A. Odd-prime synthesis and complete scope

Date: 2026-10-05. Parent: Paper_XI_Base_Realization165.zip.

This is a synthesis of the independently reviewed proofs in this continuation. The result is an ordinary mathematical proof, using the stated classical simple-type theorems. It is not a Lean certificate. Universal claim 66 remains open. Original H5 remains the original conditional premise about the same fixed candidate; no unconditional pair-factor comparison is asserted here.

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Let \(p\) be odd, \(F=\mathbb F_p((t))\), and

For every such tuple there is an explicit matrix \(\beta\in M_N(F)\) whose field \(K=F[\beta]\) is totally ramified of degree \(N\), normalizes the complete lattice chain, and gives a maximal simple stratum. It has a literal defining sequence of degrees \(p,p^2,\ldots,p^d\), with leading level \(n_1\) and successive critical levels \(n_2,\ldots,n_d\) given by

Every native parameter in the construction, every amplified approximation, and every consecutive correction has matrix trace zero over its base field.

For this actual stratum there exists a full simple character \(\theta\) with \(\theta|_{U_F^1}=1\). Retaining its canonical beta extension, the residue character and noncompact extension can be selected so that the resulting irreducible admissible supercuspidal \(\pi\) has any prescribed smooth central quasicharacter \(\omega\) trivial on \(U_F^1\). These completion choices must be free or already compatible; incompatible immutably supplied choices are not replaced.

The same candidate has

These numerical identities do not identify a Weil representation, an endo-class, or the pair epsilon factors required by H5.

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At a recursive base \((L,t_L)\), introduce

This is a separable totally ramified extension of degree \(p\), with \(v_E(\alpha)=-q_d\) and \(\operatorname{Tr}_{E/L}\alpha=0\). Choose \(1\le b<p\) with \(q_db\equiv-1\pmod p\), set \(a=(q_db+1)/p\), and put \(z=t_L^a\alpha^b\). Then \(z\) is an explicit uniformizer. Write its Eisenstein polynomial as \(f(X)=X^p+\cdots+a_0\).

Let \(P_0\) extract the constant coefficient in the basis \(1,z,\ldots,z^{p-1}\). The normalized corestriction \(s=\kappa\operatorname{pr}\), where \(\kappa=-f'(z)/z^{p-1}\), satisfies

The lift preserves the exact filtration valuation. Its crucial additional identity is

Define

with the degree-\(p\) multiplication matrix as the base case. Induction proves trace zero at every native rank. This is stronger than the automatic zero trace obtained by multiplying any matrix multiplicity by \(p\) in characteristic \(p\).

For the inductive field step \(d\ge2\), put \(n=p^{d-1}q_d\) and \(R=p^{d-2}q_{d-1}\). The derived matrix is the recursively constructed maximal-field parameter, with \(k_0=-q_1\). The inequalities \(n>pR>2R\) imply \(-2R>-n-q_1\). Bushnell–Kutzko Remark 2.2.7 therefore applies to the actual displayed matrix. It supplies its maximal field, normalization, and critical exponent. Valuation preservation and literal truncation of the recursion supply the whole defining sequence. The full details, quotient count, and conductor dependencies are in tower/TRACE_ZERO_AS_TOWER166.md and tower/RECURSIVE_TOWER166.md.

The full-character argument uses a specific order of operations. First construct a full character at native rank \(p^j\), trivial on \(U_F^1\). Transfer its full beta-determinant unit character to multiplicity \(p\), using BK 3.6.13(ii) and 3.6.14. On a scalar \(u\in U_F^1\), this gives \(\theta_j(u^p)=1\). Next add the actual trace-zero correction, using BK 3.3.18. It remains trivial on the scalar intersection of its domain. Finally extend at the next maximal native parameter field: the forced deep character and the trivial character of \(U_F^1\) agree on their intersection, so they extend through a finite abelian unit quotient. BK 3.2.5 constructs the full simple character.

This procedure never substitutes an unjustified extension through a nonmaximal centralizer. It proves full-character compatibility, including shallow units, rather than merely cancellation of a leading trace. The residue factor fixes \(\omega\) on \(\mathbb F_p^\times\); an \(N\)-th-root choice for the noncompact generator fixes \(\omega(t)\). Every choice is recorded before H5. See central/TRACE_ZERO_CHARACTER_INDUCTION166.md for the complete induction.

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Assume the one-vertex scalar-pushout contract: there are \(d\) labelled, actually supplied full character extensions with whole symplectic quotients \(V_i\) of dimension two over \(\mathbb F_p\). Their balanced \(\mathbb C^\times\)-central product has quotient \(V_{\mathrm{src}}=\bigoplus_iV_i\) and Heisenberg carrier dimension \(p^d=N\). Initially take the declared stabilizer \(C=\{1\}\).

For \(p=3\), this applies directly to the retained cubic family with whole rank-three break-type quotients. For \(p>3\), it applies only when the supplied plane data genuinely satisfy the stated original bridge contract. A native minimal rank-\(p\) type quotient has dimension \(p-1\), so it must not automatically be called a plane.

The actual candidate from Section 1 has a nondegenerate leading subspace

Here \(S\) is the image of the concrete congruence subgroup at half the leading even level. BK's high-depth equality preserves this subgroup and its commutator pairing through every later approximation; see capacity/LEADING_LAYER_CAPACITY166.md, Section 1. This part of that note applies to the direct constructor and does not require its separate classical arithmetic-induction application.

