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[<0e9d864536f570d7480471e0f99695cb><0e9d864536f570d7480471e0f99695cb>]}\n\\begin{abstract}\nLet \\(K\\) be a nonarchimedean local field. We resolve several finite layers that occur inside the local Langlands correspondence for general linear groups. Frobenius semisimple Weil Deligne parameters first split into irreducible Weil atoms and finite monodromy chains. In the tame case, character projectors show that every irreducible atom is one transitive Frobenius orbit, and Mackey theory gives an exact greatest common divisor and least common multiple rule for tensor packets. Regular finite field characters produce primitive cuspidal projectors and finite Gauss phases. For wild data, projective intertwiners recover an alternating commutator form. Its radical free quotient determines a finite He","The local Langlands correspondence for \\(\\GL_n(K)\\) is an established theorem. It matches irreducible smooth representations of \\(\\GL_n(K)\\) with Frobenius semisimple Weil Deligne representations of dimension \\(n\\). Its proof and normalization draw on global methods, harmonic analysis, types, local factors, and the geometry of Shimura varieties \\cite{Henniart2000,HarrisTaylor2001,BushnellHenniart2006}. The present article asks a narrower structural question. Which parts of the local correspondence admit finite packet coordinates whose projectors, transfers, phases, and gluing laws can be proved directly?","Let \\(W_K\\) be the Weil group of \\(K\\), let \\(I_K\\) be inertia, let \\(P_K\\) be wild inertia, and choose a geometric Frobenius lift \\(\\Phi\\). A Weil Deligne representation is a pair \\((r,N)\\) with\n\\[\nr(w)Nr(w)^{-1}=\\abs{w}_K N.\n\\]","\\begin{standardfact}[Indecomposable monodromy blocks]\\label{fact:spdecomp}\nEvery Frobenius semisimple Weil Deligne representation is a direct sum of blocks\n\\[\n\\operatorname{Sp}_m(\\tau),\n\\]\nwhere \\(\\tau\\) is an irreducible Weil representation and the nilpotent operator is one Jordan chain of length \\(m\\).\n\\end{standardfact}","\\begin{definition}[Monodromy packet chain]\\label{def:monodromy_chain}\nLet \\(V_0,\\ldots,V_{m-1}\\) be copies of the space of \\(\\tau\\). Put\n\\[\nV=\\bigoplus_{j=0}^{m-1}V_j.\n\\]\nA monodromy packet chain is a nilpotent operator \\(N\\) with\n\\[\nN(V_j)\\subseteq V_{j-1}\n\\]\nfor \\(j>0\\), with \\(N(V_0)=0\\), together with a Frobenius operator whose conjugation scales \\(N\\) by \\(q^{-1}\\).\n\\end{definition}","\\begin{proposition}[Exact block reconstruction]\\label{prop:monodromy_reconstruction}\nLet \\(P_j\\) be the projector onto \\(V_j\\). Then\n\\[\nN=\\sum_{j=1}^{m-1}P_{j-1}NP_j,\n\\qquad\nN^m=0.\n\\]\nEvery power \\(N^r\\) is the sum of the packet paths of length \\(r\\) along the chain.\n\\end{proposition}","\\begin{proof}\nThe projectors resolve the identity. Since \\(N\\) maps only from \\(V_j\\) to \\(V_{j-1}\\), every block except the displayed nearest neighbor blocks vanishes. A product of \\(r\\) nonzero blocks moves down exactly \\(r\\) levels. No path of length \\(m\\) exists, so \\(N^m=0\\).\n\\end{proof}","\\begin{proposition}[Scalar covariance model]\\label{prop:scalar_covariance}\nLet\n\\[\nF=\\operatorname{diag}(1,q,q^2,\\ldots,q^{m-1})\n\\]\nand let \\(N_{j-1,j}=1\\). Then\n\\[\nFNF^{-1}=q^{-1}N.\n\\]\n\\end{proposition}","\\begin{proof}\nThe only nonzero entry in column \\(j\\) and row \\(j-1\\) is multiplied by \\(q^{j-1}q^{-j}=q^{-1}\\) under conjugation.\n\\end{proof}","\\begin{theorem}[Formal atomic reconstruction]\\label{thm:formal_atomic}\nAssume that every irreducible Weil representation \\(\\tau\\) has been assigned a supercuspidal representation \\(\\sigma_\\tau\\), compatibly with isomorphism and unramified twist. Then the rules\n\\[\n\\operatorname{Sp}_m(\\tau)\n\\longmapsto\n\\operatorname{St}_m(\\sigma_\\tau)\n\\]\nand normalized parabolic induction determine a candidate correspondence on all Frobenius semisimple Weil Deligne parameters.\n\\end{theorem}","\\begin{remark}\\label{rem:atomic_boundary}\n\\Cref{thm:formal_atomic} is a reconstruction theorem. It does not produce the atomic assignment \\(\\tau\\mapsto\\sigma_\\tau\\), and it does not prove the required local factor identities.\n\\end{remark}","Assume that \\(P_K\\) acts trivially. After passing to a finite quotient, tame inertia is cyclic. Let \\(t\\) generate a quotient of order \\(M\\), and put \\(\\zeta_M=\\e^{2\\pi i/M}\\). For \\(j\\in\\ZZ/M\\ZZ\\), define\n\\[\nP_j\n=\n\\frac1M\n\\sum_{a=0}^{M-1}\n\\zeta_M^{-ja}r(t)^a.\n\\]","\\begin{proposition}[Tame character projectors]\\label{prop:tame_projectors}\nThe operators \\(P_j\\) are pairwise orthogonal idempotents whose sum is the identity. Frobenius permutes them by\n\\[\n\\Phi P_j\\Phi^{-1}=P_{qj}.\n\\]\n\\end{proposition}","\\begin{proof}\nThe first assertion is the Fourier idempotent calculation for a cyclic group. For the geometric Frobenius convention of \\cref{sec:wd}, the relation is \\(\\Phi^{-1}t\\Phi=t^q\\). Equivalently, \\(\\Phi t\\Phi^{-1}=t^{q^{-1}}\\), with inverse taken modulo \\(M\\) since \\(\\gcd(q,M)=1\\). Conjugating the Fourier sum and changing its summation index therefore gives \\(\\Phi P_j\\Phi^{-1}=P_{qj}\\). The inverse Frobenius convention reverses the label direction while preserving the same orbits.\n\\end{proof}","\\begin{theorem}[Tame orbit induction]\\label{thm:tame_induction}\nLet \\(\\tau\\) be an irreducible complex representation of \\(W_K\\) with trivial wild inertia. Then the nonzero tame character packets form one transitive Frobenius orbit of length \\(f\\). If \\(K_f/K\\) is the unramified extension of degree \\(f\\), there is a character \\(\\xi\\) of \\(W_{K_f}\\) such that\n\\[\n\\boxed{\n\\tau\\cong\\Ind_{W_{K_f}}^{W_K}\\xi.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nThe nonzero tame packets are permuted by Frobenius. A union of Frobenius orbits is stable under \\(W_K\\), so irreducibility forces one orbit. Choose one packet \\(V_\\theta\\). Its stabilizer is \\(W_{K_f}=\\langle I_K,\\Phi^f\\rangle\\). A nonzero proper stable subspace of \\(V_\\theta\\) would generate a nonzero proper \\(W_K\\) stable subspace after summing its Frobenius translates. Thus \\(V_\\theta\\) is irreducible for the stabilizer. Inertia acts on it by the scalar character \\(\\theta\\), and the remaining action is generated by \\(\\Phi^f\\). The stabilizer action on this packet is therefore one dimensional. Induction reconstructs the complete orbit.\n\\end{proof}","orbits is stable under \\(W_K\\), so irreducibility forces one orbit. Choose one packet \\(V_\\theta\\). Its stabilizer is \\(W_{K_f}=\\langle I_K,\\Phi^f\\rangle\\). A nonzero proper stable subspace of \\(V_\\theta\\) would generate a nonzero proper \\(W_K\\) stable subspace after summing its Frobenius translates. Thus \\(V_\\theta\\) is irreducible for the stabilizer. Inertia acts on it by the scalar character \\(\\theta\\), and the remaining action is generated by \\(\\Phi^f\\). The stabilizer action on this packet is therefore one dimensional. Induction reconstructs the complete orbit.\n\\end{proof}","ces one orbit. Choose one packet \\(V_\\theta\\). Its stabilizer is \\(W_{K_f}=\\langle I_K,\\Phi^f\\rangle\\). A nonzero proper stable subspace of \\(V_\\theta\\) would generate a nonzero proper \\(W_K\\) stable subspace after summing its Frobenius translates. Thus \\(V_\\theta\\) is irreducible for the stabilizer. Inertia acts on it by the scalar character \\(\\theta\\), and the remaining action is generated by \\(\\Phi^f\\). The stabilizer action on this packet is therefore one dimensional. Induction reconstructs the complete orbit.\n\\end{proof}","\\begin{corollary}[Dimension and regularity]\\label{cor:tame_dimension}\nThe dimension of \\(\\tau\\) equals the Frobenius orbit length of its tame inertia character. The inducing character is regular, meaning that no smaller positive power of Frobenius fixes it.\n\\end{corollary}","\\begin{proof}\nThe induced representation has one line for every Frobenius translate of the chosen character. A shorter orbit would produce a proper stabilizer and contradict the minimality of \\(f\\).\n\\end{proof}","\\begin{proposition}[Dual and self dual tame packets]\\label{prop:tame_dual}\nFor a tame atom\n\\[\n\\tau_\\xi=\\Ind_{W_{K_f}}^{W_K}\\xi,\n\\]\nwe have\n\\[\n\\tau_\\xi^\\vee\n\\cong\n\\Ind_{W_{K_f}}^{W_K}\\xi^{-1}.\n\\]\nThe atom is self dual exactly when\n\\[\n\\xi^{-1}=\\xi^{\\Phi^j}\n\\]\nfor some \\(j\\in\\ZZ/f\\ZZ\\).\n\\end{proposition}","\\begin{proof}\nDuality commutes with finite index induction, giving the first statement. Two regular induced tame atoms are isomorphic exactly when their inducing characters lie in the same Frobenius orbit. Applying this to \\(\\xi\\) and \\(\\xi^{-1}\\) gives the criterion.\n\\end{proof}","\\begin{proposition}[Unramified twist]\\label{prop:tame_twist}\nLet \\(\\nu\\) be an unramified character of \\(W_K\\). Then\n\\[\n\\tau_\\xi\\otimes\\nu\n\\cong\n\\Ind_{W_{K_f}}^{W_K}\n\\left(\n\\xi\\,\\nu|_{W_{K_f}}\n\\right).\n\\]\nThe tame inertia orbit is unchanged. If the original Frobenius cycle has closing scalar \\(a\\), the twisted cycle has closing scalar\n\\[\na\\nu(\\Phi)^f.\n\\]\n\\end{proposition}","\\begin{proof}\nThe tensor identity follows from the projection formula for induction. Since \\(\\nu\\) is trivial on inertia, it does not change any tame character packet. It multiplies every Frobenius step by \\(\\nu(\\Phi)\\), so a closed cycle of length \\(f\\) gains the factor \\(\\nu(\\Phi)^f\\).\n\\end{proof}","Characters of \\(\\FF_{q^n}^{\\times}\\) are indexed by exponents modulo \\(q^n-1\\), and Frobenius acts by multiplication by \\(q\\).","\\begin{proposition}[Möbius orbit count]\\label{prop:mobius_count}\nThe number of regular Frobenius orbits of length \\(n\\) is\n\\[\n\\boxed{\nc_n(q)\n=\n\\frac1n\n\\sum_{d\\mid n}\n\\mu(d)\n\\bigl(q^{n/d}-1\\bigr).\n}\n\\]\n\\end{proposition}","\\begin{proof}\nFor every divisor \\(r\\mid n\\), the characters fixed by \\(\\Frob^r\\) are exactly those descending to \\(\\FF_{q^r}^{\\times}\\), and there are \\(q^r-1\\) of them. Möbius inversion gives the number of characters of exact orbit length \\(n\\). Division by \\(n\\) gives the number of orbits.\n\\end{proof}","\\begin{example}[Degree four over \\(\\FF_2\\)]\\label{ex:degree_four}\nThe regular exponent orbits modulo fifteen are\n\\[\n(1,2,4,8),\n\\qquad\n(3,6,12,9),\n\\qquad\n(7,14,13,11).\n\\]\nThe formula gives\n\\[\n\\frac14\\bigl((2^4-1)-(2^2-1)\\bigr)=3.\n\\]\n\\end{example}","Let\n\\[\nG_n=\\GL_n(\\FF_q),\n\\qquad\nT_n\\cong\\FF_{q^n}^{\\times}\n\\]\nwith \\(T_n\\) an elliptic maximal torus.","\\begin{standardfact}[Regular elliptic torus characters]\\label{fact:deligne_lusztig}\nA regular character \\(\\theta:T_n\\to\\CC^{\\times}\\) determines an irreducible cuspidal representation \\(\\kappa_\\theta\\) of \\(G_n\\). Frobenius conjugate characters determine isomorphic representations, and every irreducible cuspidal representation of \\(G_n\\) occurs in this way.\n\\end{standardfact}","This is the finite field input from Deligne and Lusztig theory \\cite{DeligneLusztig1976}. We use standard finite group character theory for the central idempotents \\cite{Serre1977,Isaacs1976}. Once \\(\\kappa_\\theta\\) is known, its block is selected by a central idempotent.","\\begin{proposition}[Primitive cuspidal idempotent]\\label{prop:cuspidal_idempotent}\nLet \\(\\chi_\\theta\\) be the irreducible character of \\(\\kappa_\\theta\\), and let \\(d_\\theta=\\chi_\\theta(1)\\). Then\n\\[\ne_\\theta\n=\n\\frac{d_\\theta}{\\abs{G_n}}\n\\sum_{g\\in G_n}\n\\chi_\\theta(g^{-1})g\n\\]\nis the primitive central idempotent selecting the \\(\\kappa_\\theta\\) block in \\(\\CC[G_n]\\).\n\\end{proposition}","\\begin{proof}\nOn an irreducible representation of character \\(\\psi\\), the central element acts by a scalar. Taking traces shows that the scalar is\n\\[\n\\frac{d_\\theta}{\\abs{G_n}\\psi(1)}\n\\sum_{g\\in G_n}\n\\chi_\\theta(g^{-1})\\psi(g).\n\\]\nCharacter orthogonality makes this scalar one for \\(\\psi=\\chi_\\theta\\) and zero otherwise. The element is therefore central, idempotent, and primitive.\n\\end{proof}","\\begin{proposition}[Parabolic annihilation]\\label{prop:parabolic_annihilation}\nLet \\(P=LU\\) be a proper parabolic subgroup of \\(G_n\\), and put\n\\[\ne_U=\\frac1{\\abs U}\\sum_{u\\in U}u.\n\\]\nIf \\(\\kappa_\\theta\\) is cuspidal, then\n\\[\n\\boxed{\ne_Ue_\\theta=e_\\theta e_U=0.\n}\n\\]\n\\end{proposition}","\\begin{proof}\nThe operator \\(e_U\\) projects onto the subspace of \\(U\\) fixed vectors. A cuspidal representation has no nonzero fixed vector under the unipotent radical of a proper parabolic. Thus \\(e_U\\) acts by zero on the \\(\\kappa_\\theta\\) block. Since \\(e_\\theta\\) is central, both products vanish.\n\\end{proof}","The depth zero construction inflates a finite cuspidal representation to \\(\\GL_n(\\OO_K)\\), extends it to the standard compact modulo center subgroup, and then compactly induces it to \\(\\GL_n(K)\\). This is classical type theory \\cite{BushnellKutzko1993,BushnellHenniart2006}.","The induced tame Weil atom has an elementary cycle invariant. In an orbit basis \\(e_0,\\ldots,e_{n-1}\\), write\n\\[\n\\tau(\\Phi)e_j=e_{j+1}\n\\]\nfor \\(j<n-1\\), and\n\\[\n\\tau(\\Phi)e_{n-1}=ae_0.\n\\]","\\begin{proposition}[Frobenius cycle polynomial]\\label{prop:frobenius_cycle}\nThe Frobenius transfer matrix has characteristic polynomial\n\\[\nX^n-a\n\\]\nand determinant\n\\[\n\\boxed{\n\\det\\tau(\\Phi)=(-1)^{n-1}a.\n}\n\\]\n\\end{proposition}","\\begin{proof}\nThe matrix is the companion matrix of \\(X^n-a\\). Its only nonzero determinant term is the closed \\(n\\) cycle. That cycle has sign \\((-1)^{n-1}\\), and the closing edge contributes \\(a\\).\n\\end{proof}","\\begin{corollary}[Unramified orientation character]\\label{cor:orientation}\nThe sign \\((-1)^{n-1}\\) is the orientation of the closed Frobenius packet cycle. It is unchanged by rescaling the individual orbit basis vectors.\n\\end{corollary}","Let\n\\[\n\\tau_\\xi=\\Ind_{W_{K_n}}^{W_K}\\xi,\n\\qquad\n\\tau_\\eta=\\Ind_{W_{K_m}}^{W_K}\\eta,\n\\]\nwhere \\(K_n/K\\) and \\(K_m/K\\) are unramified. Put\n\\[\nd=\\gcd(n,m),\n\\qquad\nL=\\lcm(n,m).\n\\]","\\begin{theorem}[Greatest common divisor and least common multiple packets]\\label{thm:gcd_lcm}\nThere is a decomposition\n\\[\n\\boxed{\n\\tau_\\xi\\otimes\\tau_\\eta\n\\cong\n\\bigoplus_{i=0}^{d-1}\n\\Ind_{W_{K_L}}^{W_K}\n\\left(\n\\xi_{L,n}\\eta_{L,m}^{\\Phi^i}\n\\right),\n}\n\\]\nwhere \\(\\xi_{L,n}\\) and \\(\\eta_{L,m}\\) are the norm lifts to \\(K_L\\).\n\\end{theorem}","\\begin{proof}\nApply Mackey decomposition \\cite{Mackey1952}. The double cosets\n\\[\nW_{K_n}\\backslash W_K/W_{K_m}\n\\]\nare the orbits of simultaneous translation on \\(\\ZZ/n\\ZZ\\times\\ZZ/m\\ZZ\\). There are \\(d\\) orbits, each of length \\(L\\). The stabilizer of a point in each orbit is \\(W_{K_L}\\). The second character is shifted by the chosen Frobenius representative. Summing the induced terms over the double cosets gives the decomposition.\n\\end{proof}","\\begin{corollary}[Packet count and dimension]\\label{cor:gcd_lcm_count}\nThe tensor product has \\(d\\) induced interaction packets, each with carrier dimension and Frobenius coset-cycle length \\(L\\). The interaction character may have a shorter inertia orbit, in which case a packet may split further into irreducible atoms. The total dimension is preserved because\n\\[\ndL=nm.\n\\]\n\\end{corollary}\n\\begin{proof}\nThe double coset calculation in \\cref{thm:gcd_lcm} gives exactly \\(d\\) summands, and every stabilizer is \\(W_{K_L}\\), so every induced packet has dimension \\(L\\). The identity \\(dL=nm\\) follows from the definitions of greatest common divisor and least common multiple.\n\\end{proof}","e tensor product has \\(d\\) induced interaction packets, each with carrier dimension and Frobenius coset-cycle length \\(L\\). The interaction character may have a shorter inertia orbit, in which case a packet may split further into irreducible atoms. The total dimension is preserved because\n\\[\ndL=nm.\n\\]\n\\end{corollary}\n\\begin{proof}\nThe double coset calculation in \\cref{thm:gcd_lcm} gives exactly \\(d\\) summands, and every stabilizer is \\(W_{K_L}\\), so every induced packet has dimension \\(L\\). The identity \\(dL=nm\\) follows from the definitions of greatest common divisor and least common multiple.\n\\end{proof}","action packets, each with carrier dimension and Frobenius coset-cycle length \\(L\\). The interaction character may have a shorter inertia orbit, in which case a packet may split further into irreducible atoms. The total dimension is preserved because\n\\[\ndL=nm.\n\\]\n\\end{corollary}\n\\begin{proof}\nThe double coset calculation in \\cref{thm:gcd_lcm} gives exactly \\(d\\) summands, and every stabilizer is \\(W_{K_L}\\), so every induced packet has dimension \\(L\\). The identity \\(dL=nm\\) follows from the definitions of greatest common divisor and least common multiple.\n\\end{proof}","benius coset-cycle length \\(L\\). The interaction character may have a shorter inertia orbit, in which case a packet may split further into irreducible atoms. The total dimension is preserved because\n\\[\ndL=nm.\n\\]\n\\end{corollary}\n\\begin{proof}\nThe double coset calculation in \\cref{thm:gcd_lcm} gives exactly \\(d\\) summands, and every stabilizer is \\(W_{K_L}\\), so every induced packet has dimension \\(L\\). The identity \\(dL=nm\\) follows from the definitions of greatest common divisor and least common multiple.\n\\end{proof}","ength \\(L\\). The interaction character may have a shorter inertia orbit, in which case a packet may split further into irreducible atoms. The total dimension is preserved because\n\\[\ndL=nm.\n\\]\n\\end{corollary}\n\\begin{proof}\nThe double coset calculation in \\cref{thm:gcd_lcm} gives exactly \\(d\\) summands, and every stabilizer is \\(W_{K_L}\\), so every induced packet has dimension \\(L\\). The identity \\(dL=nm\\) follows from the definitions of greatest common divisor and least common multiple.\n\\end{proof}","Represent the residue characters by exponents\n\\[\na\\pmod{q^n-1},\n\\qquad\nb\\pmod{q^m-1}.\n\\]","\\begin{proposition}[Refined exponent formula]\\label{prop:refined_exponent}\nThe \\(i\\)th interaction packet on \\(\\FF_{q^L}^{\\times}\\) has exponent\n\\[\n\\boxed{\nc_i\n=\na\\frac{q^L-1}{q^n-1}\n+bq^i\\frac{q^L-1}{q^m-1}\n\\pmod{q^L-1}.\n}\n\\]\nIt is trivial on tame inertia exactly when \\(c_i=0\\).\n\\end{proposition}","\\begin{proof}\nNorm lifting a character exponent from \\(\\FF_{q^n}^{\\times}\\) to \\(\\FF_{q^L}^{\\times}\\) multiplies the exponent by \\((q^L-1)/(q^n-1)\\). The Frobenius shift multiplies the second exponent by \\(q^i\\). Multiplication of characters adds exponents. Triviality is equivalent to the resulting exponent being zero modulo \\(q^L-1\\).\n\\end{proof}","Let \\(V\\) be one interaction packet. If its inertia character is nontrivial, then \\(V^{I_K}=0\\). If the inertia character is trivial, Frobenius acts on the invariant packet by a companion cycle.","\\begin{proposition}[Local polynomial of one tame packet]\\label{prop:tame_local_polynomial}\nSuppose an interaction packet has orbit length \\(r\\) and closing scalar \\(a\\). Its local polynomial is\n\\[\n\\det\\left(I-u\\Phi\\mid V^{I_K}\\right)\n=\n\\begin{cases}\n1-a u^r,&\\text{if the inertia character is trivial},","\\begin{proposition}[Local polynomial of one tame packet]\\label{prop:tame_local_polynomial}\nSuppose an interaction packet has orbit length \\(r\\) and closing scalar \\(a\\). Its local polynomial is\n\\[\n\\det\\left(I-u\\Phi\\mid V^{I_K}\\right)\n=\n\\begin{cases}\n1-a u^r,&\\text{if the inertia character is trivial},\\\\\n1,&\\text{otherwise}.\n\\end{cases}\n\\]\n\\end{proposition}","\\begin{proof}\nFor nontrivial inertia, the invariant space is zero. In the trivial case, Frobenius on the invariant packet is the companion matrix of \\(X^r-a\\). Substituting \\(X=u^{-1}\\) and multiplying by \\(u^r\\) gives \\(1-a u^r\\).\n\\end{proof}","\\begin{corollary}[Tame tensor local factor]\\label{cor:tame_tensor_factor}\nThe local factor of the tame tensor product is the product of\n\\[\n(1-a_i u^{r_i})^{-1}\n\\]\nover precisely those refined packets whose inertia exponent vanishes.\n\\end{corollary}","\\begin{proposition}[Trivial tensor packet criterion]\\label{prop:trivial_tensor_packet}\nFor every refined interaction packet, let \\(r_i\\) be its cycle length and \\(a_i\\) its closing scalar. The dimension of the \\(W_K\\) fixed space in \\(\\tau_\\xi\\otimes\\tau_\\eta\\) equals the number of packets satisfying both\n\\[\nc_i=0\n\\qquad\\text{and}\\qquad\na_i=1.\n\\]\n\\end{proposition}","\\begin{proof}\nA nonzero fixed vector must lie in an inertia trivial packet, so \\(c_i=0\\). On that packet, Frobenius is a companion cycle with characteristic polynomial \\(X^{r_i}-a_i\\). Its fixed space is the eigenspace for eigenvalue one. This eigenspace is one dimensional exactly when \\(a_i=1\\), and it is zero otherwise. Summing over the refined packets proves the formula.\n\\end{proof}","\\begin{corollary}[Tame Hom test]\\label{cor:tame_hom}\nFor irreducible tame atoms,\n\\[\n\\dim\\Hom_{W_K}(\\tau_\\xi^\\vee,\\tau_\\eta)\n\\]\nis zero or one, and it is one exactly when the refined tensor packet test in \\cref{prop:trivial_tensor_packet} finds one trivial packet.\n\\end{corollary}","\\begin{proof}\nThe Hom space is the fixed space of \\(\\tau_\\xi\\otimes\\tau_\\eta\\). Schur's lemma gives dimension zero or one, and \\cref{prop:trivial_tensor_packet} gives the packet criterion.\n\\end{proof}","\\begin{proposition}[Conductor of a nontrivial tame atom]\\label{prop:tame_conductor}\nLet \\(\\tau\\) be an irreducible tame Weil representation of dimension \\(n>1\\). Then\n\\[\n\\Swan(\\tau)=0,\n\\qquad\n\\dim\\tau^{I_K}=0,\n\\qquad\n\\Art(\\tau)=n.\n\\]\n\\end{proposition}","\\begin{proof}\nWild inertia acts trivially, so the Swan conductor is zero. An inertia fixed vector would generate a nonzero Frobenius stable sum of trivial character packets. Irreducibility would then make inertia act trivially on all of \\(\\tau\\). An irreducible representation of the quotient generated by Frobenius would be one dimensional, contrary to \\(n>1\\). The Artin conductor is therefore\n\\[\n\\dim\\tau-\\dim\\tau^{I_K}+\\Swan(\\tau)=n.\n\\]\n\\end{proof}","Let \\(k=\\FF_q\\), let \\(\\psi:k\\to\\CC^{\\times}\\) be a nontrivial additive character, and let \\(\\chi:k^{\\times}\\to\\CC^{\\times}\\) be a multiplicative character. Define\n\\[\nG(\\chi,\\psi)\n=\n\\sum_{x\\in k^{\\times}}\\chi(x)\\psi(x).\n\\]\nThe standard theory of Gauss and Jacobi sums will be used in the form recorded in \\cite{DavenportHasse1935,BerndtEvansWilliams1998}.","\\begin{proposition}[Gauss magnitude]\\label{prop:gauss_magnitude}\nIf \\(\\chi\\) is nontrivial, then\n\\[\n\\boxed{\n\\abs{G(\\chi,\\psi)}=q^{1/2}.\n}\n\\]\n\\end{proposition}","\\begin{proof}\nCompute\n\\[\n\\abs{G(\\chi,\\psi)}^2\n=\n\\sum_{x,y\\in k^{\\times}}\n\\chi(xy^{-1})\\psi(x-y).\n\\]\nWrite \\(x=ty\\). The inner sum over \\(y\\) equals \\(q-1\\) when \\(t=1\\) and equals \\(-1\\) otherwise. Since \\(\\sum_{t\\in k^{\\times}}\\chi(t)=0\\), the result is \\(q\\).\n\\end{proof}","\\begin{definition}[Normalized Gauss phase]\\label{def:gauss_phase}\nFor nontrivial \\(\\chi\\), define\n\\[\n\\gamma(\\chi,\\psi)\n=\nq^{-1/2}G(\\chi,\\psi).\n\\]\nThen \\(\\abs{\\gamma(\\chi,\\psi)}=1\\).\n\\end{definition}","\\begin{proposition}[Duality of Gauss phases]\\label{prop:gauss_duality}\nFor nontrivial \\(\\chi\\),\n\\[\n\\boxed{\n\\gamma(\\chi,\\psi)\\gamma(\\chi^{-1},\\psi)\n=\n\\chi(-1).\n}\n\\]\n\\end{proposition}","\\begin{proof}\nThe standard Gauss identity gives\n\\[\nG(\\chi,\\psi)G(\\chi^{-1},\\psi)=\\chi(-1)q.\n\\]\nDivide by \\(q\\).\n\\end{proof}","\\begin{standardfact}[Hasse and Davenport lifting]\\label{fact:hasse_davenport}\nLet \\(k_r/k\\) be the extension of degree \\(r\\). Lift \\(\\chi\\) by the norm and \\(\\psi\\) by the trace. Then the lifted Gauss sum satisfies the Hasse and Davenport product relation, with a sign determined by the chosen additive character normalization.\n\\end{standardfact}","Let \\(p\\) be odd. Let \\(X\\) be an \\(r\\) dimensional vector space over \\(\\FF_p\\), and put\n\\[\nV=X\\oplus X^*.\n\\]\nDefine the alternating form\n\\[\nb\\bigl((x,\\xi),(y,\\eta)\\bigr)\n=\n\\eta(x)-\\xi(y).\n\\]\nLet \\(Z=\\FF_p\\), and define the Heisenberg group\n\\[\n\\frakH(V)=V\\times Z\n\\]\nwith multiplication\n\\[\n(v,z)(w,t)\n=\n\\left(v+w,z+t+\\frac12 b(v,w)\\right).\n\\]\nFix the additive character\n\\[\n\\psi(z)=\\e^{2\\pi iz/p}.\n\\]","\\begin{definition}[Schrödinger operators]\\label{def:schrodinger}\nOn \\(\\calH=\\CC^X\\), define\n\\[\n\\bigl(\\rho(x,\\xi,z)f\\bigr)(u)\n=\n\\psi\\left(z+\\xi(u)+\\frac12\\xi(x)\\right)f(u+x).\n\\]\n\\end{definition}","\\begin{theorem}[Finite Schrödinger representation]\\label{thm:schrodinger_rep}\nThe map \\(\\rho:\\frakH(V)\\to\\GL(\\calH)\\) is a representation. It is irreducible, its center acts by \\(\\psi\\), and\n\\[\n\\dim\\calH=p^r=\\sqrt{\\abs V}.\n\\]\n\\end{theorem}","\\begin{proof}\nA direct calculation gives\n\\[\n\\rho(v,z)\\rho(w,t)\n=\n\\rho\\left(v+w,z+t+\\frac12b(v,w)\\right).\n\\]\nTranslations by \\(X\\) and multiplications by characters from \\(X^*\\) generate the full matrix algebra on \\(\\CC^X\\). Indeed, the character multipliers give all diagonal projectors by finite Fourier inversion, and translations move each coordinate line to every other coordinate line. The commutant is therefore scalar, so the representation is irreducible. The dimension is \\(\\abs X=p^r\\).\n\\end{proof}","\\frac12b(v,w)\\right).\n\\]\nTranslations by \\(X\\) and multiplications by characters from \\(X^*\\) generate the full matrix algebra on \\(\\CC^X\\). Indeed, the character multipliers give all diagonal projectors by finite Fourier inversion, and translations move each coordinate line to every other coordinate line. The commutant is therefore scalar, so the representation is irreducible. The dimension is \\(\\abs X=p^r\\).\n\\end{proof}","\\begin{proposition}[Character support]\\label{prop:heisenberg_trace}\nThe character of the Schrödinger representation satisfies\n\\[\n\\boxed{\n\\Tr\\rho(v,z)\n=\n\\begin{cases}\np^r\\psi(z),&v=0,\\\\\n0,&v\\ne0.\n\\end{cases}\n}\n\\]\n\\end{proposition}","\\begin{proof}\nIf the translation component \\(x\\) is nonzero, the operator permutes the standard basis without a fixed point, so its trace is zero. If \\(x=0\\) but \\(\\xi\\ne0\\), the trace is\n\\[\n\\psi(z)\\sum_{u\\in X}\\psi(\\xi(u))=0.\n\\]\nWhen \\(x=\\xi=0\\), the operator is the scalar \\(\\psi(z)\\).\n\\end{proof}","Let \\(L\\subseteq V\\) be maximal isotropic. Choose a subgroup lift \\(\\widetilde L\\subseteq\\frakH(V)\\) on which the central coordinate is zero.","\\begin{proposition}[Isotropic averaging projector]\\label{prop:isotropic_selector}\nThe operator\n\\[\nP_L\n=\n\\frac1{\\abs L}\n\\sum_{\\ell\\in\\widetilde L}\n\\rho(\\ell)\n\\]\nis an orthogonal projector. In the Schrödinger model with \\(L=X\\), it has rank one and selects the constant functions.\n\\end{proposition}","\\begin{proof}\nThe lifted isotropic subgroup is abelian, so group averaging is an idempotent. Every operator in the average is unitary, and the subgroup is closed under inversion, so the average is self adjoint. For \\(L=X\\), the average is the translation average\n\\[\n(P_Lf)(u)\n=\n\\frac1{\\abs X}\n\\sum_{x\\in X}f(u+x),\n\\]\nwhich is the constant function with value equal to the mean of \\(f\\). Its range is one dimensional.\n\\end{proof}","\\begin{proof}\nThe projector rank equals its trace. The identity element contributes \\(p^r\\), while every nonidentity element of the lifted isotropic subgroup is noncentral and has trace zero by \\cref{prop:heisenberg_trace}. Since \\(\\abs L=p^r\\), we get\n\\[\n\\Tr P_L\n=\n\\frac{p^r}{p^r}=1.\n\\]\nThus the projector has rank one.\n\\end{proof}","Let \\(A\\) be a finite abelian group, and let\n\\[\nc:A\\times A\\to\\CC^{\\times}\n\\]\nbe a normalized two cocycle. The twisted group algebra \\(\\CC^c[A]\\) has basis \\((u_a)_{a\\in A}\\) and multiplication\n\\[\nu_au_b=c(a,b)u_{a+b}.\n\\]\nDefine the commutator bicharacter\n\\[\nb_c(a,b)=c(a,b)c(b,a)^{-1}.\n\\]","\\begin{proposition}[Scalar choice invariance]\\label{prop:cocycle_choice}\nReplacing \\(u_a\\) by \\(\\lambda_au_a\\), with \\(\\lambda_a\\in\\CC^{\\times}\\), changes \\(c\\) by a coboundary and leaves \\(b_c\\) unchanged.\n\\end{proposition}","\\begin{proof}\nThe new cocycle is\n\\[\nc'(a,b)\n=\n\\lambda_a\\lambda_b\\lambda_{a+b}^{-1}c(a,b).\n\\]\nThe scalar coboundary factor is symmetric in \\(a\\) and \\(b\\), so it cancels from \\(c'(a,b)c'(b,a)^{-1}\\).\n\\end{proof}","\\begin{proposition}[Alternating commutator form]\\label{prop:commutator_form}\nThe map \\(b_c\\) is an alternating bicharacter. Its radical\n\\[\nR=\\{a\\in A:b_c(a,x)=1\\text{ for every }x\\in A\\}\n\\]\nis a subgroup, and \\(b_c\\) descends to a nondegenerate alternating bicharacter on \\(A/R\\).\n\\end{proposition}","\\begin{proof}\nAssociativity of the twisted product gives multiplicativity of scalar commutators in each variable. We have \\(b_c(a,a)=1\\) and \\(b_c(b,a)=b_c(a,b)^{-1}\\). The radical is the kernel of the homomorphism from \\(A\\) to its character group defined by \\(a\\mapsto b_c(a,\\cdot)\\). The induced pairing on the quotient is nondegenerate by construction.\n\\end{proof}","\\begin{theorem}[Nondegenerate twisted group algebra]\\label{thm:twisted_matrix}\nAssume that \\(b_c\\) is nondegenerate. Then \\(\\abs A\\) is a square, and\n\\[\n\\boxed{\n\\CC^c[A]\n\\cong\nM_{\\sqrt{\\abs A}}(\\CC).\n}\n\\]\nConsequently, \\(\\CC^c[A]\\) has one irreducible module up to isomorphism, of dimension \\(\\sqrt{\\abs A}\\).\n\\end{theorem}","\\begin{proof}\nLet\n\\[\nz=\\sum_{a\\in A}z_au_a\n\\]\nbe central. Commuting with \\(u_x\\) gives\n\\[\nz_ab_c(x,a)=z_a\n\\]\nfor every \\(x\\). Nondegeneracy forces \\(z_a=0\\) for every nonzero \\(a\\). Thus the center is \\(\\CC\\). To prove semisimplicity directly, choose a linear projection \\(P\\) onto a submodule of a finite dimensional \\(\\CC^c[A]\\)-module and form\n\\[\n\\overline P=\\frac{1}{\\abs A}\\sum_{a\\in A}u_a P u_a^{-1}.\n\\]\nEvery summand maps into the submodule and is the identity there. Conjugation by any \\(u_x\\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) p","\\begin{proof}\nLet\n\\[\nz=\\sum_{a\\in A}z_au_a\n\\]\nbe central. Commuting with \\(u_x\\) gives\n\\[\nz_ab_c(x,a)=z_a\n\\]\nfor every \\(x\\). Nondegeneracy forces \\(z_a=0\\) for every nonzero \\(a\\). Thus the center is \\(\\CC\\). To prove semisimplicity directly, choose a linear projection \\(P\\) onto a submodule of a finite dimensional \\(\\CC^c[A]\\)-module and form\n\\[\n\\overline P=\\frac{1}{\\abs A}\\sum_{a\\in A}u_a P u_a^{-1}.\n\\]\nEvery summand maps into the submodule and is the identity there. Conjugation by any \\(u_x\\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) permits only one simple factor. A fi","\\begin{proof}\nLet\n\\[\nz=\\sum_{a\\in A}z_au_a\n\\]\nbe central. Commuting with \\(u_x\\) gives\n\\[\nz_ab_c(x,a)=z_a\n\\]\nfor every \\(x\\). Nondegeneracy forces \\(z_a=0\\) for every nonzero \\(a\\). Thus the center is \\(\\CC\\). To prove semisimplicity directly, choose a linear projection \\(P\\) onto a submodule of a finite dimensional \\(\\CC^c[A]\\)-module and form\n\\[\n\\overline P=\\frac{1}{\\abs A}\\sum_{a\\in A}u_a P u_a^{-1}.\n\\]\nEvery summand maps into the submodule and is the identity there. Conjugation by any \\(u_x\\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) permits only one simple factor. A finite dimensional central sim","\\begin{proof}\nLet\n\\[\nz=\\sum_{a\\in A}z_au_a\n\\]\nbe central. Commuting with \\(u_x\\) gives\n\\[\nz_ab_c(x,a)=z_a\n\\]\nfor every \\(x\\). Nondegeneracy forces \\(z_a=0\\) for every nonzero \\(a\\). Thus the center is \\(\\CC\\). To prove semisimplicity directly, choose a linear projection \\(P\\) onto a submodule of a finite dimensional \\(\\CC^c[A]\\)-module and form\n\\[\n\\overline P=\\frac{1}{\\abs A}\\sum_{a\\in A}u_a P u_a^{-1}.\n\\]\nEvery summand maps into the submodule and is the identity there. Conjugation by any \\(u_x\\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) permits only one simple factor. A finite dimensional central simple algebra over","\\begin{proof}\nLet\n\\[\nz=\\sum_{a\\in A}z_au_a\n\\]\nbe central. Commuting with \\(u_x\\) gives\n\\[\nz_ab_c(x,a)=z_a\n\\]\nfor every \\(x\\). Nondegeneracy forces \\(z_a=0\\) for every nonzero \\(a\\). Thus the center is \\(\\CC\\). To prove semisimplicity directly, choose a linear projection \\(P\\) onto a submodule of a finite dimensional \\(\\CC^c[A]\\)-module and form\n\\[\n\\overline P=\\frac{1}{\\abs A}\\sum_{a\\in A}u_a P u_a^{-1}.\n\\]\nEvery summand maps into the submodule and is the identity there. Conjugation by any \\(u_x\\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) permits only one simple factor. A finite dimensional central simple algebra over \\(\\CC\\) is a full matrix algebr","\\begin{proof}\nLet\n\\[\nz=\\sum_{a\\in A}z_au_a\n\\]\nbe central. Commuting with \\(u_x\\) gives\n\\[\nz_ab_c(x,a)=z_a\n\\]\nfor every \\(x\\). Nondegeneracy forces \\(z_a=0\\) for every nonzero \\(a\\). Thus the center is \\(\\CC\\). To prove semisimplicity directly, choose a linear projection \\(P\\) onto a submodule of a finite dimensional \\(\\CC^c[A]\\)-module and form\n\\[\n\\overline P=\\frac{1}{\\abs A}\\sum_{a\\in A}u_a P u_a^{-1}.\n\\]\nEvery summand maps into the submodule and is the identity there. Conjugation by any \\(u_x\\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) permits only one simple factor. A finite dimensional central simple algebra over \\(\\CC\\) is a full matrix algebra. Its dimension is \\(\\a","\\begin{proof}\nLet\n\\[\nz=\\sum_{a\\in A}z_au_a\n\\]\nbe central. Commuting with \\(u_x\\) gives\n\\[\nz_ab_c(x,a)=z_a\n\\]\nfor every \\(x\\). Nondegeneracy forces \\(z_a=0\\) for every nonzero \\(a\\). Thus the center is \\(\\CC\\). To prove semisimplicity directly, choose a linear projection \\(P\\) onto a submodule of a finite dimensional \\(\\CC^c[A]\\)-module and form\n\\[\n\\overline P=\\frac{1}{\\abs A}\\sum_{a\\in A}u_a P u_a^{-1}.\n\\]\nEvery summand maps into the submodule and is the identity there. Conjugation by any \\(u_x\\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) permits only one simple factor. A finite dimensional central simple algebra over \\(\\CC\\) is a full matrix algebra. Its dimension is \\(\\abs A\\), so the matrix size i","\\begin{proof}\nLet\n\\[\nz=\\sum_{a\\in A}z_au_a\n\\]\nbe central. Commuting with \\(u_x\\) gives\n\\[\nz_ab_c(x,a)=z_a\n\\]\nfor every \\(x\\). Nondegeneracy forces \\(z_a=0\\) for every nonzero \\(a\\). Thus the center is \\(\\CC\\). To prove semisimplicity directly, choose a linear projection \\(P\\) onto a submodule of a finite dimensional \\(\\CC^c[A]\\)-module and form\n\\[\n\\overline P=\\frac{1}{\\abs A}\\sum_{a\\in A}u_a P u_a^{-1}.\n\\]\nEvery summand maps into the submodule and is the identity there. Conjugation by any \\(u_x\\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) permits only one simple factor. A finite dimensional central simple algebra over \\(\\CC\\) is a full matrix algebra. Its dimension is \\(\\abs A\\), so the matrix size is \\(\\sqrt{\\abs A}\\).\n\\end{proof}","(x,a)=z_a\n\\]\nfor every \\(x\\). Nondegeneracy forces \\(z_a=0\\) for every nonzero \\(a\\). Thus the center is \\(\\CC\\). To prove semisimplicity directly, choose a linear projection \\(P\\) onto a submodule of a finite dimensional \\(\\CC^c[A]\\)-module and form\n\\[\n\\overline P=\\frac{1}{\\abs A}\\sum_{a\\in A}u_a P u_a^{-1}.\n\\]\nEvery summand maps into the submodule and is the identity there. Conjugation by any \\(u_x\\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) permits only one simple factor. A finite dimensional central simple algebra over \\(\\CC\\) is a full matrix algebra. Its dimension is \\(\\abs A\\), so the matrix size is \\(\\sqrt{\\abs A}\\).\n\\end{proof}","s \\(\\CC\\). To prove semisimplicity directly, choose a linear projection \\(P\\) onto a submodule of a finite dimensional \\(\\CC^c[A]\\)-module and form\n\\[\n\\overline P=\\frac{1}{\\abs A}\\sum_{a\\in A}u_a P u_a^{-1}.\n\\]\nEvery summand maps into the submodule and is the identity there. Conjugation by any \\(u_x\\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) permits only one simple factor. A finite dimensional central simple algebra over \\(\\CC\\) is a full matrix algebra. Its dimension is \\(\\abs A\\), so the matrix size is \\(\\sqrt{\\abs A}\\).\n\\end{proof}","e and form\n\\[\n\\overline P=\\frac{1}{\\abs A}\\sum_{a\\in A}u_a P u_a^{-1}.\n\\]\nEvery summand maps into the submodule and is the identity there. Conjugation by any \\(u_x\\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) permits only one simple factor. A finite dimensional central simple algebra over \\(\\CC\\) is a full matrix algebra. Its dimension is \\(\\abs A\\), so the matrix size is \\(\\sqrt{\\abs A}\\).\n\\end{proof}","aps into the submodule and is the identity there. Conjugation by any \\(u_x\\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) permits only one simple factor. A finite dimensional central simple algebra over \\(\\CC\\) is a full matrix algebra. Its dimension is \\(\\abs A\\), so the matrix size is \\(\\sqrt{\\abs A}\\).\n\\end{proof}","onjugation by any \\(u_x\\) permutes the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) permits only one simple factor. A finite dimensional central simple algebra over \\(\\CC\\) is a full matrix algebra. Its dimension is \\(\\abs A\\), so the matrix size is \\(\\sqrt{\\abs A}\\).\n\\end{proof}","the summands: the cocycle scalars cancel between each operator and its inverse. Hence \\(\\overline P\\) is an equivariant projection, and its kernel is an invariant complement. The algebra is therefore semisimple. Its center \\(\\CC\\) permits only one simple factor. A finite dimensional central simple algebra over \\(\\CC\\) is a full matrix algebra. Its dimension is \\(\\abs A\\), so the matrix size is \\(\\sqrt{\\abs A}\\).