The resulting estimate depends on the induced norm of the discarded superoperator transfer and the integration time.","\\subsection{Projective measurement}\nFor projectors $\\{P_\\alpha\\}$, the probability of outcome $\\alpha$ is\n\\[\np_\\alpha\n=\n\\Tr(P_\\alpha\\rho),\n\\]\nand the conditioned state is\n\\[\n\\rho_\\alpha\n=\n\\frac{P_\\alpha\\rho P_\\alpha}{p_\\alpha}\n\\]\nwhen $p_\\alpha>0$. Measurement is therefore packet selection followed by normalization.","\\subsection{Dynamic programming on packet states}\nSuppose each branch is represented by a pair $(\\alpha,r)$, where $\\alpha$ is a packet label and $r$ is a compact internal state. If polynomially many distinct pairs occur at every depth, identical pairs can be merged and their probabilities added. This gives a dynamic program instead of an exponential branch tree.","\\subsection{Projector block form}\nA block encoding places a generally nonunitary operator $A$ inside a larger unitary $U$ so that\n\\[\n(\\bra0\\otimes I)\nU\n(\\ket0\\otimes I)\n=\n\\frac{A}{\\alpha}.\n\\]\nWith\n\\[\nP_0=\\ket0\\bra0\\otimes I,\n\\]\nthis is the selected block\n\\[\nP_0UP_0\n=\n\\frac{A}{\\alpha}\n\\]\nafter identifying the encoded subspace.","\\subsection{Why block encoding does not imply classical simulation}\nA block encoding can be efficient on a quantum computer because $U$ is supplied by an oracle or a succinct circuit. A classical algorithm may not have efficient access to the entries of $A$, to the block state, or to the polynomial transform. The selected block may still have exponential dimension.","\\begin{scopebox}\nThe statement ``$A$ is a block of $U$'' is structural. A classical collapse requires a polynomial classical description of the active encoded state, efficient application of the selected block, and efficient evaluation of the output.\n\\end{scopebox}","\\subsection{Symmetry-resolved block encodings}\nIf $A$ commutes with a symmetry group, both the encoded system and the ancilla can be decomposed into packets. A symmetry-resolved block encoding may reduce normalization, memory, or simulation cost for selected sectors. The relevant objects are\n\\[\nP_\\chi P_0 U P_0P_\\psi.\n\\]\nExact symmetry removes off-diagonal $\\chi\\neq\\psi$ blocks. Approximate symmetry yields leakage terms that must be included in the error budget.","Let $\\mathcal R_n$ be a class of classical descriptions, and let\n\\[\n\\operatorname{decode}:\\mathcal R_n\\to P_n\\Hh_n\n\\]\nassociate a quantum state with each description.","\\begin{theorem}[Compressed-packet simulation]\n\\label{s:thm:compressed-packet}\nA polynomial-time uniform quantum circuit family lies in $P$ if there is a polynomial-size classical representation satisfying:","\\begin{enumerate}\n\\item the initial state has an efficiently constructible description $r_0\\in\\mathcal R_n$;\n\\item every gate induces a polynomial-time update $r\\mapsto T_j(r)$ with\n\\[\n\\operatorname{decode}(T_j(r))\n=\nG_j\\operatorname{decode}(r);\n\\]\n\\item all intermediate description sizes remain polynomial;\n\\item the acceptance probability is computable from the final description in polynomial time.\n\\end{enumerate}\n\\end{theorem}","\\subsection{Two-axis complexity}\nThe records suggest the pair\n\\[\n(D_{\\rm packet},\\chi_{\\rm internal}),\n\\]\nwhere $D_{\\rm packet}$ measures active packet size or count and $\\chi_{\\rm internal}$ measures the representation cost inside the packet. Efficient simulation may follow from either coordinate:","\\begin{itemize}\n\\item fixed weight has polynomial $D_{\\rm packet}$ and can use explicit amplitudes;\n\\item stabilizer states may have exponential support but polynomial $\\chi_{\\rm internal}$;\n\\item Gaussian states may occupy large number sectors but have polynomial covariance descriptions;\n\\item half-filled generic states may have both coordinates exponential.\n\\end{itemize}","\\subsection{Hybrid transfer graphs}\nSuppose packets are nodes of a sparse graph and each node carries an internal representation. A gate may update a node internally or transfer a compact component to neighboring nodes. The simulation cost depends on\n\\[\n\\sum_{\\alpha\\in A_t}\\operatorname{size}(r_\\alpha)\n+\n\\#\\{\\text{nonzero transfers at time }t\\}.