G Dyadic synthesis and all-length assembly scope
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Date: 2026-10-05. Base: the final reviewed Block166, pinned in TARGET167.md. Evidence: ordinary mathematical proof and independent component reviews. No Lean verification or universal claim66 closure is asserted.
G.1 1. Combined arithmetic theorem
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Let \(F=\mathbb F_2((t))\), fix an additive character \(\psi_F\) of conductor \(t\mathcal O_F\), and take increasing odd positive integers \(h_1<\cdots<h_d\). Put
There is an explicit \(\beta\in M_N(F)\) generating a separable totally ramified degree-\(N\) field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level \(n_1\), final critical exponent \(-q_1\), and successive approximation-field degrees \(2,4,\ldots,2^d\), with critical levels \(n_2,\ldots,n_d\).
The actual stratum admits a full simple character \(\theta\) trivial on \(U_F^1\). Its canonical beta extension can be completed to an actual irreducible admissible supercuspidal \(\pi\) of \(\mathrm{GL}_N(F)\) with any prescribed smooth central quasicharacter \(\omega\) trivial on \(U_F^1\), provided the noncompact completion is free or already compatible. Over this residue field, such \(\omega\) is unramified.
For this same constructed candidate,
Matching these profiles does not identify the source, its endo-class, or the pair factors in original H5.
G.2 2. The explicit construction and its depth invariant
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At a recursive coefficient field \(L\) of characteristic two, with residue field \(\mathbb F_2\) and uniformizer \(t_L\), put
The extension is separable and totally ramified of degree two; \(z\) is a uniformizer and
For constant-coefficient projection \(P_0\) in the basis \(1,z\), the normalized trace corestriction and exact filtered right inverse are
They satisfy \(s(\operatorname{Lift}_E(C))=C\), preserve the exact filtration valuation, and give the identity
The actual recursive matrix is
with multiplication by \(\alpha\) as the rank-two base case.
Every native constructor has trace one. At every next native rank, the preceding parameter amplified by two has trace zero, so the native correction also has trace one. At full ambient rank, the earlier amplified approximations instead have trace zero and the final one has trace one. These are different statements; no amplified trace is used as a substitute for the native trace needed in character construction.
At an inductive field step \(d\ge2\), set \(n=2^{d-1}q_d\), \(R=2^{d-2}q_{d-1}\). Then \(n>2R\) and \(-2R>-n-q_1\). BK Remark 2.2.7 gives the maximal field generated by the actual displayed matrix. Literal truncation and the filtered lift give the defining sequence. Nonzero native trace also proves separability: the irreducible characteristic polynomial has a nonzero coefficient of \(X^{N-1}\), so its derivative is nonzero. See tower/DYADIC_AS_TOWER167.md for all field and filtration details.
The replacement for trace zero is the depth inequality
It makes the additive correction trivial on scalar \(U_F^s\), because \((u-1)\operatorname{tr}\delta\in t\mathcal O_F\). Our native corrections have trace one, so the inequality holds at \(s=1\). This is the exact source-to-target transfer of the weighted-visibility principle; the Euler coefficient theorem is motivation, not an input to the arithmetic proof. In characteristic two, the weights are uniformizer depths, rather than integer powers of two in the coefficient field.
G.3 3. Full characters, including the odd critical endpoint
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At the minimal rank-two base, extend the forced character on \(U_E^{(q+1)/2}\) to \(U_E^1\), imposing triviality on \(U_F^1\). The prescriptions agree on their intersection because the parameter trace is one and \(\psi_F(u-1)=1\). This is an actual finite abelian character extension and gives the full minimal simple character.
For a later native step, first transfer the full beta-determinant unit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units,
For odd critical level \(r\), use \(m=\lfloor r/2\rfloor\) and \(D=H^{m+1}(\beta)=H^{m+1}(\gamma)\). The endpoint is valid because \(2(m+1)-r=1\): the additive-character cross term lies in the radical on which the matrix additive character is trivial. BK 3.3.18 supplies \(\vartheta=\Theta_j|_D\,\psi_{\beta-\gamma}\). Integral trace makes it trivial on \(D\cap U_F^1\).