\n\\end{proof}","\\begin{corollary}[Radical free envelope]\\label{cor:radical_free}\nFor an arbitrary scalar projective action of \\(A\\), the quotient\n\\[\n\\boxed{\nA^{\\mathrm{symp}}=A/\\rad b_c\n}\n\\]\nis the maximal quotient detected by noncommuting projective intertwiners. If \\(A\\) is an elementary abelian \\(p\\) group, then \\(A^{\\mathrm{symp}}\\) is a symplectic \\(\\FF_p\\) space.\n\\end{corollary}","\\begin{proof}\nEvery radical element commutes with all projective generators, so it contributes only central data. Quotienting by the radical leaves the nondegenerate commutator form. In the elementary abelian case, multiplicative values in the group of \\(p\\)th roots can be written as an alternating bilinear form over \\(\\FF_p\\).\n\\end{proof}","We now isolate a finite Clifford setting in which the quotient projective action is defined on a multiplicity space. This avoids assigning a quotient action directly to arbitrary representatives of \\(G/N\\).","\\begin{definition}[One break Clifford datum]\\label{def:one_break_datum}\nA one break Clifford datum consists of finite groups and representations\n\\[\nN\\triangleleft G,\n\\qquad\nA=G/N,\n\\qquad\n\\Pi:G\\to\\GL(\\calX),\n\\qquad\n\\rho:N\\to\\GL(W),\n\\]\nwith the following properties.","\\begin{enumerate}\n\\item The quotient \\(A\\) is a finite elementary abelian \\(p\\) group.\n\\item The representations \\(\\Pi\\) and \\(\\rho\\) are irreducible.\n\\item The isomorphism class of \\(\\rho\\) is fixed by \\(G\\).\n\\item The restriction of \\(\\Pi\\) to \\(N\\) is \\(\\rho\\) isotypic.\n\\item The quotient records one declared ramification break.\n\\end{enumerate}","The multiplicity space is\n\\[\nM_\\rho=\\Hom_N(W,\\calX).\n\\]\n\\end{definition}","The isotypic condition gives an \\(N\\) equivariant evaluation isomorphism\n\\[\nW\\otimes M_\\rho\n\\longrightarrow\n\\calX.\n\\]\nChoose a section \\(s:A\\to G\\). For every \\(a\\in A\\), choose an intertwiner \\(I_a\\) from \\(\\rho\\) to its conjugate by \\(s(a)\\). Under the evaluation isomorphism, there is a unique operator \\(T_a\\) on \\(M_\\rho\\) such that\n\\[\n\\Pi(s(a))=I_a\\otimes T_a.\n\\]\nThe operators \\(T_a\\) form a projective representation of \\(A\\).","n \\(N\\) equivariant evaluation isomorphism\n\\[\nW\\otimes M_\\rho\n\\longrightarrow\n\\calX.\n\\]\nChoose a section \\(s:A\\to G\\). For every \\(a\\in A\\), choose an intertwiner \\(I_a\\) from \\(\\rho\\) to its conjugate by \\(s(a)\\). Under the evaluation isomorphism, there is a unique operator \\(T_a\\) on \\(M_\\rho\\) such that\n\\[\n\\Pi(s(a))=I_a\\otimes T_a.\n\\]\nThe operators \\(T_a\\) form a projective representation of \\(A\\).","\\begin{proposition}[Clifford multiplicity action]\\label{prop:clifford_action}\nThe projective equivalence class of the action \\((T_a)_{a\\in A}\\) is independent of the section, the intertwiners, and the chosen isotypic evaluation isomorphism. The projective action is irreducible.\n\\end{proposition}","\\begin{proof}\nChanging an intertwiner \\(I_a\\) by a scalar changes \\(T_a\\) by the inverse scalar. Changing the isotypic evaluation isomorphism conjugates all \\(T_a\\) by one element of \\(\\GL(M_\\rho)\\). Changing the section multiplies \\(s(a)\\) by an element of \\(N\\). That factor is absorbed by the \\(\\rho\\) action on \\(W\\), leaving a scalar change on the multiplicity operator by Schur's lemma. Thus all choices give projectively equivalent actions.","If a nonzero proper subspace of \\(M_\\rho\\) were stable under all projective operators \\(T_a\\), then its tensor product with \\(W\\) would give a nonzero proper \\(G\\) stable subspace of \\(\\calX\\). This contradicts irreducibility of \\(\\Pi\\).\n\\end{proof}","\\begin{definition}[Clifford cocycle class]\\label{def:clifford_class}\nChoose scalars \\(c(a,b)\\) such that\n\\[\nT_aT_b=c(a,b)T_{a+b}.\n\\]\nThe cohomology class\n\\[\n[c_\\rho]\\in H^2(A,\\CC^{\\times})\n\\]\nis the Clifford cocycle class of the datum.\n\\end{definition}","\\begin{proposition}[Intrinsic commutator pairing]\\label{prop:one_break_form}\nThe pairing\n\\[\nb_\\rho(a,b)\n=\nc(a,b)c(b,a)^{-1}\n\\]\nis independent of every choice in \\cref{def:one_break_datum}. It is an alternating bicharacter\n\\[\nb_\\rho:A\\times A\\to\\mu_p.\n\\]\n\\end{proposition}","\\begin{proof}\nThe projective equivalence class is intrinsic by \\cref{prop:clifford_action}. Conjugation does not change scalar commutators, and scalar changes alter the cocycle by a coboundary. The commutator pairing is therefore unchanged by \\cref{prop:cocycle_choice}. Its alternating and bilinear properties follow from \\cref{prop:commutator_form}. Since every element of \\(A\\) has order \\(p\\), every value has order dividing \\(p\\).\n\\end{proof}","uivalence class is intrinsic by \\cref{prop:clifford_action}. Conjugation does not change scalar commutators, and scalar changes alter the cocycle by a coboundary. The commutator pairing is therefore unchanged by \\cref{prop:cocycle_choice}. Its alternating and bilinear properties follow from \\cref{prop:commutator_form}. Since every element of \\(A\\) has order \\(p\\), every value has order dividing \\(p\\).\n\\end{proof}","\\begin{definition}[One break symplectic envelope]\\label{def:one_break_envelope}\nThe one break symplectic envelope is\n\\[\n\\boxed{\nV_\\rho=A/\\rad b_\\rho.\n}\n\\]\n\\end{definition}","\\begin{theorem}[Nondegenerate one break packet]\\label{thm:one_break_matrix}\nIf \\(b_\\rho\\) is nondegenerate, then\n\\[\n\\CC^{c_\\rho}[A]\n\\cong\n\\End(M_\\rho),\n\\qquad\n\\dim M_\\rho=\\sqrt{\\abs A}.\n\\]\n\\end{theorem}","\\begin{proof}\nThe projective action gives a nonzero representation of the twisted group algebra on \\(M_\\rho\\). The algebra is a full matrix algebra by \\cref{thm:twisted_matrix}. Its unique irreducible module has dimension \\(\\sqrt{\\abs A}\\). The projective multiplicity action is irreducible by \\cref{prop:clifford_action}, so it is that module.\n\\end{proof}","\\begin{proposition}[Isomorphism invariance]\\label{prop:one_break_invariance}\nThe cocycle class and the equivariant isometry class of \\(V_\\rho\\) depend only on the isomorphism class of the one break Clifford datum.\n\\end{proposition}","\\begin{proposition}[Tame stabilizer action]\\label{prop:tame_stabilizer}\nSuppose a finite group \\(C\\) acts by automorphisms of the one break Clifford datum. Then \\(C\\) acts on \\(V_\\rho\\) by symplectic automorphisms.\n\\end{proposition}","\\begin{corollary}[Heisenberg representation attached to one break]\\label{cor:one_break_heisenberg}\nFix a nontrivial central character. The envelope \\(V_\\rho\\) determines one irreducible finite Heisenberg representation up to isomorphism.\n\\end{corollary}","Let \\(\\theta\\) be a maximal simple character for \\(\\GL_n(K)\\), with compact open subgroups\n\\[\nH_\\theta^1\\subseteq J_\\theta^1\\subseteq J_\\theta.\n\\]\nThe quotient\n\\[\nV_\\theta=J_\\theta^1/H_\\theta^1\n\\]\nis regarded as a finite vector space over \\(\\FF_p\\), where \\(p\\) is the residue characteristic. Fix a nontrivial additive character \\(\\psi_0:\\FF_p\\to\\mu_p\\subset\\CC^\\times\\), where \\(\\mu_p\\) is the group of \\(p\\)-th roots of unity. The simple-character commutator takes values in \\(\\mu_p\\). Since \\(\\psi_0\\) is injective, the equation\n\\[\n\\psi_0\\bigl(h_\\theta(\\bar x,\\bar y)\\bigr)\n=\n\\theta([x,y])\n\\]\nuniquely defines the \\(\\FF_p\\)-valued commutator pairing \\(h_\\theta\\). We use the same identification of \\(\\mu_p\\) w","Let \\(\\theta\\) be a maximal simple character for \\(\\GL_n(K)\\), with compact open subgroups\n\\[\nH_\\theta^1\\subseteq J_\\theta^1\\subseteq J_\\theta.\n\\]\nThe quotient\n\\[\nV_\\theta=J_\\theta^1/H_\\theta^1\n\\]\nis regarded as a finite vector space over \\(\\FF_p\\), where \\(p\\) is the residue characteristic. Fix a nontrivial additive character \\(\\psi_0:\\FF_p\\to\\mu_p\\subset\\CC^\\times\\), where \\(\\mu_p\\) is the group of \\(p\\)-th roots of unity. The simple-character commutator takes values in \\(\\mu_p\\). Since \\(\\psi_0\\) is injective, the equation\n\\[\n\\psi_0\\bigl(h_\\theta(\\bar x,\\bar y)\\bigr)\n=\n\\theta([x,y])\n\\]\nuniquely defines the \\(\\FF_p\\)-valued commutator pairing \\(h_\\theta\\). We use the same identification of \\(\\mu_p\\) with the additive group of \\(\\FF_p\\) on the Galo","Let \\(\\theta\\) be a maximal simple character for \\(\\GL_n(K)\\), with compact open subgroups\n\\[\nH_\\theta^1\\subseteq J_\\theta^1\\subseteq J_\\theta.\n\\]\nThe quotient\n\\[\nV_\\theta=J_\\theta^1/H_\\theta^1\n\\]\nis regarded as a finite vector space over \\(\\FF_p\\), where \\(p\\) is the residue characteristic. Fix a nontrivial additive character \\(\\psi_0:\\FF_p\\to\\mu_p\\subset\\CC^\\times\\), where \\(\\mu_p\\) is the group of \\(p\\)-th roots of unity. The simple-character commutator takes values in \\(\\mu_p\\). Since \\(\\psi_0\\) is injective, the equation\n\\[\n\\psi_0\\bigl(h_\\theta(\\bar x,\\bar y)\\bigr)\n=\n\\theta([x,y])\n\\]\nuniquely defines the \\(\\FF_p\\)-valued commutator pairing \\(h_\\theta\\). We use the same identification of \\(\\mu_p\\) with the additive group of \\(\\FF_p\\) on the Galois side when comparing the two symplectic spaces.","\\theta\\) be a maximal simple character for \\(\\GL_n(K)\\), with compact open subgroups\n\\[\nH_\\theta^1\\subseteq J_\\theta^1\\subseteq J_\\theta.\n\\]\nThe quotient\n\\[\nV_\\theta=J_\\theta^1/H_\\theta^1\n\\]\nis regarded as a finite vector space over \\(\\FF_p\\), where \\(p\\) is the residue characteristic. Fix a nontrivial additive character \\(\\psi_0:\\FF_p\\to\\mu_p\\subset\\CC^\\times\\), where \\(\\mu_p\\) is the group of \\(p\\)-th roots of unity. The simple-character commutator takes values in \\(\\mu_p\\). Since \\(\\psi_0\\) is injective, the equation\n\\[\n\\psi_0\\bigl(h_\\theta(\\bar x,\\bar y)\\bigr)\n=\n\\theta([x,y])\n\\]\nuniquely defines the \\(\\FF_p\\)-valued commutator pairing \\(h_\\theta\\). We use the same identification of \\(\\mu_p\\) with the additive group of \\(\\FF_p\\) on the Galois side when comparing the two symplectic spaces.","n subgroups\n\\[\nH_\\theta^1\\subseteq J_\\theta^1\\subseteq J_\\theta.\n\\]\nThe quotient\n\\[\nV_\\theta=J_\\theta^1/H_\\theta^1\n\\]\nis regarded as a finite vector space over \\(\\FF_p\\), where \\(p\\) is the residue characteristic. Fix a nontrivial additive character \\(\\psi_0:\\FF_p\\to\\mu_p\\subset\\CC^\\times\\), where \\(\\mu_p\\) is the group of \\(p\\)-th roots of unity. The simple-character commutator takes values in \\(\\mu_p\\). Since \\(\\psi_0\\) is injective, the equation\n\\[\n\\psi_0\\bigl(h_\\theta(\\bar x,\\bar y)\\bigr)\n=\n\\theta([x,y])\n\\]\nuniquely defines the \\(\\FF_p\\)-valued commutator pairing \\(h_\\theta\\). We use the same identification of \\(\\mu_p\\) with the additive group of \\(\\FF_p\\) on the Galois side when comparing the two symplectic spaces.","\\theta^1\\subseteq J_\\theta^1\\subseteq J_\\theta.\n\\]\nThe quotient\n\\[\nV_\\theta=J_\\theta^1/H_\\theta^1\n\\]\nis regarded as a finite vector space over \\(\\FF_p\\), where \\(p\\) is the residue characteristic. Fix a nontrivial additive character \\(\\psi_0:\\FF_p\\to\\mu_p\\subset\\CC^\\times\\), where \\(\\mu_p\\) is the group of \\(p\\)-th roots of unity. The simple-character commutator takes values in \\(\\mu_p\\). Since \\(\\psi_0\\) is injective, the equation\n\\[\n\\psi_0\\bigl(h_\\theta(\\bar x,\\bar y)\\bigr)\n=\n\\theta([x,y])\n\\]\nuniquely defines the \\(\\FF_p\\)-valued commutator pairing \\(h_\\theta\\). We use the same identification of \\(\\mu_p\\) with the additive group of \\(\\FF_p\\) on the Galois side when comparing the two symplectic spaces.","J_\\theta.\n\\]\nThe quotient\n\\[\nV_\\theta=J_\\theta^1/H_\\theta^1\n\\]\nis regarded as a finite vector space over \\(\\FF_p\\), where \\(p\\) is the residue characteristic. Fix a nontrivial additive character \\(\\psi_0:\\FF_p\\to\\mu_p\\subset\\CC^\\times\\), where \\(\\mu_p\\) is the group of \\(p\\)-th roots of unity. The simple-character commutator takes values in \\(\\mu_p\\). Since \\(\\psi_0\\) is injective, the equation\n\\[\n\\psi_0\\bigl(h_\\theta(\\bar x,\\bar y)\\bigr)\n=\n\\theta([x,y])\n\\]\nuniquely defines the \\(\\FF_p\\)-valued commutator pairing \\(h_\\theta\\). We use the same identification of \\(\\mu_p\\) with the additive group of \\(\\FF_p\\) on the Galois side when comparing the two symplectic spaces.","e vector space over \\(\\FF_p\\), where \\(p\\) is the residue characteristic. Fix a nontrivial additive character \\(\\psi_0:\\FF_p\\to\\mu_p\\subset\\CC^\\times\\), where \\(\\mu_p\\) is the group of \\(p\\)-th roots of unity. The simple-character commutator takes values in \\(\\mu_p\\). Since \\(\\psi_0\\) is injective, the equation\n\\[\n\\psi_0\\bigl(h_\\theta(\\bar x,\\bar y)\\bigr)\n=\n\\theta([x,y])\n\\]\nuniquely defines the \\(\\FF_p\\)-valued commutator pairing \\(h_\\theta\\). We use the same identification of \\(\\mu_p\\) with the additive group of \\(\\FF_p\\) on the Galois side when comparing the two symplectic spaces.","\\(p\\) is the residue characteristic. Fix a nontrivial additive character \\(\\psi_0:\\FF_p\\to\\mu_p\\subset\\CC^\\times\\), where \\(\\mu_p\\) is the group of \\(p\\)-th roots of unity. The simple-character commutator takes values in \\(\\mu_p\\). Since \\(\\psi_0\\) is injective, the equation\n\\[\n\\psi_0\\bigl(h_\\theta(\\bar x,\\bar y)\\bigr)\n=\n\\theta([x,y])\n\\]\nuniquely defines the \\(\\FF_p\\)-valued commutator pairing \\(h_\\theta\\). We use the same identification of \\(\\mu_p\\) with the additive group of \\(\\FF_p\\) on the Galois side when comparing the two symplectic spaces.","rivial additive character \\(\\psi_0:\\FF_p\\to\\mu_p\\subset\\CC^\\times\\), where \\(\\mu_p\\) is the group of \\(p\\)-th roots of unity. The simple-character commutator takes values in \\(\\mu_p\\). Since \\(\\psi_0\\) is injective, the equation\n\\[\n\\psi_0\\bigl(h_\\theta(\\bar x,\\bar y)\\bigr)\n=\n\\theta([x,y])\n\\]\nuniquely defines the \\(\\FF_p\\)-valued commutator pairing \\(h_\\theta\\). We use the same identification of \\(\\mu_p\\) with the additive group of \\(\\FF_p\\) on the Galois side when comparing the two symplectic spaces.","F_p\\to\\mu_p\\subset\\CC^\\times\\), where \\(\\mu_p\\) is the group of \\(p\\)-th roots of unity. The simple-character commutator takes values in \\(\\mu_p\\). Since \\(\\psi_0\\) is injective, the equation\n\\[\n\\psi_0\\bigl(h_\\theta(\\bar x,\\bar y)\\bigr)\n=\n\\theta([x,y])\n\\]\nuniquely defines the \\(\\FF_p\\)-valued commutator pairing \\(h_\\theta\\). We use the same identification of \\(\\mu_p\\) with the additive group of \\(\\FF_p\\) on the Galois side when comparing the two symplectic spaces.","\\begin{standardfact}[Type side Heisenberg quotient]\\label{fact:type_heisenberg}\nThe pairing \\(h_\\theta\\) is nondegenerate and alternating. There is one irreducible representation \\(\\eta_\\theta\\) of \\(J_\\theta^1\\) containing \\(\\theta\\), and its dimension is\n\\[\n\\sqrt{\\abs{J_\\theta^1/H_\\theta^1}}.\n\\]\n\\end{standardfact}","This is part of the simple character and Heisenberg representation theory used in the construction of types \\cite{BushnellKutzko1993}. The extension of \\(\\eta_\\theta\\) from \\(J_\\theta^1\\) to \\(J_\\theta\\) is not unique before a normalization is chosen.","\\begin{proposition}[Type side projective commutator]\\label{prop:type_commutator}\nThe projective action of \\(J_\\theta/J_\\theta^1\\) on the Heisenberg representation preserves the commutator form on \\(V_\\theta\\). Any two choices of projective lifts give the same alternating form.\n\\end{proposition}","\\begin{proof}\nConjugation by \\(J_\\theta\\) preserves \\(\\theta\\), so it preserves the commutator pairing defined from \\(\\theta([x,y])\\). Two choices of projective lifts differ by scalars. The scalar commutator is therefore unchanged by \\cref{prop:cocycle_choice}.\n\\end{proof}","The full bridge would construct an intrinsic Galois envelope for every wild atom and identify it canonically with \\(V_\\theta\\). We prove the comparison after the required symplectic identification has been supplied.","\\begin{enumerate}\n\\item a one break Galois envelope \\((V_\\rho,b_\\rho)\\)\n\\item a type side quotient \\((V_\\theta,h_\\theta)\\)\n\\item an equivariant symplectic isometry\n\\[\n\\iota:V_\\rho\\longrightarrow V_\\theta\n\\]\nfor a finite cyclic stabilizer \\(C=\\langle c\\rangle\\)\n\\item the same nontrivial central character on the two Heisenberg groups\n\\item when \\(p=2\\), compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensions lifting \\(\\iota\\).\n\\end{enumerate}\n\\end{definition}","_\\rho)\\)\n\\item a type side quotient \\((V_\\theta,h_\\theta)\\)\n\\item an equivariant symplectic isometry\n\\[\n\\iota:V_\\rho\\longrightarrow V_\\theta\n\\]\nfor a finite cyclic stabilizer \\(C=\\langle c\\rangle\\)\n\\item the same nontrivial central character on the two Heisenberg groups\n\\item when \\(p=2\\), compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensions lifting \\(\\iota\\).\n\\end{enumerate}\n\\end{definition}","\\begin{theorem}[Restricted Heisenberg bridge]\\label{thm:restricted_bridge}\nA restricted bridge datum determines an isomorphism between the two Heisenberg representations, unique up to a nonzero scalar. More precisely, there is an invertible map\n\\[\nU:\\calH_\\rho\\longrightarrow\\calH_\\theta\n\\]\nsuch that\n\\[\nU\\omega_\\rho(h)\n=\n\\omega_\\theta(\\iota_H(h))U\n\\]\nfor every Heisenberg element \\(h\\), and every other such map is a scalar multiple of \\(U\\).\n\\end{theorem}","ridge]\\label{thm:restricted_bridge}\nA restricted bridge datum determines an isomorphism between the two Heisenberg representations, unique up to a nonzero scalar. More precisely, there is an invertible map\n\\[\nU:\\calH_\\rho\\longrightarrow\\calH_\\theta\n\\]\nsuch that\n\\[\nU\\omega_\\rho(h)\n=\n\\omega_\\theta(\\iota_H(h))U\n\\]\nfor every Heisenberg element \\(h\\), and every other such map is a scalar multiple of \\(U\\).\n\\end{theorem}","\\begin{proof}\nWhen \\(p\\) is odd, the symplectic isometry and the common central character induce the standard isomorphism \\(\\iota_H\\) of the two Heisenberg groups. When \\(p=2\\), the required lift of \\(\\iota\\) is part of the bridge datum because the alternating form alone does not determine the central extension. Pulling the type side representation back through \\(\\iota_H\\) gives an irreducible Heisenberg representation with the same central character as the Galois side representation. The finite Stone von Neumann theorem gives an isomorphism. Schur's lemma gives uniqueness up to scalar.\n\\end{proof}","The stabilizer actions need not agree after choosing \\(U\\). Their discrepancy is scalar because both actions implement the same automorphisms of the irreducible Heisenberg representation.","\\begin{proposition}[Cyclic discrepancy character]\\label{prop:cyclic_discrepancy}\nAssume that the two projective stabilizer actions have been linearized to honest representations of the cyclic group \\(C\\). Then there is a character\n\\[\n\\delta:C\\to\\CC^{\\times}\n\\]\nsuch that\n\\[\nU\\Omega_\\rho(g)\n=\n\\delta(g)\\Omega_\\theta(g)U\n\\]\nfor every \\(g\\in C\\).\n\\end{proposition}","\\begin{proof}\nFor each \\(g\\), the operator\n\\[\n\\Omega_\\theta(g)^{-1}U\\Omega_\\rho(g)U^{-1}\n\\]\ncommutes with the type side Heisenberg representation. It is therefore scalar. Since both stabilizer actions are honest representations, multiplication of group elements shows that the scalars form a character.\n\\end{proof}","\\begin{lemma}[Linearization over a cyclic stabilizer]\\label{lem:cyclic_linearization}\nEvery finite dimensional projective representation of a finite cyclic group over \\(\\CC\\) is projectively equivalent to an honest representation.\n\\end{lemma}","\\begin{proof}\nLet \\(C=\\langle c\\rangle\\) have order \\(m\\), and let \\(T\\) be a lift of the projective generator. Since \\(c^m=1\\), the operator \\(T^m\\) is scalar. Choose \\(\\lambda\\in\\CC^{\\times}\\) with \\(\\lambda^mT^m=I\\). Replacing \\(T\\) by \\(\\lambda T\\) gives an honest cyclic representation.\n\\end{proof}","\\begin{theorem}[One phase closes the cyclic bridge]\\label{thm:one_phase_bridge}\nIn a restricted bridge datum, choose linearizations of the cyclic stabilizer actions. If the two actions have the same scalar phase on the generator \\(c\\), then the Heisenberg intertwiner can be chosen to intertwine the full semidirect product action.\n\\end{theorem}","\\begin{proof}\nBy \\cref{prop:cyclic_discrepancy}, the discrepancy is a character \\(\\delta\\). Equality of the generator phase gives \\(\\delta(c)=1\\). Since \\(c\\) generates \\(C\\), the character is trivial. The chosen map \\(U\\) then intertwines every stabilizer element as well as the Heisenberg group.\n\\end{proof}","\\begin{remark}\\label{rem:bridge_scope}\nThe theorem does not construct \\(\\iota\\). It proves that, once an equivariant symplectic isometry exists, the remaining finite comparison is one dimensional for a cyclic stabilizer.\n\\end{remark}","Let \\(p\\) be odd, let \\((V,b)\\) be a finite symplectic \\(\\FF_p\\) space of dimension \\(2r\\), and let\n\\[\ng\\in\\Sp(V)\n\\]\nhave no fixed vector. Fix one of the standard finite Weil normalizations used in the literature \\cite{Gerardin1977,Neuhauser2002,GurevichHadani2007}.","\\begin{standardfact}[Fixed point free Weil trace]\\label{fact:weil_trace}\nUnder the declared normalization,\n\\[\n\\boxed{\n\\Tr\\omega(g)\n=\n\\left(\\frac{-1}{p}\\right)^r\n\\left(\\frac{\\det(g-I)}{p}\\right).\n}\n\\]\nHere \\((\\frac{\\cdot}{p})\\) denotes the quadratic character of \\(\\FF_p^{\\times}\\).\n\\end{standardfact}","\\begin{definition}[Odd finite Weil phase]\\label{def:odd_phase}\nFor fixed point free \\(g\\), define\n\\[\n\\Phi_p(g,V)\n=\n\\left(\\frac{-1}{p}\\right)^r\n\\left(\\frac{\\det(g-I)}{p}\\right).\n\\]\n\\end{definition}","\\begin{proposition}[Symplectic invariance]\\label{prop:phase_invariance}\nIf\n\\[\n\\iota:(V,b)\\longrightarrow(V',b')\n\\]\nis a symplectic isometry and \\(g'=\\iota g\\iota^{-1}\\), then\n\\[\n\\Phi_p(g,V)=\\Phi_p(g',V').\n\\]\n\\end{proposition}","\\begin{proof}\nThe dimensions agree, and conjugate matrices have the same determinant of \\(g-I\\).\n\\end{proof}","\\begin{proposition}[Orthogonal multiplicativity]\\label{prop:odd_multiplicativity}\nSuppose\n\\[\nV=V_1\\perpbig\\cdots\\perpbig V_s\n\\]\nand \\(g\\) preserves every summand. If no \\(g|_{V_j}\\) has a fixed vector, then\n\\[\n\\boxed{\n\\Phi_p(g,V)\n=\n\\prod_{j=1}^{s}\n\\Phi_p(g|_{V_j},V_j).\n}\n\\]\n\\end{proposition}","\\begin{proof}\nThe half dimensions add, and\n\\[\n\\det(g-I\\mid V)\n=\n\\prod_j\\det(g-I\\mid V_j).\n\\]\nThe quadratic character is multiplicative.\n\\end{proof}","\\begin{corollary}[Characteristic polynomial certificate]\\label{cor:characteristic_phase}\nLet\n\\[\nP_g(X)=\\det(XI-g\\mid V).\n\\]\nThen\n\\[\n\\boxed{\n\\Phi_p(g,V)\n=\n\\left(\\frac{-1}{p}\\right)^r\n\\left(\\frac{P_g(1)}{p}\\right).\n}\n\\]\n\\end{corollary}","\\begin{proof}\nSince \\(\\dim V\\) is even,\n\\[\n\\det(g-I)=\\det(I-g)=P_g(1).\n\\]\n\\end{proof}","\\begin{proposition}[Fixed and moving decomposition]\\label{prop:fixed_moving}\nAssume that \\(g\\in\\Sp(V)\\) has order prime to \\(p\\). Then\n\\[\nV=\\ker(g-I)\\perpbig\\im(g-I).\n\\]\nThe restriction of the symplectic form to \\(\\im(g-I)\\) is nondegenerate, and \\(g-I\\) is invertible on that moving summand.\n\\end{proposition}","\\begin{proof}\nThe order hypothesis makes \\(g\\) semisimple, so\n\\[\nV=\\ker(g-I)\\oplus\\im(g-I).\n\\]\nIf \\(x\\in\\ker(g-I)\\) and \\(y=(g-I)z\\), then\n\\[\nb(x,y)=b(x,gz)-b(x,z)=b(g^{-1}x,z)-b(x,z)=0.\n\\]\nThus the two summands are orthogonal. Nondegeneracy on each summand follows from nondegeneracy on \\(V\\). The last statement is immediate from the direct sum decomposition.\n\\end{proof}","\\begin{definition}[Moving phase]\\label{def:moving_phase}\nUnder the hypotheses of \\cref{prop:fixed_moving}, define\n\\[\n\\Phi_p^{\\mathrm{mov}}(g,V)\n=\n\\Phi_p\\left(g|_{\\im(g-I)},\\im(g-I)\\right).\n\\]\n\\end{definition}","\\begin{proposition}[Hyperbolic phase formula]\\label{prop:hyperbolic_phase}\nLet \\(U\\) be a finite dimensional \\(\\FF_p\\) space, let \\(a\\in\\GL(U)\\) have no eigenvalue one, and let \\(g\\) act on\n\\[\nH(U)=U\\oplus U^*\n\\]\nby\n\\[\ng=(a,(a^{-1})^*).\n\\]\nThen\n\\[\n\\boxed{\n\\Phi_p(g,H(U))\n=\n\\left(\\frac{\\det a}{p}\\right).\n}\n\\]\n\\end{proposition}","\\begin{proof}\nPut \\(d=\\dim U\\). We have\n\\[\n\\det\\bigl((a^{-1})^*-I\\bigr)\n=\n(-1)^d(\\det a)^{-1}\\det(a-I).\n\\]\nHence\n\\[\n\\det(g-I)\n=\n(-1)^d(\\det a)^{-1}\\det(a-I)^2.\n\\]\nMultiplying its quadratic character by the normalization factor \\((\\frac{-1}{p})^d\\) cancels the sign and the square. Since a quadratic character equals its inverse, the result is \\((\\frac{\\det a}{p})\\).\n\\end{proof}","\\begin{corollary}[Phase only bridge test]\\label{cor:phase_only_test}\nTwo odd characteristic moving packets have the same normalized phase whenever their moving half dimensions agree and the determinants of \\(g-I\\) lie in the same square class.\n\\end{corollary}","\\begin{proof}\nApply the definition of \\(\\Phi_p\\).\n\\end{proof}","\\begin{remark}\\label{rem:fixed_space}\nWhen \\(g\\) has fixed vectors, one first separates the fixed part and applies the formula to the moving quotient. Additional normalization data may enter the full trace on the fixed part.\n\\end{remark}","Let \\((V,b)\\) be a symplectic vector space over \\(\\FF_2\\). The alternating form alone is not enough to determine the finite phase.","\\begin{definition}[Quadratic refinement]\\label{def:quadratic_refinement}\nA quadratic refinement of \\(b\\) is a map\n\\[\nq:V\\to\\FF_2\n\\]\nsatisfying\n\\[\nq(x+y)-q(x)-q(y)=b(x,y).\n\\]\n\\end{definition}","\\begin{proposition}[Affine space of refinements]\\label{prop:refinement_affine}\nIf one quadratic refinement exists, then the set of all quadratic refinements is an affine space over \\(V^*\\).\n\\end{proposition}","\\begin{proof}\nThe difference of two refinements has zero polar form, so it is additive and hence linear over \\(\\FF_2\\). Conversely, adding a linear form does not change the polar form.\n\\end{proof}","\\begin{definition}[Arf invariant]\\label{def:arf}\nChoose a symplectic basis \\((e_i,f_i)_{i=1}^{r}\\). Define\n\\[\n\\Arf(q)\n=\n\\sum_{i=1}^{r}q(e_i)q(f_i)\n\\in\\FF_2.\n\\]\n\\end{definition}","This is the classical classification of nonsingular quadratic forms over \\(\\FF_2\\) \\cite{MilnorHusemoller1973}.","\\begin{theorem}[Dyadic Gauss sign]\\label{thm:arf_gauss}\nLet \\(\\dim V=2r\\), and let \\(q\\) be nondegenerate. Then\n\\[\n\\boxed{\n\\sum_{x\\in V}(-1)^{q(x)}\n=\n(-1)^{\\Arf(q)}2^r.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nOrthogonally decompose \\((V,q)\\) into two dimensional nonsingular planes. On a plane with Arf invariant zero, the four values of \\((-1)^q\\) sum to \\(2\\). On a plane with Arf invariant one, they sum to \\(-2\\). The total sum is the product of the plane sums. The parity of the number of anisotropic planes is the Arf invariant.\n\\end{proof}","\\begin{corollary}[Orthogonal dyadic multiplicativity]\\label{cor:arf_multiplicative}\nFor an orthogonal sum\n\\[\n(V,q)=(V_1,q_1)\\perpbig(V_2,q_2),\n\\]\nwe have\n\\[\n\\Arf(q)=\\Arf(q_1)+\\Arf(q_2)\n\\]\nand the normalized Gauss sign is the product of the two normalized signs.\n\\end{corollary}","\\begin{proposition}[The form alone does not determine the phase]\\label{prop:dyadic_need_q}\nOn every nonzero symplectic \\(\\FF_2\\) space, there are quadratic refinements of both Arf invariants. Their polar forms agree, but their normalized Gauss signs are opposite.\n\\end{proposition}","\\begin{proposition}[Translation of quadratic refinements]\\label{prop:quadratic_translation}\nFor \\(v\\in V\\), put\n\\[\nq_v(x)=q(x)+b(v,x).\n\\]\nThen\n\\[\n\\boxed{\n\\Arf(q_v)=\\Arf(q)+q(v).\n}\n\\]\n\\end{proposition}","\\begin{proof}\nUsing the quadratic identity,\n\\[\nq_v(x)=q(x+v)+q(v).\n\\]\nTherefore\n\\[\n\\sum_x(-1)^{q_v(x)}\n=\n(-1)^{q(v)}\n\\sum_x(-1)^{q(x)}.\n\\]\nCompare the signs in \\cref{thm:arf_gauss}.\n\\end{proof}","\\begin{theorem}[Number of dyadic refinements]\\label{thm:quadratic_counts}\nLet \\(\\dim V=2r\\). The number of quadratic refinements with Arf invariant zero is\n\\[\n\\boxed{\n2^{r-1}(2^r+1),\n}\n\\]\nand the number with Arf invariant one is\n\\[\n\\boxed{\n2^{r-1}(2^r-1).\n}\n\\]\n\\end{theorem}","\\begin{proof}\nEvery refinement is \\(q_v\\) for a unique \\(v\\in V\\), because \\(b\\) identifies \\(V\\) with \\(V^*\\). By \\cref{prop:quadratic_translation}, the Arf invariant changes according to \\(q(v)\\). Let \\(Z\\) and \\(O\\) be the numbers of zeros and ones of \\(q\\). Then\n\\[\nZ+O=2^{2r}\n\\]\nand \\cref{thm:arf_gauss} gives\n\\[\nZ-O=(-1)^{\\Arf(q)}2^r.\n\\]\nSolving gives the two displayed numbers, with their roles exchanged when the starting refinement has Arf invariant one. Since translation runs through every refinement, the final counts are independent of the starting choice.\n\\end{proof}","Let \\(\\Gamma\\) be a connected finite graph. At every vertex \\(v\\), place finite dimensional complex vector spaces \\(V_v\\) and \\(W_v\\) of the same dimension. For every oriented edge \\(e:u\\to v\\), let\n\\[\nA_e:V_u\\longrightarrow V_v,\n\\qquad\nB_e:W_u\\longrightarrow W_v\n\\]\nbe invertible. The reversed edge carries the inverse map.","\\begin{definition}[Path transport]\\label{def:path_transport}\nFor an oriented path\n\\[\n\\gamma=(v_0\\to v_1\\to\\cdots\\to v_m),\n\\]\ndefine\n\\[\nA_\\gamma=A_{v_m v_{m-1}}\\cdots A_{v_1v_0},\n\\qquad\nB_\\gamma=B_{v_m v_{m-1}}\\cdots B_{v_1v_0}.\n\\]\nFor a loop based at \\(r\\), these are the based holonomies.\n\\end{definition}","\\begin{theorem}[Matrix holonomy criterion]\\label{thm:matrix_holonomy}\nFix a root \\(r\\) and an isomorphism\n\\[\nU_r:V_r\\longrightarrow W_r.\n\\]\nThe following conditions are equivalent.","\\begin{enumerate}\n\\item There are isomorphisms \\(U_v:V_v\\to W_v\\), with the declared \\(U_r\\), such that\n\\[\nU_vA_e=B_eU_u\n\\]\nfor every edge \\(e:u\\to v\\).\n\\item For one fundamental cycle basis based at \\(r\\),\n\\[\nU_rA_\\gamma=B_\\gamma U_r.\n\\]\n\\item The same identity holds for every based loop \\(\\gamma\\).\n\\end{enumerate}\n\\end{theorem}","Conversely, choose a spanning tree. Starting from \\(U_r\\), define \\(U_v\\) by transport along the unique tree path from \\(r\\) to \\(v\\). The edge equations hold on every tree edge by construction. Every non tree edge closes one fundamental cycle. The edge equation on that edge is equivalent to the root holonomy identity for its fundamental cycle. Condition two therefore gives all remaining edge equations.\n\\end{proof}","\\begin{corollary}[Unbased gauge classification]\\label{cor:unbased_holonomy}\nThere exists some family of vertex isomorphisms if and only if the two holonomy representations\n\\[\n\\pi_1(\\Gamma,r)\\longrightarrow\\GL(V_r),\n\\qquad\n\\pi_1(\\Gamma,r)\\longrightarrow\\GL(W_r)\n\\]\nare conjugate.\n\\end{corollary}","\\begin{proof}\nApply \\cref{thm:matrix_holonomy} and allow \\(U_r\\) to vary.\n\\end{proof}","\\begin{corollary}[Scalar and projective gluing]\\label{cor:scalar_holonomy}\nIf every based loop acts by a scalar, finite gluing is equivalent to equality of the scalar holonomies on a fundamental cycle basis. The number of independent tests is\n\\[\n\\boxed{\n\\abs E-\\abs V+1.\n}\n\\]\n\\end{corollary}","\\begin{proof}\nA connected graph has a spanning tree with \\(\\abs V-1\\) edges. Every remaining edge gives one fundamental cycle. In the scalar case, conjugacy has no effect.\n\\end{proof}","\\begin{proposition}[Exact edge residual from cycle defect]\\label{prop:holonomy_residual}\nFix a spanning tree and construct the vertex maps by tree transport from \\(U_r\\). Let \\(e:u\\to v\\) be a non tree edge. Let \\(p_u\\) and \\(p_v\\) be the tree paths from the root to \\(u\\) and \\(v\\). Define the fundamental cycle defect\n\\[\nD_e\n=\nU_rA_{p_v}^{-1}A_eA_{p_u}\n-\nB_{p_v}^{-1}B_eB_{p_u}U_r.\n\\]\nThen the edge residual is exactly\n\\[\n\\boxed{\nU_vA_e-B_eU_u\n=\nB_{p_v}D_eA_{p_u}^{-1}.\n}\n\\]\n\\end{proposition}","ct edge residual from cycle defect]\\label{prop:holonomy_residual}\nFix a spanning tree and construct the vertex maps by tree transport from \\(U_r\\). Let \\(e:u\\to v\\) be a non tree edge. Let \\(p_u\\) and \\(p_v\\) be the tree paths from the root to \\(u\\) and \\(v\\). Define the fundamental cycle defect\n\\[\nD_e\n=\nU_rA_{p_v}^{-1}A_eA_{p_u}\n-\nB_{p_v}^{-1}B_eB_{p_u}U_r.\n\\]\nThen the edge residual is exactly\n\\[\n\\boxed{\nU_vA_e-B_eU_u\n=\nB_{p_v}D_eA_{p_u}^{-1}.\n}\n\\]\n\\end{proposition}","\\begin{proof}\nSubstitute\n\\[\nU_u=B_{p_u}U_rA_{p_u}^{-1},\n\\qquad\nU_v=B_{p_v}U_rA_{p_v}^{-1}\n\\]\ninto the left side and factor \\(B_{p_v}\\) on the left and \\(A_{p_u}^{-1}\\) on the right.\n\\end{proof}","\\begin{corollary}[Quantitative finite gluing]\\label{cor:quantitative_holonomy}\nWith the notation of \\cref{prop:holonomy_residual},\n\\[\n\\norm{U_vA_e-B_eU_u}\n\\leq\n\\norm{B_{p_v}}\\norm{D_e}\\norm{A_{p_u}^{-1}}.\n\\]\nThus small fundamental cycle defects give small non tree edge residuals whenever the tree transports are uniformly conditioned.\n\\end{corollary}","\\begin{definition}[Orthogonal break decomposition]\\label{def:break_decomposition}\nAn orthogonal break decomposition is a symplectic direct sum\n\\[\nV=\\perpbig_{s\\in S}V_s\n\\]\nthat is preserved by the declared tame stabilizer. Each \\(V_s\\) is the finite symplectic packet assigned to one break.\n\\end{definition}","\\begin{proposition}[Heisenberg tensor factorization]\\label{prop:heisenberg_tensor}\nFix compatible central characters. The irreducible Heisenberg representation of an orthogonal sum is the tensor product of the irreducible Heisenberg representations of its summands:\n\\[\n\\boxed{\n\\omega_V\n\\cong\n\\bigotimes_{s\\in S}\\omega_{V_s}.\n}\n\\]\n\\end{proposition}","\\begin{proof}\nThe Heisenberg group of an orthogonal sum is the central product of the Heisenberg groups of the summands. The tensor product representation has the declared central character and has dimension\n\\[\n\\prod_s\\sqrt{\\abs{V_s}}\n=\n\\sqrt{\\abs V}.\n\\]\nIt is irreducible by the finite Stone von Neumann theorem, so it is the required representation.\n\\end{proof}","\\begin{corollary}[Breakwise phase product]\\label{cor:break_phase}\nIn odd residue characteristic, let \\(g\\in\\Sp(V)\\) preserve every break packet and suppose its restriction to \\(M=\\im(g-I)\\) is fixed point free. Put \\(g_s=g|_{V_s}\\) and \\(M_s=\\im(g_s-I)\\). The moving restrictions are symplectic, and\n\\[\n\\boxed{\n\\Phi_p(g|_M,M)\n=\n\\prod_{s\\in S}\\Phi_p(g_s|_{M_s},M_s).\n}\n\\]\nIn residue characteristic two, the analogous normalized Gauss sign is the product of the signs determined by the breakwise quadratic refinements.\n\\end{corollary}","\\begin{proof}\nThe invariant orthogonal decomposition gives \\(M=\\mathop{\\perp}\\limits_{s\\in S}M_s\\). Symplectic invariance gives \\(M^\\perp=\\ker(g-I)\\). Fixed point freeness on \\(M\\) therefore makes the restricted form nondegenerate on \\(M\\) and on each \\(M_s\\). Each restriction of \\(g_s-I\\) to \\(M_s\\) is invertible, so \\cref{prop:odd_multiplicativity} applies to the moving spaces. A zero moving space contributes one. In characteristic two, apply \\cref{cor:arf_multiplicative}.\n\\end{proof}","Local class field theory identifies one dimensional Weil characters with characters of \\(K^{\\times}\\) \\cite{Serre1967,Tate1979}. The tame induction theorem extends this explicit picture to higher dimensional tame atoms. The local polynomial and conductor calculations in \\cref{prop:tame_local_polynomial,prop:tame_conductor} therefore give explicit finite invariants for the declared tame packets. Finite Gauss sums give the residue field phase. The full local factor, however, also includes ramification depth, Haar measure normalization, the additive character conductor, and wild interaction data.","\\begin{proof}\nThe dimension and orbit follow from \\cref{thm:tame_induction}. The cycle polynomial is \\cref{prop:frobenius_cycle}. For rank greater than one, the conductor is \\cref{prop:tame_conductor}. In rank one, the Swan conductor is zero, and the Artin conductor is zero for trivial inertia and one otherwise, by the formula \\(\\operatorname{Art}=\\dim V-\\dim V^{I_K}+\\operatorname{Sw}\\). The local polynomial is \\cref{prop:tame_local_polynomial}. Tensor refinement is \\cref{thm:gcd_lcm}. The Gauss sum is its defining finite sum; for a nontrivial multiplicative character its unit phase is \\cref{def:gauss_phase}. For the trivial character and a nontrivial additive character \\(\\psi\\), the sum is \\(G(1,\\psi)=-1\\), because the sum of \\(\\psi\\) over the whole field is zero.\n\\end{proof}","ivial inertia and one otherwise, by the formula \\(\\operatorname{Art}=\\dim V-\\dim V^{I_K}+\\operatorname{Sw}\\). The local polynomial is \\cref{prop:tame_local_polynomial}. Tensor refinement is \\cref{thm:gcd_lcm}. The Gauss sum is its defining finite sum; for a nontrivial multiplicative character its unit phase is \\cref{def:gauss_phase}. For the trivial character and a nontrivial additive character \\(\\psi\\), the sum is \\(G(1,\\psi)=-1\\), because the sum of \\(\\psi\\) over the whole field is zero.\n\\end{proof}","otherwise, by the formula \\(\\operatorname{Art}=\\dim V-\\dim V^{I_K}+\\operatorname{Sw}\\). The local polynomial is \\cref{prop:tame_local_polynomial}. Tensor refinement is \\cref{thm:gcd_lcm}. The Gauss sum is its defining finite sum; for a nontrivial multiplicative character its unit phase is \\cref{def:gauss_phase}. For the trivial character and a nontrivial additive character \\(\\psi\\), the sum is \\(G(1,\\psi)=-1\\), because the sum of \\(\\psi\\) over the whole field is zero.\n\\end{proof}","rname{Art}=\\dim V-\\dim V^{I_K}+\\operatorname{Sw}\\). The local polynomial is \\cref{prop:tame_local_polynomial}. Tensor refinement is \\cref{thm:gcd_lcm}. The Gauss sum is its defining finite sum; for a nontrivial multiplicative character its unit phase is \\cref{def:gauss_phase}. For the trivial character and a nontrivial additive character \\(\\psi\\), the sum is \\(G(1,\\psi)=-1\\), because the sum of \\(\\psi\\) over the whole field is zero.