Choose a labelled symplectic embedding \(V_{\mathrm{src}}\to S\). For each original full character form

The balanced product and the corresponding host restriction have the same alternating commutator. The center-fixing extension isomorphism, obtained with matched Darboux lifts and complex roots, therefore preserves every supplied value \(\theta_i(h)\), including the shallow values.

Write \(W=V_{\mathrm{src}}\perp V_{\mathrm{src}}^\perp\). Finite Stone–von Neumann gives \(\eta_W\simeq\eta_{\mathrm{src}}\otimes\eta_\perp\). Choose an isotropic eigenspace giving a line in the complementary factor. Its actual Heisenberg image-algebra projector \(e\) has rank \(N\). The retained induction/idempotent theorem recovers the same inducing representation \(\Lambda\) from \(e\Lambda\), and then the same \(\pi\) by compact induction. No candidate is changed at recovery.

This attachment preserves the full source characters as scalar-pushout data. It does not identify the old matrix groups with literal subgroups of the new type, retain the old lower summand conjugacy classes, or place each source label at its old numerical host depth. Those stronger requirements have not been proved.

When the retained independent extraspecial tensor source has \(d\ge2\), its full determinant is trivial on wild inertia: each factor determinant has order dividing \(p\), and tensor determinants raise it to \(p^{d-1}\). Thus its central character belongs to the class in Section 1. For \(d=1\), wild triviality of the source determinant must be checked separately. Independence of the tensor factors is an input, not a consequence of distinct \(h_i\).

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There is also a positive result beyond \(C=\{1\}\). Suppose the finite prime-to-\(p\) stabilizer acts trivially on the source scalar extension, and its prescribed honest operators on the entire rank-\(N\) Weil carrier are \(\chi(c)I_N\), each lying in the actual inertia image. Assume the candidate's central character is the determinant of that same source under the fixed rank-one reciprocity convention.

Then a finite cover \(\widetilde C\twoheadrightarrow C\) acts by actual central matrices in the unchanged candidate. Its represented operators descend to exactly \(\chi\) on \(C\). Hence the attachment and recovery above are equivariant for that action. A direct matrix homomorphism from \(C\) itself is not asserted and can fail.

Indeed, for residue cardinality \(q\), let \(B=\omega(\operatorname{Teich}(k^\times))\). The actual inertia hypothesis gives \(\chi(C)^N\subset B\). Since \(N\) is a power of \(p\) and \(\chi(C)\) has order prime to \(p\), this implies \(\chi(C)\subset B\). Choose \(k_0\) inverse to \(N\) modulo \(q-1\), and set

The matrix \(\operatorname{Teich}(u^{k_0})I_N\) acts by \(\chi(c)\). The kernel acts trivially on the entire sector induction and ambient compact induction, because these matrices are central. This preserves the fixed \(\Lambda\), its canonical beta-extension factor, and all completion choices. It does not assert that the canonical beta extension alone has scalar character \(\chi\).

The full theorem and the nonsplitting example are in stabilizer/SCALAR_NATIVE_ATTACHMENT166.md, with an independent review in audit/SCALAR_NATIVE_REVIEW166.md. An arbitrary abstract phase or a phase on a proper multiplicity space does not satisfy its hypotheses merely because it is scalar.

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The old fixed-field central argument genuinely failed. At \(p=3\), \((h_1,h_2)=(1,5)\), its naturally transferred full character takes the value \(\zeta_3\ne1\) on \(1+t^2\), although the desired wild center is trivial. A shallow character choice cannot repair a forced value. The two-stage changing-field repair is retained in central/CENTRAL_REPAIR166.md; the trace-zero tower above resolves this central issue uniformly in every length.

The construction does not use an inverse established correspondence or the classical arithmetic-induction/descent candidate to select \(\pi\). It does use the named BK construction and type theorems, the retained type/formal-degree conductor formulas, and exact sector recovery. The capacity note separately retains an alternative consequence for a candidate supplied by the substantial classical arithmetic constructor; that is a different route and is not an input to Section 1.

The unresolved universal obligations are precise:

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1. Nontrivial prescribed symplectic stabilizer actions. For this totally ramified full-degree host, the stronger inner-normalizer interpretation forces prime-to-\(p\) actions on the quotient to be trivial. It therefore cannot implement a genuinely nontrivial supplied symplectic action. The raw original contract does not automatically require that stronger interpretation; it still needs an actual valid arithmetic comparison, not relabelling of external operators.

2. Arbitrary admissible source packets and transport diagrams. The proved attachment has the whole-plane, one-vertex, finite scalar-pushout hypotheses above. It does not establish all source classes, arbitrary graphs, or prescribed numerical layer placement.

3. The dyadic construction with all markings. A separate field-level recursion and some capacity statements exist at \(p=2\). The odd-prime trace-zero/full-character theorem here does not settle that case.

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Original H5 stays exactly where it was: if supplied for this fixed candidate and the stipulated tests, the retained conditional endpoint applies. No claim that the displayed profiles imply H5, or that one of the noncompact root choices must satisfy H5, is made. The unresolved construction issues above are not renamed epsilon-factor obligations.

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The arbitrary-length theorem is proved symbolically and independently reviewed. The exact \(p=3,d=2,(h_1,h_2)=(1,2)\) computation in tower/as_rank9check/ verifies the displayed matrix, trace identities, corestriction, and levels; it is a finite check, not the universal proof. The separate rank-27 run in tower/run_rank27/ checks the earlier non-trace-zero recursion and must not be described as an AS central completion example. No Lean build was run.

STATUS166.json records the closed and open obligations. The component audits identify the exact reviewed files and hashes. The dependency file pins prior checkpoints and the authorized primary-source copy without redistributing that publication.

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