The next parameter field is maximal. Extend the forced character on its deep units together with the trivial character of \(U_F^1\), then apply BK 3.2.5 to obtain the next full character. All new extensions occur at maximal native rank. No unverified determinant-image extension in a nonmaximal centralizer is used.
For the final totally ramified field, \(J/J^1=\mathbb F_2^\times=1\), so \(J=J^1\) and the canonical beta extension is \(\eta\). Extend to \(K^\times J\), and choose the noncompact scalar to match \(\omega(t)\). The needed \(N\)-th root exists in \(\mathbb C^\times\). The compact type stays fixed. All choices precede H5. See central/DYADIC_FULL_CHARACTER167.md for the complete proof.
G.4 4. Exact packet capacity and marked attachment
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The dyadic floors cannot be replaced by the odd-prime dimension formula. At stage \(j\), the centralizer grade has dimension \(2^{d-j}\). Its contribution to \(J^1/H^1\) is
Every \(n_j\) for \(j<d\) is even; \(n_d=q_1\) is odd and contributes zero. The sum is exactly \(N-2\). BK nondegeneracy makes this a symplectic space. A whole source packet of \(d\) planes fits if and only if \(N-2\ge2d\), equivalently \(d\ge3\).
| Ambient rank | Host quotient dimension | Host Heisenberg carrier | Required source carrier | Outcome |
|---|---|---|---|---|
| 2 | 0 | 1 | 2 | Insufficient |
| 4 | 2 | 2 | 4 | Insufficient |
| 8 | 6 | 8 | 8 | Exact capacity |
| 16 | 14 | 128 | 16 | Spare capacity |
Now assume \(d\ge3\), the one-vertex contract, declared \(C=\{1\}\), and actually supplied full character extensions with labelled whole-plane quotients. Retain their complete dyadic finite central-extension markings, including their specified central embeddings, lifts, square values, refinements, and phases. Any central-character kernel already factored out must be recorded; a scalar pushout cannot recover invisible kernel elements.
Under the scalar-pushout realization contract, with no independently fixed conflicting host marking, choose a labelled symplectic embedding into \(W\). Matching commutators gives a center-fixing isomorphism between the source \(\mathbb C^\times\) extension and the corresponding host restriction. Retain the image of the whole marked finite subgroup and every lift. For this isomorphism \(F_0\),
Thus its square bits, refinement, and Gauss sign survive, along with all supplied full character values. They are not inferred from the unmarked pushout. Stone–von Neumann gives the source factor inside \(\eta\); a complementary line gives an actual rank-\(N\) image-algebra projector. The retained induction/recovery idempotent recovers the same \(\Lambda\) and the same \(\pi\).
This gives an actual represented marked finite subgroup in the scalar extension of the actual type. It permits complex scalars times type operators, as the declared contract does. It does not assert literal inclusion of old matrix groups, numerical depth placement, or identification with a separately fixed canonical native finite subgroup.
If a genuine host-native quadratic refinement and its finite \(\mu_2\)-central extension are independently supplied, quadratic embedding has a sharper condition. With otherwise free embeddings, at rank eight the equal-dimensional source and host must have equal Arf invariants. For \(d\ge4\), the host has at least one spare symplectic plane, so either source Arf type embeds. If the original embedding is preassigned, it must instead satisfy the pointwise equation \(q_W(f(v))=q_{\rm src}(v)\). These conclusions require the actual host marking as input; dimension does not manufacture it.
The full finite proofs are in markings/DYADIC_MARKINGS167.md. In particular, the D8 and Q8 unmarked complex pushouts coincide while their whole marked finite \(\mu_2\)-central extensions differ. Merely recording the central \(\{\pm1\}\) subgroup would lose that distinction.
G.5 5. Failed repairs and boundaries
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The rank-two and rank-four carrier failures cannot be repaired by changing the maximal simple parameter while keeping the specified Swan exponent and torsion number. At rank two, odd Swan forces a minimal odd-level stratum and \(J^1=H^1\). At rank four, Swan \(6h_2\) with \(h_2\) odd forces the degree pattern \(4\to2\); the maximal quadratic derived field forces an odd lower critical level. Every such maximal-type carrier has dimension two, below the required four. Scalar twists preserve this capacity.