\n\\end{proof}","Fix the paper's reciprocity, additive-character, local-factor, and phase\nnormalizations. Let \\(\\mathcal A\\) be the declared class of irreducible wild\nWeil atoms under consideration. For \\(\\rho\\in\\mathcal A\\), the supplied\nassembled finite datum \\(d_\\rho\\) consists of the following inputs,\nhistorically denoted H1--H4:\n\\begin{enumerate}\n\\item an intrinsic finite break decomposition on the Galois side;\n\\item the declared equivariant symplectic bridges to the supplied breakwise\ntype-side quotients;\n\\item compatible odd or dyadic central-extension and phase data on every\nbreak, including every prescribed finite marking;\n\\item a compatible transport system with vanishing matrix holonomy\nobstruction.\n\\end{enumerate}\nThese are finite comparison inputs. Supplying separate breakwise target\ntypes does not","Fix the paper's reciprocity, additive-character, local-factor, and phase\nnormalizations. Let \\(\\mathcal A\\) be the declared class of irreducible wild\nWeil atoms under consideration. For \\(\\rho\\in\\mathcal A\\), the supplied\nassembled finite datum \\(d_\\rho\\) consists of the following inputs,\nhistorically denoted H1--H4:\n\\begin{enumerate}\n\\item an intrinsic finite break decomposition on the Galois side;\n\\item the declared equivariant symplectic bridges to the supplied breakwise\ntype-side quotients;\n\\item compatible odd or dyadic central-extension and phase data on every\nbreak, including every prescribed finite marking;\n\\item a compatible transport system with vanishing matrix holonomy\nobstruction.\n\\end{enumerate}\nThese are finite comparison inputs. Supplying separate breakwise target\ntypes does not itself supply one common arithmetic candidate. If a common\ncandidate is already supplied, that is a","Fix the paper's reciprocity, additive-character, local-factor, and phase\nnormalizations. Let \\(\\mathcal A\\) be the declared class of irreducible wild\nWeil atoms under consideration. For \\(\\rho\\in\\mathcal A\\), the supplied\nassembled finite datum \\(d_\\rho\\) consists of the following inputs,\nhistorically denoted H1--H4:\n\\begin{enumerate}\n\\item an intrinsic finite break decomposition on the Galois side;\n\\item the declared equivariant symplectic bridges to the supplied breakwise\ntype-side quotients;\n\\item compatible odd or dyadic central-extension and phase data on every\nbreak, including every prescribed finite marking;\n\\item a compatible transport system with vanishing matrix holonomy\nobstruction.\n\\end{enumerate}\nThese are finite comparison inputs. Supplying separate breakwise target\ntypes does not itself supply one common arithmetic candidate. If a common\ncandidate is already supplied, that is a stronger restricted hypothesis.","An \\emph{arithmetic realization witness}\n\\(\\mathcal R(d_\\rho,\\pi)\\) consists of actual arithmetic inducing data\nfor an admissible supercuspidal \\(\\pi\\), together with explicit\ncomparison and recovery maps carrying the prescribed assembled finite\npacket. The maps preserve its declared stabilizer action, all character values\nexplicitly included in the supplied datum, central-extension and phase\ndata, and compatible transports.\nRecovery returns that same \\(\\pi\\), with its fixed central character and\nnoncompact completion. An enlarged actual inducing carrier is permitted;\nthe whole packet is not required to be the original maximal-simple-type\nquotient. Any stronger native-marking or numerical-layer requirement is\npart of the declared datum and must be verifie","An \\emph{arithmetic realization witness}\n\\(\\mathcal R(d_\\rho,\\pi)\\) consists of actual arithmetic inducing data\nfor an admissible supercuspidal \\(\\pi\\), together with explicit\ncomparison and recovery maps carrying the prescribed assembled finite\npacket. The maps preserve its declared stabilizer action, all character values\nexplicitly included in the supplied datum, central-extension and phase\ndata, and compatible transports.\nRecovery returns that same \\(\\pi\\), with its fixed central character and\nnoncompact completion. An enlarged actual inducing carrier is permitted;\nthe whole packet is not required to be the original maximal-simple-type\nquotient. Any stronger native-marking or numerical-layer requirement is\npart of the declared datum and must be verified separately.","ho,\\pi)\\) consists of actual arithmetic inducing data\nfor an admissible supercuspidal \\(\\pi\\), together with explicit\ncomparison and recovery maps carrying the prescribed assembled finite\npacket. The maps preserve its declared stabilizer action, all character values\nexplicitly included in the supplied datum, central-extension and phase\ndata, and compatible transports.\nRecovery returns that same \\(\\pi\\), with its fixed central character and\nnoncompact completion. An enlarged actual inducing carrier is permitted;\nthe whole packet is not required to be the original maximal-simple-type\nquotient. Any stronger native-marking or numerical-layer requirement is\npart of the declared datum and must be verified separately.","\\begingroup\n\\renewcommand{\\theconjecture}{66}\n\\begin{conjecture}[Arithmetic packet realization]\n\\label{conj:arithmetic_realization}\\label{thm:conditional_atomic}\nFor every \\(\\rho\\in\\mathcal A\\) with its supplied H1--H4 assembled\nfinite datum \\(d_\\rho\\), there exists an actual admissible supercuspidal\nrepresentation \\(\\pi\\) of \\(\\GL_{\\dim\\rho}(K)\\) and an arithmetic\nrealization witness \\(\\mathcal R(d_\\rho,\\pi)\\).\n\\end{conjecture}\n\\endgroup","\\begin{proposition}[Comparison of one realized candidate]\n\\label{prop:realized_candidate_comparison}\nLet \\(\\rho\\in\\mathcal A\\), and let \\(\\pi\\) be one actual admissible\nsupercuspidal representation of \\(\\GL_{\\dim\\rho}(K)\\) with\n\\(\\mathcal R(d_\\rho,\\pi)\\). Suppose H5 holds for this same \\(\\pi\\):\nits required pair local factors equal those prescribed for \\(\\rho\\)\non the stipulated characterizing family of twists, with the fixed\nnormalizations. Assume the stated local-factor characterization applies\nto that family and class. Then \\(\\pi\\) is isomorphic to the representation\nassigned to \\(\\rho\\) by the established local Langlands correspondence.\n\\end{proposition}","\\(\\pi\\) be one actual admissible\nsupercuspidal representation of \\(\\GL_{\\dim\\rho}(K)\\) with\n\\(\\mathcal R(d_\\rho,\\pi)\\). Suppose H5 holds for this same \\(\\pi\\):\nits required pair local factors equal those prescribed for \\(\\rho\\)\non the stipulated characterizing family of twists, with the fixed\nnormalizations. Assume the stated local-factor characterization applies\nto that family and class. Then \\(\\pi\\) is isomorphic to the representation\nassigned to \\(\\rho\\) by the established local Langlands correspondence.\n\\end{proposition}","tation of \\(\\GL_{\\dim\\rho}(K)\\) with\n\\(\\mathcal R(d_\\rho,\\pi)\\). Suppose H5 holds for this same \\(\\pi\\):\nits required pair local factors equal those prescribed for \\(\\rho\\)\non the stipulated characterizing family of twists, with the fixed\nnormalizations. Assume the stated local-factor characterization applies\nto that family and class. Then \\(\\pi\\) is isomorphic to the representation\nassigned to \\(\\rho\\) by the established local Langlands correspondence.\n\\end{proposition}","\\begin{enumerate}\n\\item Construct a choice free finite symplectic envelope from an arbitrary irreducible wild inertia atom, with its ramification filtration and tame stabilizer action.\n\\item Prove a canonical equivariant symplectic isometry from that envelope to the type side quotient \\(J_\\theta^1/H_\\theta^1\\).\n\\item Recover the canonical beta extension phase in every break, including the required dyadic quadratic refinement.\n\\item Prove matrix holonomy compatibility for the resulting packet transport system.\n\\item Compare the full family of pair \\(L\\), epsilon, and gamma factors without invoking the established correspondence.\n\\end{enumerate}","n and tame stabilizer action.\n\\item Prove a canonical equivariant symplectic isometry from that envelope to the type side quotient \\(J_\\theta^1/H_\\theta^1\\).\n\\item Recover the canonical beta extension phase in every break, including the required dyadic quadratic refinement.\n\\item Prove matrix holonomy compatibility for the resulting packet transport system.\n\\item Compare the full family of pair \\(L\\), epsilon, and gamma factors without invoking the established correspondence.\n\\end{enumerate}","\\begin{example}[A degree four tame orbit]\\label{ex:tame_four_again}\nOver \\(\\FF_2\\), the exponent orbit\n\\[\n1\\longmapsto2\\longmapsto4\\longmapsto8\\longmapsto1\n\\]\nhas length four. It therefore gives a tame atom induced from the unramified extension of degree four. The Frobenius transfer has one four cycle, and the number of regular degree four orbits is three by \\cref{prop:mobius_count}.\n\\end{example}","\\begin{example}[Tensor packets of lengths four and six]\\label{ex:tensor_46}\nFor \\(n=4\\) and \\(m=6\\),\n\\[\n\\gcd(4,6)=2,\n\\qquad\n\\lcm(4,6)=12.\n\\]\nThe tensor product splits into two interaction packets of depth twelve. Their total dimension is \\(2\\cdot12=24\\), equal to the product \\(4\\cdot6\\).\n\\end{example}","\\begin{example}[Rank one Heisenberg group]\\label{ex:heisenberg_rank_one}\nTake \\(V=\\FF_3^2\\). The Heisenberg group has order \\(3^3=27\\), and its irreducible representation with a fixed nontrivial central character has dimension three. The trace vanishes on every noncentral element. Averaging the translation subgroup gives a rank one projector.\n\\end{example}","\\begin{example}[Nondegenerate twisted algebra]\\label{ex:twisted_plane}\nLet \\(A=\\FF_p^2\\) with its standard alternating form. A cocycle with that commutator form gives\n\\[\n\\CC^c[A]\\cong M_p(\\CC).\n\\]\nThe unique irreducible projective representation has dimension \\(p\\).\n\\end{example}","\\begin{example}[Dyadic phase ambiguity]\\label{ex:dyadic_plane}\nOn a symplectic plane over \\(\\FF_2\\), choose refinements \\(q_0\\) and \\(q_1\\) with Arf invariants zero and one. Their polar forms are the same, while\n\\[\n\\sum_x(-1)^{q_0(x)}=2,\n\\qquad\n\\sum_x(-1)^{q_1(x)}=-2.\n\\]\nThus the alternating packet alone cannot select the dyadic sign.\n\\end{example}","The later constructions provide actual arithmetic candidates and exact\nfinite packet recovery in substantial restricted families. They do not\nestablish \\cref{conj:arithmetic_realization} for every admitted source.\nTheir detailed ordinary proofs are typeset in the appendices, and the\noriginal reviewed Markdown files accompany this edition. In these\nappendices \\(F\\) denotes the base local field, while \\(K=F[\\beta]\\)\ndenotes a parameter field; these are local notation conventions.\nAll references to ``claim 66'' in the retained notes mean the general\nendpoint now isolated as Conjecture 66.","ater constructions provide actual arithmetic candidates and exact\nfinite packet recovery in substantial restricted families. They do not\nestablish \\cref{conj:arithmetic_realization} for every admitted source.\nTheir detailed ordinary proofs are typeset in the appendices, and the\noriginal reviewed Markdown files accompany this edition. In these\nappendices \\(F\\) denotes the base local field, while \\(K=F[\\beta]\\)\ndenotes a parameter field; these are local notation conventions.\nAll references to ``claim 66'' in the retained notes mean the general\nendpoint now isolated as Conjecture 66.","\\begin{theorem}[Restricted odd-prime arithmetic realization]\n\\label{thm:odd_arithmetic}\nLet \\(p\\) be an odd prime, \\(F=\\FF_p((t))\\), fix \\(d\\ge1\\), and let\n\\[\n0<h_1<\\cdots<h_d,\\qquad p\\nmid h_i,\n\\qquad q_i=(p+1)h_i,\\qquad N=p^d.\n\\]\nThere is an explicit maximal simple stratum in \\(M_N(F)\\) with totally\nramified full-degree parameter field, a literal defining sequence of\ndegrees \\(p,p^2,\\ldots,p^d\\), and levels\n\\(n_j=p^{d-j}q_{d-j+1}\\). Every native parameter and consecutive\ncorrection has trace zero over its recursive base field.\nThe stratum admits a full simple character trivial on \\(U_F^1\\).\nIts fixed canonical beta extension can be completed to an actual\nirreducible admissible supercuspidal \\(\\pi\\) with any prescribed smooth\ncentral quasicharacter trivial on \\(U_F^1\\), when its comp","\\begin{theorem}[Restricted odd-prime arithmetic realization]\n\\label{thm:odd_arithmetic}\nLet \\(p\\) be an odd prime, \\(F=\\FF_p((t))\\), fix \\(d\\ge1\\), and let\n\\[\n0<h_1<\\cdots<h_d,\\qquad p\\nmid h_i,\n\\qquad q_i=(p+1)h_i,\\qquad N=p^d.\n\\]\nThere is an explicit maximal simple stratum in \\(M_N(F)\\) with totally\nramified full-degree parameter field, a literal defining sequence of\ndegrees \\(p,p^2,\\ldots,p^d\\), and levels\n\\(n_j=p^{d-j}q_{d-j+1}\\). Every native parameter and consecutive\ncorrection has trace zero over its recursive base field.\nThe stratum admits a full simple character trivial on \\(U_F^1\\).\nIts fixed canonical beta extension can be completed to an actual\nirreducible admissible supercuspidal \\(\\pi\\) with any prescribed smooth\ncentral quasicharacter trivial on \\(U_F^1\\), when its completion choices\nare free or already","\\begin{theorem}[Restricted odd-prime arithmetic realization]\n\\label{thm:odd_arithmetic}\nLet \\(p\\) be an odd prime, \\(F=\\FF_p((t))\\), fix \\(d\\ge1\\), and let\n\\[\n0<h_1<\\cdots<h_d,\\qquad p\\nmid h_i,\n\\qquad q_i=(p+1)h_i,\\qquad N=p^d.\n\\]\nThere is an explicit maximal simple stratum in \\(M_N(F)\\) with totally\nramified full-degree parameter field, a literal defining sequence of\ndegrees \\(p,p^2,\\ldots,p^d\\), and levels\n\\(n_j=p^{d-j}q_{d-j+1}\\). Every native parameter and consecutive\ncorrection has trace zero over its recursive base field.\nThe stratum admits a full simple character trivial on \\(U_F^1\\).\nIts fixed canonical beta extension can be completed to an actual\nirreducible admissible supercuspidal \\(\\pi\\) with any prescribed smooth\ncentral quasicharacter trivial on \\(U_F^1\\), when its completion choices\nare free or already compatible.","[Restricted odd-prime arithmetic realization]\n\\label{thm:odd_arithmetic}\nLet \\(p\\) be an odd prime, \\(F=\\FF_p((t))\\), fix \\(d\\ge1\\), and let\n\\[\n0<h_1<\\cdots<h_d,\\qquad p\\nmid h_i,\n\\qquad q_i=(p+1)h_i,\\qquad N=p^d.\n\\]\nThere is an explicit maximal simple stratum in \\(M_N(F)\\) with totally\nramified full-degree parameter field, a literal defining sequence of\ndegrees \\(p,p^2,\\ldots,p^d\\), and levels\n\\(n_j=p^{d-j}q_{d-j+1}\\). Every native parameter and consecutive\ncorrection has trace zero over its recursive base field.\nThe stratum admits a full simple character trivial on \\(U_F^1\\).\nIts fixed canonical beta extension can be completed to an actual\nirreducible admissible supercuspidal \\(\\pi\\) with any prescribed smooth\ncentral quasicharacter trivial on \\(U_F^1\\), when its completion choices\nare free or already compatible.","_i,\n\\qquad q_i=(p+1)h_i,\\qquad N=p^d.\n\\]\nThere is an explicit maximal simple stratum in \\(M_N(F)\\) with totally\nramified full-degree parameter field, a literal defining sequence of\ndegrees \\(p,p^2,\\ldots,p^d\\), and levels\n\\(n_j=p^{d-j}q_{d-j+1}\\). Every native parameter and consecutive\ncorrection has trace zero over its recursive base field.\nThe stratum admits a full simple character trivial on \\(U_F^1\\).\nIts fixed canonical beta extension can be completed to an actual\nirreducible admissible supercuspidal \\(\\pi\\) with any prescribed smooth\ncentral quasicharacter trivial on \\(U_F^1\\), when its completion choices\nare free or already compatible.","amified full-degree parameter field, a literal defining sequence of\ndegrees \\(p,p^2,\\ldots,p^d\\), and levels\n\\(n_j=p^{d-j}q_{d-j+1}\\). Every native parameter and consecutive\ncorrection has trace zero over its recursive base field.\nThe stratum admits a full simple character trivial on \\(U_F^1\\).\nIts fixed canonical beta extension can be completed to an actual\nirreducible admissible supercuspidal \\(\\pi\\) with any prescribed smooth\ncentral quasicharacter trivial on \\(U_F^1\\), when its completion choices\nare free or already compatible.","egrees \\(p,p^2,\\ldots,p^d\\), and levels\n\\(n_j=p^{d-j}q_{d-j+1}\\). Every native parameter and consecutive\ncorrection has trace zero over its recursive base field.\nThe stratum admits a full simple character trivial on \\(U_F^1\\).\nIts fixed canonical beta extension can be completed to an actual\nirreducible admissible supercuspidal \\(\\pi\\) with any prescribed smooth\ncentral quasicharacter trivial on \\(U_F^1\\), when its completion choices\nare free or already compatible.","Assume actually supplied labelled full-character extensions with whole\ntwo-dimensional symplectic quotients whose balanced complex-scalar\ncentral product has \\(d\\) planes, one vertex, and trivial stabilizer.\nThen the same \\(\\pi\\) admits a full scalar-pushout realization of that\npacket and exact recovery of its unchanged inducing representation.\nThe prescribed central character must lie in the covered tame class.\nNo literal embedding of the old matrix groups, canonical native marking,\nor prescribed numerical host-depth placement is inferred.\n\\end{theorem}","\\begin{proof}\nThe explicit Artin--Schreier recursion and its normalized filtered lift\nconstruct the maximal parameter and literal defining sequence; see\n\\cref{app:odd_tower}. Its trace transport identity makes every native\ntrace zero, rather than relying on amplification by \\(p\\). Native-rank\nfull-character construction, full beta-determinant transfer, trace-zero\ncorrection, and maximal-field character extension give the scalar\ntriviality and central completion; see \\cref{app:odd_character}.\nThe actual leading symplectic layer has dimension\n\\((p-1)p^{d-1}\\ge2d\\) and survives every later approximation\n(\\cref{app:odd_capacity}). A labelled symplectic embedding gives a\ncenter-fixing full scalar-pushout isomorphism. Finite Stone--von Neumann\ndecomposition separates the source factor from its orthogonal\ncomplement. Maximal-isotropic averaging on the complementary Heisenberg\nfactor selects a one-dimensional representation line and hence a\nsource-sized sector. The retain","see\n\\cref{app:odd_tower}. Its trace transport identity makes every native\ntrace zero, rather than relying on amplification by \\(p\\). Native-rank\nfull-character construction, full beta-determinant transfer, trace-zero\ncorrection, and maximal-field character extension give the scalar\ntriviality and central completion; see \\cref{app:odd_character}.\nThe actual leading symplectic layer has dimension\n\\((p-1)p^{d-1}\\ge2d\\) and survives every later approximation\n(\\cref{app:odd_capacity}). A labelled symplectic embedding gives a\ncenter-fixing full scalar-pushout isomorphism. Finite Stone--von Neumann\ndecomposition separates the source factor from its orthogonal\ncomplement. Maximal-isotropic averaging on the complementary Heisenberg\nfactor selects a one-dimensional representation line and hence a\nsource-sized sector. The retained induction/idempotent recovery returns the same\n\\(\\Lambda\\) and \\(\\pi\\), as proved in \\cref{app:odd_synthesis}.\n\\end{proof}","y}). A labelled symplectic embedding gives a\ncenter-fixing full scalar-pushout isomorphism. Finite Stone--von Neumann\ndecomposition separates the source factor from its orthogonal\ncomplement. Maximal-isotropic averaging on the complementary Heisenberg\nfactor selects a one-dimensional representation line and hence a\nsource-sized sector. The retained induction/idempotent recovery returns the same\n\\(\\Lambda\\) and \\(\\pi\\), as proved in \\cref{app:odd_synthesis}.\n\\end{proof}","symplectic embedding gives a\ncenter-fixing full scalar-pushout isomorphism. Finite Stone--von Neumann\ndecomposition separates the source factor from its orthogonal\ncomplement. Maximal-isotropic averaging on the complementary Heisenberg\nfactor selects a one-dimensional representation line and hence a\nsource-sized sector. The retained induction/idempotent recovery returns the same\n\\(\\Lambda\\) and \\(\\pi\\), as proved in \\cref{app:odd_synthesis}.\n\\end{proof}","For \\(p>3\\), a native minimal rank-\\(p\\) type quotient has dimension\n\\(p-1\\), so it is not automatically a plane. The theorem requires\ngenuinely supplied whole-plane data. Independence of source tensor\nfactors is also an input. An arbitrary single-factor wild determinant\nneed not be tame.","\\begin{theorem}[Restricted dyadic marked assembly realization]\n\\label{thm:dyadic_arithmetic}\nLet \\(F=\\FF_2((t))\\), fix an additive character of conductor\n\\(t\\OO_F\\), fix \\(d\\ge1\\), and let \\(h_1<\\cdots<h_d\\) be positive odd integers.\nPut \\(q_i=3h_i\\), \\(N=2^d\\), and\n\\(n_j=2^{d-j}q_{d-j+1}\\). The explicit recursive matrix generates a\nseparable totally ramified degree-\\(N\\) maximal parameter field,\nnormalizes the complete lattice chain, and has the literal defining\nsequence of degrees \\(2,4,\\ldots,2^d\\). It admits a full simple character\ntrivial on \\(U_F^1\\) and an actual supercuspidal completion with any\nprescribed central quasicharacter trivial on \\(U_F^1\\), when the\nnoncompact completion is free or already compatible.","c}\nLet \\(F=\\FF_2((t))\\), fix an additive character of conductor\n\\(t\\OO_F\\), fix \\(d\\ge1\\), and let \\(h_1<\\cdots<h_d\\) be positive odd integers.\nPut \\(q_i=3h_i\\), \\(N=2^d\\), and\n\\(n_j=2^{d-j}q_{d-j+1}\\). The explicit recursive matrix generates a\nseparable totally ramified degree-\\(N\\) maximal parameter field,\nnormalizes the complete lattice chain, and has the literal defining\nsequence of degrees \\(2,4,\\ldots,2^d\\). It admits a full simple character\ntrivial on \\(U_F^1\\) and an actual supercuspidal completion with any\nprescribed central quasicharacter trivial on \\(U_F^1\\), when the\nnoncompact completion is free or already compatible.","(t\\OO_F\\), fix \\(d\\ge1\\), and let \\(h_1<\\cdots<h_d\\) be positive odd integers.\nPut \\(q_i=3h_i\\), \\(N=2^d\\), and\n\\(n_j=2^{d-j}q_{d-j+1}\\). The explicit recursive matrix generates a\nseparable totally ramified degree-\\(N\\) maximal parameter field,\nnormalizes the complete lattice chain, and has the literal defining\nsequence of degrees \\(2,4,\\ldots,2^d\\). It admits a full simple character\ntrivial on \\(U_F^1\\) and an actual supercuspidal completion with any\nprescribed central quasicharacter trivial on \\(U_F^1\\), when the\nnoncompact completion is free or already compatible.","d\\), and\n\\(n_j=2^{d-j}q_{d-j+1}\\). The explicit recursive matrix generates a\nseparable totally ramified degree-\\(N\\) maximal parameter field,\nnormalizes the complete lattice chain, and has the literal defining\nsequence of degrees \\(2,4,\\ldots,2^d\\). It admits a full simple character\ntrivial on \\(U_F^1\\) and an actual supercuspidal completion with any\nprescribed central quasicharacter trivial on \\(U_F^1\\), when the\nnoncompact completion is free or already compatible.","For actually supplied full marked whole-plane scalar-pushout data with\none vertex, trivial stabilizer, and prescribed source central character\ntrivial on \\(U_F^1\\), an arithmetic assembly carrier exists\nat every length and recovers the same candidate: inside its original\ntype for \\(d\\ge3\\), and in explicit enlarged inducing carriers for\n\\(d=1,2\\). Every supplied full character value and transported whole\nfinite subgroup, lift, square, and phase marking is retained at the\nrepresented marking level. Any central-character kernel already factored\nout remains recorded separately; scalar pushout does not recover its\ninvisible elements. This does\nnot identify the transported marking with an independently prescribed\ncanonical native marking.\n\\end{theorem}","\\begin{proof}\nThe Artin--Schreier filtered lift has native trace one. Consequently\neach native correction \\(\\delta\\) satisfies\n\\[\nv_F(\\operatorname{tr}\\delta)+s\\ge1 \\quad (s=1),\n\\]\nwhich makes its additive scalar correction trivial on \\(U_F^1\\).\nThe actual field and defining-sequence proof is\n\\cref{app:dyadic_tower}. Full beta-determinant transfer and character\nextension at each new maximal native field prove the full-character\nclaim, including the odd-critical endpoint\n\\(2(\\lfloor r/2\\rfloor+1)-r=1\\); see \\cref{app:dyadic_character}.\nThe exact original quotient has dimension \\(2^d-2\\), which accommodates\nthe required \\(2d\\) source dimensions precisely for \\(d\\ge3\\)\n(\\cref{app:dyadic_capacity}).","ently\neach native correction \\(\\delta\\) satisfies\n\\[\nv_F(\\operatorname{tr}\\delta)+s\\ge1 \\quad (s=1),\n\\]\nwhich makes its additive scalar correction trivial on \\(U_F^1\\).\nThe actual field and defining-sequence proof is\n\\cref{app:dyadic_tower}. Full beta-determinant transfer and character\nextension at each new maximal native field prove the full-character\nclaim, including the odd-critical endpoint\n\\(2(\\lfloor r/2\\rfloor+1)-r=1\\); see \\cref{app:dyadic_character}.\nThe exact original quotient has dimension \\(2^d-2\\), which accommodates\nthe required \\(2d\\) source dimensions precisely for \\(d\\ge3\\)\n(\\cref{app:dyadic_capacity}).","h makes its additive scalar correction trivial on \\(U_F^1\\).\nThe actual field and defining-sequence proof is\n\\cref{app:dyadic_tower}. Full beta-determinant transfer and character\nextension at each new maximal native field prove the full-character\nclaim, including the odd-critical endpoint\n\\(2(\\lfloor r/2\\rfloor+1)-r=1\\); see \\cref{app:dyadic_character}.\nThe exact original quotient has dimension \\(2^d-2\\), which accommodates\nthe required \\(2d\\) source dimensions precisely for \\(d\\ge3\\)\n(\\cref{app:dyadic_capacity}).","ial on \\(U_F^1\\).\nThe actual field and defining-sequence proof is\n\\cref{app:dyadic_tower}. Full beta-determinant transfer and character\nextension at each new maximal native field prove the full-character\nclaim, including the odd-critical endpoint\n\\(2(\\lfloor r/2\\rfloor+1)-r=1\\); see \\cref{app:dyadic_character}.\nThe exact original quotient has dimension \\(2^d-2\\), which accommodates\nthe required \\(2d\\) source dimensions precisely for \\(d\\ge3\\)\n(\\cref{app:dyadic_capacity}).","defining-sequence proof is\n\\cref{app:dyadic_tower}. Full beta-determinant transfer and character\nextension at each new maximal native field prove the full-character\nclaim, including the odd-critical endpoint\n\\(2(\\lfloor r/2\\rfloor+1)-r=1\\); see \\cref{app:dyadic_character}.\nThe exact original quotient has dimension \\(2^d-2\\), which accommodates\nthe required \\(2d\\) source dimensions precisely for \\(d\\ge3\\)\n(\\cref{app:dyadic_capacity}).","At rank two, with \\(q=2m+1\\), enlarge\n\\(J=U_K^1U_A^{m+1}\\) to \\(P=U_K^1U_A^m\\). The actual character\nratio \\(\\lambda=\\theta^g/\\theta\\) for \\(g\\notin J\\) is nontrivial\nand sign-valued. With \\(H_0=\\ker\\lambda\\), the quotient is\n\\(P/H_0\\simeq\\FF_2^2\\), and \\(\\Ind_J^P\\theta\\) is an\nirreducible two-dimensional Heisenberg carrier\n(\\cref{app:rank2}). At rank four, the actual outer quadratic\ncentralizer gives \\(P=J U_B^m\\), \\([P:J]=2\\), and\n\\(P/H_0\\simeq\\FF_2^4\\), with carrier \\(\\Ind_J^P\\eta\\) of\ndimension four (\\cref{app:rank4}). The finite scalar center can be\nlarger than \\(\\mu_2\\); no literal extraspecial quotient is claimed.","two, with \\(q=2m+1\\), enlarge\n\\(J=U_K^1U_A^{m+1}\\) to \\(P=U_K^1U_A^m\\). The actual character\nratio \\(\\lambda=\\theta^g/\\theta\\) for \\(g\\notin J\\) is nontrivial\nand sign-valued. With \\(H_0=\\ker\\lambda\\), the quotient is\n\\(P/H_0\\simeq\\FF_2^2\\), and \\(\\Ind_J^P\\theta\\) is an\nirreducible two-dimensional Heisenberg carrier\n(\\cref{app:rank2}). At rank four, the actual outer quadratic\ncentralizer gives \\(P=J U_B^m\\), \\([P:J]=2\\), and\n\\(P/H_0\\simeq\\FF_2^4\\), with carrier \\(\\Ind_J^P\\eta\\) of\ndimension four (\\cref{app:rank4}). The finite scalar center can be\nlarger than \\(\\mu_2\\); no literal extraspecial quotient is claimed.","2m+1\\), enlarge\n\\(J=U_K^1U_A^{m+1}\\) to \\(P=U_K^1U_A^m\\). The actual character\nratio \\(\\lambda=\\theta^g/\\theta\\) for \\(g\\notin J\\) is nontrivial\nand sign-valued. With \\(H_0=\\ker\\lambda\\), the quotient is\n\\(P/H_0\\simeq\\FF_2^2\\), and \\(\\Ind_J^P\\theta\\) is an\nirreducible two-dimensional Heisenberg carrier\n(\\cref{app:rank2}). At rank four, the actual outer quadratic\ncentralizer gives \\(P=J U_B^m\\), \\([P:J]=2\\), and\n\\(P/H_0\\simeq\\FF_2^4\\), with carrier \\(\\Ind_J^P\\eta\\) of\ndimension four (\\cref{app:rank4}). The finite scalar center can be\nlarger than \\(\\mu_2\\); no literal extraspecial quotient is claimed.","(J=U_K^1U_A^{m+1}\\) to \\(P=U_K^1U_A^m\\). The actual character\nratio \\(\\lambda=\\theta^g/\\theta\\) for \\(g\\notin J\\) is nontrivial\nand sign-valued. With \\(H_0=\\ker\\lambda\\), the quotient is\n\\(P/H_0\\simeq\\FF_2^2\\), and \\(\\Ind_J^P\\theta\\) is an\nirreducible two-dimensional Heisenberg carrier\n(\\cref{app:rank2}). At rank four, the actual outer quadratic\ncentralizer gives \\(P=J U_B^m\\), \\([P:J]=2\\), and\n\\(P/H_0\\simeq\\FF_2^4\\), with carrier \\(\\Ind_J^P\\eta\\) of\ndimension four (\\cref{app:rank4}). The finite scalar center can be\nlarger than \\(\\mu_2\\); no literal extraspecial quotient is claimed.","\\). The actual character\nratio \\(\\lambda=\\theta^g/\\theta\\) for \\(g\\notin J\\) is nontrivial\nand sign-valued. With \\(H_0=\\ker\\lambda\\), the quotient is\n\\(P/H_0\\simeq\\FF_2^2\\), and \\(\\Ind_J^P\\theta\\) is an\nirreducible two-dimensional Heisenberg carrier\n(\\cref{app:rank2}). At rank four, the actual outer quadratic\ncentralizer gives \\(P=J U_B^m\\), \\([P:J]=2\\), and\n\\(P/H_0\\simeq\\FF_2^4\\), with carrier \\(\\Ind_J^P\\eta\\) of\ndimension four (\\cref{app:rank4}). The finite scalar center can be\nlarger than \\(\\mu_2\\); no literal extraspecial quotient is claimed.","-valued. With \\(H_0=\\ker\\lambda\\), the quotient is\n\\(P/H_0\\simeq\\FF_2^2\\), and \\(\\Ind_J^P\\theta\\) is an\nirreducible two-dimensional Heisenberg carrier\n(\\cref{app:rank2}). At rank four, the actual outer quadratic\ncentralizer gives \\(P=J U_B^m\\), \\([P:J]=2\\), and\n\\(P/H_0\\simeq\\FF_2^4\\), with carrier \\(\\Ind_J^P\\eta\\) of\ndimension four (\\cref{app:rank4}). The finite scalar center can be\nlarger than \\(\\mu_2\\); no literal extraspecial quotient is claimed.","At both ranks \\(K^\\times\\) normalizes \\(P\\). Thus\n\\(\\mathcal N=K^\\times J\\) is normal of index two in\n\\(\\mathcal M=K^\\times P\\), and\n\\[\n\\Sigma=\\Ind_{\\mathcal N}^{\\mathcal M}\\Lambda,\n\\qquad\n\\pi\\simeq\\cInd_{\\mathcal M}^{\\GL_N(F)}\\Sigma.\n\\]\nInduction in stages preserves the full original \\(\\Lambda\\), its\ncentral character, and its noncompact root; see\n\\cref{app:rank4_stages,app:lowrank_contract}. The full marked\nscalar-pushout comparison applies to these actual enlarged pairs as\nwell as to the original higher-rank types (\\cref{app:dyadic_markings}).\nThe full synthesis and scope are in \\cref{app:dyadic_synthesis}.\n\\end{proof}","The original low-rank maximal-type capacity obstruction remains true.\nThe new pair is a common arithmetic assembly carrier; it is not renamed\nas the original maximal-simple-character pair. For an independently\nfixed native quadratic marking, a preassigned embedding must satisfy\n\\(q_W(f(v))=q_{\\rm src}(v)\\). At equal capacity, a free embedding\nrequires equal Arf invariant. Spare capacity for \\(d\\ge4\\) admits\neither source Arf type when the embedding is free. Retaining only the\ncentral \\(\\{\\pm1\\}\\) would erase the distinction between whole marked\nfinite extensions such as \\(D_8\\) and \\(Q_8\\).","mon arithmetic assembly carrier; it is not renamed\nas the original maximal-simple-character pair. For an independently\nfixed native quadratic marking, a preassigned embedding must satisfy\n\\(q_W(f(v))=q_{\\rm src}(v)\\). At equal capacity, a free embedding\nrequires equal Arf invariant. Spare capacity for \\(d\\ge4\\) admits\neither source Arf type when the embedding is free. Retaining only the\ncentral \\(\\{\\pm1\\}\\) would erase the distinction between whole marked\nfinite extensions such as \\(D_8\\) and \\(Q_8\\).","ter pair. For an independently\nfixed native quadratic marking, a preassigned embedding must satisfy\n\\(q_W(f(v))=q_{\\rm src}(v)\\). At equal capacity, a free embedding\nrequires equal Arf invariant. Spare capacity for \\(d\\ge4\\) admits\neither source Arf type when the embedding is free. Retaining only the\ncentral \\(\\{\\pm1\\}\\) would erase the distinction between whole marked\nfinite extensions such as \\(D_8\\) and \\(Q_8\\).","r an independently\nfixed native quadratic marking, a preassigned embedding must satisfy\n\\(q_W(f(v))=q_{\\rm src}(v)\\). At equal capacity, a free embedding\nrequires equal Arf invariant. Spare capacity for \\(d\\ge4\\) admits\neither source Arf type when the embedding is free. Retaining only the\ncentral \\(\\{\\pm1\\}\\) would erase the distinction between whole marked\nfinite extensions such as \\(D_8\\) and \\(Q_8\\).","For the same candidates the constructions give\n\\[\n\\begin{array}{c|c|c}\n & p\\text{ odd} & p=2\\\\ \\hline\n\\dim_{\\FF_p}(J^1/H^1) & p^d-1 & 2^d-2\\\\\n\\Swan(\\pi)&p^{d-1}(p+1)h_d&3\\cdot2^{d-1}h_d\n\\end{array}\n\\]\nand torsion number one. Here \\(\\Swan\\) denotes the Swan conductor,\nand \\(a\\) below denotes the ordinary pair conductor. An epsilon exponent\nfor a specified additive character is a different quantity. The separate\nprogress paper, Sections 22--24, reconciles these conventions and the\nactual-order hypotheses. Their self-pair conductor formulas are\n\\[\n\\begin{aligned}\na(\\pi^\\vee\\times\\pi)\n &=p^{2d}-1+\\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i\n &&(p\\text{ odd}),","\\Swan(\\pi)&p^{d-1}(p+1)h_d&3\\cdot2^{d-1}h_d\n\\end{array}\n\\]\nand torsion number one. Here \\(\\Swan\\) denotes the Swan conductor,\nand \\(a\\) below denotes the ordinary pair conductor. An epsilon exponent\nfor a specified additive character is a different quantity. The separate\nprogress paper, Sections 22--24, reconciles these conventions and the\nactual-order hypotheses. Their self-pair conductor formulas are\n\\[\n\\begin{aligned}\na(\\pi^\\vee\\times\\pi)\n &=p^{2d}-1+\\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i\n &&(p\\text{ odd}),","i)&p^{d-1}(p+1)h_d&3\\cdot2^{d-1}h_d\n\\end{array}\n\\]\nand torsion number one. Here \\(\\Swan\\) denotes the Swan conductor,\nand \\(a\\) below denotes the ordinary pair conductor. An epsilon exponent\nfor a specified additive character is a different quantity. The separate\nprogress paper, Sections 22--24, reconciles these conventions and the\nactual-order hypotheses. Their self-pair conductor formulas are\n\\[\n\\begin{aligned}\na(\\pi^\\vee\\times\\pi)\n &=p^{2d}-1+\\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i\n &&(p\\text{ odd}),\\\\\na(\\pi^\\vee\\times\\pi)\n &=N^2-1+\\sum_{i=1}^d3\\cdot4^{i-1}h_i\n &&(p=2).\n\\end{aligned}\n\\]\nThese identities do not prove H5, identify an endo-class, or construct\na canonical source-to-host marking. At rank eight, the retained\nskipped-degree host has the same displayed numerical profiles as the\nfull tower and a smaller Heisenberg quotient. The explicit construction,\nrather than numerical matching alone, supplies the arithmetic object.","Nontrivial prescribed symplectic stabilizer actions, independently fixed\ncanonical native markings, arbitrary source/transport diagrams, and any\nrequired numerical layer placement remain outside the restricted results.\nThe current totally ramified host cannot implement nontrivial\nprime-to-\\(p\\) symplectic actions by the stronger character-preserving\ninner-normalizer mechanism; this obstruction is stated precisely in\n\\cref{app:stabilizer_contract}. That stronger mechanism is not silently\nimposed on the original conjecture. H5 remains conditional for the same\nfixed candidate.","general & 2\\^{}d & 2\\^{}d−2 & \\(2^{2^{d-1}-1}\\) & 2\\^{}d &\nSufficient iff d\\textgreater=3","The H staircase uses the corresponding t\\_j. Hence the exact\ncontribution of transition j to \\(J^1/H^1\\) is","One can also obtain this by subtracting the exact logarithmic indices of\nH and J in 1+P. It is an equality of the group index\n\\textbar \\(J^1/H^1\\)\\textbar, not a guess from the visible\nleading matrix support. BK 3.4.1 makes the resulting commutator pairing\nnondegenerate.","The exponents coincide because N is odd. Hence \\(J^1=H^1\\),\neta is one-dimensional, and no two-dimensional symplectic source plane\ncan embed into this quotient.","The construction retains the actual canonical beta extension and can be\ncompleted with any prescribed central quasicharacter omega\\_F trivial on\nU\\_F\\^{}1. Because \\(k_F^\\times=\\FF_2^\\times\\) is trivial,\nsuch an omega\\_F is unramified. Thus this covers every tamely ramified\ncentral quasicharacter in this particular residue-field setting.","They are the compatible native truncations of (5), because the outer\nextension, uniformizer, and lift depend on the same retained largest h\nat each stage. For \\(1\\le j<d\\), in rank 2\\^{}(j+1), put","It is well defined by the intersection condition. Choose M\\textgreater=a\nwith chi\\textbar U\\_K\\^{}M=1. Then chi\\_0 is a character of a subgroup\nof the finite abelian group \\(U_K^1/U_K^M\\). Extend it by\nadjoining generators and choosing the required complex roots, and pull\nit back. The resulting chi\\_K extends chi and is trivial on U\\_F\\^{}1.","Section4 constructs chi\\_E on U\\_E\\^{}1 extending phi and trivial on\nU\\_F\\^{}1. Define theta\\_1 on \\(H^1=U_E^1\\) U\\_(A\\_1)\\^{}a\nby chi\\_E on the first factor and psi\\_alpha on the second. The\nrestrictions agree on the intersection. BK3.2.1 gives the actual full\nsimple character, with theta\\_1\\textbar U\\_F\\^{}1=1.","Here \\(\\OO_K^\\times=U_K^1\\) because the residue field is\nF\\_2. Since U\\_K\\^{}1 is contained in J\\^{}1, we have","Let omega\\_F be any smooth quasicharacter trivial on U\\_F\\^{}1. Since\n\\(\\OO_F^\\times=U_F^1\\), it is determined by its value\nomega\\_F(t). The actual maximal type extends to K\\^{}times J.