19 Restricted arithmetic realization theorems
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The later constructions provide actual arithmetic candidates and exact finite packet recovery in substantial restricted families. They do not establish Conjecture 66 for every admitted source. Their detailed ordinary proofs are typeset in the appendices, and the original reviewed Markdown files accompany this edition. In these appendices \(F\) denotes the base local field, while \(K=F[\beta]\) denotes a parameter field; these are local notation conventions. All references to “claim 66” in the retained notes mean the general endpoint now isolated as Conjecture 66.
19.1 The odd-prime trace-zero construction
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Theorem 19.1: Restricted odd-prime arithmetic realization
Let \(p\) be an odd prime, \(F={\mathbb F}_p((t))\), fix \(d\ge1\), and let
There is an explicit maximal simple stratum in \(M_N(F)\) with totally ramified full-degree parameter field, a literal defining sequence of degrees \(p,p^2,\ldots,p^d\), and levels \(n_j=p^{d-j}q_{d-j+1}\). Every native parameter and consecutive correction has trace zero over its recursive base field. The stratum admits a full simple character trivial on \(U_F^1\). Its fixed canonical beta extension can be completed to an actual irreducible admissible supercuspidal \(\pi\) with any prescribed smooth central quasicharacter trivial on \(U_F^1\), when its completion choices are free or already compatible.
Assume actually supplied labelled full-character extensions with whole two-dimensional symplectic quotients whose balanced complex-scalar central product has \(d\) planes, one vertex, and trivial stabilizer. Then the same \(\pi\) admits a full scalar-pushout realization of that packet and exact recovery of its unchanged inducing representation. The prescribed central character must lie in the covered tame class. No literal embedding of the old matrix groups, canonical native marking, or prescribed numerical host-depth placement is inferred.
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Proof of Theorem 19.1
The explicit Artin–Schreier recursion and its normalized filtered lift construct the maximal parameter and literal defining sequence; see Section B. Its trace transport identity makes every native trace zero, rather than relying on amplification by \(p\). Native-rank full-character construction, full beta-determinant transfer, trace-zero correction, and maximal-field character extension give the scalar triviality and central completion; see Section C. The actual leading symplectic layer has dimension \((p-1)p^{d-1}\ge2d\) and survives every later approximation (Section D). A labelled symplectic embedding gives a center-fixing full scalar-pushout isomorphism. Finite Stone–von Neumann decomposition separates the source factor from its orthogonal complement. Maximal-isotropic averaging on the complementary Heisenberg factor selects a one-dimensional representation line and hence a source-sized sector. The retained induction/idempotent recovery returns the same \(\Lambda\) and \(\pi\), as proved in Section A.
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For \(p>3\), a native minimal rank-\(p\) type quotient has dimension \(p-1\), so it is not automatically a plane. The theorem requires genuinely supplied whole-plane data. Independence of source tensor factors is also an input. An arbitrary single-factor wild determinant need not be tame.
Under additional actual-inertia hypotheses, genuine scalar phases on the entire source carrier are realized by actual central matrices of a finite cover and descend to the stipulated operators. A direct matrix homomorphism from the stabilizer itself need not exist; the exact statement and descent proof are in Section E.
19.2 The weighted dyadic construction and enlarged carriers
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Theorem 19.2: Restricted dyadic marked assembly realization
Let \(F={\mathbb F}_2((t))\), fix an additive character of conductor \(t{\mathcal O}_F\), fix \(d\ge1\), and let \(h_1<\cdots<h_d\) be positive odd integers. Put \(q_i=3h_i\), \(N=2^d\), and \(n_j=2^{d-j}q_{d-j+1}\). The explicit recursive matrix generates a separable totally ramified degree-\(N\) maximal parameter field, normalizes the complete lattice chain, and has the literal defining sequence of degrees \(2,4,\ldots,2^d\). It admits a full simple character trivial on \(U_F^1\) and an actual supercuspidal completion with any prescribed central quasicharacter trivial on \(U_F^1\), when the noncompact completion is free or already compatible.