This is an obstruction to the whole-source sector inside that maximal-type carrier. It is not a counterexample to the original theorem or a proof that a different arithmetic carrier in the same representation cannot work. Section 7 constructs such carriers and proves their exact recovery.
At rank eight, numerical profiles alone also fail to determine capacity. capacity/DYADIC_CAPACITY167.md, Section 6, constructs an actual skipped-degree \(8\to2\) host with the same Swan, torsion and self-pair conductor but quotient dimension four rather than six. Our successful result uses the explicit full tower, rather than inferring its profile from invariants.
Thus the completed dyadic scope is:
Actual fields, full characters and central completion in every rank \(2^d\).
Full marked scalar-pushout packet realization for the stated family when \(d\ge3\), with \(C=1\), one vertex, and transported markings retained.
A conditional quadratic embedding into an independently supplied native marked host, with the exact Arf or pointwise condition above.
Section 7 constructs different actual arithmetic carriers at the two low ranks and recovers the same candidate from them. The maximal-type capacity failure above remains true. Original H5 stays the original same-candidate conditional premise; these conductor identities and noncompact root choices do not prove it.
G.6 6. Evidence
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The arbitrary-length field and full-character arguments are ordinary proofs. The exact rank-eight run for \(h=(1,3,5)\) checks actual matrix identities, normalized lifts, leading and adjacent-correction grades \(-60,-18,-3\), native traces one and ambient traces \(0,0,1\). It outputs a matrix with 25 nonzero Laurent-polynomial entries. The corresponding critical exponents are proved symbolically. The run does not prove simplicity, full-character existence, or universal marked realization. Those conclusions use the separate proofs and their stated scope. No Lean build was run.
The base synthesis before the low-rank continuation is preserved in history/PROOF167_BEFORE_ENLARGEMENT.md, together with its review. Its previously open enlarged-carrier route is executed next.
G.7 7. The low-rank repair: enlarge the actual inducing carrier
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The maximal-type carrier is not the only arithmetic carrier available in the constructed representation. The following constructions keep the original \(\beta,\theta,\Lambda\) and \(\pi\) fixed.
G.7.1 Rank two
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Write the odd minimal level as \(q=2m+1\), with \(m\ge1\), and let \(J=H^1=J^1=U_K^1U_A^{m+1}\). Define
This is an actual compact open group, with \(J\triangleleft P\), \([P:J]=2\) and \(P\cap(K^\times J)=J\). For \(g\in P\setminus J\), the actual intertwining theorem makes \(\lambda=\theta^g/\theta\) nontrivial. The commutator \([P,J]\subset U_A^{m+1}\) lies in the forced additive-character domain, so \(\lambda\) has values \(\{\pm1\}\). Put \(H_0=\ker\lambda\). Then
The actual represented generators are a diagonal sign and an off-diagonal matrix that anticommute. Thus their commutator form is nondegenerate, and \(\theta|_{H_0}\) is the full character of an actual new arithmetic Heisenberg pair. No abstract operators create that group action.
The identity compact-induction double coset realizes \(E\) inside the same \(\pi\), as functions supported on \((K^\times J)P\). The complete group and recovery proof is lowrank/RANK2_ENLARGED_CARRIER167.md.
G.7.2 Rank four
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Use the actual rank-four outer approximation field \(E_0/F\) of degree two and its centralizer \(B=\operatorname{End}_{E_0}(F^4)\). The odd lower critical level is \(r=3h_1=2m+1\); the half leading level is \(a=3h_2>r\). Define
where \(J=J^1\) is the actual compact type group. The filtered centralizer estimate for the actual final field, without assuming \(E_0\subset K\), proves that \(P\) is a compact group with \(J\) normal of index two. For a concrete new coset representative \(u\), \(\lambda=\theta^u/\theta\) on the actual \(H=H^1\) is nontrivial with values \(\{\pm1\}\). Put \(H_0=\ker\lambda\subset H\). The calculation proves
The new represented commutators are scalar, its quotient has exponent two, and nondegeneracy follows from its exact scalar preimage and irreducibility. The finite scalar center may be larger than \(\mu_2\); a literal extraspecial quotient is not asserted. The full character values are retained by the actual finite Heisenberg extension. Again \(E\) occurs inside the same \(\pi\) through the identity double coset. See lowrank/RANK4_ENLARGED_CARRIER167.md.