\nConcretely, (K\\^{}times J)/J is infinite cyclic, generated by a\nK-uniformizer. An intertwiner for its conjugation action on eta gives an\nextension Lambda\\_0. Denote its central character by omega\\_0.","This note does not compute \\(J^1/H^1\\) or prove that it\ncontains a particular dyadic source packet. It does not establish a\nquadratic refinement, canonical phase matching, nontrivial stabilizer\naction, general transport graph, source admission, or H5. In particular,\nan actual supercuspidal host with the required central character is not\nasserted to be a completed realization of the original dyadic packet\nwithout those separate proofs.","Let \\(F=\\mathbb F_2((t))\\), fix an additive character \\(\\psi_F\\) of\nconductor \\(t\\mathcal O_F\\), and take increasing odd positive integers\n\\(h_1<\\cdots<h_d\\). Put \\[\nq_i=3h_i,\\qquad N=2^d,\\qquad n_j=2^{d-j}q_{d-j+1}.\n\\] There is an explicit \\(\\beta\\in M_N(F)\\) generating a separable\ntotally ramified degree-\\(N\\) field that normalizes the complete lattice\nchain. It gives a maximal simple stratum with leading level \\(n_1\\),\nfinal critical exponent \\(-q_1\\), and successive approximation-field\ndegrees \\(2,4,\\ldots,2^d\\), with critical levels \\(n_2,\\ldots,n_d\\).","((t))\\), fix an additive character \\(\\psi_F\\) of\nconductor \\(t\\mathcal O_F\\), and take increasing odd positive integers\n\\(h_1<\\cdots<h_d\\). Put \\[\nq_i=3h_i,\\qquad N=2^d,\\qquad n_j=2^{d-j}q_{d-j+1}.\n\\] There is an explicit \\(\\beta\\in M_N(F)\\) generating a separable\ntotally ramified degree-\\(N\\) field that normalizes the complete lattice\nchain. It gives a maximal simple stratum with leading level \\(n_1\\),\nfinal critical exponent \\(-q_1\\), and successive approximation-field\ndegrees \\(2,4,\\ldots,2^d\\), with critical levels \\(n_2,\\ldots,n_d\\).","r \\(\\psi_F\\) of\nconductor \\(t\\mathcal O_F\\), and take increasing odd positive integers\n\\(h_1<\\cdots<h_d\\). Put \\[\nq_i=3h_i,\\qquad N=2^d,\\qquad n_j=2^{d-j}q_{d-j+1}.\n\\] There is an explicit \\(\\beta\\in M_N(F)\\) generating a separable\ntotally ramified degree-\\(N\\) field that normalizes the complete lattice\nchain. It gives a maximal simple stratum with leading level \\(n_1\\),\nfinal critical exponent \\(-q_1\\), and successive approximation-field\ndegrees \\(2,4,\\ldots,2^d\\), with critical levels \\(n_2,\\ldots,n_d\\).","increasing odd positive integers\n\\(h_1<\\cdots<h_d\\). Put \\[\nq_i=3h_i,\\qquad N=2^d,\\qquad n_j=2^{d-j}q_{d-j+1}.\n\\] There is an explicit \\(\\beta\\in M_N(F)\\) generating a separable\ntotally ramified degree-\\(N\\) field that normalizes the complete lattice\nchain. It gives a maximal simple stratum with leading level \\(n_1\\),\nfinal critical exponent \\(-q_1\\), and successive approximation-field\ndegrees \\(2,4,\\ldots,2^d\\), with critical levels \\(n_2,\\ldots,n_d\\).","cdots<h_d\\). Put \\[\nq_i=3h_i,\\qquad N=2^d,\\qquad n_j=2^{d-j}q_{d-j+1}.\n\\] There is an explicit \\(\\beta\\in M_N(F)\\) generating a separable\ntotally ramified degree-\\(N\\) field that normalizes the complete lattice\nchain. It gives a maximal simple stratum with leading level \\(n_1\\),\nfinal critical exponent \\(-q_1\\), and successive approximation-field\ndegrees \\(2,4,\\ldots,2^d\\), with critical levels \\(n_2,\\ldots,n_d\\).","The actual stratum admits a full simple character \\(\\theta\\) trivial on\n\\(U_F^1\\). Its canonical beta extension can be completed to an actual\nirreducible admissible supercuspidal \\(\\pi\\) of \\(\\mathrm{GL}_N(F)\\)\nwith any prescribed smooth central quasicharacter \\(\\omega\\) trivial on\n\\(U_F^1\\), provided the noncompact completion is free or already\ncompatible. Over this residue field, such \\(\\omega\\) is unramified.","For this same constructed candidate, \\[\n\\begin{aligned}\nt(\\pi)&=1,& \\operatorname{Sw}(\\pi)&=3\\,2^{d-1}h_d,\\\\\n\\dim_{\\mathbb F_2}W&=N-2,&\n\\dim\\eta&=2^{N/2-1},\\qquad W=J^1/H^1,\\\\\na(\\pi^\\vee\\times\\pi)\n &=N^2-1+\\sum_{i=1}^d3\\,4^{i-1}h_i.\n\\end{aligned}\n\\] Matching these profiles does not identify the source, its endo-class,\nor the pair factors in original H5.","At a recursive coefficient field \\(L\\) of characteristic two, with\nresidue field \\(\\mathbb F_2\\) and uniformizer \\(t_L\\), put \\[\nE=L(\\alpha),\\qquad \\alpha^2-\\alpha=t_L^{-q},\\qquad\na=(q+1)/2,\\qquad z=t_L^a\\alpha.\n\\] The extension is separable and totally ramified of degree two; \\(z\\)\nis a uniformizer and \\[\nz^2+t_L^az+t_L=0,\\qquad \\operatorname{Tr}_{E/L}(\\alpha)=1.\n\\] For constant-coefficient projection \\(P_0\\) in the basis \\(1,z\\), the\nnormalized trace corestriction and exact filtered right inverse are \\[\n\\kappa=\\alpha^{-1},\\qquad s=\\kappa\\operatorname{pr},\\qquad\ns(P_0)=t_L/z^2,\\qquad\n\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\] They satisfy \\(s(\\operatorname{Lift}_E(C))=C\\), preserve the exact\nfiltration valuation, and give the","At a recursive coefficient field \\(L\\) of characteristic two, with\nresidue field \\(\\mathbb F_2\\) and uniformizer \\(t_L\\), put \\[\nE=L(\\alpha),\\qquad \\alpha^2-\\alpha=t_L^{-q},\\qquad\na=(q+1)/2,\\qquad z=t_L^a\\alpha.\n\\] The extension is separable and totally ramified of degree two; \\(z\\)\nis a uniformizer and \\[\nz^2+t_L^az+t_L=0,\\qquad \\operatorname{Tr}_{E/L}(\\alpha)=1.\n\\] For constant-coefficient projection \\(P_0\\) in the basis \\(1,z\\), the\nnormalized trace corestriction and exact filtered right inverse are \\[\n\\kappa=\\alpha^{-1},\\qquad s=\\kappa\\operatorname{pr},\\qquad\ns(P_0)=t_L/z^2,\\qquad\n\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\] They satisfy \\(s(\\operatorname{Lift}_E(C))=C\\), preserve the exact\nfiltration valuation, and give the identity \\[\n\\operatorname{tr}_L\\operatorname{Lift}_E(C)","At a recursive coefficient field \\(L\\) of characteristic two, with\nresidue field \\(\\mathbb F_2\\) and uniformizer \\(t_L\\), put \\[\nE=L(\\alpha),\\qquad \\alpha^2-\\alpha=t_L^{-q},\\qquad\na=(q+1)/2,\\qquad z=t_L^a\\alpha.\n\\] The extension is separable and totally ramified of degree two; \\(z\\)\nis a uniformizer and \\[\nz^2+t_L^az+t_L=0,\\qquad \\operatorname{Tr}_{E/L}(\\alpha)=1.\n\\] For constant-coefficient projection \\(P_0\\) in the basis \\(1,z\\), the\nnormalized trace corestriction and exact filtered right inverse are \\[\n\\kappa=\\alpha^{-1},\\qquad s=\\kappa\\operatorname{pr},\\qquad\ns(P_0)=t_L/z^2,\\qquad\n\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\] They satisfy \\(s(\\operatorname{Lift}_E(C))=C\\), preserve the exact\nfiltration valuation, and give the identity \\[\n\\operatorname{tr}_L\\operatorname{Lift}_E(C)\n =\\operatorname{Tr}_{E/L}","At a recursive coefficient field \\(L\\) of characteristic two, with\nresidue field \\(\\mathbb F_2\\) and uniformizer \\(t_L\\), put \\[\nE=L(\\alpha),\\qquad \\alpha^2-\\alpha=t_L^{-q},\\qquad\na=(q+1)/2,\\qquad z=t_L^a\\alpha.\n\\] The extension is separable and totally ramified of degree two; \\(z\\)\nis a uniformizer and \\[\nz^2+t_L^az+t_L=0,\\qquad \\operatorname{Tr}_{E/L}(\\alpha)=1.\n\\] For constant-coefficient projection \\(P_0\\) in the basis \\(1,z\\), the\nnormalized trace corestriction and exact filtered right inverse are \\[\n\\kappa=\\alpha^{-1},\\qquad s=\\kappa\\operatorname{pr},\\qquad\ns(P_0)=t_L/z^2,\\qquad\n\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\] They satisfy \\(s(\\operatorname{Lift}_E(C))=C\\), preserve the exact\nfiltration valuation, and give the identity \\[\n\\operatorname{tr}_L\\operatorname{Lift}_E(C)\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\alpha\\operatorname{tr}_E C\\bigr).\n\\] The actual recursive matrix is \\[\nB(L,t_L;h_1,","At a recursive coefficient field \\(L\\) of characteristic two, with\nresidue field \\(\\mathbb F_2\\) and uniformizer \\(t_L\\), put \\[\nE=L(\\alpha),\\qquad \\alpha^2-\\alpha=t_L^{-q},\\qquad\na=(q+1)/2,\\qquad z=t_L^a\\alpha.\n\\] The extension is separable and totally ramified of degree two; \\(z\\)\nis a uniformizer and \\[\nz^2+t_L^az+t_L=0,\\qquad \\operatorname{Tr}_{E/L}(\\alpha)=1.\n\\] For constant-coefficient projection \\(P_0\\) in the basis \\(1,z\\), the\nnormalized trace corestriction and exact filtered right inverse are \\[\n\\kappa=\\alpha^{-1},\\qquad s=\\kappa\\operatorname{pr},\\qquad\ns(P_0)=t_L/z^2,\\qquad\n\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\] They satisfy \\(s(\\operatorname{Lift}_E(C))=C\\), preserve the exact\nfiltration valuation, and give the identity \\[\n\\operatorname{tr}_L\\operatorname{Lift}_E(C)\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\alpha\\operatorname{tr}_E C\\bigr).\n\\] The actual recursive matrix is \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{2^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z","At a recursive coefficient field \\(L\\) of characteristic two, with\nresidue field \\(\\mathbb F_2\\) and uniformizer \\(t_L\\), put \\[\nE=L(\\alpha),\\qquad \\alpha^2-\\alpha=t_L^{-q},\\qquad\na=(q+1)/2,\\qquad z=t_L^a\\alpha.\n\\] The extension is separable and totally ramified of degree two; \\(z\\)\nis a uniformizer and \\[\nz^2+t_L^az+t_L=0,\\qquad \\operatorname{Tr}_{E/L}(\\alpha)=1.\n\\] For constant-coefficient projection \\(P_0\\) in the basis \\(1,z\\), the\nnormalized trace corestriction and exact filtered right inverse are \\[\n\\kappa=\\alpha^{-1},\\qquad s=\\kappa\\operatorname{pr},\\qquad\ns(P_0)=t_L/z^2,\\qquad\n\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\] They satisfy \\(s(\\operatorname{Lift}_E(C))=C\\), preserve the exact\nfiltration valuation, and give the identity \\[\n\\operatorname{tr}_L\\operatorname{Lift}_E(C)\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\alpha\\operatorname{tr}_E C\\bigr).\n\\] The actual recursive matrix is \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{2^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with multiplication by \\(\\alpha\\) as the rank-two base","ursive coefficient field \\(L\\) of characteristic two, with\nresidue field \\(\\mathbb F_2\\) and uniformizer \\(t_L\\), put \\[\nE=L(\\alpha),\\qquad \\alpha^2-\\alpha=t_L^{-q},\\qquad\na=(q+1)/2,\\qquad z=t_L^a\\alpha.\n\\] The extension is separable and totally ramified of degree two; \\(z\\)\nis a uniformizer and \\[\nz^2+t_L^az+t_L=0,\\qquad \\operatorname{Tr}_{E/L}(\\alpha)=1.\n\\] For constant-coefficient projection \\(P_0\\) in the basis \\(1,z\\), the\nnormalized trace corestriction and exact filtered right inverse are \\[\n\\kappa=\\alpha^{-1},\\qquad s=\\kappa\\operatorname{pr},\\qquad\ns(P_0)=t_L/z^2,\\qquad\n\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\] They satisfy \\(s(\\operatorname{Lift}_E(C))=C\\), preserve the exact\nfiltration valuation, and give the identity \\[\n\\operatorname{tr}_L\\operatorname{Lift}_E(C)\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\alpha\\operatorname{tr}_E C\\bigr).\n\\] The actual recursive matrix is \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{2^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with multiplication by \\(\\alpha\\) as the rank-two base case.","eld \\(L\\) of characteristic two, with\nresidue field \\(\\mathbb F_2\\) and uniformizer \\(t_L\\), put \\[\nE=L(\\alpha),\\qquad \\alpha^2-\\alpha=t_L^{-q},\\qquad\na=(q+1)/2,\\qquad z=t_L^a\\alpha.\n\\] The extension is separable and totally ramified of degree two; \\(z\\)\nis a uniformizer and \\[\nz^2+t_L^az+t_L=0,\\qquad \\operatorname{Tr}_{E/L}(\\alpha)=1.\n\\] For constant-coefficient projection \\(P_0\\) in the basis \\(1,z\\), the\nnormalized trace corestriction and exact filtered right inverse are \\[\n\\kappa=\\alpha^{-1},\\qquad s=\\kappa\\operatorname{pr},\\qquad\ns(P_0)=t_L/z^2,\\qquad\n\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\] They satisfy \\(s(\\operatorname{Lift}_E(C))=C\\), preserve the exact\nfiltration valuation, and give the identity \\[\n\\operatorname{tr}_L\\operatorname{Lift}_E(C)\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\alpha\\operatorname{tr}_E C\\bigr).\n\\] The actual recursive matrix is \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{2^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with multiplication by \\(\\alpha\\) as the rank-two base case.","er \\(t_L\\), put \\[\nE=L(\\alpha),\\qquad \\alpha^2-\\alpha=t_L^{-q},\\qquad\na=(q+1)/2,\\qquad z=t_L^a\\alpha.\n\\] The extension is separable and totally ramified of degree two; \\(z\\)\nis a uniformizer and \\[\nz^2+t_L^az+t_L=0,\\qquad \\operatorname{Tr}_{E/L}(\\alpha)=1.\n\\] For constant-coefficient projection \\(P_0\\) in the basis \\(1,z\\), the\nnormalized trace corestriction and exact filtered right inverse are \\[\n\\kappa=\\alpha^{-1},\\qquad s=\\kappa\\operatorname{pr},\\qquad\ns(P_0)=t_L/z^2,\\qquad\n\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\] They satisfy \\(s(\\operatorname{Lift}_E(C))=C\\), preserve the exact\nfiltration valuation, and give the identity \\[\n\\operatorname{tr}_L\\operatorname{Lift}_E(C)\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\alpha\\operatorname{tr}_E C\\bigr).\n\\] The actual recursive matrix is \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{2^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with multiplication by \\(\\alpha\\) as the rank-two base case.","lly ramified of degree two; \\(z\\)\nis a uniformizer and \\[\nz^2+t_L^az+t_L=0,\\qquad \\operatorname{Tr}_{E/L}(\\alpha)=1.\n\\] For constant-coefficient projection \\(P_0\\) in the basis \\(1,z\\), the\nnormalized trace corestriction and exact filtered right inverse are \\[\n\\kappa=\\alpha^{-1},\\qquad s=\\kappa\\operatorname{pr},\\qquad\ns(P_0)=t_L/z^2,\\qquad\n\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\] They satisfy \\(s(\\operatorname{Lift}_E(C))=C\\), preserve the exact\nfiltration valuation, and give the identity \\[\n\\operatorname{tr}_L\\operatorname{Lift}_E(C)\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\alpha\\operatorname{tr}_E C\\bigr).\n\\] The actual recursive matrix is \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{2^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with multiplication by \\(\\alpha\\) as the rank-two base case.","}_{E/L}(\\alpha)=1.\n\\] For constant-coefficient projection \\(P_0\\) in the basis \\(1,z\\), the\nnormalized trace corestriction and exact filtered right inverse are \\[\n\\kappa=\\alpha^{-1},\\qquad s=\\kappa\\operatorname{pr},\\qquad\ns(P_0)=t_L/z^2,\\qquad\n\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\] They satisfy \\(s(\\operatorname{Lift}_E(C))=C\\), preserve the exact\nfiltration valuation, and give the identity \\[\n\\operatorname{tr}_L\\operatorname{Lift}_E(C)\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\alpha\\operatorname{tr}_E C\\bigr).\n\\] The actual recursive matrix is \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{2^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with multiplication by \\(\\alpha\\) as the rank-two base case.","erse are \\[\n\\kappa=\\alpha^{-1},\\qquad s=\\kappa\\operatorname{pr},\\qquad\ns(P_0)=t_L/z^2,\\qquad\n\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\] They satisfy \\(s(\\operatorname{Lift}_E(C))=C\\), preserve the exact\nfiltration valuation, and give the identity \\[\n\\operatorname{tr}_L\\operatorname{Lift}_E(C)\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\alpha\\operatorname{tr}_E C\\bigr).\n\\] The actual recursive matrix is \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{2^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with multiplication by \\(\\alpha\\) as the rank-two base case.","ey satisfy \\(s(\\operatorname{Lift}_E(C))=C\\), preserve the exact\nfiltration valuation, and give the identity \\[\n\\operatorname{tr}_L\\operatorname{Lift}_E(C)\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\alpha\\operatorname{tr}_E C\\bigr).\n\\] The actual recursive matrix is \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{2^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with multiplication by \\(\\alpha\\) as the rank-two base case.","At an inductive field step \\(d\\ge2\\), set \\(n=2^{d-1}q_d\\),\n\\(R=2^{d-2}q_{d-1}\\). Then \\(n>2R\\) and \\(-2R>-n-q_1\\). BK Remark 2.2.7\ngives the maximal field generated by the actual displayed matrix.\nLiteral truncation and the filtered lift give the defining sequence.\nNonzero native trace also proves separability: the irreducible\ncharacteristic polynomial has a nonzero coefficient of \\(X^{N-1}\\), so\nits derivative is nonzero. See \\path{tower/DYADIC_AS_TOWER167.md}\nfor all field and filtration details.","The replacement for trace zero is the depth inequality \\[\n\\boxed{v_F(\\operatorname{tr}\\delta)+s\\ge1.}\n\\] It makes the additive correction trivial on scalar \\(U_F^s\\), because\n\\((u-1)\\operatorname{tr}\\delta\\in t\\mathcal O_F\\). Our native\ncorrections have trace one, so the inequality holds at \\(s=1\\). This is\nthe exact source-to-target transfer of the weighted-visibility\nprinciple; the Euler coefficient theorem is motivation, not an input to\nthe arithmetic proof. In characteristic two, the weights are uniformizer\ndepths, rather than integer powers of two in the coefficient field.","At the minimal rank-two base, extend the forced character on\n\\(U_E^{(q+1)/2}\\) to \\(U_E^1\\), imposing triviality on \\(U_F^1\\). The\nprescriptions agree on their intersection because the parameter trace is\none and \\(\\psi_F(u-1)=1\\). This is an actual finite abelian character\nextension and gives the full minimal simple character.","For a later native step, first transfer the full beta-determinant unit\ncharacter to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar\nunits, \\[\n\\Theta_j(uI)=\\theta_j(u^2)=1.\n\\] For odd critical level \\(r\\), use \\(m=\\lfloor r/2\\rfloor\\) and\n\\(D=H^{m+1}(\\beta)=H^{m+1}(\\gamma)\\). The endpoint is valid because\n\\(2(m+1)-r=1\\): the additive-character cross term lies in the radical on\nwhich the matrix additive character is trivial. BK 3.3.18 supplies\n\\(\\vartheta=\\Theta_j|_D\\,\\psi_{\\beta-\\gamma}\\). Integral trace makes it\ntrivial on \\(D\\cap U_F^1\\).","minant unit\ncharacter to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar\nunits, \\[\n\\Theta_j(uI)=\\theta_j(u^2)=1.\n\\] For odd critical level \\(r\\), use \\(m=\\lfloor r/2\\rfloor\\) and\n\\(D=H^{m+1}(\\beta)=H^{m+1}(\\gamma)\\). The endpoint is valid because\n\\(2(m+1)-r=1\\): the additive-character cross term lies in the radical on\nwhich the matrix additive character is trivial. BK 3.3.18 supplies\n\\(\\vartheta=\\Theta_j|_D\\,\\psi_{\\beta-\\gamma}\\). Integral trace makes it\ntrivial on \\(D\\cap U_F^1\\).","units, \\[\n\\Theta_j(uI)=\\theta_j(u^2)=1.\n\\] For odd critical level \\(r\\), use \\(m=\\lfloor r/2\\rfloor\\) and\n\\(D=H^{m+1}(\\beta)=H^{m+1}(\\gamma)\\). The endpoint is valid because\n\\(2(m+1)-r=1\\): the additive-character cross term lies in the radical on\nwhich the matrix additive character is trivial. BK 3.3.18 supplies\n\\(\\vartheta=\\Theta_j|_D\\,\\psi_{\\beta-\\gamma}\\). Integral trace makes it\ntrivial on \\(D\\cap U_F^1\\).","The next parameter field is maximal. Extend the forced character on its\ndeep units together with the trivial character of \\(U_F^1\\), then apply\nBK 3.2.5 to obtain the next full character. All new extensions occur at\nmaximal native rank. No unverified determinant-image extension in a\nnonmaximal centralizer is used.","For the final totally ramified field, \\(J/J^1=\\mathbb F_2^\\times=1\\), so\n\\(J=J^1\\) and the canonical beta extension is \\(\\eta\\). Extend to\n\\(K^\\times J\\), and choose the noncompact scalar to match \\(\\omega(t)\\).\nThe needed \\(N\\)-th root exists in \\(\\mathbb C^\\times\\). The compact\ntype stays fixed. All choices precede H5. See\n\\path{central/DYADIC_FULL_CHARACTER167.md} for the complete proof.","The dyadic floors cannot be replaced by the odd-prime dimension formula.\nAt stage \\(j\\), the centralizer grade has dimension \\(2^{d-j}\\). Its\ncontribution to \\(J^1/H^1\\) is \\[\n\\bigl(2^{d-j+1}-2^{d-j}\\bigr)\\,\\mathbf1_{n_j\\text{ even}}.\n\\] Every \\(n_j\\) for \\(j<d\\) is even; \\(n_d=q_1\\) is odd and contributes\nzero. The sum is exactly \\(N-2\\). BK nondegeneracy makes this a\nsymplectic space. A whole source packet of \\(d\\) planes fits if and only\nif \\(N-2\\ge2d\\), equivalently \\(d\\ge3\\).","cannot be replaced by the odd-prime dimension formula.\nAt stage \\(j\\), the centralizer grade has dimension \\(2^{d-j}\\). Its\ncontribution to \\(J^1/H^1\\) is \\[\n\\bigl(2^{d-j+1}-2^{d-j}\\bigr)\\,\\mathbf1_{n_j\\text{ even}}.\n\\] Every \\(n_j\\) for \\(j<d\\) is even; \\(n_d=q_1\\) is odd and contributes\nzero. The sum is exactly \\(N-2\\). BK nondegeneracy makes this a\nsymplectic space. A whole source packet of \\(d\\) planes fits if and only\nif \\(N-2\\ge2d\\), equivalently \\(d\\ge3\\).","ime dimension formula.\nAt stage \\(j\\), the centralizer grade has dimension \\(2^{d-j}\\). Its\ncontribution to \\(J^1/H^1\\) is \\[\n\\bigl(2^{d-j+1}-2^{d-j}\\bigr)\\,\\mathbf1_{n_j\\text{ even}}.\n\\] Every \\(n_j\\) for \\(j<d\\) is even; \\(n_d=q_1\\) is odd and contributes\nzero. The sum is exactly \\(N-2\\). BK nondegeneracy makes this a\nsymplectic space. A whole source packet of \\(d\\) planes fits if and only\nif \\(N-2\\ge2d\\), equivalently \\(d\\ge3\\).","tage \\(j\\), the centralizer grade has dimension \\(2^{d-j}\\). Its\ncontribution to \\(J^1/H^1\\) is \\[\n\\bigl(2^{d-j+1}-2^{d-j}\\bigr)\\,\\mathbf1_{n_j\\text{ even}}.\n\\] Every \\(n_j\\) for \\(j<d\\) is even; \\(n_d=q_1\\) is odd and contributes\nzero. The sum is exactly \\(N-2\\). BK nondegeneracy makes this a\nsymplectic space. A whole source packet of \\(d\\) planes fits if and only\nif \\(N-2\\ge2d\\), equivalently \\(d\\ge3\\).","Now assume \\(d\\ge3\\), the one-vertex contract, declared \\(C=\\{1\\}\\), and\nactually supplied full character extensions with labelled whole-plane\nquotients. Retain their complete dyadic finite central-extension\nmarkings, including their specified central embeddings, lifts, square\nvalues, refinements and phases. Any central-character kernel already\nfactored out must be recorded; a scalar pushout cannot recover invisible\nkernel elements.","Under the scalar-pushout realization contract, with no independently\nfixed conflicting host marking, choose a labelled symplectic embedding\ninto \\(W\\). Matching commutators gives a center-fixing isomorphism\nbetween the source \\(\\mathbb C^\\times\\) extension and the corresponding\nhost restriction. \\textbf{Retain the image of the whole marked finite\nsubgroup and every lift.} For this isomorphism \\(F_0\\), \\[\nF_0(a)^2=F_0(a^2).\n\\] Thus its square bits, refinement and Gauss sign survive, along with\nall supplied full character values. They are not inferred from the\nunmarked pushout. Stone--von Neumann gives the source factor inside\n\\(\\eta\\); a complementary line gives an actual rank-\\(N\\) image-algebra\nprojector. The retained induction/recovery idempotent recovers the same\n\\(\\Lambda\\) and the same \\(\\pi\\).","he scalar-pushout realization contract, with no independently\nfixed conflicting host marking, choose a labelled symplectic embedding\ninto \\(W\\). Matching commutators gives a center-fixing isomorphism\nbetween the source \\(\\mathbb C^\\times\\) extension and the corresponding\nhost restriction. \\textbf{Retain the image of the whole marked finite\nsubgroup and every lift.} For this isomorphism \\(F_0\\), \\[\nF_0(a)^2=F_0(a^2).\n\\] Thus its square bits, refinement and Gauss sign survive, along with\nall supplied full character values. They are not inferred from the\nunmarked pushout. Stone--von Neumann gives the source factor inside\n\\(\\eta\\); a complementary line gives an actual rank-\\(N\\) image-algebra\nprojector. The retained induction/recovery idempotent recovers the same\n\\(\\Lambda\\) and the same \\(\\pi\\).","C^\\times\\) extension and the corresponding\nhost restriction. \\textbf{Retain the image of the whole marked finite\nsubgroup and every lift.} For this isomorphism \\(F_0\\), \\[\nF_0(a)^2=F_0(a^2).\n\\] Thus its square bits, refinement and Gauss sign survive, along with\nall supplied full character values. They are not inferred from the\nunmarked pushout. Stone--von Neumann gives the source factor inside\n\\(\\eta\\); a complementary line gives an actual rank-\\(N\\) image-algebra\nprojector. The retained induction/recovery idempotent recovers the same\n\\(\\Lambda\\) and the same \\(\\pi\\).","triction. \\textbf{Retain the image of the whole marked finite\nsubgroup and every lift.} For this isomorphism \\(F_0\\), \\[\nF_0(a)^2=F_0(a^2).\n\\] Thus its square bits, refinement and Gauss sign survive, along with\nall supplied full character values. They are not inferred from the\nunmarked pushout. Stone--von Neumann gives the source factor inside\n\\(\\eta\\); a complementary line gives an actual rank-\\(N\\) image-algebra\nprojector. The retained induction/recovery idempotent recovers the same\n\\(\\Lambda\\) and the same \\(\\pi\\).","this isomorphism \\(F_0\\), \\[\nF_0(a)^2=F_0(a^2).\n\\] Thus its square bits, refinement and Gauss sign survive, along with\nall supplied full character values. They are not inferred from the\nunmarked pushout. Stone--von Neumann gives the source factor inside\n\\(\\eta\\); a complementary line gives an actual rank-\\(N\\) image-algebra\nprojector. The retained induction/recovery idempotent recovers the same\n\\(\\Lambda\\) and the same \\(\\pi\\).","\\[\nF_0(a)^2=F_0(a^2).\n\\] Thus its square bits, refinement and Gauss sign survive, along with\nall supplied full character values. They are not inferred from the\nunmarked pushout. Stone--von Neumann gives the source factor inside\n\\(\\eta\\); a complementary line gives an actual rank-\\(N\\) image-algebra\nprojector. The retained induction/recovery idempotent recovers the same\n\\(\\Lambda\\) and the same \\(\\pi\\).","If a genuine host-native quadratic refinement and its finite\n\\(\\mu_2\\)-central extension are independently supplied, quadratic\nembedding has a sharper condition. With otherwise free embeddings, at\nrank eight the equal-dimensional source and host must have equal Arf\ninvariants. For \\(d\\ge4\\), the host has at least one spare symplectic\nplane, so either source Arf type embeds. If the original embedding is\npreassigned, it must instead satisfy the pointwise equation\n\\(q_W(f(v))=q_{\\rm src}(v)\\). These conclusions require the actual host\nmarking as input; dimension does not manufacture it.","\\)-central extension are independently supplied, quadratic\nembedding has a sharper condition. With otherwise free embeddings, at\nrank eight the equal-dimensional source and host must have equal Arf\ninvariants. For \\(d\\ge4\\), the host has at least one spare symplectic\nplane, so either source Arf type embeds. If the original embedding is\npreassigned, it must instead satisfy the pointwise equation\n\\(q_W(f(v))=q_{\\rm src}(v)\\). These conclusions require the actual host\nmarking as input; dimension does not manufacture it.","The full finite proofs are in \\path{markings/DYADIC_MARKINGS167.md}.\nIn particular, the D8 and Q8 unmarked complex pushouts coincide while\ntheir whole marked finite \\(\\mu_2\\)-central extensions differ. Merely\nrecording the central \\(\\{\\pm1\\}\\) subgroup would lose that distinction.","The rank-two and rank-four carrier failures cannot be repaired by\nchanging the maximal simple parameter while keeping the specified Swan\nexponent and torsion number. At rank two, odd Swan forces a minimal\nodd-level stratum and \\(J^1=H^1\\). At rank four, Swan \\(6h_2\\) with\n\\(h_2\\) odd forces the degree pattern \\(4\\to2\\); the maximal quadratic\nderived field forces an odd lower critical level. Every such\nmaximal-type carrier has dimension two, below the required four. Scalar\ntwists preserve this capacity.","At rank eight, numerical profiles alone also fail to determine capacity.\n\\path{capacity/DYADIC_CAPACITY167.md}, Section 6, constructs an\nactual skipped-degree \\(8\\to2\\) host with the same Swan, torsion and\nself-pair conductor but quotient dimension four rather than six. Our\nsuccessful result uses the explicit full tower, rather than inferring\nits profile from invariants.","\\begin{itemize}\n\\tightlist\n\\item\n  Actual fields, full characters and central completion in every rank\n  \\(2^d\\).\n\\item\n  Full marked scalar-pushout packet realization for the stated family\n  when \\(d\\ge3\\), with \\(C=1\\), one vertex, and transported markings\n  retained.\n\\item\n  A conditional quadratic embedding into an independently supplied\n  native marked host, with the exact Arf or pointwise condition above.\n\\end{itemize}","The arbitrary-length field and full-character arguments are ordinary\nproofs. The exact rank-eight run for \\(h=(1,3,5)\\) checks actual matrix\nidentities, normalized lifts, leading and adjacent-correction grades\n\\(-60,-18,-3\\), native traces one and ambient traces \\(0,0,1\\). It\noutputs a matrix with 25 nonzero Laurent-polynomial entries. The\ncorresponding critical exponents are proved symbolically. The run does\nnot prove simplicity, full-character existence, or universal marked\nrealization. Those conclusions use the separate proofs and their stated\nscope. No Lean build was run.","The maximal-type carrier is not the only arithmetic carrier available in\nthe constructed representation. The following constructions keep the\noriginal \\(\\beta,\\theta,\\Lambda\\) and \\(\\pi\\) fixed.","Write the odd minimal level as \\(q=2m+1\\), with \\(m\\ge1\\), and let\n\\(J=H^1=J^1=U_K^1U_A^{m+1}\\). Define \\[\nP=U_K^1U_A^m.\n\\] This is an actual compact open group, with \\(J\\triangleleft P\\),\n\\([P:J]=2\\) and \\(P\\cap(K^\\times J)=J\\). For \\(g\\in P\\setminus J\\), the\nactual intertwining theorem makes \\(\\lambda=\\theta^g/\\theta\\)\nnontrivial. The commutator \\([P,J]\\subset U_A^{m+1}\\) lies in the forced\nadditive-character domain, so \\(\\lambda\\) has values \\(\\{\\pm1\\}\\). Put\n\\(H_0=\\ker\\lambda\\). Then \\[\nP/H_0\\simeq\\mathbb F_2^2,\\qquad\nE=\\operatorname{Ind}_J^P\\theta\\text{ is irreducible of dimension }2.\n\\] The actual represented generators are a diagonal sign and an\noff-diagonal matrix that anticommute. Thus their commutator form is\nnondegener","Write the odd minimal level as \\(q=2m+1\\), with \\(m\\ge1\\), and let\n\\(J=H^1=J^1=U_K^1U_A^{m+1}\\). Define \\[\nP=U_K^1U_A^m.\n\\] This is an actual compact open group, with \\(J\\triangleleft P\\),\n\\([P:J]=2\\) and \\(P\\cap(K^\\times J)=J\\). For \\(g\\in P\\setminus J\\), the\nactual intertwining theorem makes \\(\\lambda=\\theta^g/\\theta\\)\nnontrivial. The commutator \\([P,J]\\subset U_A^{m+1}\\) lies in the forced\nadditive-character domain, so \\(\\lambda\\) has values \\(\\{\\pm1\\}\\). Put\n\\(H_0=\\ker\\lambda\\). Then \\[\nP/H_0\\simeq\\mathbb F_2^2,\\qquad\nE=\\operatorname{Ind}_J^P\\theta\\text{ is irreducible of dimension }2.\n\\] The actual represented generators are a diagonal sign and an\noff-diagonal matrix that anticommute. Thus their commutator form is\nnondegenerate, and \\(\\thet","Write the odd minimal level as \\(q=2m+1\\), with \\(m\\ge1\\), and let\n\\(J=H^1=J^1=U_K^1U_A^{m+1}\\). Define \\[\nP=U_K^1U_A^m.\n\\] This is an actual compact open group, with \\(J\\triangleleft P\\),\n\\([P:J]=2\\) and \\(P\\cap(K^\\times J)=J\\). For \\(g\\in P\\setminus J\\), the\nactual intertwining theorem makes \\(\\lambda=\\theta^g/\\theta\\)\nnontrivial. The commutator \\([P,J]\\subset U_A^{m+1}\\) lies in the forced\nadditive-character domain, so \\(\\lambda\\) has values \\(\\{\\pm1\\}\\). Put\n\\(H_0=\\ker\\lambda\\). Then \\[\nP/H_0\\simeq\\mathbb F_2^2,\\qquad\nE=\\operatorname{Ind}_J^P\\theta\\text{ is irreducible of dimension }2.\n\\] The actual represented generators are a diagonal sign and an\noff-diagonal matrix that anticommute. Thus their commutator form is\nnondegenerate, and \\(\\theta|_{H_0}\\) is the full character of an","Write the odd minimal level as \\(q=2m+1\\), with \\(m\\ge1\\), and let\n\\(J=H^1=J^1=U_K^1U_A^{m+1}\\). Define \\[\nP=U_K^1U_A^m.\n\\] This is an actual compact open group, with \\(J\\triangleleft P\\),\n\\([P:J]=2\\) and \\(P\\cap(K^\\times J)=J\\). For \\(g\\in P\\setminus J\\), the\nactual intertwining theorem makes \\(\\lambda=\\theta^g/\\theta\\)\nnontrivial. The commutator \\([P,J]\\subset U_A^{m+1}\\) lies in the forced\nadditive-character domain, so \\(\\lambda\\) has values \\(\\{\\pm1\\}\\). Put\n\\(H_0=\\ker\\lambda\\). Then \\[\nP/H_0\\simeq\\mathbb F_2^2,\\qquad\nE=\\operatorname{Ind}_J^P\\theta\\text{ is irreducible of dimension }2.\n\\] The actual represented generators are a diagonal sign and an\noff-diagonal matrix that anticommute. Thus their commutator form is\nnondegenerate, and \\(\\theta|_{H_0}\\) is the full character of an actual\nnew arithmetic Heise","Write the odd minimal level as \\(q=2m+1\\), with \\(m\\ge1\\), and let\n\\(J=H^1=J^1=U_K^1U_A^{m+1}\\). Define \\[\nP=U_K^1U_A^m.\n\\] This is an actual compact open group, with \\(J\\triangleleft P\\),\n\\([P:J]=2\\) and \\(P\\cap(K^\\times J)=J\\). For \\(g\\in P\\setminus J\\), the\nactual intertwining theorem makes \\(\\lambda=\\theta^g/\\theta\\)\nnontrivial. The commutator \\([P,J]\\subset U_A^{m+1}\\) lies in the forced\nadditive-character domain, so \\(\\lambda\\) has values \\(\\{\\pm1\\}\\). Put\n\\(H_0=\\ker\\lambda\\). Then \\[\nP/H_0\\simeq\\mathbb F_2^2,\\qquad\nE=\\operatorname{Ind}_J^P\\theta\\text{ is irreducible of dimension }2.\n\\] The actual represented generators are a diagonal sign and an\noff-diagonal matrix that anticommute. Thus their commutator form is\nnondegenerate, and \\(\\theta|_{H_0}\\) is the full character of an actual\nnew arithmetic Heisenberg pair. No abstract operators create that group\naction.","as \\(q=2m+1\\), with \\(m\\ge1\\), and let\n\\(J=H^1=J^1=U_K^1U_A^{m+1}\\). Define \\[\nP=U_K^1U_A^m.\n\\] This is an actual compact open group, with \\(J\\triangleleft P\\),\n\\([P:J]=2\\) and \\(P\\cap(K^\\times J)=J\\). For \\(g\\in P\\setminus J\\), the\nactual intertwining theorem makes \\(\\lambda=\\theta^g/\\theta\\)\nnontrivial. The commutator \\([P,J]\\subset U_A^{m+1}\\) lies in the forced\nadditive-character domain, so \\(\\lambda\\) has values \\(\\{\\pm1\\}\\). Put\n\\(H_0=\\ker\\lambda\\). Then \\[\nP/H_0\\simeq\\mathbb F_2^2,\\qquad\nE=\\operatorname{Ind}_J^P\\theta\\text{ is irreducible of dimension }2.\n\\] The actual represented generators are a diagonal sign and an\noff-diagonal matrix that anticommute. Thus their commutator form is\nnondegenerate, and \\(\\theta|_{H_0}\\) is the full character of an actual\nnew arithmetic Heisenberg pair. No abstract operators create that group\naction.","\\ge1\\), and let\n\\(J=H^1=J^1=U_K^1U_A^{m+1}\\). Define \\[\nP=U_K^1U_A^m.\n\\] This is an actual compact open group, with \\(J\\triangleleft P\\),\n\\([P:J]=2\\) and \\(P\\cap(K^\\times J)=J\\). For \\(g\\in P\\setminus J\\), the\nactual intertwining theorem makes \\(\\lambda=\\theta^g/\\theta\\)\nnontrivial. The commutator \\([P,J]\\subset U_A^{m+1}\\) lies in the forced\nadditive-character domain, so \\(\\lambda\\) has values \\(\\{\\pm1\\}\\). Put\n\\(H_0=\\ker\\lambda\\). Then \\[\nP/H_0\\simeq\\mathbb F_2^2,\\qquad\nE=\\operatorname{Ind}_J^P\\theta\\text{ is irreducible of dimension }2.\n\\] The actual represented generators are a diagonal sign and an\noff-diagonal matrix that anticommute. Thus their commutator form is\nnondegenerate, and \\(\\theta|_{H_0}\\) is the full character of an actual\nnew arithmetic Heisenberg pair. No abstract operators create that group\naction.","J=H^1=J^1=U_K^1U_A^{m+1}\\). Define \\[\nP=U_K^1U_A^m.\n\\] This is an actual compact open group, with \\(J\\triangleleft P\\),\n\\([P:J]=2\\) and \\(P\\cap(K^\\times J)=J\\). For \\(g\\in P\\setminus J\\), the\nactual intertwining theorem makes \\(\\lambda=\\theta^g/\\theta\\)\nnontrivial. The commutator \\([P,J]\\subset U_A^{m+1}\\) lies in the forced\nadditive-character domain, so \\(\\lambda\\) has values \\(\\{\\pm1\\}\\). Put\n\\(H_0=\\ker\\lambda\\). Then \\[\nP/H_0\\simeq\\mathbb F_2^2,\\qquad\nE=\\operatorname{Ind}_J^P\\theta\\text{ is irreducible of dimension }2.\n\\] The actual represented generators are a diagonal sign and an\noff-diagonal matrix that anticommute. Thus their commutator form is\nnondegenerate, and \\(\\theta|_{H_0}\\) is the full character of an actual\nnew arithmetic Heisenberg pair. No abstract operators create that group\naction.",". Define \\[\nP=U_K^1U_A^m.\n\\] This is an actual compact open group, with \\(J\\triangleleft P\\),\n\\([P:J]=2\\) and \\(P\\cap(K^\\times J)=J\\). For \\(g\\in P\\setminus J\\), the\nactual intertwining theorem makes \\(\\lambda=\\theta^g/\\theta\\)\nnontrivial. The commutator \\([P,J]\\subset U_A^{m+1}\\) lies in the forced\nadditive-character domain, so \\(\\lambda\\) has values \\(\\{\\pm1\\}\\). Put\n\\(H_0=\\ker\\lambda\\). Then \\[\nP/H_0\\simeq\\mathbb F_2^2,\\qquad\nE=\\operatorname{Ind}_J^P\\theta\\text{ is irreducible of dimension }2.\n\\] The actual represented generators are a diagonal sign and an\noff-diagonal matrix that anticommute. Thus their commutator form is\nnondegenerate, and \\(\\theta|_{H_0}\\) is the full character of an actual\nnew arithmetic Heisenberg pair. No abstract operators create that group\naction.","\\([P,J]\\subset U_A^{m+1}\\) lies in the forced\nadditive-character domain, so \\(\\lambda\\) has values \\(\\{\\pm1\\}\\). Put\n\\(H_0=\\ker\\lambda\\). Then \\[\nP/H_0\\simeq\\mathbb F_2^2,\\qquad\nE=\\operatorname{Ind}_J^P\\theta\\text{ is irreducible of dimension }2.