For actually supplied full marked whole-plane scalar-pushout data with one vertex, trivial stabilizer, and prescribed source central character trivial on \(U_F^1\), an arithmetic assembly carrier exists at every length and recovers the same candidate: inside its original type for \(d\ge3\), and in explicit enlarged inducing carriers for \(d=1,2\). Every supplied full character value and transported whole finite subgroup, lift, square, and phase marking is retained at the represented marking level. Any central-character kernel already factored out remains recorded separately; scalar pushout does not recover its invisible elements. This does not identify the transported marking with an independently prescribed canonical native marking.
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Proof of Theorem 19.2
The Artin–Schreier filtered lift has native trace one. Consequently each native correction \(\delta\) satisfies
which makes its additive scalar correction trivial on \(U_F^1\). The actual field and defining-sequence proof is Section H. Full beta-determinant transfer and character extension at each new maximal native field prove the full-character claim, including the odd-critical endpoint \(2(\lfloor r/2\rfloor+1)-r=1\); see Section I. The exact original quotient has dimension \(2^d-2\), which accommodates the required \(2d\) source dimensions precisely for \(d\ge3\) (Section J).
At rank two, with \(q=2m+1\), enlarge \(J=U_K^1U_A^{m+1}\) to \(P=U_K^1U_A^m\). The actual character ratio \(\lambda=\theta^g/\theta\) for \(g\notin J\) is nontrivial and sign-valued. With \(H_0=\ker\lambda\), the quotient is \(P/H_0\simeq{\mathbb F}_2^2\), and \({\operatorname{Ind}}_J^P\theta\) is an irreducible two-dimensional Heisenberg carrier (Section L). At rank four, the actual outer quadratic centralizer gives \(P=J U_B^m\), \([P:J]=2\), and \(P/H_0\simeq{\mathbb F}_2^4\), with carrier \({\operatorname{Ind}}_J^P\eta\) of dimension four (Section M). The finite scalar center can be larger than \(\mu_2\); no literal extraspecial quotient is claimed.
At both ranks \(K^\times\) normalizes \(P\). Thus \(\mathcal N=K^\times J\) is normal of index two in \(\mathcal M=K^\times P\), and
Induction in stages preserves the full original \(\Lambda\), its central character, and its noncompact root; see Section N and Section O. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original higher-rank types (Section K). The full synthesis and scope are in Section G.
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The original low-rank maximal-type capacity obstruction remains true. The new pair is a common arithmetic assembly carrier; it is not renamed as the original maximal-simple-character pair. For an independently fixed native quadratic marking, a preassigned embedding must satisfy \(q_W(f(v))=q_{\rm src}(v)\). At equal capacity, a free embedding requires equal Arf invariant. Spare capacity for \(d\ge4\) admits either source Arf type when the embedding is free. Retaining only the central \(\{\pm1\}\) would erase the distinction between whole marked finite extensions such as \(D_8\) and \(Q_8\).
19.3 Scope, numerical invariants, and evidence
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For the same candidates the constructions give
and torsion number one. Here \({\operatorname{Swan}}\) denotes the Swan conductor, and \(a\) below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22–24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are
These identities do not prove H5, identify an endo-class, or construct a canonical source-to-host marking. At rank eight, the retained skipped-degree host has the same displayed numerical profiles as the full tower and a smaller Heisenberg quotient. The explicit construction, rather than numerical matching alone, supplies the arithmetic object.
The proofs use the stated Bushnell–Kutzko simple-type theorems [11] and the retained conductor inputs. Exact source locations and hypothesis checks are recorded in the component proofs and accompanying audits. The finite rank-nine and rank-eight runs check particular matrices, traces, normalized lifts, and filtration grades. They are bounded checks, not arbitrary-length proofs. No new Lean build of these arithmetic constructions is claimed.
Nontrivial prescribed symplectic stabilizer actions, independently fixed canonical native markings, arbitrary source/transport diagrams, and any required numerical layer placement remain outside the restricted results. The current totally ramified host cannot implement nontrivial prime-to-\(p\) symplectic actions by the stronger character-preserving inner-normalizer mechanism; this obstruction is stated precisely in Section F. That stronger mechanism is not silently imposed on the original conjecture. H5 remains conditional for the same fixed candidate.
The accompanying archive retains the mathematical reviews, computational evidence, and source records underlying the displayed scope.