G.7.3 Exact recovery, with the noncompact normalization retained
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At both low ranks, \(K^\times\) normalizes the new \(P\). In rank two this follows directly from normalization of \(A\) and its radical. In rank four, a \(K\)-uniformizer is approximated by the derived chain shift \(R\) times a unit of \(B\), with relative error in \(P_A^{n-r}\). Its conjugation error on \(U_B^m\) lies in \(U_A^{n-r+m}\subset J\), which proves normalization. The exact argument is lowrank/RANK4_INDUCTION_BY_STAGES167.md.
Consequently \(\mathcal M=K^\times P\) is an actual open group compact modulo the center, \(\mathcal N=K^\times J\) is normal of index two in \(\mathcal M\), and
Restriction to \(P\) is irreducible, so \(\Sigma\) is irreducible. Induction in stages proves the last equality for the original \(\Lambda\), including its fixed central character and noncompact root. This is a genuine enlarged inducing representation; it changes neither the candidate nor its original canonical beta-extension data.
There is also a recovery from compact data explicitly recording that root. With \(A_0=\Lambda(\varpi_K)\) and \(X=\operatorname{c\!-\!Ind}_J^G\eta\), define \((Tf)(g)=A_0^{-1}f(\varpi_Kg)\). Then \(X/(T-1)X\simeq\pi\). After imposing the fixed central character, \(\mathcal N/(F^\times J)\) is cyclic of order \(N\), and the finite average \(N^{-1}\sum_{i=0}^{N-1}T^i\) selects this same extension. These operators act on complex representation spaces, so their scalar division by \(N\) is valid. This is a different recovery from the Block159 projector onto a sector of \(\eta\). The exact kernel and finite projector are proved in lowrank/ENLARGED_CARRIER_CONTRACT167.md.
G.7.4 What the low-rank repair admits
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Apply the marked scalar-pushout theorem of Section 4 to the actual new pair \((H_0,P)\) and its carrier \(E\). It has exactly the capacity of the whole source at each low rank. All full source character values and the transported whole finite subgroup, lifts, squares and phase markings can be attached and retained. The displayed \(\Sigma\) and induction in stages recover the same candidate.
The old H2 break targets remain their supplied actual simple-character quotients. The enlarged pair is their common arithmetic assembly carrier; it is not renamed as the original maximal-simple \(H^1,J^1\) pair. If a stronger reading requires the whole packet itself to be the quotient of a maximal simple character in the same candidate, the earlier dimension obstruction still applies. Likewise, agreement with an independently fixed native quadratic marking must be checked separately. The contract note proves this distinction explicitly.
G.8 8. Final scope and next construction question
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The weighted dyadic mechanism now gives actual arithmetic fields, full characters and tame central completion in every rank \(2^d\) of the stated increasing odd-break family. Under the one-vertex, \(C=1\), full marked scalar-pushout assembly contract, the packet carrier can be realized at every length: inside the original type for \(d\ge3\), and in the explicit enlarged inducing carrier at \(d=1,2\). The source's prescribed central character must belong to the covered tame class. In particular no arbitrary single-factor wild determinant is assumed to be trivial.
Remaining universal obligations are construction/comparison of independently prescribed canonical native markings, nontrivial symplectic stabilizer actions, general source/transport diagrams and any required numerical layer placement. Original H5 remains conditional for this same fixed candidate. No equality of profiles or choice among noncompact roots is promoted to H5.
The next concrete target is equivariance of the arithmetic attachment for a nontrivial prescribed symplectic action. Another matrix search inside the old low-rank maximal-type carrier would repeat its already proved obstruction. STATUS167.json and ROUTES167.md record the achieved scopes and open routes.