\n\\] The actual represented generators are a diagonal sign and an\noff-diagonal matrix that anticommute. Thus their commutator form is\nnondegenerate, and \\(\\theta|_{H_0}\\) is the full character of an actual\nnew arithmetic Heisenberg pair. No abstract operators create that group\naction.","The identity compact-induction double coset realizes \\(E\\) inside the\nsame \\(\\pi\\), as functions supported on \\((K^\\times J)P\\). The complete\ngroup and recovery proof is\n\\path{lowrank/RANK2_ENLARGED_CARRIER167.md}.","Use the actual rank-four outer approximation field \\(E_0/F\\) of degree\ntwo and its centralizer \\(B=\\operatorname{End}_{E_0}(F^4)\\). The odd\nlower critical level is \\(r=3h_1=2m+1\\); the half leading level is\n\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4","Use the actual rank-four outer approximation field \\(E_0/F\\) of degree\ntwo and its centralizer \\(B=\\operatorname{End}_{E_0}(F^4)\\). The odd\nlower critical level is \\(r=3h_1=2m+1\\); the half leading level is\n\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has expo","Use the actual rank-four outer approximation field \\(E_0/F\\) of degree\ntwo and its centralizer \\(B=\\operatorname{End}_{E_0}(F^4)\\). The odd\nlower critical level is \\(r=3h_1=2m+1\\); the half leading level is\n\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exac","Use the actual rank-four outer approximation field \\(E_0/F\\) of degree\ntwo and its centralizer \\(B=\\operatorname{End}_{E_0}(F^4)\\). The odd\nlower critical level is \\(r=3h_1=2m+1\\); the half leading level is\n\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. Th","Use the actual rank-four outer approximation field \\(E_0/F\\) of degree\ntwo and its centralizer \\(B=\\operatorname{End}_{E_0}(F^4)\\). The odd\nlower critical level is \\(r=3h_1=2m+1\\); the half leading level is\n\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may","Use the actual rank-four outer approximation field \\(E_0/F\\) of degree\ntwo and its centralizer \\(B=\\operatorname{End}_{E_0}(F^4)\\). The odd\nlower critical level is \\(r=3h_1=2m+1\\); the half leading level is\n\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may be larger than","Use the actual rank-four outer approximation field \\(E_0/F\\) of degree\ntwo and its centralizer \\(B=\\operatorname{End}_{E_0}(F^4)\\). The odd\nlower critical level is \\(r=3h_1=2m+1\\); the half leading level is\n\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may be larger than \\(\\mu_2\\); a\nliteral extraspecial quotient is not asserted. The full character values\nare retained by the actual finite Heisenberg","actual rank-four outer approximation field \\(E_0/F\\) of degree\ntwo and its centralizer \\(B=\\operatorname{End}_{E_0}(F^4)\\). The odd\nlower critical level is \\(r=3h_1=2m+1\\); the half leading level is\n\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may be larger than \\(\\mu_2\\); a\nliteral extraspecial quotient is not asserted. The full character values\nare retained by the actual finite Heisenberg extension. Again \\","imation field \\(E_0/F\\) of degree\ntwo and its centralizer \\(B=\\operatorname{End}_{E_0}(F^4)\\). The odd\nlower critical level is \\(r=3h_1=2m+1\\); the half leading level is\n\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may be larger than \\(\\mu_2\\); a\nliteral extraspecial quotient is not asserted. The full character values\nare retained by the actual finite Heisenberg extension. Again \\(E\\)\noccurs inside the same \\…","atorname{End}_{E_0}(F^4)\\). The odd\nlower critical level is \\(r=3h_1=2m+1\\); the half leading level is\n\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may be larger than \\(\\mu_2\\); a\nliteral extraspecial quotient is not asserted. The full character values\nare retained by the actual finite Heisenberg extension. Again \\(E\\)\noccurs inside the same \\(\\pi\\) through the identity double coset. See\n\\path{lowrank/RANK4_E…","e{End}_{E_0}(F^4)\\). The odd\nlower critical level is \\(r=3h_1=2m+1\\); the half leading level is\n\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may be larger than \\(\\mu_2\\); a\nliteral extraspecial quotient is not asserted. The full character values\nare retained by the actual finite Heisenberg extension. Again \\(E\\)\noccurs inside the same \\(\\pi\\) through the identity double coset. See\n\\path{lowrank/RANK4_ENLARGED…","l level is \\(r=3h_1=2m+1\\); the half leading level is\n\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may be larger than \\(\\mu_2\\); a\nliteral extraspecial quotient is not asserted. The full character values\nare retained by the actual finite Heisenberg extension. Again \\(E\\)\noccurs inside the same \\(\\pi\\) through the identity double coset. See\n\\path{lowrank/RANK4_ENLARGED_CARRIER167.md}.","leading level is\n\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may be larger than \\(\\mu_2\\); a\nliteral extraspecial quotient is not asserted. The full character values\nare retained by the actual finite Heisenberg extension. Again \\(E\\)\noccurs inside the same \\(\\pi\\) through the identity double coset. See\n\\path{lowrank/RANK4_ENLARGED_CARRIER167.md}.","\\(a=3h_2>r\\). Define \\[\nP=J\\,U_B^m,\n\\] where \\(J=J^1\\) is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may be larger than \\(\\mu_2\\); a\nliteral extraspecial quotient is not asserted. The full character values\nare retained by the actual finite Heisenberg extension. Again \\(E\\)\noccurs inside the same \\(\\pi\\) through the identity double coset. See\n\\path{lowrank/RANK4_ENLARGED_CARRIER167.md}.",") is the actual compact type group. The filtered\ncentralizer estimate for the actual final field, without assuming\n\\(E_0\\subset K\\), proves that \\(P\\) is a compact group with \\(J\\) normal\nof index two. For a concrete new coset representative \\(u\\),\n\\(\\lambda=\\theta^u/\\theta\\) on the actual \\(H=H^1\\) is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may be larger than \\(\\mu_2\\); a\nliteral extraspecial quotient is not asserted. The full character values\nare retained by the actual finite Heisenberg extension. Again \\(E\\)\noccurs inside the same \\(\\pi\\) through the identity double coset. See\n\\path{lowrank/RANK4_ENLARGED_CARRIER167.md}.","is nontrivial with\nvalues \\(\\{\\pm1\\}\\). Put \\(H_0=\\ker\\lambda\\subset H\\). The calculation\nproves \\[\nP/H_0\\simeq\\mathbb F_2^4,\\qquad\nE=\\operatorname{Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may be larger than \\(\\mu_2\\); a\nliteral extraspecial quotient is not asserted. The full character values\nare retained by the actual finite Heisenberg extension. Again \\(E\\)\noccurs inside the same \\(\\pi\\) through the identity double coset. See\n\\path{lowrank/RANK4_ENLARGED_CARRIER167.md}.","Ind}_J^P\\eta\\text{ is irreducible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may be larger than \\(\\mu_2\\); a\nliteral extraspecial quotient is not asserted. The full character values\nare retained by the actual finite Heisenberg extension. Again \\(E\\)\noccurs inside the same \\(\\pi\\) through the identity double coset. See\n\\path{lowrank/RANK4_ENLARGED_CARRIER167.md}.","ible of dimension }4.\n\\] The new represented commutators are scalar, its quotient has exponent\ntwo, and nondegeneracy follows from its exact scalar preimage and\nirreducibility. The finite scalar center may be larger than \\(\\mu_2\\); a\nliteral extraspecial quotient is not asserted. The full character values\nare retained by the actual finite Heisenberg extension. Again \\(E\\)\noccurs inside the same \\(\\pi\\) through the identity double coset. See\n\\path{lowrank/RANK4_ENLARGED_CARRIER167.md}.","At both low ranks, \\(K^\\times\\) normalizes the new \\(P\\). In rank two\nthis follows directly from normalization of \\(A\\) and its radical. In\nrank four, a \\(K\\)-uniformizer is approximated by the derived chain\nshift \\(R\\) times a unit of \\(B\\), with relative error in \\(P_A^{n-r}\\).\nIts conjugation error on \\(U_B^m\\) lies in \\(U_A^{n-r+m}\\subset J\\),\nwhich proves normalization. The exact argument is\n\\path{lowrank/RANK4_INDUCTION_BY_STAGES167.md}.","Consequently \\(\\mathcal M=K^\\times P\\) is an actual open group compact\nmodulo the center, \\(\\mathcal N=K^\\times J\\) is normal of index two in\n\\(\\mathcal M\\), and \\[\n\\Sigma=\\operatorname{Ind}_{\\mathcal N}^{\\mathcal M}\\Lambda,\n\\qquad \\Sigma|_P\\simeq E,\n\\qquad\n\\boxed{\\pi\\simeq\\operatorname{c\\!-\\!Ind}_{\\mathcal M}^G\\Sigma.}\n\\] Restriction to \\(P\\) is irreducible, so \\(\\Sigma\\) is irreducible.\nInduction in stages proves the last equality for the original\n\\(\\Lambda\\), including its fixed central character and noncompact root.\nThis is a genuine enlarged inducing representation; it changes neither\nthe candidate nor its original canonical beta-extension data.","group compact\nmodulo the center, \\(\\mathcal N=K^\\times J\\) is normal of index two in\n\\(\\mathcal M\\), and \\[\n\\Sigma=\\operatorname{Ind}_{\\mathcal N}^{\\mathcal M}\\Lambda,\n\\qquad \\Sigma|_P\\simeq E,\n\\qquad\n\\boxed{\\pi\\simeq\\operatorname{c\\!-\\!Ind}_{\\mathcal M}^G\\Sigma.}\n\\] Restriction to \\(P\\) is irreducible, so \\(\\Sigma\\) is irreducible.\nInduction in stages proves the last equality for the original\n\\(\\Lambda\\), including its fixed central character and noncompact root.\nThis is a genuine enlarged inducing representation; it changes neither\nthe candidate nor its original canonical beta-extension data.","There is also a recovery from compact data explicitly recording that\nroot. With \\(A_0=\\Lambda(\\varpi_K)\\) and\n\\(X=\\operatorname{c\\!-\\!Ind}_J^G\\eta\\), define\n\\((Tf)(g)=A_0^{-1}f(\\varpi_Kg)\\). Then \\(X/(T-1)X\\simeq\\pi\\). After\nimposing the fixed central character, \\(\\mathcal N/(F^\\times J)\\) is\ncyclic of order \\(N\\), and the finite average\n\\(N^{-1}\\sum_{i=0}^{N-1}T^i\\) selects this same extension. These\noperators act on complex representation spaces, so their scalar division\nby \\(N\\) is valid. This is a different recovery from the Block159\nprojector onto a sector of \\(\\eta\\). The exact kernel and finite\nprojector are proved in\n\\path{lowrank/ENLARGED_CARRIER_CONTRACT167.md}.","_0=\\Lambda(\\varpi_K)\\) and\n\\(X=\\operatorname{c\\!-\\!Ind}_J^G\\eta\\), define\n\\((Tf)(g)=A_0^{-1}f(\\varpi_Kg)\\). Then \\(X/(T-1)X\\simeq\\pi\\). After\nimposing the fixed central character, \\(\\mathcal N/(F^\\times J)\\) is\ncyclic of order \\(N\\), and the finite average\n\\(N^{-1}\\sum_{i=0}^{N-1}T^i\\) selects this same extension. These\noperators act on complex representation spaces, so their scalar division\nby \\(N\\) is valid. This is a different recovery from the Block159\nprojector onto a sector of \\(\\eta\\). The exact kernel and finite\nprojector are proved in\n\\path{lowrank/ENLARGED_CARRIER_CONTRACT167.md}.","1}f(\\varpi_Kg)\\). Then \\(X/(T-1)X\\simeq\\pi\\). After\nimposing the fixed central character, \\(\\mathcal N/(F^\\times J)\\) is\ncyclic of order \\(N\\), and the finite average\n\\(N^{-1}\\sum_{i=0}^{N-1}T^i\\) selects this same extension. These\noperators act on complex representation spaces, so their scalar division\nby \\(N\\) is valid. This is a different recovery from the Block159\nprojector onto a sector of \\(\\eta\\). The exact kernel and finite\nprojector are proved in\n\\path{lowrank/ENLARGED_CARRIER_CONTRACT167.md}.","Apply the marked scalar-pushout theorem of Section 4 to the\n\\textbf{actual new} pair \\((H_0,P)\\) and its carrier \\(E\\). It has\nexactly the capacity of the whole source at each low rank. All full\nsource character values and the transported whole finite subgroup,\nlifts, squares and phase markings can be attached and retained. The\ndisplayed \\(\\Sigma\\) and induction in stages recover the same candidate.","The old H2 break targets remain their supplied actual simple-character\nquotients. The enlarged pair is their common arithmetic assembly\ncarrier; it is not renamed as the original maximal-simple \\(H^1,J^1\\)\npair. If a stronger reading requires the whole packet itself to be the\nquotient of a maximal simple character in the same candidate, the\nearlier dimension obstruction still applies. Likewise, agreement with an\nindependently fixed native quadratic marking must be checked separately.\nThe contract note proves this distinction explicitly.","The weighted dyadic mechanism now gives actual arithmetic fields, full\ncharacters and tame central completion in every rank \\(2^d\\) of the\nstated increasing odd-break family. Under the one-vertex, \\(C=1\\), full\nmarked scalar-pushout \\textbf{assembly} contract, the packet carrier can\nbe realized at every length: inside the original type for \\(d\\ge3\\), and\nin the explicit enlarged inducing carrier at \\(d=1,2\\). The source's\nprescribed central character must belong to the covered tame class. In\nparticular no arbitrary single-factor wild determinant is assumed to be\ntrivial.","weighted dyadic mechanism now gives actual arithmetic fields, full\ncharacters and tame central completion in every rank \\(2^d\\) of the\nstated increasing odd-break family. Under the one-vertex, \\(C=1\\), full\nmarked scalar-pushout \\textbf{assembly} contract, the packet carrier can\nbe realized at every length: inside the original type for \\(d\\ge3\\), and\nin the explicit enlarged inducing carrier at \\(d=1,2\\). The source's\nprescribed central character must belong to the covered tame class. In\nparticular no arbitrary single-factor wild determinant is assumed to be\ntrivial.","B-bimodularity gives s(Lift\\_E(C))=C. The lift preserves exact\nvaluation: u\\_0\\^{}(-1) is an E-unit, and expansion in 1,z gives\ncoefficient valuations distinct modulo 2. Each coefficient entry has the\nsame grade in the interleaved L-chain and the E-chain. In particular\n\\(s(P^r)=Q^r\\).","Then an actual maximal simple type in pi, with Heisenberg quotient\n\\(W=J^1/H^1\\), contains a nondegenerate symplectic\nF\\_p-subspace S of dimension","For k=F\\_p this is exactly (p−1)p\\^{}(r−1). The subspace is the image of\nthe concrete congruence group U\\^{}(N/2)(A) in \\(J^1/H^1\\).\nIt is present regardless of the later critical exponents, their\nparities, or the number of intermediate approximation fields.","Put P=rad(A), \\(U^a(A)=1+P^a\\) for a\\textgreater=1, and\nm=N/2.","For H, BK 3.1.9(ii) states that\n\\(H^a(\\beta_i)=H^a(\\beta_{i+1})\\) as soon as","Here H\\^{}m means \\(H^1\\cap U^m(A)\\). Therefore the image S of\nU\\^{}m(A) in W is canonically","For u,v in U\\^{}m(A), their group commutator is in\n\\(U^{2m}(A)=U^N(A)\\), which lies in U\\^{}(m+1)(A). The\nterminal approximation satisfies","The right side is the Heisenberg commutator pairing for the minimal\nsimple stratum of gamma. Its quotient\n\\(J^1(\\gamma)/H^1(\\gamma)\\) is precisely\n\\(U^m(A)/H^m(\\gamma)\\), because\n\\(J^1(\\gamma)=U^1(B_{\\mathrm{order}})U^m(A)\\), with\n\\(U^1(B_{\\mathrm{order}})\\subset H^1(\\gamma)\\). BK's nondegeneracy theorem for\nsimple-character commutators therefore says that the form on S is\nnondegenerate.","For general k, this is a nondegenerate alternating F\\_p-form of\ndimension \\([k:\\FF_p](n-n/p)\\). For \\(k=\\FF_p\\), it has dimension \\(n-n/p\\). This\nproves the theorem.","\\begin{enumerate}\n\\def\\labelenumi{\\arabic{enumi}.}\n\\item\n  K\\_j=F{[}b\\_j{]} is a totally ramified maximal field of degree p\\^{}j,\n  normalizing the native complete period-p\\^{}j chain A\\_j.\n\\item\n  For \\(1\\le j<d\\), in rank p\\^{}(j+1), gamma\\_j=b\\_j tensor I\\_p is a simple\n  approximation to b\\_(j+1), at the critical level\n  r\\_j=q\\_(d-j)\\textgreater0 of b\\_(j+1). They are equivalent as strata\n  at that level.\n\\item\n  For \\(1\\le j<d\\), delta\\_j=b\\_(j+1)-gamma\\_j has grade -r\\_j and F-matrix trace zero.\n\\item\n  At native rank p, b\\_1 is multiplication by alpha in the explicit\n  Artin--Schreier field","enumi{\\arabic{enumi}.}\n\\item\n  K\\_j=F{[}b\\_j{]} is a totally ramified maximal field of degree p\\^{}j,\n  normalizing the native complete period-p\\^{}j chain A\\_j.\n\\item\n  For \\(1\\le j<d\\), in rank p\\^{}(j+1), gamma\\_j=b\\_j tensor I\\_p is a simple\n  approximation to b\\_(j+1), at the critical level\n  r\\_j=q\\_(d-j)\\textgreater0 of b\\_(j+1). They are equivalent as strata\n  at that level.\n\\item\n  For \\(1\\le j<d\\), delta\\_j=b\\_(j+1)-gamma\\_j has grade -r\\_j and F-matrix trace zero.\n\\item\n  At native rank p, b\\_1 is multiplication by alpha in the explicit\n  Artin--Schreier field","Indeed, on U\\_K\\^{}a U\\_F\\^{}1 define chi\\_0(ku)=chi(k). Agreement on\nthe intersection makes this a character. Choose M\\textgreater=a such\nthat chi is trivial on U\\_K\\^{}M. The character chi\\_0 descends to a\nsubgroup of the finite abelian group \\(U_K^1/U_K^M\\). It\nextends to that whole group by successively adjoining generators and\nselecting complex roots. Pulling it back proves the assertion. This also\ndescribes an actual finite procedure for making the extension choices.","For the native minimal stratum, \\(H^1=U_E^1\\)\nU\\_(A\\_1)\\^{}(a). Define theta\\_1 by chi\\_E on U\\_E\\^{}1 and by\npsi\\_alpha on U\\_(A\\_1)\\^{}a. They agree on the intersection, so BK3.2.1\nsupplies the actual full simple character. It satisfies\ntheta\\_1\\textbar U\\_F\\^{}1=1.","Let \\(p\\) be odd, \\(F=\\mathbb F_p((t))\\), and \\[\n0<h_1<\\cdots<h_d,\\qquad p\\nmid h_i,\\qquad\nq_i=(p+1)h_i,\\qquad N=p^d.\n\\] For every such tuple there is an explicit matrix \\(\\beta\\in M_N(F)\\)\nwhose field \\(K=F[\\beta]\\) is totally ramified of degree \\(N\\),\nnormalizes the complete lattice chain, and gives a maximal simple\nstratum. It has a literal defining sequence of degrees\n\\(p,p^2,\\ldots,p^d\\), with leading level \\(n_1\\) and successive critical\nlevels \\(n_2,\\ldots,n_d\\) given by \\[\nn_j=p^{d-j}q_{d-j+1}\\quad(1\\le j\\le d).\n\\] Every native parameter in the construction, every amplified\napproximation, and every consecutive correction has matrix trace zero\nover its base field.","\\(F=\\mathbb F_p((t))\\), and \\[\n0<h_1<\\cdots<h_d,\\qquad p\\nmid h_i,\\qquad\nq_i=(p+1)h_i,\\qquad N=p^d.\n\\] For every such tuple there is an explicit matrix \\(\\beta\\in M_N(F)\\)\nwhose field \\(K=F[\\beta]\\) is totally ramified of degree \\(N\\),\nnormalizes the complete lattice chain, and gives a maximal simple\nstratum. It has a literal defining sequence of degrees\n\\(p,p^2,\\ldots,p^d\\), with leading level \\(n_1\\) and successive critical\nlevels \\(n_2,\\ldots,n_d\\) given by \\[\nn_j=p^{d-j}q_{d-j+1}\\quad(1\\le j\\le d).\n\\] Every native parameter in the construction, every amplified\napproximation, and every consecutive correction has matrix trace zero\nover its base field.","dots<h_d,\\qquad p\\nmid h_i,\\qquad\nq_i=(p+1)h_i,\\qquad N=p^d.\n\\] For every such tuple there is an explicit matrix \\(\\beta\\in M_N(F)\\)\nwhose field \\(K=F[\\beta]\\) is totally ramified of degree \\(N\\),\nnormalizes the complete lattice chain, and gives a maximal simple\nstratum. It has a literal defining sequence of degrees\n\\(p,p^2,\\ldots,p^d\\), with leading level \\(n_1\\) and successive critical\nlevels \\(n_2,\\ldots,n_d\\) given by \\[\nn_j=p^{d-j}q_{d-j+1}\\quad(1\\le j\\le d).\n\\] Every native parameter in the construction, every amplified\napproximation, and every consecutive correction has matrix trace zero\nover its base field.","qquad\nq_i=(p+1)h_i,\\qquad N=p^d.\n\\] For every such tuple there is an explicit matrix \\(\\beta\\in M_N(F)\\)\nwhose field \\(K=F[\\beta]\\) is totally ramified of degree \\(N\\),\nnormalizes the complete lattice chain, and gives a maximal simple\nstratum. It has a literal defining sequence of degrees\n\\(p,p^2,\\ldots,p^d\\), with leading level \\(n_1\\) and successive critical\nlevels \\(n_2,\\ldots,n_d\\) given by \\[\nn_j=p^{d-j}q_{d-j+1}\\quad(1\\le j\\le d).\n\\] Every native parameter in the construction, every amplified\napproximation, and every consecutive correction has matrix trace zero\nover its base field.","For this actual stratum there exists a full simple character \\(\\theta\\)\nwith \\(\\theta|_{U_F^1}=1\\). Retaining its canonical beta extension, the\nresidue character and noncompact extension can be selected so that the\nresulting irreducible admissible supercuspidal \\(\\pi\\) has any\nprescribed smooth central quasicharacter \\(\\omega\\) trivial on\n\\(U_F^1\\). These completion choices must be free or already compatible;\nincompatible immutably supplied choices are not replaced.","The same candidate has \\[\n\\begin{aligned}\n\\operatorname{rank}(\\pi)&=N, &t(\\pi)&=1,\\\\\n\\operatorname{Sw}(\\pi)&=p^{d-1}(p+1)h_d, &\n\\dim_{\\mathbb F_p}(J^1/H^1)&=N-1,\\\\\n\\dim\\eta&=p^{(N-1)/2},\\\\\na(\\pi^\\vee\\times\\pi)\n &=p^{2d}-1+\\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i.\n\\end{aligned}\n\\] These numerical identities do not identify a Weil representation, an\nendo-class, or the pair epsilon factors required by H5.","At a recursive base \\((L,t_L)\\), introduce \\[\nE=L(\\alpha),\\qquad \\alpha^p-\\alpha=t_L^{-q_d}.\n\\] This is a separable totally ramified extension of degree \\(p\\), with\n\\(v_E(\\alpha)=-q_d\\) and \\(\\operatorname{Tr}_{E/L}\\alpha=0\\). Choose\n\\(1\\le b<p\\) with \\(q_db\\equiv-1\\pmod p\\), set \\(a=(q_db+1)/p\\), and put\n\\(z=t_L^a\\alpha^b\\). Then \\(z\\) is an explicit uniformizer. Write its\nEisenstein polynomial as \\(f(X)=X^p+\\cdots+a_0\\).","recursive base \\((L,t_L)\\), introduce \\[\nE=L(\\alpha),\\qquad \\alpha^p-\\alpha=t_L^{-q_d}.\n\\] This is a separable totally ramified extension of degree \\(p\\), with\n\\(v_E(\\alpha)=-q_d\\) and \\(\\operatorname{Tr}_{E/L}\\alpha=0\\). Choose\n\\(1\\le b<p\\) with \\(q_db\\equiv-1\\pmod p\\), set \\(a=(q_db+1)/p\\), and put\n\\(z=t_L^a\\alpha^b\\). Then \\(z\\) is an explicit uniformizer. Write its\nEisenstein polynomial as \\(f(X)=X^p+\\cdots+a_0\\).","Let \\(P_0\\) extract the constant coefficient in the basis\n\\(1,z,\\ldots,z^{p-1}\\). The normalized corestriction\n\\(s=\\kappa\\operatorname{pr}\\), where \\(\\kappa=-f'(z)/z^{p-1}\\),\nsatisfies \\[\nu_0=s(P_0)=a_0/z^p\\in\\mathcal O_E^\\times,\n\\qquad\n\\operatorname{Lift}_E(C)=u_0^{-1}CP_0,\n\\qquad s(\\operatorname{Lift}_E(C))=C.\n\\] The lift preserves the exact filtration valuation. Its crucial\nadditional identity is \\[\n\\operatorname{tr}_L(\\operatorname{Lift}_E(C))\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\kappa^{-1}\\operatorname{tr}_E(C)\\bigr).\n\\] Define \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{p^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with the degree-\\(p\\) multiplication matrix as the base","Let \\(P_0\\) extract the constant coefficient in the basis\n\\(1,z,\\ldots,z^{p-1}\\). The normalized corestriction\n\\(s=\\kappa\\operatorname{pr}\\), where \\(\\kappa=-f'(z)/z^{p-1}\\),\nsatisfies \\[\nu_0=s(P_0)=a_0/z^p\\in\\mathcal O_E^\\times,\n\\qquad\n\\operatorname{Lift}_E(C)=u_0^{-1}CP_0,\n\\qquad s(\\operatorname{Lift}_E(C))=C.\n\\] The lift preserves the exact filtration valuation. Its crucial\nadditional identity is \\[\n\\operatorname{tr}_L(\\operatorname{Lift}_E(C))\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\kappa^{-1}\\operatorname{tr}_E(C)\\bigr).\n\\] Define \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{p^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with the degree-\\(p\\) multiplication matrix as the base case.\nInduction proves trace zero at every native rank. This is stron","Let \\(P_0\\) extract the constant coefficient in the basis\n\\(1,z,\\ldots,z^{p-1}\\). The normalized corestriction\n\\(s=\\kappa\\operatorname{pr}\\), where \\(\\kappa=-f'(z)/z^{p-1}\\),\nsatisfies \\[\nu_0=s(P_0)=a_0/z^p\\in\\mathcal O_E^\\times,\n\\qquad\n\\operatorname{Lift}_E(C)=u_0^{-1}CP_0,\n\\qquad s(\\operatorname{Lift}_E(C))=C.\n\\] The lift preserves the exact filtration valuation. Its crucial\nadditional identity is \\[\n\\operatorname{tr}_L(\\operatorname{Lift}_E(C))\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\kappa^{-1}\\operatorname{tr}_E(C)\\bigr).\n\\] Define \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{p^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with the degree-\\(p\\) multiplication matrix as the base case.\nInduction proves trace zero at every native rank. This is stronger than\nthe automatic zero trace obtained by multiplying an","Let \\(P_0\\) extract the constant coefficient in the basis\n\\(1,z,\\ldots,z^{p-1}\\). The normalized corestriction\n\\(s=\\kappa\\operatorname{pr}\\), where \\(\\kappa=-f'(z)/z^{p-1}\\),\nsatisfies \\[\nu_0=s(P_0)=a_0/z^p\\in\\mathcal O_E^\\times,\n\\qquad\n\\operatorname{Lift}_E(C)=u_0^{-1}CP_0,\n\\qquad s(\\operatorname{Lift}_E(C))=C.\n\\] The lift preserves the exact filtration valuation. Its crucial\nadditional identity is \\[\n\\operatorname{tr}_L(\\operatorname{Lift}_E(C))\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\kappa^{-1}\\operatorname{tr}_E(C)\\bigr).\n\\] Define \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{p^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with the degree-\\(p\\) multiplication matrix as the base case.\nInduction proves trace zero at every native rank. This is stronger than\nthe automatic zero trace obtained by multiplying any matrix multiplicity\nby \\(p\\) in","Let \\(P_0\\) extract the constant coefficient in the basis\n\\(1,z,\\ldots,z^{p-1}\\). The normalized corestriction\n\\(s=\\kappa\\operatorname{pr}\\), where \\(\\kappa=-f'(z)/z^{p-1}\\),\nsatisfies \\[\nu_0=s(P_0)=a_0/z^p\\in\\mathcal O_E^\\times,\n\\qquad\n\\operatorname{Lift}_E(C)=u_0^{-1}CP_0,\n\\qquad s(\\operatorname{Lift}_E(C))=C.\n\\] The lift preserves the exact filtration valuation. Its crucial\nadditional identity is \\[\n\\operatorname{tr}_L(\\operatorname{Lift}_E(C))\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\kappa^{-1}\\operatorname{tr}_E(C)\\bigr).\n\\] Define \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{p^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with the degree-\\(p\\) multiplication matrix as the base case.\nInduction proves trace zero at every native rank. This is stronger than\nthe automatic zero trace obtained by multiplying any matrix multiplicity\nby \\(p\\) in characteristic \\(p\\).","(P_0\\) extract the constant coefficient in the basis\n\\(1,z,\\ldots,z^{p-1}\\). The normalized corestriction\n\\(s=\\kappa\\operatorname{pr}\\), where \\(\\kappa=-f'(z)/z^{p-1}\\),\nsatisfies \\[\nu_0=s(P_0)=a_0/z^p\\in\\mathcal O_E^\\times,\n\\qquad\n\\operatorname{Lift}_E(C)=u_0^{-1}CP_0,\n\\qquad s(\\operatorname{Lift}_E(C))=C.\n\\] The lift preserves the exact filtration valuation. Its crucial\nadditional identity is \\[\n\\operatorname{tr}_L(\\operatorname{Lift}_E(C))\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\kappa^{-1}\\operatorname{tr}_E(C)\\bigr).\n\\] Define \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{p^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with the degree-\\(p\\) multiplication matrix as the base case.\nInduction proves trace zero at every native rank. This is stronger than\nthe automatic zero trace obtained by multiplying any matrix multiplicity\nby \\(p\\) in characteristic \\(p\\).",", where \\(\\kappa=-f'(z)/z^{p-1}\\),\nsatisfies \\[\nu_0=s(P_0)=a_0/z^p\\in\\mathcal O_E^\\times,\n\\qquad\n\\operatorname{Lift}_E(C)=u_0^{-1}CP_0,\n\\qquad s(\\operatorname{Lift}_E(C))=C.\n\\] The lift preserves the exact filtration valuation. Its crucial\nadditional identity is \\[\n\\operatorname{tr}_L(\\operatorname{Lift}_E(C))\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\kappa^{-1}\\operatorname{tr}_E(C)\\bigr).\n\\] Define \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{p^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with the degree-\\(p\\) multiplication matrix as the base case.\nInduction proves trace zero at every native rank. This is stronger than\nthe automatic zero trace obtained by multiplying any matrix multiplicity\nby \\(p\\) in characteristic \\(p\\).","P_0,\n\\qquad s(\\operatorname{Lift}_E(C))=C.\n\\] The lift preserves the exact filtration valuation. Its crucial\nadditional identity is \\[\n\\operatorname{tr}_L(\\operatorname{Lift}_E(C))\n =\\operatorname{Tr}_{E/L}\n   \\bigl(\\kappa^{-1}\\operatorname{tr}_E(C)\\bigr).\n\\] Define \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{p^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with the degree-\\(p\\) multiplication matrix as the base case.\nInduction proves trace zero at every native rank. This is stronger than\nthe automatic zero trace obtained by multiplying any matrix multiplicity\nby \\(p\\) in characteristic \\(p\\).","me{Tr}_{E/L}\n   \\bigl(\\kappa^{-1}\\operatorname{tr}_E(C)\\bigr).\n\\] Define \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{p^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with the degree-\\(p\\) multiplication matrix as the base case.\nInduction proves trace zero at every native rank. This is stronger than\nthe automatic zero trace obtained by multiplying any matrix multiplicity\nby \\(p\\) in characteristic \\(p\\).","appa^{-1}\\operatorname{tr}_E(C)\\bigr).\n\\] Define \\[\nB(L,t_L;h_1,\\ldots,h_d)\n =\\alpha I_{p^{d-1}}+\n   \\operatorname{Lift}_E\\bigl(B(E,z;h_1,\\ldots,h_{d-1})\\bigr),\n\\] with the degree-\\(p\\) multiplication matrix as the base case.\nInduction proves trace zero at every native rank. This is stronger than\nthe automatic zero trace obtained by multiplying any matrix multiplicity\nby \\(p\\) in characteristic \\(p\\).","For the inductive field step \\(d\\ge2\\), put \\(n=p^{d-1}q_d\\) and\n\\(R=p^{d-2}q_{d-1}\\). The derived matrix is the recursively constructed\nmaximal-field parameter, with \\(k_0=-q_1\\). The inequalities \\(n>pR>2R\\)\nimply \\(-2R>-n-q_1\\). Bushnell--Kutzko Remark 2.2.7 therefore applies to\nthe actual displayed matrix. It supplies its maximal field,\nnormalization, and critical exponent. Valuation preservation and literal\ntruncation of the recursion supply the whole defining sequence. The full\ndetails, quotient count, and conductor dependencies are in\n\\path{tower/TRACE_ZERO_AS_TOWER166.md} and\n\\path{tower/RECURSIVE_TOWER166.md}.","The full-character argument uses a specific order of operations. First\nconstruct a full character at native rank \\(p^j\\), trivial on \\(U_F^1\\).\nTransfer its full beta-determinant unit character to multiplicity \\(p\\),\nusing BK 3.6.13(ii) and 3.6.14. On a scalar \\(u\\in U_F^1\\), this gives\n\\(\\theta_j(u^p)=1\\). Next add the actual trace-zero correction, using BK\n3.3.18. It remains trivial on the scalar intersection of its domain.\nFinally extend at the next \\textbf{maximal native parameter field}: the\nforced deep character and the trivial character of \\(U_F^1\\) agree on\ntheir intersection, so they extend through a finite abelian unit\nquotient. BK 3.2.5 constructs the full simple character.","s full beta-determinant unit character to multiplicity \\(p\\),\nusing BK 3.6.13(ii) and 3.6.14. On a scalar \\(u\\in U_F^1\\), this gives\n\\(\\theta_j(u^p)=1\\). Next add the actual trace-zero correction, using BK\n3.3.18. It remains trivial on the scalar intersection of its domain.\nFinally extend at the next \\textbf{maximal native parameter field}: the\nforced deep character and the trivial character of \\(U_F^1\\) agree on\ntheir intersection, so they extend through a finite abelian unit\nquotient. BK 3.2.5 constructs the full simple character.","This procedure never substitutes an unjustified extension through a\nnonmaximal centralizer. It proves full-character compatibility,\nincluding shallow units, rather than merely cancellation of a leading\ntrace. The residue factor fixes \\(\\omega\\) on \\(\\mathbb F_p^\\times\\); an\n\\(N\\)-th-root choice for the noncompact generator fixes \\(\\omega(t)\\).\nEvery choice is recorded before H5. See\n\\path{central/TRACE_ZERO_CHARACTER_INDUCTION166.md} for the\ncomplete induction.","Assume the one-vertex scalar-pushout contract: there are \\(d\\) labelled,\nactually supplied full character extensions with whole symplectic\nquotients \\(V_i\\) of dimension two over \\(\\mathbb F_p\\). Their balanced\n\\(\\mathbb C^\\times\\)-central product has quotient\n\\(V_{\\mathrm{src}}=\\bigoplus_iV_i\\) and Heisenberg carrier dimension\n\\(p^d=N\\). Initially take the declared stabilizer \\(C=\\{1\\}\\).","For \\(p=3\\), this applies directly to the retained cubic family with\nwhole rank-three break-type quotients. For \\(p>3\\), it applies only when\nthe supplied plane data genuinely satisfy the stated original bridge\ncontract. A native minimal rank-\\(p\\) type quotient has dimension\n\\(p-1\\), so it must not automatically be called a plane.","The actual candidate from Section 1 has a nondegenerate leading subspace\n\\[\nS\\subset J^1/H^1,\\qquad\n\\dim_{\\mathbb F_p}S=(p-1)p^{d-1}\\ge2d.\n\\] Here \\(S\\) is the image of the concrete congruence subgroup at half\nthe leading even level. BK's high-depth equality preserves this subgroup\nand its commutator pairing through every later approximation; see\n\\path{capacity/LEADING_LAYER_CAPACITY166.md}, Section 1. This part\nof that note applies to the direct constructor and does not require its\nseparate classical arithmetic-induction application.","Choose a labelled symplectic embedding \\(V_{\\mathrm{src}}\\to S\\). For\neach original full character form \\[\n\\mathcal P_i=(\\mathbb C^\\times\\times J_i^1)/\n \\{(\\theta_i(h)^{-1},h):h\\in H_i^1\\}.\n\\] The balanced product and the corresponding host restriction have the\nsame alternating commutator. The center-fixing extension isomorphism,\nobtained with matched Darboux lifts and complex roots, therefore\npreserves every supplied value \\(\\theta_i(h)\\), including the shallow\nvalues.","mbedding \\(V_{\\mathrm{src}}\\to S\\). For\neach original full character form \\[\n\\mathcal P_i=(\\mathbb C^\\times\\times J_i^1)/\n \\{(\\theta_i(h)^{-1},h):h\\in H_i^1\\}.\n\\] The balanced product and the corresponding host restriction have the\nsame alternating commutator. The center-fixing extension isomorphism,\nobtained with matched Darboux lifts and complex roots, therefore\npreserves every supplied value \\(\\theta_i(h)\\), including the shallow\nvalues.","Write \\(W=V_{\\mathrm{src}}\\perp V_{\\mathrm{src}}^\\perp\\). Finite\nStone--von Neumann gives\n\\(\\eta_W\\simeq\\eta_{\\mathrm{src}}\\otimes\\eta_\\perp\\). Choose an\nisotropic eigenspace giving a line in the complementary factor. Its\nactual Heisenberg image-algebra projector \\(e\\) has rank \\(N\\). The\nretained induction/idempotent theorem recovers the \\textbf{same}\ninducing representation \\(\\Lambda\\) from \\(e\\Lambda\\), and then the same\n\\(\\pi\\) by compact induction. No candidate is changed at recovery.","p V_{\\mathrm{src}}^\\perp\\). Finite\nStone--von Neumann gives\n\\(\\eta_W\\simeq\\eta_{\\mathrm{src}}\\otimes\\eta_\\perp\\). Choose an\nisotropic eigenspace giving a line in the complementary factor. Its\nactual Heisenberg image-algebra projector \\(e\\) has rank \\(N\\). The\nretained induction/idempotent theorem recovers the \\textbf{same}\ninducing representation \\(\\Lambda\\) from \\(e\\Lambda\\), and then the same\n\\(\\pi\\) by compact induction. No candidate is changed at recovery.","When the retained independent extraspecial tensor source has \\(d\\ge2\\),\nits full determinant is trivial on wild inertia: each factor determinant\nhas order dividing \\(p\\), and tensor determinants raise it to\n\\(p^{d-1}\\). Thus its central character belongs to the class in Section\n1. For \\(d=1\\), wild triviality of the source determinant must be\nchecked separately. Independence of the tensor factors is an input, not\na consequence of distinct \\(h_i\\).","nsor source has \\(d\\ge2\\),\nits full determinant is trivial on wild inertia: each factor determinant\nhas order dividing \\(p\\), and tensor determinants raise it to\n\\(p^{d-1}\\). Thus its central character belongs to the class in Section\n1. For \\(d=1\\), wild triviality of the source determinant must be\nchecked separately. Independence of the tensor factors is an input, not\na consequence of distinct \\(h_i\\).","There is also a positive result beyond \\(C=\\{1\\}\\). Suppose the finite\nprime-to-\\(p\\) stabilizer acts trivially on the source scalar extension,\nand its prescribed honest operators on the \\textbf{entire} rank-\\(N\\)\nWeil carrier are \\(\\chi(c)I_N\\), each lying in the actual inertia image.\nAssume the candidate's central character is the determinant of that same\nsource under the fixed rank-one reciprocity convention.","Then a finite cover \\(\\widetilde C\\twoheadrightarrow C\\) acts by actual\ncentral matrices in the unchanged candidate. Its represented operators\ndescend to exactly \\(\\chi\\) on \\(C\\). Hence the attachment and recovery\nabove are equivariant for that action. A direct matrix homomorphism from\n\\(C\\) itself is not asserted and can fail.","Indeed, for residue cardinality \\(q\\), let\n\\(B=\\omega(\\operatorname{Teich}(k^\\times))\\). The actual inertia\nhypothesis gives \\(\\chi(C)^N\\subset B\\). Since \\(N\\) is a power of \\(p\\)\nand \\(\\chi(C)\\) has order prime to \\(p\\), this implies\n\\(\\chi(C)\\subset B\\). Choose \\(k_0\\) inverse to \\(N\\) modulo \\(q-1\\),\nand set \\[\n\\delta(u)=\\omega(\\operatorname{Teich}(u))^{k_0},\\qquad\n\\widetilde C=\\{(c,u):\\chi(c)=\\delta(u)\\}.\n\\] The matrix \\(\\operatorname{Teich}(u^{k_0})I_N\\) acts by \\(\\chi(c)\\).\nThe kernel acts trivially on the entire sector induction and ambient\ncompact induction, because these matrices are central. This preserves\nthe fixed \\(\\Lambda\\), its canonical beta-extension factor, and all\ncompletion choices. It does not assert tha","Indeed, for residue cardinality \\(q\\), let\n\\(B=\\omega(\\operatorname{Teich}(k^\\times))\\). The actual inertia\nhypothesis gives \\(\\chi(C)^N\\subset B\\). Since \\(N\\) is a power of \\(p\\)\nand \\(\\chi(C)\\) has order prime to \\(p\\), this implies\n\\(\\chi(C)\\subset B\\). Choose \\(k_0\\) inverse to \\(N\\) modulo \\(q-1\\),\nand set \\[\n\\delta(u)=\\omega(\\operatorname{Teich}(u))^{k_0},\\qquad\n\\widetilde C=\\{(c,u):\\chi(c)=\\delta(u)\\}.\n\\] The matrix \\(\\operatorname{Teich}(u^{k_0})I_N\\) acts by \\(\\chi(c)\\).\nThe kernel acts trivially on the entire sector induction and ambient\ncompact induction, because these matrices are central. This preserves\nthe fixed \\(\\Lambda\\), its canonical beta-extension factor, and all\ncompletion choices. It does not assert that the canonical beta extension\nalone has scalar ch","Indeed, for residue cardinality \\(q\\), let\n\\(B=\\omega(\\operatorname{Teich}(k^\\times))\\). The actual inertia\nhypothesis gives \\(\\chi(C)^N\\subset B\\). Since \\(N\\) is a power of \\(p\\)\nand \\(\\chi(C)\\) has order prime to \\(p\\), this implies\n\\(\\chi(C)\\subset B\\). Choose \\(k_0\\) inverse to \\(N\\) modulo \\(q-1\\),\nand set \\[\n\\delta(u)=\\omega(\\operatorname{Teich}(u))^{k_0},\\qquad\n\\widetilde C=\\{(c,u):\\chi(c)=\\delta(u)\\}.