\n\\]\nThis model combines symmetry selection rules with compact state classes.","\\subsection{Forrelation amplitude}\nLet $f,g:\\{0,1\\}^n\\to\\{-1,1\\}$ and $N=2^n$. The circuit with alternating Hadamards and phase oracles has zero-state amplitude\n\\begin{equation}\n\\Phi(f,g)\n=\n\\frac1{N^{3/2}}\n\\sum_{x,y}\n f(x)(-1)^{x\\cdot y}g(y).\n\\label{s:eq:forrelation}\n\\end{equation}\nThe circuit does not evaluate the $N^2$ terms separately. It preserves phases and lets them interfere.","\\subsection{Why Fourier structure is not automatically easy}\nThe Walsh transform is explicit and highly symmetric. That does not imply that the bilinear expression in \\cref{s:eq:forrelation} is classically polynomial-time computable when $f$ and $g$ are supplied through restricted query access. The difficulty lies in the access model and the coherent global relation, not in the absence of algebraic structure.","\\subsection{Truth tables versus succinct access}\nIf the complete truth tables of $f$ and $g$ are part of the classical input, their length is already $N$, and an $O(N\\log N)$ Walsh transform is polynomial in the explicit input length. In a query or succinct-circuit model, $n$ is the parameter and $N=2^n$. The same formula then represents exponentially many values.","\\subsection{Cross-packet coherence}\nSuppose a projector family gives many packets $P_\\alpha$. Even if each diagonal block is simple, the output can depend on the phases of\n\\[\nP_\\alpha\\ket\\psi\n\\quad\\text{relative to}\\quad\nP_\\beta\\ket\\psi\n\\]\nfor exponentially many pairs. A simulator that retains only populations\n\\[\n\\|P_\\alpha\\ket\\psi\\|^2\n\\]\nloses this information. The hard-side invariant must measure classical compression of coherent relations, not merely active dimension.","\\subsection{Why simulator lower bounds are insufficient}\nShowing that a circuit defeats one simulation strategy does not prove that its decision problem is outside $P$. A classical algorithm may compute the answer without reconstructing the quantum state. It is therefore insufficient to rule out only:","A proof of $P\\neq BQP$ would need a lower bound against every deterministic polynomial-time classical algorithm for an explicit language.","\\subsection{A representation-relative invariant}\nFor a declared classical representation class $\\mathcal R$, define\n\\[\n\\operatorname{TC}^{\\mathcal R}_\\varepsilon(C)\n=\n\\min\\left\\{\n\\operatorname{cost}(R):\nR\\in\\mathcal R,\n\\ |R(C)-p_C|\\le\\varepsilon\n\\right\\},\n\\]\nwhere $p_C$ is the target acceptance probability. This permits a disciplined ladder:","\\begin{enumerate}\n\\item exact polynomial-packet simulation for easy sectors;\n\\item lower bounds for local packet-query methods;\n\\item lower bounds for broader tensor, sampling, low-degree, and communication representations;\n\\item a representation-independent lower bound;\n\\item a separation of $P$ and $BQP$.\n\\end{enumerate}","If true, \\cref{s:thm:poly-packet,s:thm:compressed-packet} would imply $BQP\\subseteq P$, and hence $P=BQP$. The records do not establish this conjecture and explicitly identify global interference as an obstacle.","\\subsection{The opposite route}\nAn explicit uniform family with a proved representation-independent classical lower bound and a polynomial quantum algorithm would establish $P\\neq BQP$. The candidate succinct transfer-interference problem in the records illustrates the form of such a target, but the required lower bound is the unresolved theorem itself.","\\subsection{Nonrelativizing issues}\nOracle separations show that query complexity can distinguish quantum and classical access. They also show why a black-box argument alone cannot settle ordinary $P$ versus $BQP$. A full separation would have to use explicit features such as uniform circuit generation, arithmetic structure, algebraic degree, communication constraints, or another nonrelativizing mechanism.","\\subsection{Metric collapse and existence collapse}\nThe simulation theorems compare costs of two realizations that both exist. A broader restricted-realization framework also allows the restricted object to be absent. If\n\\[\n\\mathcal