\n\\] The matrix \\(\\operatorname{Teich}(u^{k_0})I_N\\) acts by \\(\\chi(c)\\).\nThe kernel acts trivially on the entire sector induction and ambient\ncompact induction, because these matrices are central. This preserves\nthe fixed \\(\\Lambda\\), its canonical beta-extension factor, and all\ncompletion choices. It does not assert that the canonical beta extension\nalone has scalar character \\(\\chi\\).","y \\(q\\), let\n\\(B=\\omega(\\operatorname{Teich}(k^\\times))\\). The actual inertia\nhypothesis gives \\(\\chi(C)^N\\subset B\\). Since \\(N\\) is a power of \\(p\\)\nand \\(\\chi(C)\\) has order prime to \\(p\\), this implies\n\\(\\chi(C)\\subset B\\). Choose \\(k_0\\) inverse to \\(N\\) modulo \\(q-1\\),\nand set \\[\n\\delta(u)=\\omega(\\operatorname{Teich}(u))^{k_0},\\qquad\n\\widetilde C=\\{(c,u):\\chi(c)=\\delta(u)\\}.\n\\] The matrix \\(\\operatorname{Teich}(u^{k_0})I_N\\) acts by \\(\\chi(c)\\).\nThe kernel acts trivially on the entire sector induction and ambient\ncompact induction, because these matrices are central. This preserves\nthe fixed \\(\\Lambda\\), its canonical beta-extension factor, and all\ncompletion choices. It does not assert that the canonical beta extension\nalone has scalar character \\(\\chi\\).","k^\\times))\\). The actual inertia\nhypothesis gives \\(\\chi(C)^N\\subset B\\). Since \\(N\\) is a power of \\(p\\)\nand \\(\\chi(C)\\) has order prime to \\(p\\), this implies\n\\(\\chi(C)\\subset B\\). Choose \\(k_0\\) inverse to \\(N\\) modulo \\(q-1\\),\nand set \\[\n\\delta(u)=\\omega(\\operatorname{Teich}(u))^{k_0},\\qquad\n\\widetilde C=\\{(c,u):\\chi(c)=\\delta(u)\\}.\n\\] The matrix \\(\\operatorname{Teich}(u^{k_0})I_N\\) acts by \\(\\chi(c)\\).\nThe kernel acts trivially on the entire sector induction and ambient\ncompact induction, because these matrices are central. This preserves\nthe fixed \\(\\Lambda\\), its canonical beta-extension factor, and all\ncompletion choices. It does not assert that the canonical beta extension\nalone has scalar character \\(\\chi\\).","(\\chi(C)\\subset B\\). Choose \\(k_0\\) inverse to \\(N\\) modulo \\(q-1\\),\nand set \\[\n\\delta(u)=\\omega(\\operatorname{Teich}(u))^{k_0},\\qquad\n\\widetilde C=\\{(c,u):\\chi(c)=\\delta(u)\\}.\n\\] The matrix \\(\\operatorname{Teich}(u^{k_0})I_N\\) acts by \\(\\chi(c)\\).\nThe kernel acts trivially on the entire sector induction and ambient\ncompact induction, because these matrices are central. This preserves\nthe fixed \\(\\Lambda\\), its canonical beta-extension factor, and all\ncompletion choices. It does not assert that the canonical beta extension\nalone has scalar character \\(\\chi\\).","hi(c)=\\delta(u)\\}.\n\\] The matrix \\(\\operatorname{Teich}(u^{k_0})I_N\\) acts by \\(\\chi(c)\\).\nThe kernel acts trivially on the entire sector induction and ambient\ncompact induction, because these matrices are central. This preserves\nthe fixed \\(\\Lambda\\), its canonical beta-extension factor, and all\ncompletion choices. It does not assert that the canonical beta extension\nalone has scalar character \\(\\chi\\).","The old fixed-field central argument genuinely failed. At \\(p=3\\),\n\\((h_1,h_2)=(1,5)\\), its naturally transferred full character takes the\nvalue \\(\\zeta_3\\ne1\\) on \\(1+t^2\\), although the desired wild center is\ntrivial. A shallow character choice cannot repair a forced value. The\ntwo-stage changing-field repair is retained in\n\\path{central/CENTRAL_REPAIR166.md}; the trace-zero tower above\nresolves this central issue uniformly in every length.","The construction does not use an inverse established correspondence or\nthe classical arithmetic-induction/descent candidate to select \\(\\pi\\).\nIt does use the named BK construction and type theorems, the retained\ntype/formal-degree conductor formulas, and exact sector recovery. The\ncapacity note separately retains an alternative consequence for a\ncandidate supplied by the substantial classical arithmetic constructor;\nthat is a different route and is not an input to Section 1.","\\begin{enumerate}\n\\def\\labelenumi{\\arabic{enumi}.}\n\\tightlist\n\\item\n  \\textbf{Nontrivial prescribed symplectic stabilizer actions.} For this\n  totally ramified full-degree host, the stronger inner-normalizer\n  interpretation forces prime-to-\\(p\\) actions on the quotient to be\n  trivial. It therefore cannot implement a genuinely nontrivial supplied\n  symplectic action. The raw original contract does not automatically\n  require that stronger interpretation; it still needs an actual valid\n  arithmetic comparison, not relabelling of external operators.\n\\item\n  \\textbf{Arbitrary admissible source packets and transport diagrams.}\n  The proved attachment has the whole-plane, one-vertex, finite\n  scalar-pushout hypotheses above. It does not establish all source\n  classes, arbitrary graphs, or prescribed numerical layer placement.\n\\item\n  \\textbf{The dyadic construction with all markings.} A separate\n  field-level recursion and some capaci","f{Arbitrary admissible source packets and transport diagrams.}\n  The proved attachment has the whole-plane, one-vertex, finite\n  scalar-pushout hypotheses above. It does not establish all source\n  classes, arbitrary graphs, or prescribed numerical layer placement.\n\\item\n  \\textbf{The dyadic construction with all markings.} A separate\n  field-level recursion and some capacity statements exist at \\(p=2\\).\n  The odd-prime trace-zero/full-character theorem here does not settle\n  that case.\n\\end{enumerate}","The arbitrary-length theorem is proved symbolically and independently\nreviewed. The exact \\(p=3,d=2,(h_1,h_2)=(1,2)\\) computation in\n\\texttt{tower/as\\_rank9check/} verifies the displayed matrix, trace\nidentities, corestriction, and levels; it is a finite check, not the\nuniversal proof. The separate rank-27 run in \\texttt{tower/run\\_rank27/}\nchecks the earlier non-trace-zero recursion and must not be described as\nan AS central completion example. No Lean build was run.","Then v\\_E(z)=pa-qb=1, so z is a uniformizer and E=L(z). Its monic\nminimal polynomial \\(f(X)=X^p+a_{p-1}X^{p-1}+\\cdots+a_0\\)\nis Eisenstein. It is an explicit polynomial: compute the product over\nthe p conjugates","and reduce coefficients using \\(\\alpha^p-\\alpha=t_L^{-q}\\).\nThe invariant coefficients belong to L. This is a finite algebraic\ninstruction, with no choice of an unknown extension or matrix.","Indeed v\\_E(a\\_k)\\textgreater=p for 1\\textless=k\\textless p, so the\ncontribution indexed by a\\_k has valuation at least k greater than the\nfirst contribution. Applying s entrywise on E\\^{}m gives a B-bimodule\ncorestriction from End\\_L(E\\^{}m) to B=End\\_E(E\\^{}m). With the complete\ninterleaved period-pm L-chain and period-m E-chain, it preserves grades\nand satisfies \\(s(P^r)=Q^r\\).","equal to its E-grade, and distinct i have valuations distinct modulo p.\nThus the minimum cannot cancel. This also proves the stated surjectivity\n\\(s(P^r)=Q^r\\) without a compactness or lifting-existence\nassumption.","The field degrees and critical levels of \\(B^{[j]}\\) are exactly","Write R=rad(A), \\(U_A^a=1+R^a\\) for a\\textgreater=1, and\n\\(U_K^a=1+\\mathfrak p_K^a\\). Let theta be a full simple character.\nSince q is odd and the residue field is F2, the actual maximal simple\ntype group and character groups coincide:","The new Heisenberg pair is an actual pair of compact subgroups of\nGL2(F), but is not asserted to be the BK \\(H^1/J^1\\) pair of\na maximal simple type. Whether the original H2 realization relation\nallows this enlargement is a separate obligation.","K× normalizes A and every power of its radical, so U\\_K\\^{}1 normalizes\nU\\_A\\^{}m. Therefore P is a subgroup and is compact open. It is\ncontained in U\\_A\\^{}1 because m\\textgreater=1 and\n\\(U_K^1\\subset U_A^1\\).","For if ku∈U\\_A\\^{}m with k∈U\\_K\\^{}1 and u∈U\\_A\\^{}(m+1), then\n\\(k\\in K^\\times\\cap U_A^m=U_K^m\\). The reverse inclusion is immediate.\nThe complete period-two A-grade \\(R^m/R^{m+1}\\) has\nF2-dimension2, and its field grade \\(\\mathfrak p_K^m/\\mathfrak p_K^{m+1}\\) has\ndimension1. Thus","Finally, if kj∈P with k∈K× and j∈J, then \\(k\\in Pj^{-1}\\subset U_A^1\\).\nHence k∈U\\_K\\^{}1 and kj∈J. Therefore","As g²∈J, conjugation by g² fixes the character theta. Thus\n\\(\\lambda^g=\\lambda^{-1}=\\lambda\\). It follows that","The previously reviewed negative statement concerned eta of the original\nBK \\(J^1/H^1\\). That quotient remains zero. The new positive\nstatement concerns a different, explicitly constructed arithmetic pair\ninside the same ambient representation, with exact\nnoncompact-normalization-aware recovery.","Use \\(U_A^j=1+P_A^j\\) and \\(U_B^j=1+Q^j\\).\nBK3.1.7--9 give the actual groups","The quotient \\(U_B^m/U_B^{m+1}\\) has dimension two. By\nSection2 the image of J cap U\\_B\\^{}m on this grade is exactly its\ncentralizing line, of dimension one. For the converse inclusion needed\nhere, the B-approximation of a K-element of grade m differs from it by\ngrade m+n-r\\textgreater a; that error lies in U\\_A\\^{}a subset J. Thus\nthe whole centralizing line really occurs in the intersection, not only\nin an abstract associated graded space. Consequently","Its leading grade is outside the centralizing line, so u represents the\nnontrivial coset of P/J. Since \\(u^2=1+x^2\\) in\ncharacteristic two, we have u\\^{}2 in U\\_B\\^{}(2m) subset U\\_B\\^{}(m+1)\nsubset H.","First, lambda is invariant under u. Indeed u\\^{}2 belongs to J and theta\nis J-invariant, so \\(\\lambda^u=\\lambda^{-1}=\\lambda\\) by (3).\nConsequently H0 is normal in P, and theta0 is P-invariant.","Write J=H U\\_A\\^{}a. Since lambda is a character and J-invariant, it\nkills {[}H,H{]} and {[}H,U\\_A\\^{}a{]}. It also kills\n\\([U_A^a,U_A^a]\\), which lies in U\\_A\\^{}(2a). Hence\nlambda({[}J,J{]})=1. It kills every square in J as well: squares of H\nhave lambda-value one, squares of U\\_A\\^{}a lie in U\\_A\\^{}(2a), and the\nmixed commutator contributions have just been handled.","The quotient \\(K^\\times/F^\\times\\) is compact because K/F is a\nfinite totally ramified extension, and P is compact. Hence M is compact\nmodulo F\\^{}times. It is open in G because it contains the open subgroup\nJ.","Appendix B restores the approximation index \\(B^{[j]}\\) that was converted into a footnote. The earlier dimension-factor and 44 superscript repairs remain in place. All 15 original construction notes and the exact source diff accompany the release. These source restorations and local mathematical clarifications do not supply a general arithmetic witness.","The symbol $K$ denotes the original local field in the finite-theory discussion. In the explicit arithmetic constructions, $F=\\F_p((t))$ is the base field and $K=F[\\beta]$ is its totally ramified parameter field. A symplectic quotient dimension and a Heisenberg representation dimension are different quantities.","Monodromy organizes an atom into a finite Weil--Deligne chain. It does not supply an atom that is missing. Likewise a finite Heisenberg comparison identifies already supplied finite representations, rather than automatically constructing a representation of $\\GL_n(K)$.\n\\takeaway{The final existence arrow is the content of Conjecture 66. Finite comparison, arithmetic construction, and local-factor characterization are separate steps.}","\\lesson{PART I / FINITE TAME COORDINATES}{Monodromy is a finite transfer chain}{Original Section 2}\n\\plate{02_monodromy}{The arrows point from $V_j$ to $V_{j-1}$, the direction of $N$.}\nPut $V=\\bigoplus_{j=0}^{m-1}V_j$ and let $P_j$ be the coordinate projector. If $N(V_j)\\subseteq V_{j-1}$ and $N(V_0)=0$, then\n\\[\n N=\\sum_{j=1}^{m-1}P_{j-1}NP_j,\\qquad N^m=0.\n\\]\nEvery term in $N^r$ follows a path of length $r$ down the chain. There is no path of length $m$. With $F=\\operatorname{diag}(1,q,\\ldots,q^{m-1})$, each nonzero transfer block gains the conjugation factor $q^{j-1}q^{-j}=q^{-1}$.\n\\takeaway{Nilpotence and covariance follow from the displayed block structure. Assigning the initial supercuspidal atom is a separate arithmetic input.}","ART I / FINITE TAME COORDINATES}{Monodromy is a finite transfer chain}{Original Section 2}\n\\plate{02_monodromy}{The arrows point from $V_j$ to $V_{j-1}$, the direction of $N$.}\nPut $V=\\bigoplus_{j=0}^{m-1}V_j$ and let $P_j$ be the coordinate projector. If $N(V_j)\\subseteq V_{j-1}$ and $N(V_0)=0$, then\n\\[\n N=\\sum_{j=1}^{m-1}P_{j-1}NP_j,\\qquad N^m=0.\n\\]\nEvery term in $N^r$ follows a path of length $r$ down the chain. There is no path of length $m$. With $F=\\operatorname{diag}(1,q,\\ldots,q^{m-1})$, each nonzero transfer block gains the conjugation factor $q^{j-1}q^{-j}=q^{-1}$.\n\\takeaway{Nilpotence and covariance follow from the displayed block structure. Assigning the initial supercuspidal atom is a separate arithmetic input.}","finite transfer chain}{Original Section 2}\n\\plate{02_monodromy}{The arrows point from $V_j$ to $V_{j-1}$, the direction of $N$.}\nPut $V=\\bigoplus_{j=0}^{m-1}V_j$ and let $P_j$ be the coordinate projector. If $N(V_j)\\subseteq V_{j-1}$ and $N(V_0)=0$, then\n\\[\n N=\\sum_{j=1}^{m-1}P_{j-1}NP_j,\\qquad N^m=0.\n\\]\nEvery term in $N^r$ follows a path of length $r$ down the chain. There is no path of length $m$. With $F=\\operatorname{diag}(1,q,\\ldots,q^{m-1})$, each nonzero transfer block gains the conjugation factor $q^{j-1}q^{-j}=q^{-1}$.\n\\takeaway{Nilpotence and covariance follow from the displayed block structure. Assigning the initial supercuspidal atom is a separate arithmetic input.}","ansfer chain}{Original Section 2}\n\\plate{02_monodromy}{The arrows point from $V_j$ to $V_{j-1}$, the direction of $N$.}\nPut $V=\\bigoplus_{j=0}^{m-1}V_j$ and let $P_j$ be the coordinate projector. If $N(V_j)\\subseteq V_{j-1}$ and $N(V_0)=0$, then\n\\[\n N=\\sum_{j=1}^{m-1}P_{j-1}NP_j,\\qquad N^m=0.\n\\]\nEvery term in $N^r$ follows a path of length $r$ down the chain. There is no path of length $m$. With $F=\\operatorname{diag}(1,q,\\ldots,q^{m-1})$, each nonzero transfer block gains the conjugation factor $q^{j-1}q^{-j}=q^{-1}$.\n\\takeaway{Nilpotence and covariance follow from the displayed block structure. Assigning the initial supercuspidal atom is a separate arithmetic input.}","irection of $N$.}\nPut $V=\\bigoplus_{j=0}^{m-1}V_j$ and let $P_j$ be the coordinate projector. If $N(V_j)\\subseteq V_{j-1}$ and $N(V_0)=0$, then\n\\[\n N=\\sum_{j=1}^{m-1}P_{j-1}NP_j,\\qquad N^m=0.\n\\]\nEvery term in $N^r$ follows a path of length $r$ down the chain. There is no path of length $m$. With $F=\\operatorname{diag}(1,q,\\ldots,q^{m-1})$, each nonzero transfer block gains the conjugation factor $q^{j-1}q^{-j}=q^{-1}$.\n\\takeaway{Nilpotence and covariance follow from the displayed block structure. Assigning the initial supercuspidal atom is a separate arithmetic input.}","\\lesson{PART I / FINITE TAME COORDINATES}{Fourier projectors isolate tame inertia}{Original Section 3; geometric Frobenius convention}\nLet $t$ generate a finite tame quotient of order $M$, with $(q,M)=1$. For $j\\in\\mathbb Z/M\\mathbb Z$ define\n\\[\n P_j=\\frac1M\\sum_{a=0}^{M-1}\\zeta_M^{-ja}r(t)^a,\n \\qquad \\zeta_M=e^{2\\pi i/M}.\n\\]\nThe finite Fourier identities give $P_jP_k=\\delta_{jk}P_j$ and $\\sum_jP_j=I$. Thus $P_jV$ is the $\\zeta_M^j$ character space for $r(t)$.","I / FINITE TAME COORDINATES}{Fourier projectors isolate tame inertia}{Original Section 3; geometric Frobenius convention}\nLet $t$ generate a finite tame quotient of order $M$, with $(q,M)=1$. For $j\\in\\mathbb Z/M\\mathbb Z$ define\n\\[\n P_j=\\frac1M\\sum_{a=0}^{M-1}\\zeta_M^{-ja}r(t)^a,\n \\qquad \\zeta_M=e^{2\\pi i/M}.\n\\]\nThe finite Fourier identities give $P_jP_k=\\delta_{jk}P_j$ and $\\sum_jP_j=I$. Thus $P_jV$ is the $\\zeta_M^j$ character space for $r(t)$.","ME COORDINATES}{Fourier projectors isolate tame inertia}{Original Section 3; geometric Frobenius convention}\nLet $t$ generate a finite tame quotient of order $M$, with $(q,M)=1$. For $j\\in\\mathbb Z/M\\mathbb Z$ define\n\\[\n P_j=\\frac1M\\sum_{a=0}^{M-1}\\zeta_M^{-ja}r(t)^a,\n \\qquad \\zeta_M=e^{2\\pi i/M}.\n\\]\nThe finite Fourier identities give $P_jP_k=\\delta_{jk}P_j$ and $\\sum_jP_j=I$. Thus $P_jV$ is the $\\zeta_M^j$ character space for $r(t)$.","rs isolate tame inertia}{Original Section 3; geometric Frobenius convention}\nLet $t$ generate a finite tame quotient of order $M$, with $(q,M)=1$. For $j\\in\\mathbb Z/M\\mathbb Z$ define\n\\[\n P_j=\\frac1M\\sum_{a=0}^{M-1}\\zeta_M^{-ja}r(t)^a,\n \\qquad \\zeta_M=e^{2\\pi i/M}.\n\\]\nThe finite Fourier identities give $P_jP_k=\\delta_{jk}P_j$ and $\\sum_jP_j=I$. Thus $P_jV$ is the $\\zeta_M^j$ character space for $r(t)$.","The guide uses geometric Frobenius with\n\\[\n \\Phi^{-1}r(t)\\Phi=r(t)^q.\n\\]\nEquivalently $\\Phi r(t)\\Phi^{-1}=r(t)^{q^{-1}}$, where the inverse is taken modulo $M$. Substituting in the Fourier sum gives\n\\[\n \\Phi P_j\\Phi^{-1}=P_{qj}.\n\\]\n\\boundary{If the forward relation $\\Phi r(t)\\Phi^{-1}=r(t)^q$ is used instead, the projector label is $q^{-1}j$. The two directions have the same orbit sets but different conjugation identities. The release corrects the historical mixed-convention sentence.}\n\\takeaway{A packet is an actual eigenspace selected by an idempotent. Frobenius carries it to the next character packet.}","\\lesson{PART I / FINITE TAME COORDINATES}{One orbit gives one tame atom}{Original Theorem 3.2}\n\\plate{03_tame_orbit}{For $q=2$ modulo 15, the displayed geometric projector orbit is $(1,2,4,8)$.}\nFor an irreducible tame Weil representation, a union of Frobenius orbits of nonzero inertia packets would give a stable subspace. Irreducibility therefore forces one orbit, of some length $f$.","The stabilizer of a packet is $W_{K_f}=\\langle I_K,\\Phi^f\\rangle$, where $K_f/K$ is the unramified extension of degree $f$. Inertia acts on the packet by one scalar character. Its remaining stabilizer action is generated by $\\Phi^f$; an irreducible complex action of this cyclic quotient is one dimensional. Hence\n\\[\n \\tau\\simeq\\Ind_{W_{K_f}}^{W_K}\\xi,\\qquad \\dim\\tau=f.\n\\]\n\\takeaway{Orbit transitivity supplies both the dimension and the induced model. It is a theorem about the tame case.}","\\lesson{PART I / FINITE TAME COORDINATES}{Regular orbits have exact counts}{Original Proposition 3.6}\n\\plate{04_orbit_counts}{Exact integer counts; the vertical scale is logarithmic.}\nCharacters of $\\F_{q^n}^{\\times}$ are indexed modulo $q^n-1$. Characters fixed by a Frobenius power of degree $r\\mid n$ number $q^r-1$. Möbius inversion removes those belonging to smaller orbits:\n\\[\n c_n(q)=\\frac1n\\sum_{d\\mid n}\\mu(d)(q^{n/d}-1).\n\\]\nFor $q=2,n=4$, the regular exponent orbits modulo 15 are\n\\[\n (1,2,4,8),\\quad(3,6,12,9),\\quad(7,14,13,11).\n\\]\nTheir number is $\\tfrac14(15-3)=3$. The plotted values are computed by the exact divisor sum and retained in \\texttt{figure\\_data.json}.\n\\takeaway{Regularity is an exact orbit property. The plot illustrates an integer formula rather than supplying its proof.}","S}{Regular orbits have exact counts}{Original Proposition 3.6}\n\\plate{04_orbit_counts}{Exact integer counts; the vertical scale is logarithmic.}\nCharacters of $\\F_{q^n}^{\\times}$ are indexed modulo $q^n-1$. Characters fixed by a Frobenius power of degree $r\\mid n$ number $q^r-1$. Möbius inversion removes those belonging to smaller orbits:\n\\[\n c_n(q)=\\frac1n\\sum_{d\\mid n}\\mu(d)(q^{n/d}-1).\n\\]\nFor $q=2,n=4$, the regular exponent orbits modulo 15 are\n\\[\n (1,2,4,8),\\quad(3,6,12,9),\\quad(7,14,13,11).\n\\]\nTheir number is $\\tfrac14(15-3)=3$. The plotted values are computed by the exact divisor sum and retained in \\texttt{figure\\_data.json}.\n\\takeaway{Regularity is an exact orbit property. The plot illustrates an integer formula rather than supplying its proof.}","ion 3.6}\n\\plate{04_orbit_counts}{Exact integer counts; the vertical scale is logarithmic.}\nCharacters of $\\F_{q^n}^{\\times}$ are indexed modulo $q^n-1$. Characters fixed by a Frobenius power of degree $r\\mid n$ number $q^r-1$. Möbius inversion removes those belonging to smaller orbits:\n\\[\n c_n(q)=\\frac1n\\sum_{d\\mid n}\\mu(d)(q^{n/d}-1).\n\\]\nFor $q=2,n=4$, the regular exponent orbits modulo 15 are\n\\[\n (1,2,4,8),\\quad(3,6,12,9),\\quad(7,14,13,11).\n\\]\nTheir number is $\\tfrac14(15-3)=3$. The plotted values are computed by the exact divisor sum and retained in \\texttt{figure\\_data.json}.\n\\takeaway{Regularity is an exact orbit property. The plot illustrates an integer formula rather than supplying its proof.}","scale is logarithmic.}\nCharacters of $\\F_{q^n}^{\\times}$ are indexed modulo $q^n-1$. Characters fixed by a Frobenius power of degree $r\\mid n$ number $q^r-1$. Möbius inversion removes those belonging to smaller orbits:\n\\[\n c_n(q)=\\frac1n\\sum_{d\\mid n}\\mu(d)(q^{n/d}-1).\n\\]\nFor $q=2,n=4$, the regular exponent orbits modulo 15 are\n\\[\n (1,2,4,8),\\quad(3,6,12,9),\\quad(7,14,13,11).\n\\]\nTheir number is $\\tfrac14(15-3)=3$. The plotted values are computed by the exact divisor sum and retained in \\texttt{figure\\_data.json}.\n\\takeaway{Regularity is an exact orbit property. The plot illustrates an integer formula rather than supplying its proof.}","\\lesson{PART I / FINITE TAME COORDINATES}{A primitive projector selects a cuspidal block}{Original Section 4}\nLet $G_n=\\GL_n(\\F_q)$ and $T_n\\simeq\\F_{q^n}^{\\times}$ be an elliptic torus. A regular torus character determines an irreducible cuspidal representation $\\kappa_\\theta$ by the classical finite-field construction. If its character is $\\chi_\\theta$ and degree is $d_\\theta$, then\n\\[\n e_\\theta=\\frac{d_\\theta}{|G_n|}\\sum_{g\\in G_n}\\chi_\\theta(g^{-1})g\n\\]\nis its primitive central idempotent in $\\C[G_n]$.","tion 4}\nLet $G_n=\\GL_n(\\F_q)$ and $T_n\\simeq\\F_{q^n}^{\\times}$ be an elliptic torus. A regular torus character determines an irreducible cuspidal representation $\\kappa_\\theta$ by the classical finite-field construction. If its character is $\\chi_\\theta$ and degree is $d_\\theta$, then\n\\[\n e_\\theta=\\frac{d_\\theta}{|G_n|}\\sum_{g\\in G_n}\\chi_\\theta(g^{-1})g\n\\]\nis its primitive central idempotent in $\\C[G_n]$.","On an irreducible representation with character $\\psi$, this central element acts by a scalar. Taking traces and using character orthogonality makes the scalar 1 on $\\kappa_\\theta$ and 0 on every other irreducible block.","For a proper parabolic $P=LU$, averaging over its unipotent radical gives\n\\[\n e_U=|U|^{-1}\\sum_{u\\in U}u,\\qquad e_Ue_\\theta=e_\\theta e_U=0.\n\\]\nThe average selects $U$-fixed vectors; the cuspidal block has none.\n\\boundary{The existence and classification of $\\kappa_\\theta$ are classical inputs. The idempotent calculation explains its exact finite selector.}","\\lesson{PART I / FINITE TAME COORDINATES}{Tensor refinement is a diagonal orbit problem}{Original Theorem 5.1}\n\\plate{05_tensor_grid}{The $4\\times6$ grid separates into two diagonal translation orbits, labelled 0 and 1.}\nThe simultaneous translation action on $\\mathbb Z/n\\mathbb Z\\times\\mathbb Z/m\\mathbb Z$ has\n\\[\n d=\\gcd(n,m)\\quad\\text{orbits, each of length}\\quad L=\\operatorname{lcm}(n,m).\n\\]\nMackey decomposition then gives\n\\[\n \\Ind_{W_{K_n}}^{W_K}\\xi\\otimes\\Ind_{W_{K_m}}^{W_K}\\eta\n \\simeq\\bigoplus_{i=0}^{d-1}\n \\Ind_{W_{K_L}}^{W_K}\\bigl(\\xi_{L,n}\\eta_{L,m}^{\\Phi^i}\\bigr).\n\\]\nHere the subscripts denote norm lifts and the Frobenius superscript follows the fixed source convention. The dimension identity is exact: $dL=nm$.\n\\boundary{These are interaction packets before combining isomorphic summands. A refined induced summand need not rem","\\lesson{PART I / FINITE TAME COORDINATES}{Tensor refinement is a diagonal orbit problem}{Original Theorem 5.1}\n\\plate{05_tensor_grid}{The $4\\times6$ grid separates into two diagonal translation orbits, labelled 0 and 1.}\nThe simultaneous translation action on $\\mathbb Z/n\\mathbb Z\\times\\mathbb Z/m\\mathbb Z$ has\n\\[\n d=\\gcd(n,m)\\quad\\text{orbits, each of length}\\quad L=\\operatorname{lcm}(n,m).\n\\]\nMackey decomposition then gives\n\\[\n \\Ind_{W_{K_n}}^{W_K}\\xi\\otimes\\Ind_{W_{K_m}}^{W_K}\\eta\n \\simeq\\bigoplus_{i=0}^{d-1}\n \\Ind_{W_{K_L}}^{W_K}\\bigl(\\xi_{L,n}\\eta_{L,m}^{\\Phi^i}\\bigr).\n\\]\nHere the subscripts denote norm lifts and the Frobenius superscript follows the fixed source convention. The dimension identity is exact: $dL=nm$.\n\\boundary{These are interaction packets before combining isomorphic summands. A refined induced summand need not remain irreducible.}","RDINATES}{Tensor refinement is a diagonal orbit problem}{Original Theorem 5.1}\n\\plate{05_tensor_grid}{The $4\\times6$ grid separates into two diagonal translation orbits, labelled 0 and 1.}\nThe simultaneous translation action on $\\mathbb Z/n\\mathbb Z\\times\\mathbb Z/m\\mathbb Z$ has\n\\[\n d=\\gcd(n,m)\\quad\\text{orbits, each of length}\\quad L=\\operatorname{lcm}(n,m).\n\\]\nMackey decomposition then gives\n\\[\n \\Ind_{W_{K_n}}^{W_K}\\xi\\otimes\\Ind_{W_{K_m}}^{W_K}\\eta\n \\simeq\\bigoplus_{i=0}^{d-1}\n \\Ind_{W_{K_L}}^{W_K}\\bigl(\\xi_{L,n}\\eta_{L,m}^{\\Phi^i}\\bigr).\n\\]\nHere the subscripts denote norm lifts and the Frobenius superscript follows the fixed source convention. The dimension identity is exact: $dL=nm$.\n\\boundary{These are interaction packets before combining isomorphic summands. A refined induced summand need not remain irreducible.}","(n,m)\\quad\\text{orbits, each of length}\\quad L=\\operatorname{lcm}(n,m).\n\\]\nMackey decomposition then gives\n\\[\n \\Ind_{W_{K_n}}^{W_K}\\xi\\otimes\\Ind_{W_{K_m}}^{W_K}\\eta\n \\simeq\\bigoplus_{i=0}^{d-1}\n \\Ind_{W_{K_L}}^{W_K}\\bigl(\\xi_{L,n}\\eta_{L,m}^{\\Phi^i}\\bigr).\n\\]\nHere the subscripts denote norm lifts and the Frobenius superscript follows the fixed source convention. The dimension identity is exact: $dL=nm$.\n\\boundary{These are interaction packets before combining isomorphic summands. A refined induced summand need not remain irreducible.}","\\lesson{PART I / FINITE TAME COORDINATES}{Exponent coordinates test inertia triviality}{Original Proposition 5.3}\nWrite the residue character exponents as $a\\bmod(q^n-1)$ and $b\\bmod(q^m-1)$. The $i$th refined packet has exponent\n\\[\n c_i=a\\frac{q^L-1}{q^n-1}+bq^i\\frac{q^L-1}{q^m-1}\n \\pmod{q^L-1}.\n\\]\nNorm lifting multiplies an exponent by the quotient of group orders. Shifting the second character by Frobenius multiplies its exponent by $q^i$. Multiplication of characters adds the two terms.","{Exponent coordinates test inertia triviality}{Original Proposition 5.3}\nWrite the residue character exponents as $a\\bmod(q^n-1)$ and $b\\bmod(q^m-1)$. The $i$th refined packet has exponent\n\\[\n c_i=a\\frac{q^L-1}{q^n-1}+bq^i\\frac{q^L-1}{q^m-1}\n \\pmod{q^L-1}.\n\\]\nNorm lifting multiplies an exponent by the quotient of group orders. Shifting the second character by Frobenius multiplies its exponent by $q^i$. Multiplication of characters adds the two terms.","For $q=2,n=4,m=6$, the common modulus is $2^{12}-1=4095$. Taking $a=b=1$, the two packets have\n\\[\n c_0=273+65=338,\\qquad c_1=273+130=403.\n\\]\nBoth are nonzero modulo 4095, so neither packet contributes inertia invariants.","\\takeaway{The test is $c_i=0$ in the common modulus. A numerical-looking residue formula is an exact character calculation.}","\\lesson{PART I / FINITE TAME COORDINATES}{The closing scalar carries noncompact information}{Original Propositions 3.5 and 4.4}\n\\plate{06_cycle_scalar}{The closing scalar is basis-invariant after multiplication around the cycle.}\nIn an orbit basis let $\\Phi e_j=e_{j+1}$ for $j<n-1$ and $\\Phi e_{n-1}=ae_0$. Then\n\\[\n \\det(XI-\\Phi)=X^n-a,\\qquad \\det\\Phi=(-1)^{n-1}a.\n\\]\nRescaling the individual basis vectors changes edge coefficients, but their product remains $a$. An unramified twist $\\nu$ does not change the inertia orbit and sends the closing scalar to $a\\nu(\\Phi)^n$.\n\\takeaway{Compact inertia information alone does not determine the unramified completion. The cyclic scalar is additional data.}","arries noncompact information}{Original Propositions 3.5 and 4.4}\n\\plate{06_cycle_scalar}{The closing scalar is basis-invariant after multiplication around the cycle.}\nIn an orbit basis let $\\Phi e_j=e_{j+1}$ for $j<n-1$ and $\\Phi e_{n-1}=ae_0$. Then\n\\[\n \\det(XI-\\Phi)=X^n-a,\\qquad \\det\\Phi=(-1)^{n-1}a.\n\\]\nRescaling the individual basis vectors changes edge coefficients, but their product remains $a$. An unramified twist $\\nu$ does not change the inertia orbit and sends the closing scalar to $a\\nu(\\Phi)^n$.\n\\takeaway{Compact inertia information alone does not determine the unramified completion. The cyclic scalar is additional data.}","tion}{Original Propositions 3.5 and 4.4}\n\\plate{06_cycle_scalar}{The closing scalar is basis-invariant after multiplication around the cycle.}\nIn an orbit basis let $\\Phi e_j=e_{j+1}$ for $j<n-1$ and $\\Phi e_{n-1}=ae_0$. Then\n\\[\n \\det(XI-\\Phi)=X^n-a,\\qquad \\det\\Phi=(-1)^{n-1}a.\n\\]\nRescaling the individual basis vectors changes edge coefficients, but their product remains $a$. An unramified twist $\\nu$ does not change the inertia orbit and sends the closing scalar to $a\\nu(\\Phi)^n$.\n\\takeaway{Compact inertia information alone does not determine the unramified completion. The cyclic scalar is additional data.}","sing scalar is basis-invariant after multiplication around the cycle.}\nIn an orbit basis let $\\Phi e_j=e_{j+1}$ for $j<n-1$ and $\\Phi e_{n-1}=ae_0$. Then\n\\[\n \\det(XI-\\Phi)=X^n-a,\\qquad \\det\\Phi=(-1)^{n-1}a.\n\\]\nRescaling the individual basis vectors changes edge coefficients, but their product remains $a$. An unramified twist $\\nu$ does not change the inertia orbit and sends the closing scalar to $a\\nu(\\Phi)^n$.\n\\takeaway{Compact inertia information alone does not determine the unramified completion. The cyclic scalar is additional data.}","\\lesson{PART I / FINITE TAME COORDINATES}{Local polynomials and fixed vectors}{Original Propositions 5.4--5.8}\nFor a packet of orbit length $r$ and closing scalar $a$,\n\\[\n \\det(I-u\\Phi\\mid V^{I_K})=\n \\begin{cases}1-au^r,&\\text{if inertia acts trivially},","\\lesson{PART I / FINITE TAME COORDINATES}{Local polynomials and fixed vectors}{Original Propositions 5.4--5.8}\nFor a packet of orbit length $r$ and closing scalar $a$,\n\\[\n \\det(I-u\\Phi\\mid V^{I_K})=\n \\begin{cases}1-au^r,&\\text{if inertia acts trivially},\\\\1,&\\text{otherwise}.\\end{cases}\n\\]\nThus tame tensor local factors multiply $(1-a_iu^{r_i})^{-1}$ over precisely the refined packets whose inertia exponent vanishes.","1,&\\text{otherwise}.\\end{cases}\n\\]\nThus tame tensor local factors multiply $(1-a_iu^{r_i})^{-1}$ over precisely the refined packets whose inertia exponent vanishes.","An inertia-trivial companion cycle has a Weil-fixed vector exactly when $a_i=1$, and then its fixed space is one dimensional. Therefore\n\\[\n \\dim(\\tau_\\xi\\otimes\\tau_\\eta)^{W_K}\n =\\#\\{i:c_i=0,\\ a_i=1\\}.\n\\]\nFor an irreducible nontrivial tame atom of dimension $n>1$, wild inertia acts trivially and inertia has no fixed vectors. Hence\n\\[\n \\Sw(\\tau)=0,\\qquad \\operatorname{Art}(\\tau)=n.\n\\]\n\\boundary{These exact tame formulas do not establish the full wild pair-factor identities required by H5.}","\\lesson{PART I / FINITE TAME COORDINATES}{Finite Gauss phases are unit scalars}{Original Section 6}\nFor a nontrivial additive character $\\psi$ of $\\F_q$ and a nontrivial multiplicative character $\\chi$ of $\\F_q^\\times$, define\n\\[\n G(\\chi,\\psi)=\\sum_{x\\in\\F_q^\\times}\\chi(x)\\psi(x),\\qquad\n \\gamma(\\chi,\\psi)=q^{-1/2}G(\\chi,\\psi).\n\\]\nThe exact identities are\n\\[\n |G(\\chi,\\psi)|^2=q,\\qquad\n \\gamma(\\chi,\\psi)\\gamma(\\chi^{-1},\\psi)=\\chi(-1).\n\\]\nThe magnitude calculation substitutes $x=ty$ in the double sum. The inner sum is $q-1$ for $t=1$ and $-1$ otherwise; multiplicative-character orthogonality supplies the result.","Original Section 6}\nFor a nontrivial additive character $\\psi$ of $\\F_q$ and a nontrivial multiplicative character $\\chi$ of $\\F_q^\\times$, define\n\\[\n G(\\chi,\\psi)=\\sum_{x\\in\\F_q^\\times}\\chi(x)\\psi(x),\\qquad\n \\gamma(\\chi,\\psi)=q^{-1/2}G(\\chi,\\psi).\n\\]\nThe exact identities are\n\\[\n |G(\\chi,\\psi)|^2=q,\\qquad\n \\gamma(\\chi,\\psi)\\gamma(\\chi^{-1},\\psi)=\\chi(-1).\n\\]\nThe magnitude calculation substitutes $x=ty$ in the double sum. The inner sum is $q-1$ for $t=1$ and $-1$ otherwise; multiplicative-character orthogonality supplies the result.","ve character $\\psi$ of $\\F_q$ and a nontrivial multiplicative character $\\chi$ of $\\F_q^\\times$, define\n\\[\n G(\\chi,\\psi)=\\sum_{x\\in\\F_q^\\times}\\chi(x)\\psi(x),\\qquad\n \\gamma(\\chi,\\psi)=q^{-1/2}G(\\chi,\\psi).\n\\]\nThe exact identities are\n\\[\n |G(\\chi,\\psi)|^2=q,\\qquad\n \\gamma(\\chi,\\psi)\\gamma(\\chi^{-1},\\psi)=\\chi(-1).\n\\]\nThe magnitude calculation substitutes $x=ty$ in the double sum. The inner sum is $q-1$ for $t=1$ and $-1$ otherwise; multiplicative-character orthogonality supplies the result.","er $\\psi$ of $\\F_q$ and a nontrivial multiplicative character $\\chi$ of $\\F_q^\\times$, define\n\\[\n G(\\chi,\\psi)=\\sum_{x\\in\\F_q^\\times}\\chi(x)\\psi(x),\\qquad\n \\gamma(\\chi,\\psi)=q^{-1/2}G(\\chi,\\psi).\n\\]\nThe exact identities are\n\\[\n |G(\\chi,\\psi)|^2=q,\\qquad\n \\gamma(\\chi,\\psi)\\gamma(\\chi^{-1},\\psi)=\\chi(-1).\n\\]\nThe magnitude calculation substitutes $x=ty$ in the double sum. The inner sum is $q-1$ for $t=1$ and $-1$ otherwise; multiplicative-character orthogonality supplies the result.","of $\\F_q$ and a nontrivial multiplicative character $\\chi$ of $\\F_q^\\times$, define\n\\[\n G(\\chi,\\psi)=\\sum_{x\\in\\F_q^\\times}\\chi(x)\\psi(x),\\qquad\n \\gamma(\\chi,\\psi)=q^{-1/2}G(\\chi,\\psi).\n\\]\nThe exact identities are\n\\[\n |G(\\chi,\\psi)|^2=q,\\qquad\n \\gamma(\\chi,\\psi)\\gamma(\\chi^{-1},\\psi)=\\chi(-1).\n\\]\nThe magnitude calculation substitutes $x=ty$ in the double sum. The inner sum is $q-1$ for $t=1$ and $-1$ otherwise; multiplicative-character orthogonality supplies the result.","\\lesson{PART II / FINITE WILD COMPARISON}{Translations and multipliers create Heisenberg packets}{Original Section 7}\n\\plate{07_heisenberg}{The commutator is central and nontrivial when the pairing is nonzero.}\nFor odd $p$ let $V=X\\oplus X^*$ and\n\\[\n b((x,\\xi),(y,\\eta))=\\eta(x)-\\xi(y).\n\\]\nThe group $H(V)=V\\times\\F_p$ has product\n\\[\n (v,z)(w,t)=(v+w,z+t+\\tfrac12b(v,w)).\n\\]\nOn $\\mathcal H=\\C^X$ the Schrödinger operator is\n\\[\n (\\rho(x,\\xi,z)f)(u)\n =\\psi(z+\\xi(u)+\\tfrac12\\xi(x))f(u+x).\n\\]\nTranslations and character multipliers generate every matrix unit by finite Fourier inversion. Thus the representation is irreducible and has dimension $|X|=\\sqrt{|V|}$.","ILD COMPARISON}{Translations and multipliers create Heisenberg packets}{Original Section 7}\n\\plate{07_heisenberg}{The commutator is central and nontrivial when the pairing is nonzero.}\nFor odd $p$ let $V=X\\oplus X^*$ and\n\\[\n b((x,\\xi),(y,\\eta))=\\eta(x)-\\xi(y).\n\\]\nThe group $H(V)=V\\times\\F_p$ has product\n\\[\n (v,z)(w,t)=(v+w,z+t+\\tfrac12b(v,w)).\n\\]\nOn $\\mathcal H=\\C^X$ the Schrödinger operator is\n\\[\n (\\rho(x,\\xi,z)f)(u)\n =\\psi(z+\\xi(u)+\\tfrac12\\xi(x))f(u+x).\n\\]\nTranslations and character multipliers generate every matrix unit by finite Fourier inversion. Thus the representation is irreducible and has dimension $|X|=\\sqrt{|V|}$.","and\n\\[\n b((x,\\xi),(y,\\eta))=\\eta(x)-\\xi(y).\n\\]\nThe group $H(V)=V\\times\\F_p$ has product\n\\[\n (v,z)(w,t)=(v+w,z+t+\\tfrac12b(v,w)).\n\\]\nOn $\\mathcal H=\\C^X$ the Schrödinger operator is\n\\[\n (\\rho(x,\\xi,z)f)(u)\n =\\psi(z+\\xi(u)+\\tfrac12\\xi(x))f(u+x).\n\\]\nTranslations and character multipliers generate every matrix unit by finite Fourier inversion. Thus the representation is irreducible and has dimension $|X|=\\sqrt{|V|}$.","\\lesson{PART II / FINITE WILD COMPARISON}{An isotropic average selects one line}{Original Propositions 7.3--7.6}\nThe Schrödinger character is supported on the center:\n\\[\n \\Tr\\rho(v,z)=\\begin{cases}|X|\\psi(z),&v=0,\\\\0,&v\\ne0.\\end{cases}\n\\]\nA nonzero translation has no fixed coordinate. A nontrivial multiplier has vanishing character sum.","For a maximal isotropic subspace $L$ and a subgroup lift $\\widetilde L$ with trivial central coordinate, the average\n\\[\n P_L=\\frac1{|L|}\\sum_{\\ell\\in\\widetilde L}\\rho(\\ell)\n\\]\nis a self-adjoint idempotent. Its trace is $|X|/|L|=1$, because only the identity contributes. Its range is therefore one dimensional.","For $L=X$ this range consists of constant functions:\n\\[\n (P_Lf)(u)=|X|^{-1}\\sum_{x\\in X}f(u+x).\n\\]\n\\takeaway{The complementary-line projector later used in arithmetic attachment is an exact operator in the Heisenberg image algebra. Its existence is finite representation theory.}","\\lesson{PART II / FINITE WILD COMPARISON}{Projective choices leave a commutator invariant}{Original Section 8}\n\\plate{08_radical}{A cocycle contains choices; its alternating commutator forgets scalar rescaling.}\nFor $T_aT_b=c(a,b)T_{a+b}$ define $b_c(a,b)=c(a,b)c(b,a)^{-1}$. Replacing $T_a$ by $\\lambda_aT_a$ multiplies $c$ by the symmetric coboundary $\\lambda_a\\lambda_b\\lambda_{a+b}^{-1}$, so $b_c$ is unchanged.","The radical consists of $a$ commuting projectively with every $b$. Its quotient carries a nondegenerate alternating pairing. In the nondegenerate case the twisted group algebra has scalar center and is a full matrix algebra:\n\\[\n \\C_c[A]\\simeq M_{\\sqrt{|A|}}(\\C).\n\\]\n\\boundary{This constructs the finite envelope for a supplied projective action. It does not produce a canonical action for every arbitrary wild inertia atom.}","\\lesson{PART II / FINITE WILD COMPARISON}{Clifford multiplicity is the correct action space}{Original Section 9}\nSupply finite groups $N\\triangleleft G$, elementary abelian quotient $A=G/N$, and irreducibles $\\Pi$ and $\\rho$, with $\\rho$ fixed up to isomorphism by $G$ and $\\Pi|_N$ isotypic. The multiplicity space is\n\\[\n M_\\rho=\\operatorname{Hom}_N(W,X),\\qquad W\\otimes M_\\rho\\simeq X.\n\\]\nFor a section $s:A\\to G$ and intertwiners $I_a$ of $\\rho$, write\n\\[\n \\Pi(s(a))=I_a\\otimes T_a.\n\\]\nThe operators $T_a$ give a projective action on $M_\\rho$. Section and intertwiner changes alter it only by scalar changes and conjugation. Its commutator, radical, and symplectic quotient are intrinsic to this supplied datum.","on{PART II / FINITE WILD COMPARISON}{Clifford multiplicity is the correct action space}{Original Section 9}\nSupply finite groups $N\\triangleleft G$, elementary abelian quotient $A=G/N$, and irreducibles $\\Pi$ and $\\rho$, with $\\rho$ fixed up to isomorphism by $G$ and $\\Pi|_N$ isotypic. The multiplicity space is\n\\[\n M_\\rho=\\operatorname{Hom}_N(W,X),\\qquad W\\otimes M_\\rho\\simeq X.\n\\]\nFor a section $s:A\\to G$ and intertwiners $I_a$ of $\\rho$, write\n\\[\n \\Pi(s(a))=I_a\\otimes T_a.\n\\]\nThe operators $T_a$ give a projective action on $M_\\rho$. Section and intertwiner changes alter it only by scalar changes and conjugation. Its commutator, radical, and symplectic quotient are intrinsic to this supplied datum.","PARISON}{Clifford multiplicity is the correct action space}{Original Section 9}\nSupply finite groups $N\\triangleleft G$, elementary abelian quotient $A=G/N$, and irreducibles $\\Pi$ and $\\rho$, with $\\rho$ fixed up to isomorphism by $G$ and $\\Pi|_N$ isotypic. The multiplicity space is\n\\[\n M_\\rho=\\operatorname{Hom}_N(W,X),\\qquad W\\otimes M_\\rho\\simeq X.\n\\]\nFor a section $s:A\\to G$ and intertwiners $I_a$ of $\\rho$, write\n\\[\n \\Pi(s(a))=I_a\\otimes T_a.\n\\]\nThe operators $T_a$ give a projective action on $M_\\rho$. Section and intertwiner changes alter it only by scalar changes and conjugation. Its commutator, radical, and symplectic quotient are intrinsic to this supplied datum.","Clifford multiplicity is the correct action space}{Original Section 9}\nSupply finite groups $N\\triangleleft G$, elementary abelian quotient $A=G/N$, and irreducibles $\\Pi$ and $\\rho$, with $\\rho$ fixed up to isomorphism by $G$ and $\\Pi|_N$ isotypic. The multiplicity space is\n\\[\n M_\\rho=\\operatorname{Hom}_N(W,X),\\qquad W\\otimes M_\\rho\\simeq X.\n\\]\nFor a section $s:A\\to G$ and intertwiners $I_a$ of $\\rho$, write\n\\[\n \\Pi(s(a))=I_a\\otimes T_a.\n\\]\nThe operators $T_a$ give a projective action on $M_\\rho$. Section and intertwiner changes alter it only by scalar changes and conjugation. Its commutator, radical, and symplectic quotient are intrinsic to this supplied datum.","licity is the correct action space}{Original Section 9}\nSupply finite groups $N\\triangleleft G$, elementary abelian quotient $A=G/N$, and irreducibles $\\Pi$ and $\\rho$, with $\\rho$ fixed up to isomorphism by $G$ and $\\Pi|_N$ isotypic. The multiplicity space is\n\\[\n M_\\rho=\\operatorname{Hom}_N(W,X),\\qquad W\\otimes M_\\rho\\simeq X.\n\\]\nFor a section $s:A\\to G$ and intertwiners $I_a$ of $\\rho$, write\n\\[\n \\Pi(s(a))=I_a\\otimes T_a.\n\\]\nThe operators $T_a$ give a projective action on $M_\\rho$. Section and intertwiner changes alter it only by scalar changes and conjugation. Its commutator, radical, and symplectic quotient are intrinsic to this supplied datum.","ection 9}\nSupply finite groups $N\\triangleleft G$, elementary abelian quotient $A=G/N$, and irreducibles $\\Pi$ and $\\rho$, with $\\rho$ fixed up to isomorphism by $G$ and $\\Pi|_N$ isotypic. The multiplicity space is\n\\[\n M_\\rho=\\operatorname{Hom}_N(W,X),\\qquad W\\otimes M_\\rho\\simeq X.\n\\]\nFor a section $s:A\\to G$ and intertwiners $I_a$ of $\\rho$, write\n\\[\n \\Pi(s(a))=I_a\\otimes T_a.\n\\]\nThe operators $T_a$ give a projective action on $M_\\rho$. Section and intertwiner changes alter it only by scalar changes and conjugation. Its commutator, radical, and symplectic quotient are intrinsic to this supplied datum.","triangleleft G$, elementary abelian quotient $A=G/N$, and irreducibles $\\Pi$ and $\\rho$, with $\\rho$ fixed up to isomorphism by $G$ and $\\Pi|_N$ isotypic. The multiplicity space is\n\\[\n M_\\rho=\\operatorname{Hom}_N(W,X),\\qquad W\\otimes M_\\rho\\simeq X.\n\\]\nFor a section $s:A\\to G$ and intertwiners $I_a$ of $\\rho$, write\n\\[\n \\Pi(s(a))=I_a\\otimes T_a.\n\\]\nThe operators $T_a$ give a projective action on $M_\\rho$. Section and intertwiner changes alter it only by scalar changes and conjugation. Its commutator, radical, and symplectic quotient are intrinsic to this supplied datum.","If the commutator is nondegenerate, $\\dim M_\\rho=\\sqrt{|A|}$. A tame stabilizer acting by automorphisms of the datum preserves the induced symplectic form.\n\\boundary{The normal group, isotypic constituent, and one-break interpretation are assumptions of the restricted construction. They have not been constructed for every wild atom.}","\\lesson{PART II / FINITE WILD COMPARISON}{The type-side quotient is an actual group quotient}{Original Section 10}\nFor an actual maximal simple character $\\theta$ of $\\GL_n(K)$, classical simple-type theory supplies\n\\[\n H_\\theta^1\\subseteq J_\\theta^1\\subseteq J_\\theta,\n \\qquad V_\\theta=J_\\theta^1/H_\\theta^1.\n\\]\nThe commutator pairing is encoded by\n\\[\n \\psi_0(h_\\theta(\\bar x,\\bar y))=\\theta([x,y]).\n\\]\nIt is nondegenerate and alternating. The corresponding irreducible Heisenberg representation $\\eta_\\theta$ of $J_\\theta^1$ contains $\\theta$ and satisfies\n\\[\n \\dim\\eta_\\theta=\\sqrt{|J_\\theta^1/H_\\theta^1|}.\n\\]\nThe canonical beta extension supplies a normalized extension of this representation using the declared type-theory data and finite corrections.\n\\boundary{An arbitrary abstract symplectic space is not automatically a quotient of an actual simple ch","\\lesson{PART II / FINITE WILD COMPARISON}{The type-side quotient is an actual group quotient}{Original Section 10}\nFor an actual maximal simple character $\\theta$ of $\\GL_n(K)$, classical simple-type theory supplies\n\\[\n H_\\theta^1\\subseteq J_\\theta^1\\subseteq J_\\theta,\n \\qquad V_\\theta=J_\\theta^1/H_\\theta^1.\n\\]\nThe commutator pairing is encoded by\n\\[\n \\psi_0(h_\\theta(\\bar x,\\bar y))=\\theta([x,y]).\n\\]\nIt is nondegenerate and alternating. The corresponding irreducible Heisenberg representation $\\eta_\\theta$ of $J_\\theta^1$ contains $\\theta$ and satisfies\n\\[\n \\dim\\eta_\\theta=\\sqrt{|J_\\theta^1/H_\\theta^1|}.\n\\]\nThe canonical beta extension supplies a normalized extension of this representation using the declared type-theory data and finite corrections.\n\\boundary{An arbitrary abstract symplectic space is not automatically a quotient of an actual simple character. An eq","\\lesson{PART II / FINITE WILD COMPARISON}{The type-side quotient is an actual group quotient}{Original Section 10}\nFor an actual maximal simple character $\\theta$ of $\\GL_n(K)$, classical simple-type theory supplies\n\\[\n H_\\theta^1\\subseteq J_\\theta^1\\subseteq J_\\theta,\n \\qquad V_\\theta=J_\\theta^1/H_\\theta^1.\n\\]\nThe commutator pairing is encoded by\n\\[\n \\psi_0(h_\\theta(\\bar x,\\bar y))=\\theta([x,y]).\n\\]\nIt is nondegenerate and alternating. The corresponding irreducible Heisenberg representation $\\eta_\\theta$ of $J_\\theta^1$ contains $\\theta$ and satisfies\n\\[\n \\dim\\eta_\\theta=\\sqrt{|J_\\theta^1/H_\\theta^1|}.\n\\]\nThe canonical beta extension supplies a normalized extension of this representation using the declared type-theory data and finite corrections.\n\\boundary{An arbitrary abstract symplectic space is not automatically a quotient of an actual simple character. An equivalent inducing carrier need not have this original quotient or a newly invented canonical beta extension.}","inal Section 10}\nFor an actual maximal simple character $\\theta$ of $\\GL_n(K)$, classical simple-type theory supplies\n\\[\n H_\\theta^1\\subseteq J_\\theta^1\\subseteq J_\\theta,\n \\qquad V_\\theta=J_\\theta^1/H_\\theta^1.\n\\]\nThe commutator pairing is encoded by\n\\[\n \\psi_0(h_\\theta(\\bar x,\\bar y))=\\theta([x,y]).\n\\]\nIt is nondegenerate and alternating. The corresponding irreducible Heisenberg representation $\\eta_\\theta$ of $J_\\theta^1$ contains $\\theta$ and satisfies\n\\[\n \\dim\\eta_\\theta=\\sqrt{|J_\\theta^1/H_\\theta^1|}.\n\\]\nThe canonical beta extension supplies a normalized extension of this representation using the declared type-theory data and finite corrections.\n\\boundary{An arbitrary abstract symplectic space is not automatically a quotient of an actual simple character. An equivalent inducing carrier need not have this original quotient or a newly invented canonical beta extension.}","For an actual maximal simple character $\\theta$ of $\\GL_n(K)$, classical simple-type theory supplies\n\\[\n H_\\theta^1\\subseteq J_\\theta^1\\subseteq J_\\theta,\n \\qquad V_\\theta=J_\\theta^1/H_\\theta^1.\n\\]\nThe commutator pairing is encoded by\n\\[\n \\psi_0(h_\\theta(\\bar x,\\bar y))=\\theta([x,y]).\n\\]\nIt is nondegenerate and alternating. The corresponding irreducible Heisenberg representation $\\eta_\\theta$ of $J_\\theta^1$ contains $\\theta$ and satisfies\n\\[\n \\dim\\eta_\\theta=\\sqrt{|J_\\theta^1/H_\\theta^1|}.\n\\]\nThe canonical beta extension supplies a normalized extension of this representation using the declared type-theory data and finite corrections.\n\\boundary{An arbitrary abstract symplectic space is not automatically a quotient of an actual simple character. An equivalent inducing carrier need not have this original quotient or a newly invented canonical beta extension.}","simple character $\\theta$ of $\\GL_n(K)$, classical simple-type theory supplies\n\\[\n H_\\theta^1\\subseteq J_\\theta^1\\subseteq J_\\theta,\n \\qquad V_\\theta=J_\\theta^1/H_\\theta^1.\n\\]\nThe commutator pairing is encoded by\n\\[\n \\psi_0(h_\\theta(\\bar x,\\bar y))=\\theta([x,y]).\n\\]\nIt is nondegenerate and alternating. The corresponding irreducible Heisenberg representation $\\eta_\\theta$ of $J_\\theta^1$ contains $\\theta$ and satisfies\n\\[\n \\dim\\eta_\\theta=\\sqrt{|J_\\theta^1/H_\\theta^1|}.\n\\]\nThe canonical beta extension supplies a normalized extension of this representation using the declared type-theory data and finite corrections.\n\\boundary{An arbitrary abstract symplectic space is not automatically a quotient of an actual simple character. An equivalent inducing carrier need not have this original quotient or a newly invented canonical beta extension.}","$ of $\\GL_n(K)$, classical simple-type theory supplies\n\\[\n H_\\theta^1\\subseteq J_\\theta^1\\subseteq J_\\theta,\n \\qquad V_\\theta=J_\\theta^1/H_\\theta^1.\n\\]\nThe commutator pairing is encoded by\n\\[\n \\psi_0(h_\\theta(\\bar x,\\bar y))=\\theta([x,y]).\n\\]\nIt is nondegenerate and alternating. The corresponding irreducible Heisenberg representation $\\eta_\\theta$ of $J_\\theta^1$ contains $\\theta$ and satisfies\n\\[\n \\dim\\eta_\\theta=\\sqrt{|J_\\theta^1/H_\\theta^1|}.\n\\]\nThe canonical beta extension supplies a normalized extension of this representation using the declared type-theory data and finite corrections.\n\\boundary{An arbitrary abstract symplectic space is not automatically a quotient of an actual simple character. An equivalent inducing carrier need not have this original quotient or a newly invented canonical beta extension.}","\\lesson{PART II / FINITE WILD COMPARISON}{A supplied bridge reduces comparison to a scalar}{Original Section 11}\n\\plate{09_bridge}{The symplectic isometry is an input. Stone--von Neumann supplies the lower intertwiner.}\nSupply an equivariant symplectic isometry $\\iota:V_\\rho\\to V_\\theta$, a common nontrivial central character, and, in characteristic two, the compatible refinements and lift of central extensions.","The two pulled-back irreducible Heisenberg representations then have the same central character. Stone--von Neumann gives an invertible $U$ with\n\\[\n U\\omega_\\rho(h)=\\omega_\\theta(\\iota_H(h))U.\n\\]\nSchur's lemma makes $U$ unique up to a nonzero scalar. After honest cyclic stabilizer linearizations, their discrepancy is a character $\\delta:C\\to\\C^\\times$. Equality at one generator kills this character.\n\\boundary{Finite linearization does not identify the stabilizer action with an arithmetic action on a newly constructed common candidate.}","\\lesson{PART II / FINITE WILD COMPARISON}{Odd phases have a determinant certificate}{Original Section 12}\nFor an odd prime $p$, a symplectic space $V$ of dimension $2r$, and fixed-point-free $g\\in\\Sp(V)$, the declared finite Weil normalization gives\n\\[\n \\Phi_p(g,V)=\\left(\\frac{-1}{p}\\right)^r\n \\left(\\frac{\\det(g-I)}p\\right).\n\\]\nWith $P_g(X)=\\det(XI-g)$, even dimension gives $\\det(g-I)=P_g(1)$. The phase is therefore read from a characteristic polynomial and a square class.","For a hyperbolic space $U\\oplus U^*$ with $g=(a,(a^{-1})^*)$ and $a-I$ invertible,\n\\[\n \\Phi_p(g,U\\oplus U^*)=\\left(\\frac{\\det a}{p}\\right).\n\\]\nIf $g$ has order prime to $p$, then\n\\[\n V=\\ker(g-I)\\perp\\operatorname{im}(g-I).\n\\]\nThe moving summand is nondegenerate and fixed-point-free. Orthogonal block phases multiply.\n\\boundary{The fixed-point-free formula applies to the moving part. The full trace with a fixed part requires its declared normalization data.}","\\lesson{PART II / FINITE WILD COMPARISON}{Two dyadic refinements can have opposite signs}{Original Section 13}\n\\plate{10_quadratic_planes}{The entries are exact values in $\\F_2$; both polar forms are $xy'+x'y$.}\nA quadratic refinement satisfies $q(x+y)+q(x)+q(y)=b(x,y)$. On a plane,\n\\[\n q_0(x,y)=xy,\\qquad q_1(x,y)=xy+x+y.\n\\]\nFor $q_0$ there are three zeros and one one, so its Gauss sum is $2$. For $q_1$ there is one zero and three ones, so its sum is $-2$. The alternating forms agree but the Arf invariants differ.\n\\[\n \\sum_{x\\in V}(-1)^{q(x)}=(-1)^{\\Arf(q)}2^r,\n \\qquad \\dim V=2r.\n\\]\n\\takeaway{Keeping the alternating form alone loses dyadic phase information. The refinement and the marked central extension must travel with the packet.}","esson{PART II / FINITE WILD COMPARISON}{Two dyadic refinements can have opposite signs}{Original Section 13}\n\\plate{10_quadratic_planes}{The entries are exact values in $\\F_2$; both polar forms are $xy'+x'y$.}\nA quadratic refinement satisfies $q(x+y)+q(x)+q(y)=b(x,y)$. On a plane,\n\\[\n q_0(x,y)=xy,\\qquad q_1(x,y)=xy+x+y.\n\\]\nFor $q_0$ there are three zeros and one one, so its Gauss sum is $2$. For $q_1$ there is one zero and three ones, so its sum is $-2$. The alternating forms agree but the Arf invariants differ.\n\\[\n \\sum_{x\\in V}(-1)^{q(x)}=(-1)^{\\Arf(q)}2^r,\n \\qquad \\dim V=2r.\n\\]\n\\takeaway{Keeping the alternating form alone loses dyadic phase information. The refinement and the marked central extension must travel with the packet.}","inements can have opposite signs}{Original Section 13}\n\\plate{10_quadratic_planes}{The entries are exact values in $\\F_2$; both polar forms are $xy'+x'y$.}\nA quadratic refinement satisfies $q(x+y)+q(x)+q(y)=b(x,y)$. On a plane,\n\\[\n q_0(x,y)=xy,\\qquad q_1(x,y)=xy+x+y.\n\\]\nFor $q_0$ there are three zeros and one one, so its Gauss sum is $2$. For $q_1$ there is one zero and three ones, so its sum is $-2$. The alternating forms agree but the Arf invariants differ.\n\\[\n \\sum_{x\\in V}(-1)^{q(x)}=(-1)^{\\Arf(q)}2^r,\n \\qquad \\dim V=2r.\n\\]\n\\takeaway{Keeping the alternating form alone loses dyadic phase information. The refinement and the marked central extension must travel with the packet.}","uadratic_planes}{The entries are exact values in $\\F_2$; both polar forms are $xy'+x'y$.}\nA quadratic refinement satisfies $q(x+y)+q(x)+q(y)=b(x,y)$. On a plane,\n\\[\n q_0(x,y)=xy,\\qquad q_1(x,y)=xy+x+y.\n\\]\nFor $q_0$ there are three zeros and one one, so its Gauss sum is $2$. For $q_1$ there is one zero and three ones, so its sum is $-2$. The alternating forms agree but the Arf invariants differ.\n\\[\n \\sum_{x\\in V}(-1)^{q(x)}=(-1)^{\\Arf(q)}2^r,\n \\qquad \\dim V=2r.\n\\]\n\\takeaway{Keeping the alternating form alone loses dyadic phase information. The refinement and the marked central extension must travel with the packet.}","\\lesson{PART II / FINITE WILD COMPARISON}{Refinements form an affine family}{Original Propositions 13.2 and 13.8--13.9}\n\\plate{11_arf_counts}{Exact counts of refinements of a fixed nondegenerate alternating space.}\nThe difference between two refinements is linear. Nondegeneracy identifies every linear form with $b(v,\\cdot)$, so all refinements are\n\\[\n q_v(x)=q(x)+b(v,x),\\qquad \\Arf(q_v)=\\Arf(q)+q(v).\n\\]\nThe numbers of the two Arf types are\n\\[\n 2^{r-1}(2^r+1),\\qquad 2^{r-1}(2^r-1).\n\\]\nFor $r=1$ these are 3 and 1. For $r=3$ they are 36 and 28. Orthogonal sums add Arf invariants and multiply normalized Gauss signs.\n\\takeaway{Equal dimension and equal alternating form do not force equal refinement. An embedding fixed in advance must preserve $q$ pointwise.}","nements form an affine family}{Original Propositions 13.2 and 13.8--13.9}\n\\plate{11_arf_counts}{Exact counts of refinements of a fixed nondegenerate alternating space.}\nThe difference between two refinements is linear. Nondegeneracy identifies every linear form with $b(v,\\cdot)$, so all refinements are\n\\[\n q_v(x)=q(x)+b(v,x),\\qquad \\Arf(q_v)=\\Arf(q)+q(v).\n\\]\nThe numbers of the two Arf types are\n\\[\n 2^{r-1}(2^r+1),\\qquad 2^{r-1}(2^r-1).\n\\]\nFor $r=1$ these are 3 and 1. For $r=3$ they are 36 and 28. Orthogonal sums add Arf invariants and multiply normalized Gauss signs.\n\\takeaway{Equal dimension and equal alternating form do not force equal refinement. An embedding fixed in advance must preserve $q$ pointwise.}","ions 13.2 and 13.8--13.9}\n\\plate{11_arf_counts}{Exact counts of refinements of a fixed nondegenerate alternating space.}\nThe difference between two refinements is linear. Nondegeneracy identifies every linear form with $b(v,\\cdot)$, so all refinements are\n\\[\n q_v(x)=q(x)+b(v,x),\\qquad \\Arf(q_v)=\\Arf(q)+q(v).\n\\]\nThe numbers of the two Arf types are\n\\[\n 2^{r-1}(2^r+1),\\qquad 2^{r-1}(2^r-1).\n\\]\nFor $r=1$ these are 3 and 1. For $r=3$ they are 36 and 28. Orthogonal sums add Arf invariants and multiply normalized Gauss signs.\n\\takeaway{Equal dimension and equal alternating form do not force equal refinement. An embedding fixed in advance must preserve $q$ pointwise.}","ate{11_arf_counts}{Exact counts of refinements of a fixed nondegenerate alternating space.}\nThe difference between two refinements is linear. Nondegeneracy identifies every linear form with $b(v,\\cdot)$, so all refinements are\n\\[\n q_v(x)=q(x)+b(v,x),\\qquad \\Arf(q_v)=\\Arf(q)+q(v).\n\\]\nThe numbers of the two Arf types are\n\\[\n 2^{r-1}(2^r+1),\\qquad 2^{r-1}(2^r-1).\n\\]\nFor $r=1$ these are 3 and 1. For $r=3$ they are 36 and 28. Orthogonal sums add Arf invariants and multiply normalized Gauss signs.\n\\takeaway{Equal dimension and equal alternating form do not force equal refinement. An embedding fixed in advance must preserve $q$ pointwise.}","\\[\n q_v(x)=q(x)+b(v,x),\\qquad \\Arf(q_v)=\\Arf(q)+q(v).\n\\]\nThe numbers of the two Arf types are\n\\[\n 2^{r-1}(2^r+1),\\qquad 2^{r-1}(2^r-1).\n\\]\nFor $r=1$ these are 3 and 1. For $r=3$ they are 36 and 28. Orthogonal sums add Arf invariants and multiply normalized Gauss signs.\n\\takeaway{Equal dimension and equal alternating form do not force equal refinement. An embedding fixed in advance must preserve $q$ pointwise.}","\\lesson{PART II / FINITE WILD COMPARISON}{A spanning tree fixes the vertex gauges}{Original Theorem 14.2}\n\\plate{12_holonomy}{Four vertices and five edges leave two fundamental cycle tests.}\nLet $A_e:V_u\\to V_v$ and $B_e:W_u\\to W_v$ be invertible transports, with reverse-edge inverses. Fix a root comparison $U_r$.","Choose a spanning tree. Along its unique root path $p_v$ define\n\\[\n U_v=B_{p_v}U_rA_{p_v}^{-1}.\n\\]\nThis forces every tree-edge equation $U_vA_e=B_eU_u$. Each non-tree edge closes one fundamental cycle. Its edge equation holds exactly when the root comparison intertwines that cycle's holonomy.\n\\takeaway{All edge equations are equivalent to the fundamental cycle identities $U_rA_\\gamma=B_\\gamma U_r$. Holonomy need not be individually trivial; the two holonomy representations must agree through the root map.}","\\lesson{PART II / FINITE WILD COMPARISON}{Cycle defects measure edge residuals exactly}{Original Proposition 14.6}\nFor a non-tree edge $e:u\\to v$, let\n\\[\n D_e=U_rA_{p_v}^{-1}A_eA_{p_u}\n      -B_{p_v}^{-1}B_eB_{p_u}U_r.\n\\]\nSubstituting the tree-defined gauges gives the exact identity\n\\[\n U_vA_e-B_eU_u=B_{p_v}D_eA_{p_u}^{-1}.\n\\]\nConsequently\n\\[\n \\|U_vA_e-B_eU_u\\|\n \\le\\|B_{p_v}\\|\\,\\|D_e\\|\\,\\|A_{p_u}^{-1}\\|.\n\\]\nSmall cycle defects give small edge residuals when the tree transports are conditioned. Zero defects give exact finite gluing. In the scalar case there are $|E|-|V|+1$ independent tests on a connected graph.","}^{-1}A_eA_{p_u}\n      -B_{p_v}^{-1}B_eB_{p_u}U_r.\n\\]\nSubstituting the tree-defined gauges gives the exact identity\n\\[\n U_vA_e-B_eU_u=B_{p_v}D_eA_{p_u}^{-1}.\n\\]\nConsequently\n\\[\n \\|U_vA_e-B_eU_u\\|\n \\le\\|B_{p_v}\\|\\,\\|D_e\\|\\,\\|A_{p_u}^{-1}\\|.\n\\]\nSmall cycle defects give small edge residuals when the tree transports are conditioned. Zero defects give exact finite gluing. In the scalar case there are $|E|-|V|+1$ independent tests on a connected graph.","\\lesson{PART II / FINITE WILD COMPARISON}{Orthogonal breaks become tensor factors}{Original Section 15}\n\\plate{13_break_tensor}{Symplectic direct sum and Heisenberg tensor product are compatible.}\nFor $V=\\perp_sV_s$ and compatible central characters,\n\\[\n \\omega_V\\simeq\\bigotimes_s\\omega_{V_s},\\qquad\n \\prod_s\\sqrt{|V_s|}=\\sqrt{|V|}.\n\\]\nThe groups combine by a central product. Tensoring the individual intertwiners gives the full finite bridge, once the stabilizer actions and phases agree. Transport-graph compatibility is then the matrix holonomy condition.\n\\boundary{Separate actual type quotients at each break do not, merely by tensoring, become one actual maximal simple type in the source atom's rank. Constructing a common arithmetic carrier and recovering a candidate are the extra realization requirements.}","\\lesson{PART III / ACTUAL ARITHMETIC CONSTRUCTION}{A recursive matrix gives a literal parameter}{Blocks 166--167, tower notes}\n\\plate{14_recursive_parameter}{The recursive lift is an actual filtered right inverse of normalized corestriction.}\nStart over a coefficient field $L$ with uniformizer $t_L$ and form\n\\[\n E=L(\\alpha),\\qquad \\alpha^p-\\alpha=t_L^{-q}.\n\\]\nAt a recursive stage, $C$ is an actual matrix over $E$. The outer matrix is\n\\[\n B=\\alpha I_m+\\operatorname{Lift}_E(C).\n\\]\nThe lift has prescribed corestriction and preserves its exact filtration valuation. The field and simple-stratum proof uses the actual displayed matrix, together with the cited type-theory derived-stratum criterion.\n\\boundary{The derived field $E[C]$ is not identified with the final field $L[B]$. The recursion proves a literal defining sequence without silently declaring intermediate fields to be nested inside the final field.}","II / ACTUAL ARITHMETIC CONSTRUCTION}{A recursive matrix gives a literal parameter}{Blocks 166--167, tower notes}\n\\plate{14_recursive_parameter}{The recursive lift is an actual filtered right inverse of normalized corestriction.}\nStart over a coefficient field $L$ with uniformizer $t_L$ and form\n\\[\n E=L(\\alpha),\\qquad \\alpha^p-\\alpha=t_L^{-q}.\n\\]\nAt a recursive stage, $C$ is an actual matrix over $E$. The outer matrix is\n\\[\n B=\\alpha I_m+\\operatorname{Lift}_E(C).\n\\]\nThe lift has prescribed corestriction and preserves its exact filtration valuation. The field and simple-stratum proof uses the actual displayed matrix, together with the cited type-theory derived-stratum criterion.\n\\boundary{The derived field $E[C]$ is not identified with the final field $L[B]$. The recursion proves a literal defining sequence without silently declaring intermediate fields to be nested inside the final field.}","STRUCTION}{A recursive matrix gives a literal parameter}{Blocks 166--167, tower notes}\n\\plate{14_recursive_parameter}{The recursive lift is an actual filtered right inverse of normalized corestriction.}\nStart over a coefficient field $L$ with uniformizer $t_L$ and form\n\\[\n E=L(\\alpha),\\qquad \\alpha^p-\\alpha=t_L^{-q}.\n\\]\nAt a recursive stage, $C$ is an actual matrix over $E$. The outer matrix is\n\\[\n B=\\alpha I_m+\\operatorname{Lift}_E(C).\n\\]\nThe lift has prescribed corestriction and preserves its exact filtration valuation. The field and simple-stratum proof uses the actual displayed matrix, together with the cited type-theory derived-stratum criterion.\n\\boundary{The derived field $E[C]$ is not identified with the final field $L[B]$. The recursion proves a literal defining sequence without silently declaring intermediate fields to be nested inside the final field.}","uad \\alpha^p-\\alpha=t_L^{-q}.\n\\]\nAt a recursive stage, $C$ is an actual matrix over $E$. The outer matrix is\n\\[\n B=\\alpha I_m+\\operatorname{Lift}_E(C).\n\\]\nThe lift has prescribed corestriction and preserves its exact filtration valuation. The field and simple-stratum proof uses the actual displayed matrix, together with the cited type-theory derived-stratum criterion.\n\\boundary{The derived field $E[C]$ is not identified with the final field $L[B]$. The recursion proves a literal defining sequence without silently declaring intermediate fields to be nested inside the final field.}","e stage, $C$ is an actual matrix over $E$. The outer matrix is\n\\[\n B=\\alpha I_m+\\operatorname{Lift}_E(C).\n\\]\nThe lift has prescribed corestriction and preserves its exact filtration valuation. The field and simple-stratum proof uses the actual displayed matrix, together with the cited type-theory derived-stratum criterion.\n\\boundary{The derived field $E[C]$ is not identified with the final field $L[B]$. The recursion proves a literal defining sequence without silently declaring intermediate fields to be nested inside the final field.}","\\lesson{PART III / ACTUAL ARITHMETIC CONSTRUCTION}{The approximation degrees and critical levels}{Blocks 166--167, combined arithmetic statements}\n\\plate{15_degree_levels}{These examples were recomputed exactly; the all-length assertions are proved separately.}\nFor increasing permitted $h_1<\\cdots<h_d$, put $q_i=(p+1)h_i$ and $N=p^d$. The approximation-field degrees are $p,p^2,\\ldots,p^d$, with\n\\[\n n_j=p^{d-j}q_{d-j+1}.\n\\]\nFor odd $p$, require $p\\nmid h_i$. For $p=2$, require all $h_i$ odd and positive. The leading ambient level is $n_1$; $n_2,\\ldots,n_d$ are the successive critical approximation levels.\n\\boundary{Every amplified ambient approximation still has leading level $n_1$. The declining values plotted here are critical levels, not new leading valuations for those ambient matrices.}","CONSTRUCTION}{The approximation degrees and critical levels}{Blocks 166--167, combined arithmetic statements}\n\\plate{15_degree_levels}{These examples were recomputed exactly; the all-length assertions are proved separately.}\nFor increasing permitted $h_1<\\cdots<h_d$, put $q_i=(p+1)h_i$ and $N=p^d$. The approximation-field degrees are $p,p^2,\\ldots,p^d$, with\n\\[\n n_j=p^{d-j}q_{d-j+1}.\n\\]\nFor odd $p$, require $p\\nmid h_i$. For $p=2$, require all $h_i$ odd and positive. The leading ambient level is $n_1$; $n_2,\\ldots,n_d$ are the successive critical approximation levels.\n\\boundary{Every amplified ambient approximation still has leading level $n_1$. The declining values plotted here are critical levels, not new leading valuations for those ambient matrices.}","}{The approximation degrees and critical levels}{Blocks 166--167, combined arithmetic statements}\n\\plate{15_degree_levels}{These examples were recomputed exactly; the all-length assertions are proved separately.}\nFor increasing permitted $h_1<\\cdots<h_d$, put $q_i=(p+1)h_i$ and $N=p^d$. The approximation-field degrees are $p,p^2,\\ldots,p^d$, with\n\\[\n n_j=p^{d-j}q_{d-j+1}.\n\\]\nFor odd $p$, require $p\\nmid h_i$. For $p=2$, require all $h_i$ odd and positive. The leading ambient level is $n_1$; $n_2,\\ldots,n_d$ are the successive critical approximation levels.\n\\boundary{Every amplified ambient approximation still has leading level $n_1$. The declining values plotted here are critical levels, not new leading valuations for those ambient matrices.}","n degrees and critical levels}{Blocks 166--167, combined arithmetic statements}\n\\plate{15_degree_levels}{These examples were recomputed exactly; the all-length assertions are proved separately.}\nFor increasing permitted $h_1<\\cdots<h_d$, put $q_i=(p+1)h_i$ and $N=p^d$. The approximation-field degrees are $p,p^2,\\ldots,p^d$, with\n\\[\n n_j=p^{d-j}q_{d-j+1}.\n\\]\nFor odd $p$, require $p\\nmid h_i$. For $p=2$, require all $h_i$ odd and positive. The leading ambient level is $n_1$; $n_2,\\ldots,n_d$ are the successive critical approximation levels.\n\\boundary{Every amplified ambient approximation still has leading level $n_1$. The declining values plotted here are critical levels, not new leading valuations for those ambient matrices.}","cal levels}{Blocks 166--167, combined arithmetic statements}\n\\plate{15_degree_levels}{These examples were recomputed exactly; the all-length assertions are proved separately.}\nFor increasing permitted $h_1<\\cdots<h_d$, put $q_i=(p+1)h_i$ and $N=p^d$. The approximation-field degrees are $p,p^2,\\ldots,p^d$, with\n\\[\n n_j=p^{d-j}q_{d-j+1}.\n\\]\nFor odd $p$, require $p\\nmid h_i$. For $p=2$, require all $h_i$ odd and positive. The leading ambient level is $n_1$; $n_2,\\ldots,n_d$ are the successive critical approximation levels.\n\\boundary{Every amplified ambient approximation still has leading level $n_1$. The declining values plotted here are critical levels, not new leading valuations for those ambient matrices.}","ements}\n\\plate{15_degree_levels}{These examples were recomputed exactly; the all-length assertions are proved separately.}\nFor increasing permitted $h_1<\\cdots<h_d$, put $q_i=(p+1)h_i$ and $N=p^d$. The approximation-field degrees are $p,p^2,\\ldots,p^d$, with\n\\[\n n_j=p^{d-j}q_{d-j+1}.\n\\]\nFor odd $p$, require $p\\nmid h_i$. For $p=2$, require all $h_i$ odd and positive. The leading ambient level is $n_1$; $n_2,\\ldots,n_d$ are the successive critical approximation levels.\n\\boundary{Every amplified ambient approximation still has leading level $n_1$. The declining values plotted here are critical levels, not new leading valuations for those ambient matrices.}","\\plate{15_degree_levels}{These examples were recomputed exactly; the all-length assertions are proved separately.}\nFor increasing permitted $h_1<\\cdots<h_d$, put $q_i=(p+1)h_i$ and $N=p^d$. The approximation-field degrees are $p,p^2,\\ldots,p^d$, with\n\\[\n n_j=p^{d-j}q_{d-j+1}.\n\\]\nFor odd $p$, require $p\\nmid h_i$. For $p=2$, require all $h_i$ odd and positive. The leading ambient level is $n_1$; $n_2,\\ldots,n_d$ are the successive critical approximation levels.\n\\boundary{Every amplified ambient approximation still has leading level $n_1$. The declining values plotted here are critical levels, not new leading valuations for those ambient matrices.}","d $h_1<\\cdots<h_d$, put $q_i=(p+1)h_i$ and $N=p^d$. The approximation-field degrees are $p,p^2,\\ldots,p^d$, with\n\\[\n n_j=p^{d-j}q_{d-j+1}.\n\\]\nFor odd $p$, require $p\\nmid h_i$. For $p=2$, require all $h_i$ odd and positive. The leading ambient level is $n_1$; $n_2,\\ldots,n_d$ are the successive critical approximation levels.\n\\boundary{Every amplified ambient approximation still has leading level $n_1$. The declining values plotted here are critical levels, not new leading valuations for those ambient matrices.}","\\lesson{PART III / ACTUAL ARITHMETIC CONSTRUCTION}{Odd towers retain native trace zero}{Block 166, recursive tower and full-character proofs}\nFor odd $p$, $\\Tr_{E/L}(\\alpha)=0$. Choose $b$ with $qb\\equiv-1\\pmod p$, let $a=(qb+1)/p$, and set $z=t_L^a\\alpha^b$. Then $z$ is a uniformizer.","For its Eisenstein polynomial $f$ and constant-coordinate projection $P_0$, the normalized corestriction uses $\\kappa=-f'(z)/z^{p-1}$. Writing $u_0=s(P_0)$,\n\\[\n \\operatorname{Lift}_E(C)=u_0^{-1}CP_0,\\qquad\n \\tr_L\\operatorname{Lift}_E(C)\n =\\Tr_{E/L}(\\kappa^{-1}\\tr_EC).\n\\]\nIf $\\tr_EC=0$, the lift has trace zero. Adding $\\alpha I_m$ preserves trace zero. Induction therefore gives trace zero at every native stage and every native correction.","\\takeaway{This is stronger than the automatic ambient trace cancellation from a multiplicity divisible by $p$. Native trace controls the scalar restriction used in full-character construction.}","\\lesson{PART III / ACTUAL ARITHMETIC CONSTRUCTION}{Dyadic trace one is deep enough}{Block 167, depth invariant}\n\\plate{16_scalar_depth}{The scalar correction disappears when its argument lies in $t\\OO_F$.}\nFix $\\psi_F$ of conductor $t\\OO_F$. On a scalar unit $u\\in U_F^s$, the additive correction sees $(u-1)\\tr\\delta$. Hence\n\\[\n v_F(\\tr\\delta)+s\\ge1\n \\quad\\Longrightarrow\\quad\n \\psi_F((u-1)\\tr\\delta)=1.\n\\]\nThe dyadic native corrections have trace 1. Its valuation is 0, so $s=1$ already satisfies the inequality. Thus trace zero is sufficient but unnecessary.\n\\boundary{Euler's weighted-visibility viewpoint motivated this step. No Euler coefficient theorem is an arithmetic proof input here. In characteristic two, the weights are uniformizer depths, not integer powers of two in the coefficient field.}","enough}{Block 167, depth invariant}\n\\plate{16_scalar_depth}{The scalar correction disappears when its argument lies in $t\\OO_F$.}\nFix $\\psi_F$ of conductor $t\\OO_F$. On a scalar unit $u\\in U_F^s$, the additive correction sees $(u-1)\\tr\\delta$. Hence\n\\[\n v_F(\\tr\\delta)+s\\ge1\n \\quad\\Longrightarrow\\quad\n \\psi_F((u-1)\\tr\\delta)=1.\n\\]\nThe dyadic native corrections have trace 1. Its valuation is 0, so $s=1$ already satisfies the inequality. Thus trace zero is sufficient but unnecessary.\n\\boundary{Euler's weighted-visibility viewpoint motivated this step. No Euler coefficient theorem is an arithmetic proof input here. In characteristic two, the weights are uniformizer depths, not integer powers of two in the coefficient field.}","\\lesson{PART III / ACTUAL ARITHMETIC CONSTRUCTION}{Native trace and ambient trace are different}{Block 167, exact trace transport}\n\\plate{17_native_ambient_trace}{A trace-one native matrix amplified by an even multiplicity has ambient trace zero.}\nFor odd $q$, let $a=(q+1)/2$ and $z=t_L^a\\alpha$. Then\n\\[\n z^2+t_L^az+t_L=0,\\quad \\Tr_{E/L}\\alpha=1,\n \\quad\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\]\nThe trace identity is\n\\[\n \\tr_L\\operatorname{Lift}_E(C)=\\Tr_{E/L}(\\alpha\\tr_EC).\n\\]\nIt sends inner trace 1 to outer trace 1. At native rank $2^j$, every constructor has trace 1; its next native correction has trace $1-2\\cdot1=1$. In rank 8, the earlier amplified approximations instead have traces $0,0$, and the final matrix has trace 1.\n\\takeaway{Only the appropriate native trace establishes the scalar-character compatibility at each recursive stage.}","UAL ARITHMETIC CONSTRUCTION}{Native trace and ambient trace are different}{Block 167, exact trace transport}\n\\plate{17_native_ambient_trace}{A trace-one native matrix amplified by an even multiplicity has ambient trace zero.}\nFor odd $q$, let $a=(q+1)/2$ and $z=t_L^a\\alpha$. Then\n\\[\n z^2+t_L^az+t_L=0,\\quad \\Tr_{E/L}\\alpha=1,\n \\quad\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\]\nThe trace identity is\n\\[\n \\tr_L\\operatorname{Lift}_E(C)=\\Tr_{E/L}(\\alpha\\tr_EC).\n\\]\nIt sends inner trace 1 to outer trace 1. At native rank $2^j$, every constructor has trace 1; its next native correction has trace $1-2\\cdot1=1$. In rank 8, the earlier amplified approximations instead have traces $0,0$, and the final matrix has trace 1.\n\\takeaway{Only the appropriate native trace establishes the scalar-character compatibility at each recursive stage.}","7_native_ambient_trace}{A trace-one native matrix amplified by an even multiplicity has ambient trace zero.}\nFor odd $q$, let $a=(q+1)/2$ and $z=t_L^a\\alpha$. Then\n\\[\n z^2+t_L^az+t_L=0,\\quad \\Tr_{E/L}\\alpha=1,\n \\quad\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\]\nThe trace identity is\n\\[\n \\tr_L\\operatorname{Lift}_E(C)=\\Tr_{E/L}(\\alpha\\tr_EC).\n\\]\nIt sends inner trace 1 to outer trace 1. At native rank $2^j$, every constructor has trace 1; its next native correction has trace $1-2\\cdot1=1$. In rank 8, the earlier amplified approximations instead have traces $0,0$, and the final matrix has trace 1.\n\\takeaway{Only the appropriate native trace establishes the scalar-character compatibility at each recursive stage.}","iplicity has ambient trace zero.}\nFor odd $q$, let $a=(q+1)/2$ and $z=t_L^a\\alpha$. Then\n\\[\n z^2+t_L^az+t_L=0,\\quad \\Tr_{E/L}\\alpha=1,\n \\quad\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\]\nThe trace identity is\n\\[\n \\tr_L\\operatorname{Lift}_E(C)=\\Tr_{E/L}(\\alpha\\tr_EC).\n\\]\nIt sends inner trace 1 to outer trace 1. At native rank $2^j$, every constructor has trace 1; its next native correction has trace $1-2\\cdot1=1$. In rank 8, the earlier amplified approximations instead have traces $0,0$, and the final matrix has trace 1.\n\\takeaway{Only the appropriate native trace establishes the scalar-character compatibility at each recursive stage.}","Then\n\\[\n z^2+t_L^az+t_L=0,\\quad \\Tr_{E/L}\\alpha=1,\n \\quad\\operatorname{Lift}_E(C)=(z^2/t_L)CP_0.\n\\]\nThe trace identity is\n\\[\n \\tr_L\\operatorname{Lift}_E(C)=\\Tr_{E/L}(\\alpha\\tr_EC).\n\\]\nIt sends inner trace 1 to outer trace 1. At native rank $2^j$, every constructor has trace 1; its next native correction has trace $1-2\\cdot1=1$. In rank 8, the earlier amplified approximations instead have traces $0,0$, and the final matrix has trace 1.\n\\takeaway{Only the appropriate native trace establishes the scalar-character compatibility at each recursive stage.