R(x)=\\varnothing,\n\\qquad\n\\mathcal A(x)\\neq\\varnothing,\n\\]\nthen enlarging the realization category changes feasibility rather than merely runtime.","\\subsection{Unified terminology}\nFor a target $x$, let $c_{\\rm res}(x)$ and $c_{\\rm amb}(x)$ be minimum costs, with $+\\infty$ assigned when a realization does not exist. Then\n\\[\n\\Delta(x)\n=\nc_{\\rm res}(x)-c_{\\rm amb}(x)\n\\]\nincludes both finite metric defects and infinite existence defects. Symmetry collapse concerns the special case where the ambient quantum description has a cheaper classical realization because its reachable representation is compressed.","\\subsection{Algebraic and numerical validation paths}\nFor manageable $n$, embed a random packet state into the full space, apply identical gates in both representations, and compare final states. The included numerical validation program performs this check for structurally invariant fixed-weight circuits using complex floating-point arithmetic, with reconstruction tolerance $2\\cdot10^{-12}$. The separate rational validation uses exact fractions and rational orthogonal gates. Exact subspace invariance is a property of the model, while near-machine-precision agreement is numerical evidence about this implementation.","\\midrule\n\\endhead\nFixed charge $k=O(1)$ & Explicit subset basis of size $\\binom nk$ & Number-preserving gates and efficient measurement","Permutation-symmetric qubits & Dicke basis of size $n+1$ & Collective or permutation-invariant dynamics","Symmetric qudits, fixed $d$ & Occupation basis $\\binom{n+d-1}{d-1}$ & Efficient occupation-space updates","\\subsection{Moving-frame discovery}\n\\begin{question}\nWhich circuit families admit efficiently computable frames \\(W_t\\) that expose one fixed polynomial packet even though no fixed laboratory-frame symmetry is preserved?\n\\end{question}\nThe theorem is exact, but the discovery problem is the substance. Useful frames must have compact descriptions and restricted gate matrices that can be generated without solving the original simulation problem.","\\subsection{Minimal packet complexity}\n\\begin{question}\nGiven a circuit and error tolerance $\\varepsilon$, what is the minimum active representation size among all efficiently applicable exact or approximate packet decompositions?\n\\end{question}\nThis optimization should include internal encoding and transfer sparsity, not only projector rank.","\\subsection{Universal packetization}\n\\begin{question}\nIs every polynomial-size quantum circuit classically compressible by some efficiently discoverable representation of polynomial size?\n\\end{question}\nA positive answer would imply $P=BQP$. The record does not support this as a universal empirical principle.","polynomial. Exact charge conservation is not necessary: moving packet schedules, efficiently computable moving frames, and bounded unions of low-excitation sectors also collapse to polynomial classical simulation. Fully symmetric fixed-local-dimension dynamics is polynomial. A stronger mixed-state theorem compresses all permutation-invariant Schur sectors into a polynomial commutant record, with $O(n^3)$ density-matrix parameters for qubits under efficient block access. Polynomial-orbit walks are polynomial under quotient-access hypotheses. Few-particle bosonic and fermionic systems are polynomial under efficient updates, while growing particle number gives a sharp progression through quasipolynomial, subexponential, and exponential sector sizes. Stabilizer and Gaussian representations show that an exponential packet can also be easy when its internal states admit compact closed coordinates. Recorded number-conserving benchmarks show large memory and repeated-application gains with zero reconstruction error at the recorded precision.","The core research invariant is the complete classical representation record\n\\[\n\\bigl(\n\\text{discovery cost},\nN_{\\rm packets},\nD_{\\rm internal},\nS_{\\rm transfer},\nL_{\\rm leakage},\n\\text{measurement cost}\n\\bigr).\n\\]\nA future compiler that computes and certifies this record can decide when the full state vector is unnecessary, when a hybrid representation is preferable, and when symmetry alone is insufficient."]}