}","On a scalar $u\\in U_F^1$, multiplicity $p$ transfer gives\n\\[\n \\Theta_j(uI)=\\theta_j(u^p)=1.\n\\]\nThe correction is trivial on the scalar intersection by trace zero in the odd case, or integral trace and the depth inequality in the dyadic case. The forced deep character therefore agrees with the imposed trivial scalar character on their intersection. A finite abelian unit-character extension supplies the full next character.","For an odd dyadic critical level $r$, put $m=\\lfloor r/2\\rfloor$. The endpoint satisfies $2(m+1)-r=1$, making the additive cross term lie in the character's radical.\n\\boundary{Checking a leading trace alone does not check shallow forced values. The earlier fixed-field central argument failed on an actual scalar unit; the recursive native-field proof repairs that problem.}","\\lesson{PART III / ACTUAL ARITHMETIC CONSTRUCTION}{The same arithmetic candidate has exact profiles}{Combined statements 166--167}\nLet $N=p^d$ and retain the constructed full character, canonical beta extension, and compatible central completion. Both families give an actual irreducible admissible supercuspidal candidate of $\\GL_N(F)$ with torsion number 1.","For odd $p$,\n\\[\n \\begin{aligned}\n \\Sw(\\pi)&=p^{d-1}(p+1)h_d,","For odd $p$,\n\\[\n \\begin{aligned}\n \\Sw(\\pi)&=p^{d-1}(p+1)h_d,\\\\\n \\dim_{\\F_p}(J^1/H^1)&=N-1,\\quad\\dim\\eta=p^{(N-1)/2},\\\\\n a(\\pi^\\vee\\times\\pi)&=p^{2d}-1+\\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i.\n \\end{aligned}\n\\]\nFor $p=2$,\n\\[\n \\begin{aligned}\n \\Sw(\\pi)&=3\\,2^{d-1}h_d,","a(\\pi^\\vee\\times\\pi)&=p^{2d}-1+\\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i.\n \\end{aligned}\n\\]\nFor $p=2$,\n\\[\n \\begin{aligned}\n \\Sw(\\pi)&=3\\,2^{d-1}h_d,","a(\\pi^\\vee\\times\\pi)&=p^{2d}-1+\\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i.\n \\end{aligned}\n\\]\nFor $p=2$,\n\\[\n \\begin{aligned}\n \\Sw(\\pi)&=3\\,2^{d-1}h_d,\\\\\n \\dim_{\\F_2}(J^1/H^1)&=N-2,\\quad\\dim\\eta=2^{N/2-1},\\\\\n a(\\pi^\\vee\\times\\pi)&=N^2-1+\\sum_{i=1}^d3\\,4^{i-1}h_i.\n \\end{aligned}\n\\]\n\\boundary{These identities belong to the same constructed candidate. Matching numerical profiles does not identify a source Weil atom, an endo-class, or H5 pair factors.}","\\lesson{PART III / ACTUAL ARITHMETIC CONSTRUCTION}{Dyadic capacity has two distinct dimensions}{Block 167, capacity proof}\n\\plate{18_capacity}{Compare symplectic dimensions first. The original low-rank quotient remains too small.}\nThe host symplectic quotient has dimension $N-2$ over $\\F_2$; the supplied whole source of $d$ planes has dimension $2d$. Its original-type embedding is possible by dimension exactly when $d\\ge3$.\n\\begin{center}\\begin{tabular}{rrrrr}\\toprule\n$N$ & host quotient & source quotient & host carrier & source carrier","CTUAL ARITHMETIC CONSTRUCTION}{Dyadic capacity has two distinct dimensions}{Block 167, capacity proof}\n\\plate{18_capacity}{Compare symplectic dimensions first. The original low-rank quotient remains too small.}\nThe host symplectic quotient has dimension $N-2$ over $\\F_2$; the supplied whole source of $d$ planes has dimension $2d$. Its original-type embedding is possible by dimension exactly when $d\\ge3$.\n\\begin{center}\\begin{tabular}{rrrrr}\\toprule\n$N$ & host quotient & source quotient & host carrier & source carrier","distinct dimensions}{Block 167, capacity proof}\n\\plate{18_capacity}{Compare symplectic dimensions first. The original low-rank quotient remains too small.}\nThe host symplectic quotient has dimension $N-2$ over $\\F_2$; the supplied whole source of $d$ planes has dimension $2d$. Its original-type embedding is possible by dimension exactly when $d\\ge3$.\n\\begin{center}\\begin{tabular}{rrrrr}\\toprule\n$N$ & host quotient & source quotient & host carrier & source carrier","\\lesson{PART III / ACTUAL ARITHMETIC CONSTRUCTION}{A sector can retain the whole source packet}{Blocks 166--167, scalar-pushout attachment}\nSuppose the supplied full break characters have labelled plane quotients and satisfy the one-vertex scalar-pushout assembly contract, initially with $C=1$. Their source representation has dimension $p^d=N$.","For odd $p$, the actual leading subspace has dimension $(p-1)p^{d-1}\\ge2d$. For dyadic $d\\ge3$, use the host quotient dimension $N-2\\ge2d$. Choose the declared symplectic embedding into the actual host.","An orthogonal splitting $W=V_{\\rm src}\\perp V_{\\rm src}^{\\perp}$ gives\n\\[\n \\eta_W\\simeq\\eta_{\\rm src}\\otimes\\eta_{\\perp}.\n\\]\nA maximal-isotropic average on the complementary Heisenberg factor selects a one-dimensional representation line. Tensoring its projector with the identity gives a rank-$N$ image-algebra projector. The retained induction/recovery theorem recovers the same $\\Lambda$ and then the same $\\pi$.\n\\boundary{For $p>3$, genuinely supplied plane data are required: a native minimal rank-$p$ quotient has dimension $p-1$ and is not automatically a plane. The construction retains source labels but does not prove their old numerical depth placement.}","nal splitting $W=V_{\\rm src}\\perp V_{\\rm src}^{\\perp}$ gives\n\\[\n \\eta_W\\simeq\\eta_{\\rm src}\\otimes\\eta_{\\perp}.\n\\]\nA maximal-isotropic average on the complementary Heisenberg factor selects a one-dimensional representation line. Tensoring its projector with the identity gives a rank-$N$ image-algebra projector. The retained induction/recovery theorem recovers the same $\\Lambda$ and then the same $\\pi$.\n\\boundary{For $p>3$, genuinely supplied plane data are required: a native minimal rank-$p$ quotient has dimension $p-1$ and is not automatically a plane. The construction retains source labels but does not prove their old numerical depth placement.}","rm src}\\perp V_{\\rm src}^{\\perp}$ gives\n\\[\n \\eta_W\\simeq\\eta_{\\rm src}\\otimes\\eta_{\\perp}.\n\\]\nA maximal-isotropic average on the complementary Heisenberg factor selects a one-dimensional representation line. Tensoring its projector with the identity gives a rank-$N$ image-algebra projector. The retained induction/recovery theorem recovers the same $\\Lambda$ and then the same $\\pi$.\n\\boundary{For $p>3$, genuinely supplied plane data are required: a native minimal rank-$p$ quotient has dimension $p-1$ and is not automatically a plane. The construction retains source labels but does not prove their old numerical depth placement.}","imes\\eta_{\\perp}.\n\\]\nA maximal-isotropic average on the complementary Heisenberg factor selects a one-dimensional representation line. Tensoring its projector with the identity gives a rank-$N$ image-algebra projector. The retained induction/recovery theorem recovers the same $\\Lambda$ and then the same $\\pi$.\n\\boundary{For $p>3$, genuinely supplied plane data are required: a native minimal rank-$p$ quotient has dimension $p-1$ and is not automatically a plane. The construction retains source labels but does not prove their old numerical depth placement.}","mal-isotropic average on the complementary Heisenberg factor selects a one-dimensional representation line. Tensoring its projector with the identity gives a rank-$N$ image-algebra projector. The retained induction/recovery theorem recovers the same $\\Lambda$ and then the same $\\pi$.\n\\boundary{For $p>3$, genuinely supplied plane data are required: a native minimal rank-$p$ quotient has dimension $p-1$ and is not automatically a plane. The construction retains source labels but does not prove their old numerical depth placement.}","\\lesson{PART III / ACTUAL ARITHMETIC CONSTRUCTION}{Retain the whole marked finite extension}{Block 167, markings proof}\n\\plate{21_marked_extension}{Squares are preserved by the actual map of marked extensions.}\nFor a full character $\\theta_i$, its scalar pushout is\n\\[\n \\mathcal P_i=(\\C^\\times\\times J_i^1)/\n \\{(\\theta_i(h)^{-1},h):h\\in H_i^1\\}.\n\\]\nA center-fixing comparison retains every supplied character value. In the dyadic case retain the image of the entire marked finite subgroup, its central embedding, lifts, squares, refinements, and phases. An actual extension map $F_0$ obeys $F_0(a)^2=F_0(a^2)$.\n\\boundary{The unmarked complex pushouts of $D_8$ and $Q_8$ coincide, while their marked finite $\\mu_2$ extensions differ. Retaining only the central $\\{\\pm1\\}$ loses that distinction. Invisible kernel elements already removed by the central character cannot be recovered by pushout.}","tual map of marked extensions.}\nFor a full character $\\theta_i$, its scalar pushout is\n\\[\n \\mathcal P_i=(\\C^\\times\\times J_i^1)/\n \\{(\\theta_i(h)^{-1},h):h\\in H_i^1\\}.\n\\]\nA center-fixing comparison retains every supplied character value. In the dyadic case retain the image of the entire marked finite subgroup, its central embedding, lifts, squares, refinements, and phases. An actual extension map $F_0$ obeys $F_0(a)^2=F_0(a^2)$.\n\\boundary{The unmarked complex pushouts of $D_8$ and $Q_8$ coincide, while their marked finite $\\mu_2$ extensions differ. Retaining only the central $\\{\\pm1\\}$ loses that distinction. Invisible kernel elements already removed by the central character cannot be recovered by pushout.}","marked extensions.}\nFor a full character $\\theta_i$, its scalar pushout is\n\\[\n \\mathcal P_i=(\\C^\\times\\times J_i^1)/\n \\{(\\theta_i(h)^{-1},h):h\\in H_i^1\\}.\n\\]\nA center-fixing comparison retains every supplied character value. In the dyadic case retain the image of the entire marked finite subgroup, its central embedding, lifts, squares, refinements, and phases. An actual extension map $F_0$ obeys $F_0(a)^2=F_0(a^2)$.\n\\boundary{The unmarked complex pushouts of $D_8$ and $Q_8$ coincide, while their marked finite $\\mu_2$ extensions differ. Retaining only the central $\\{\\pm1\\}$ loses that distinction. Invisible kernel elements already removed by the central character cannot be recovered by pushout.}","pushout is\n\\[\n \\mathcal P_i=(\\C^\\times\\times J_i^1)/\n \\{(\\theta_i(h)^{-1},h):h\\in H_i^1\\}.\n\\]\nA center-fixing comparison retains every supplied character value. In the dyadic case retain the image of the entire marked finite subgroup, its central embedding, lifts, squares, refinements, and phases. An actual extension map $F_0$ obeys $F_0(a)^2=F_0(a^2)$.\n\\boundary{The unmarked complex pushouts of $D_8$ and $Q_8$ coincide, while their marked finite $\\mu_2$ extensions differ. Retaining only the central $\\{\\pm1\\}$ loses that distinction. Invisible kernel elements already removed by the central character cannot be recovered by pushout.}","\\[\n \\mathcal P_i=(\\C^\\times\\times J_i^1)/\n \\{(\\theta_i(h)^{-1},h):h\\in H_i^1\\}.\n\\]\nA center-fixing comparison retains every supplied character value. In the dyadic case retain the image of the entire marked finite subgroup, its central embedding, lifts, squares, refinements, and phases. An actual extension map $F_0$ obeys $F_0(a)^2=F_0(a^2)$.\n\\boundary{The unmarked complex pushouts of $D_8$ and $Q_8$ coincide, while their marked finite $\\mu_2$ extensions differ. Retaining only the central $\\{\\pm1\\}$ loses that distinction. Invisible kernel elements already removed by the central character cannot be recovered by pushout.}","\\{(\\theta_i(h)^{-1},h):h\\in H_i^1\\}.\n\\]\nA center-fixing comparison retains every supplied character value. In the dyadic case retain the image of the entire marked finite subgroup, its central embedding, lifts, squares, refinements, and phases. An actual extension map $F_0$ obeys $F_0(a)^2=F_0(a^2)$.\n\\boundary{The unmarked complex pushouts of $D_8$ and $Q_8$ coincide, while their marked finite $\\mu_2$ extensions differ. Retaining only the central $\\{\\pm1\\}$ loses that distinction. Invisible kernel elements already removed by the central character cannot be recovered by pushout.}","ing comparison retains every supplied character value. In the dyadic case retain the image of the entire marked finite subgroup, its central embedding, lifts, squares, refinements, and phases. An actual extension map $F_0$ obeys $F_0(a)^2=F_0(a^2)$.\n\\boundary{The unmarked complex pushouts of $D_8$ and $Q_8$ coincide, while their marked finite $\\mu_2$ extensions differ. Retaining only the central $\\{\\pm1\\}$ loses that distinction. Invisible kernel elements already removed by the central character cannot be recovered by pushout.}","\\lesson{PART III / ACTUAL ARITHMETIC CONSTRUCTION}{A native quadratic marking adds an embedding condition}{Block 167, markings theorem}\nTransported source markings and an independently prescribed native host marking are different inputs. If the native host refinement $q_W$ is independently supplied, a prescribed embedding $f$ must satisfy\n\\[\n q_W(f(v))=q_{\\rm src}(v)\\quad\\text{for every }v.\n\\]\nWith a free embedding and equal dimensions, quadratic classification requires equal Arf invariants. This is the rank-eight case: both quotient spaces have dimension 6.","For $d\\ge4$, the dyadic host has at least one spare symplectic plane. The complement's Arf type can absorb the difference, so either source Arf type admits an embedding into the supplied native quadratic host.","\\lesson{PART III / ACTUAL ARITHMETIC CONSTRUCTION}{Low-rank repair enlarges an actual compact group}{Block 167, low-rank constructions}\n\\plate{19_enlarged_carrier}{The new carrier occurs in the original candidate; it is not an abstract extra tensor factor.}\nAt rank two, write $q=2m+1$, $m\\ge1$, and use\n\\[\n J=U_K^1U_A^{m+1},\\qquad P=U_K^1U_A^m.\n\\]\nThen $J\\triangleleft P$, $[P:J]=2$, and $P\\cap(K^\\times J)=J$. The conjugate character ratio $\\lambda=\\theta^g/\\theta$ is nontrivial and sign-valued. With $H_0=\\ker\\lambda$,\n\\[\n P/H_0\\simeq\\F_2^2,\\qquad E=\\Ind_J^P\\theta,\\quad\\dim E=2.\n\\]\nThe represented diagonal sign and off-diagonal generators anticommute, proving the nondegenerate Heisenberg structure.\n\\boundary{The original quotient $J^1/H^1$ is still zero-dimensional. The new pair is a separate arithmetic assembly carrier.}","UCTION}{Low-rank repair enlarges an actual compact group}{Block 167, low-rank constructions}\n\\plate{19_enlarged_carrier}{The new carrier occurs in the original candidate; it is not an abstract extra tensor factor.}\nAt rank two, write $q=2m+1$, $m\\ge1$, and use\n\\[\n J=U_K^1U_A^{m+1},\\qquad P=U_K^1U_A^m.\n\\]\nThen $J\\triangleleft P$, $[P:J]=2$, and $P\\cap(K^\\times J)=J$. The conjugate character ratio $\\lambda=\\theta^g/\\theta$ is nontrivial and sign-valued. With $H_0=\\ker\\lambda$,\n\\[\n P/H_0\\simeq\\F_2^2,\\qquad E=\\Ind_J^P\\theta,\\quad\\dim E=2.\n\\]\nThe represented diagonal sign and off-diagonal generators anticommute, proving the nondegenerate Heisenberg structure.\n\\boundary{The original quotient $J^1/H^1$ is still zero-dimensional. The new pair is a separate arithmetic assembly carrier.}","k 167, low-rank constructions}\n\\plate{19_enlarged_carrier}{The new carrier occurs in the original candidate; it is not an abstract extra tensor factor.}\nAt rank two, write $q=2m+1$, $m\\ge1$, and use\n\\[\n J=U_K^1U_A^{m+1},\\qquad P=U_K^1U_A^m.\n\\]\nThen $J\\triangleleft P$, $[P:J]=2$, and $P\\cap(K^\\times J)=J$. The conjugate character ratio $\\lambda=\\theta^g/\\theta$ is nontrivial and sign-valued. With $H_0=\\ker\\lambda$,\n\\[\n P/H_0\\simeq\\F_2^2,\\qquad E=\\Ind_J^P\\theta,\\quad\\dim E=2.\n\\]\nThe represented diagonal sign and off-diagonal generators anticommute, proving the nondegenerate Heisenberg structure.\n\\boundary{The original quotient $J^1/H^1$ is still zero-dimensional. The new pair is a separate arithmetic assembly carrier.}","tructions}\n\\plate{19_enlarged_carrier}{The new carrier occurs in the original candidate; it is not an abstract extra tensor factor.}\nAt rank two, write $q=2m+1$, $m\\ge1$, and use\n\\[\n J=U_K^1U_A^{m+1},\\qquad P=U_K^1U_A^m.\n\\]\nThen $J\\triangleleft P$, $[P:J]=2$, and $P\\cap(K^\\times J)=J$. The conjugate character ratio $\\lambda=\\theta^g/\\theta$ is nontrivial and sign-valued. With $H_0=\\ker\\lambda$,\n\\[\n P/H_0\\simeq\\F_2^2,\\qquad E=\\Ind_J^P\\theta,\\quad\\dim E=2.\n\\]\nThe represented diagonal sign and off-diagonal generators anticommute, proving the nondegenerate Heisenberg structure.\n\\boundary{The original quotient $J^1/H^1$ is still zero-dimensional. The new pair is a separate arithmetic assembly carrier.}","U_A^m.\n\\]\nThen $J\\triangleleft P$, $[P:J]=2$, and $P\\cap(K^\\times J)=J$. The conjugate character ratio $\\lambda=\\theta^g/\\theta$ is nontrivial and sign-valued. With $H_0=\\ker\\lambda$,\n\\[\n P/H_0\\simeq\\F_2^2,\\qquad E=\\Ind_J^P\\theta,\\quad\\dim E=2.\n\\]\nThe represented diagonal sign and off-diagonal generators anticommute, proving the nondegenerate Heisenberg structure.\n\\boundary{The original quotient $J^1/H^1$ is still zero-dimensional. The new pair is a separate arithmetic assembly carrier.}","\\lesson{PART III / ACTUAL ARITHMETIC CONSTRUCTION}{Rank four gains a four-dimensional carrier}{Block 167, rank-four enlarged carrier proof}\nLet $E_0/F$ be the actual outer quadratic approximation field and $B=\\operatorname{End}_{E_0}(F^4)$. With lower odd critical level $r=3h_1=2m+1$ and half leading level $a=3h_2>r$, define\n\\[\n P=J\\,U_B^m.\n\\]\nThe filtered-centralizer estimate proves this is a compact group with $J$ normal of index two. It does not assume $E_0\\subset K$.","/ ACTUAL ARITHMETIC CONSTRUCTION}{Rank four gains a four-dimensional carrier}{Block 167, rank-four enlarged carrier proof}\nLet $E_0/F$ be the actual outer quadratic approximation field and $B=\\operatorname{End}_{E_0}(F^4)$. With lower odd critical level $r=3h_1=2m+1$ and half leading level $a=3h_2>r$, define\n\\[\n P=J\\,U_B^m.\n\\]\nThe filtered-centralizer estimate proves this is a compact group with $J$ normal of index two. It does not assume $E_0\\subset K$.","gains a four-dimensional carrier}{Block 167, rank-four enlarged carrier proof}\nLet $E_0/F$ be the actual outer quadratic approximation field and $B=\\operatorname{End}_{E_0}(F^4)$. With lower odd critical level $r=3h_1=2m+1$ and half leading level $a=3h_2>r$, define\n\\[\n P=J\\,U_B^m.\n\\]\nThe filtered-centralizer estimate proves this is a compact group with $J$ normal of index two. It does not assume $E_0\\subset K$.","For a new coset representative $u$, the ratio $\\lambda=\\theta^u/\\theta$ on the actual $H^1$ is nontrivial and sign-valued. Setting $H_0=\\ker\\lambda$ gives\n\\[\n P/H_0\\simeq\\F_2^4,\\qquad E=\\Ind_J^P\\eta,\\quad\\dim E=4.\n\\]\nIts represented commutators are scalar; the quotient has exponent two; irreducibility and the exact scalar preimage prove nondegeneracy. The finite scalar center can exceed $\\mu_2$.\n\\boundary{No literal extraspecial quotient is claimed. Full character values and marked attachment are retained through the actual finite extension.}","\\lesson{PART III / ACTUAL ARITHMETIC CONSTRUCTION}{Induction in stages recovers the same candidate}{Block 167, low-rank recovery proof}\n\\plate{20_induction_stages}{The noncompact extension $\\Lambda$ is fixed before forming $\\Sigma$.}\nAt both low ranks, $K^\\times$ normalizes $P$. Put\n\\[\n \\mathcal N=K^\\times J,\\quad\\mathcal M=K^\\times P,\n \\qquad\\Sigma=\\Ind_{\\mathcal N}^{\\mathcal M}\\Lambda.\n\\]\nThe subgroup $\\mathcal N$ is normal of index two, $\\Sigma|_P\\simeq E$, and $\\Sigma$ is irreducible. Induction in stages proves\n\\[\n \\pi\\simeq\\cInd_{\\mathcal M}^{G}\\Sigma.\n\\]\n\\takeaway{The parameter, full character, original canonical beta-extension data, central character, and noncompact scalar remain those of the original candidate. The inducing carrier changes.}","PART III / ACTUAL ARITHMETIC CONSTRUCTION}{Induction in stages recovers the same candidate}{Block 167, low-rank recovery proof}\n\\plate{20_induction_stages}{The noncompact extension $\\Lambda$ is fixed before forming $\\Sigma$.}\nAt both low ranks, $K^\\times$ normalizes $P$. Put\n\\[\n \\mathcal N=K^\\times J,\\quad\\mathcal M=K^\\times P,\n \\qquad\\Sigma=\\Ind_{\\mathcal N}^{\\mathcal M}\\Lambda.\n\\]\nThe subgroup $\\mathcal N$ is normal of index two, $\\Sigma|_P\\simeq E$, and $\\Sigma$ is irreducible. Induction in stages proves\n\\[\n \\pi\\simeq\\cInd_{\\mathcal M}^{G}\\Sigma.\n\\]\n\\takeaway{The parameter, full character, original canonical beta-extension data, central character, and noncompact scalar remain those of the original candidate. The inducing carrier changes.}","TION}{Induction in stages recovers the same candidate}{Block 167, low-rank recovery proof}\n\\plate{20_induction_stages}{The noncompact extension $\\Lambda$ is fixed before forming $\\Sigma$.}\nAt both low ranks, $K^\\times$ normalizes $P$. Put\n\\[\n \\mathcal N=K^\\times J,\\quad\\mathcal M=K^\\times P,\n \\qquad\\Sigma=\\Ind_{\\mathcal N}^{\\mathcal M}\\Lambda.\n\\]\nThe subgroup $\\mathcal N$ is normal of index two, $\\Sigma|_P\\simeq E$, and $\\Sigma$ is irreducible. Induction in stages proves\n\\[\n \\pi\\simeq\\cInd_{\\mathcal M}^{G}\\Sigma.\n\\]\n\\takeaway{The parameter, full character, original canonical beta-extension data, central character, and noncompact scalar remain those of the original candidate. The inducing carrier changes.}","recovers the same candidate}{Block 167, low-rank recovery proof}\n\\plate{20_induction_stages}{The noncompact extension $\\Lambda$ is fixed before forming $\\Sigma$.}\nAt both low ranks, $K^\\times$ normalizes $P$. Put\n\\[\n \\mathcal N=K^\\times J,\\quad\\mathcal M=K^\\times P,\n \\qquad\\Sigma=\\Ind_{\\mathcal N}^{\\mathcal M}\\Lambda.\n\\]\nThe subgroup $\\mathcal N$ is normal of index two, $\\Sigma|_P\\simeq E$, and $\\Sigma$ is irreducible. Induction in stages proves\n\\[\n \\pi\\simeq\\cInd_{\\mathcal M}^{G}\\Sigma.\n\\]\n\\takeaway{The parameter, full character, original canonical beta-extension data, central character, and noncompact scalar remain those of the original candidate. The inducing carrier changes.}","overy proof}\n\\plate{20_induction_stages}{The noncompact extension $\\Lambda$ is fixed before forming $\\Sigma$.}\nAt both low ranks, $K^\\times$ normalizes $P$. Put\n\\[\n \\mathcal N=K^\\times J,\\quad\\mathcal M=K^\\times P,\n \\qquad\\Sigma=\\Ind_{\\mathcal N}^{\\mathcal M}\\Lambda.\n\\]\nThe subgroup $\\mathcal N$ is normal of index two, $\\Sigma|_P\\simeq E$, and $\\Sigma$ is irreducible. Induction in stages proves\n\\[\n \\pi\\simeq\\cInd_{\\mathcal M}^{G}\\Sigma.\n\\]\n\\takeaway{The parameter, full character, original canonical beta-extension data, central character, and noncompact scalar remain those of the original candidate. The inducing carrier changes.}","\\lesson{PART III / ACTUAL ARITHMETIC CONSTRUCTION}{Compact recovery must remember the noncompact root}{Block 167, enlarged-carrier contract}\nLet $X=\\cInd_J^G\\eta$ and retain $A_0=\\Lambda(\\varpi_K)$. Define the normalized shift\n\\[\n (Tf)(g)=A_0^{-1}f(\\varpi_Kg).\n\\]\nThe original candidate is the quotient\n\\[\n X/(T-1)X\\simeq\\pi.\n\\]\nThis does not split before central normalization: a vector of $X$ has finite support modulo compact $J$, whereas a scalar uniformizer shifts determinant valuations. It has no nonzero eigenvector with the required central eigenvalue.","HMETIC CONSTRUCTION}{Compact recovery must remember the noncompact root}{Block 167, enlarged-carrier contract}\nLet $X=\\cInd_J^G\\eta$ and retain $A_0=\\Lambda(\\varpi_K)$. Define the normalized shift\n\\[\n (Tf)(g)=A_0^{-1}f(\\varpi_Kg).\n\\]\nThe original candidate is the quotient\n\\[\n X/(T-1)X\\simeq\\pi.\n\\]\nThis does not split before central normalization: a vector of $X$ has finite support modulo compact $J$, whereas a scalar uniformizer shifts determinant valuations. It has no nonzero eigenvector with the required central eigenvalue.","After imposing the already fixed central character, the relevant quotient is cyclic of order $N$. Then $T^N=1$ and the finite average\n\\[\n e_\\Lambda=\\frac1N\\sum_{i=0}^{N-1}T^i\n\\]\nselects the same extension. The operator acts on a complex representation space, where division by $N$ is valid.\n\\boundary{This is a different recovery mechanism from a finite sector inside the original $\\eta$. The root is retained, not reconstructed from the compact packet alone.}","\\lesson{PART III / ACTUAL ARITHMETIC CONSTRUCTION}{Central completion is a construction choice}{Blocks 166--167, full-character completion}\n\\plate{22_completion}{This diagram starts with an already constructed compatible arithmetic host.}\nThe full character is trivial on $U_F^1$. Residue data and the noncompact scalar can then match a prescribed smooth central quasicharacter $\\omega$ trivial on $U_F^1$, when completion choices are free or already compatible.","For $F=\\F_2((t))$, the residue unit group is trivial, so these covered central characters are unramified. An $N$th root in $\\C^\\times$ fixes the action required by $\\omega(t)$. All choices are recorded before invoking H5.","\\lesson{PART IV / REALIZATION AND COMPARISON}{Why nontrivial stabilizer actions need another step}{Block 166, stabilizer contract}\nFor the current totally ramified degree-$p^r$ maximal-type host, character-preserving normalizers have\n\\[\n N/(F^\\times J^1)\\simeq\\mathbb Z/p^r\\mathbb Z.\n\\]\nA native action of a finite prime-to-$p$ group $C$ on the quotient therefore factors through a $p$-group and is trivial.","The kernel of the action of center-fixing scalar-extension automorphisms on $W$ is $\\operatorname{Hom}(W,\\C^\\times)$, an elementary abelian $p$-group. Hence the full native scalar-extension action is trivial as well. Honest implementers on an irreducible Heisenberg carrier are scalar by Schur's lemma.","\\lesson{PART IV / REALIZATION AND COMPARISON}{Genuine scalar phases can descend from a finite cover}{Block 166, scalar native attachment}\nThere is a proved extension beyond $C=1$ when the prime-to-$p$ stabilizer acts trivially on the source extension and its honest operators on the entire rank-$N$ source carrier are $\\chi(c)I_N$ in the actual inertia image.","Under the same-source determinant central-character condition, the required scalar phase can be implemented by actual central matrices after a finite cover $\\widetilde C\\twoheadrightarrow C$. The kernel acts trivially on the whole recovered candidate, so the represented action descends to $C$.","\\takeaway{A cover may be needed even when the represented operator is a scalar. A direct matrix homomorphism from $C$ need not exist.}\n\\boundary{The theorem requires a genuine inertia phase on the whole carrier, the fixed reciprocity convention, and compatible completion. It does not apply merely because an arbitrary abstract phase on a multiplicity subspace is scalar. It does not change a fixed canonical beta extension to force a desired action.}","\\lesson{PART IV / REALIZATION AND COMPARISON}{Numerical matching does not construct realization}{Blocks 162, 166--167, profile and capacity audits}\nFor compact $J\\subset\\GL_n(F)$, the homomorphism $v_F\\circ\\det$ has finite image in $\\mathbb Z$, hence image zero. An unramified character\n\\[\n \\nu_a(g)=a^{v_F(\\det g)}\n\\]\nis therefore trivial on $J$. The compact packet restrictions of $\\pi$ and $\\pi\\otimes\\nu_a$ agree, while the central character changes by $a^n$ on a scalar uniformizer. Choosing $a^n\\ne1$ gives a different atom.","tching does not construct realization}{Blocks 162, 166--167, profile and capacity audits}\nFor compact $J\\subset\\GL_n(F)$, the homomorphism $v_F\\circ\\det$ has finite image in $\\mathbb Z$, hence image zero. An unramified character\n\\[\n \\nu_a(g)=a^{v_F(\\det g)}\n\\]\nis therefore trivial on $J$. The compact packet restrictions of $\\pi$ and $\\pi\\otimes\\nu_a$ agree, while the central character changes by $a^n$ on a scalar uniformizer. Choosing $a^n\\ne1$ gives a different atom.","locks 162, 166--167, profile and capacity audits}\nFor compact $J\\subset\\GL_n(F)$, the homomorphism $v_F\\circ\\det$ has finite image in $\\mathbb Z$, hence image zero. An unramified character\n\\[\n \\nu_a(g)=a^{v_F(\\det g)}\n\\]\nis therefore trivial on $J$. The compact packet restrictions of $\\pi$ and $\\pi\\otimes\\nu_a$ agree, while the central character changes by $a^n$ on a scalar uniformizer. Choosing $a^n\\ne1$ gives a different atom.","Block 167 also records a rank-eight skipped-degree $8\\to2$ host with the same torsion, Swan, and self-pair conductor as the full tower, but quotient dimension 4 rather than 6.\n\\takeaway{The finite packet, numerical profiles, and noncompact normalization contain different information. No one of these alone is a substitute for an arithmetic witness preserving the full declared datum.}","\\lesson{PART IV / REALIZATION AND COMPARISON}{Conjecture 66 isolates the missing existence claim}{Historical Theorem 16.2; current realization contract}\n\\plate{23_conjecture_boundary}{The upper existence arrow is conjectural. The lower comparison follows conditionally.}\n\\textbf{Conjecture 66 (arithmetic packet realization).} Fix the declared class $\\mathcal A$ of wild atoms and the paper's normalizations. For every $\\rho\\in\\mathcal A$ with supplied H1--H4 assembled finite datum $d_\\rho$, there is one actual admissible supercuspidal representation $\\pi$ of $\\GL_{\\dim\\rho}(F)$ and an arithmetic realization witness $R(d_\\rho,\\pi)$ recovering the packet and preserving its declared stabilizer action, central-extension and phase data, and compatible transports.","ALIZATION AND COMPARISON}{Conjecture 66 isolates the missing existence claim}{Historical Theorem 16.2; current realization contract}\n\\plate{23_conjecture_boundary}{The upper existence arrow is conjectural. The lower comparison follows conditionally.}\n\\textbf{Conjecture 66 (arithmetic packet realization).} Fix the declared class $\\mathcal A$ of wild atoms and the paper's normalizations. For every $\\rho\\in\\mathcal A$ with supplied H1--H4 assembled finite datum $d_\\rho$, there is one actual admissible supercuspidal representation $\\pi$ of $\\GL_{\\dim\\rho}(F)$ and an arithmetic realization witness $R(d_\\rho,\\pi)$ recovering the packet and preserving its declared stabilizer action, central-extension and phase data, and compatible transports.","stence claim}{Historical Theorem 16.2; current realization contract}\n\\plate{23_conjecture_boundary}{The upper existence arrow is conjectural. The lower comparison follows conditionally.}\n\\textbf{Conjecture 66 (arithmetic packet realization).} Fix the declared class $\\mathcal A$ of wild atoms and the paper's normalizations. For every $\\rho\\in\\mathcal A$ with supplied H1--H4 assembled finite datum $d_\\rho$, there is one actual admissible supercuspidal representation $\\pi$ of $\\GL_{\\dim\\rho}(F)$ and an arithmetic realization witness $R(d_\\rho,\\pi)$ recovering the packet and preserving its declared stabilizer action, central-extension and phase data, and compatible transports.","plate{23_conjecture_boundary}{The upper existence arrow is conjectural. The lower comparison follows conditionally.}\n\\textbf{Conjecture 66 (arithmetic packet realization).} Fix the declared class $\\mathcal A$ of wild atoms and the paper's normalizations. For every $\\rho\\in\\mathcal A$ with supplied H1--H4 assembled finite datum $d_\\rho$, there is one actual admissible supercuspidal representation $\\pi$ of $\\GL_{\\dim\\rho}(F)$ and an arithmetic realization witness $R(d_\\rho,\\pi)$ recovering the packet and preserving its declared stabilizer action, central-extension and phase data, and compatible transports.","conjecture_boundary}{The upper existence arrow is conjectural. The lower comparison follows conditionally.}\n\\textbf{Conjecture 66 (arithmetic packet realization).} Fix the declared class $\\mathcal A$ of wild atoms and the paper's normalizations. For every $\\rho\\in\\mathcal A$ with supplied H1--H4 assembled finite datum $d_\\rho$, there is one actual admissible supercuspidal representation $\\pi$ of $\\GL_{\\dim\\rho}(F)$ and an arithmetic realization witness $R(d_\\rho,\\pi)$ recovering the packet and preserving its declared stabilizer action, central-extension and phase data, and compatible transports.","ral. The lower comparison follows conditionally.}\n\\textbf{Conjecture 66 (arithmetic packet realization).} Fix the declared class $\\mathcal A$ of wild atoms and the paper's normalizations. For every $\\rho\\in\\mathcal A$ with supplied H1--H4 assembled finite datum $d_\\rho$, there is one actual admissible supercuspidal representation $\\pi$ of $\\GL_{\\dim\\rho}(F)$ and an arithmetic realization witness $R(d_\\rho,\\pi)$ recovering the packet and preserving its declared stabilizer action, central-extension and phase data, and compatible transports.","Here $R$ includes actual arithmetic inducing data and explicit comparison and recovery. It is not merely the existence of abstract operators with the right commutators.\n\\boundary{The conjecture is restricted to the declared admitted class. It does not assert that arbitrary coefficient targets, arbitrary partial hosts, or independently conflicting completion choices can all be extended.}","\\lesson{PART IV / REALIZATION AND COMPARISON}{H5 compares this same actual candidate}{Contract audit 162; conditional comparison proposition}\nSuppose $\\pi$ is an actual admissible supercuspidal representation of the correct rank and $R(d_\\rho,\\pi)$ is supplied. If H5 gives equality of its required pair local factors with those of $\\LLC(\\rho)$ on the stipulated characterizing twist family, with the fixed normalizations, then\n\\[\n \\pi\\simeq\\LLC(\\rho).\n\\]\nThe proof applies the supplied local-factor characterization to these two actual representations. It does not need existence for other diagrams or H5 for every alternative completion.","AND COMPARISON}{H5 compares this same actual candidate}{Contract audit 162; conditional comparison proposition}\nSuppose $\\pi$ is an actual admissible supercuspidal representation of the correct rank and $R(d_\\rho,\\pi)$ is supplied. If H5 gives equality of its required pair local factors with those of $\\LLC(\\rho)$ on the stipulated characterizing twist family, with the fixed normalizations, then\n\\[\n \\pi\\simeq\\LLC(\\rho).\n\\]\nThe proof applies the supplied local-factor characterization to these two actual representations. It does not need existence for other diagrams or H5 for every alternative completion.","There is a logical difference between\n\\[\n \\forall\\pi\\,[R(d,\\pi)\\Rightarrow\\mathrm{H5}(\\pi)]\n \\quad\\text{and}\\quad\n \\exists\\pi\\,[R(d,\\pi)\\land\\mathrm{H5}(\\pi)].\n\\]\nThe first statement can be vacuous. The second includes an actual witness. Choosing the established reference atom only supplies a profile-matching object; it leaves its recovery of $d$ unproved.\n\\takeaway{H5 remains a conditional premise about the same independently constructed candidate. It is not renamed as an unconditional factor-comparison theorem.}","\\lesson{PART IV / REALIZATION AND COMPARISON}{What the restricted realizations establish}{Blocks 166--167, final composition reviews}\n\\begin{itemize}\n\\item Actual totally ramified parameter fields, literal defining sequences, full simple characters, and covered central completions exist throughout the stated odd-prime and dyadic families.\n\\item Under the one-vertex, $C=1$, supplied full-character scalar-pushout contract, the whole marked packet has an actual arithmetic assembly carrier and same-candidate recovery.\n\\item In the dyadic low ranks, the actual enlarged carriers restore capacity without changing the original maximal-simple quotient.\n\\item Genuine scalar inertia phases have the separately stated finite-cover descent result.\n\\end{itemize}\nThe remaining general obligations include nontrivial prescribed symplectic stabilizer actions on a common arithmetic candidate, general sources and transport diagrams, comparison with independently fixed canonical native markings, and numerical layer placement when the declared contract requires it.","\\lesson{PART IV / EVIDENCE AND REPRODUCIBILITY}{Exact computation checks a bounded identity}{Block 167, computation contract; final verification records}\nThe rank-eight run for $h=(1,3,5)$ produces an actual matrix with 25 nonzero Laurent-polynomial entries. Its checked leading and adjacent-correction grades are\n\\[\n -60,\\quad-18,\\quad-3.\n\\]\nIt checks normalized lifts, actual matrix identities, native traces 1, and amplified ambient traces $0,0,1$. The odd $p=3$ example $h=(1,2,4)$ has levels $144,24,4$ and native trace zero.","TY}{Exact computation checks a bounded identity}{Block 167, computation contract; final verification records}\nThe rank-eight run for $h=(1,3,5)$ produces an actual matrix with 25 nonzero Laurent-polynomial entries. Its checked leading and adjacent-correction grades are\n\\[\n -60,\\quad-18,\\quad-3.\n\\]\nIt checks normalized lifts, actual matrix identities, native traces 1, and amplified ambient traces $0,0,1$. The odd $p=3$ example $h=(1,2,4)$ has levels $144,24,4$ and native trace zero.","tion checks a bounded identity}{Block 167, computation contract; final verification records}\nThe rank-eight run for $h=(1,3,5)$ produces an actual matrix with 25 nonzero Laurent-polynomial entries. Its checked leading and adjacent-correction grades are\n\\[\n -60,\\quad-18,\\quad-3.\n\\]\nIt checks normalized lifts, actual matrix identities, native traces 1, and amplified ambient traces $0,0,1$. The odd $p=3$ example $h=(1,2,4)$ has levels $144,24,4$ and native trace zero.","bounded identity}{Block 167, computation contract; final verification records}\nThe rank-eight run for $h=(1,3,5)$ produces an actual matrix with 25 nonzero Laurent-polynomial entries. Its checked leading and adjacent-correction grades are\n\\[\n -60,\\quad-18,\\quad-3.\n\\]\nIt checks normalized lifts, actual matrix identities, native traces 1, and amplified ambient traces $0,0,1$. The odd $p=3$ example $h=(1,2,4)$ has levels $144,24,4$ and native trace zero.","167, computation contract; final verification records}\nThe rank-eight run for $h=(1,3,5)$ produces an actual matrix with 25 nonzero Laurent-polynomial entries. Its checked leading and adjacent-correction grades are\n\\[\n -60,\\quad-18,\\quad-3.\n\\]\nIt checks normalized lifts, actual matrix identities, native traces 1, and amplified ambient traces $0,0,1$. The odd $p=3$ example $h=(1,2,4)$ has levels $144,24,4$ and native trace zero."]}