E Actual scalar inertia phases and finite-cover descent
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Evidence: ordinary proof using rank-one reciprocity, the given central character, and elementary finite group arguments. The result constructs actual central matrix operators through a finite cover and proves their descent. It does not construct a new arithmetic candidate or alter a canonical beta extension.
E.1 1. Exact statement
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Let F be a nonarchimedean local field with residue field k of cardinality q and residue characteristic p. Put Q=q-1 and n=p^r, allowing r=0. Let rho:W_F -> GL_n(C) be a smooth Weil representation with finite inertia image. Use the fixed reciprocity convention identifying its determinant with a character omega:F× -> C×.
Let C be a finite group of order prime to p, and suppose its prescribed operators on the full source carrier are
Omega_src(c)=chi(c)I_n,where chi is a character and each Omega_src(c) belongs to the actual matrix image rho(I_F). A finite cyclic scalar tame image is the intended example. This hypothesis refers to actual source operators, not freely chosen phases on an unrelated multiplicity-space factor.
Let Pi be any already constructed irreducible smooth GL_n(F) representation with central character omega. Suppose it has its supplied actual inducing representation Lambda on N, with F× contained in N, and an attached finite sector E=eLambda realizing the prescribed source Heisenberg packet. Assume the prescribed C action on the source scalar extension is trivial, and that the honest scalar phase on the packet is the same chi above.
Theorem. There is a finite group C_tilde surjecting onto C and a genuine homomorphism
g:C_tilde -> F× I_n contained in Nwhose action on Lambda and on Pi descends to the prescribed honest C action chi(c)I. Every finite sector is stable under this action. Any center-preserving Heisenberg attachment and its intertwiner are C-equivariant. The retained sector-induction recovery maps recover the same Lambda and Pi equivariantly. All actual inducing data, including the fixed canonical beta extension, remain unchanged.
The arithmetic action is supplied by actual central matrices, with an explicit finite-cover descent. The theorem does not assert that C itself embeds as a matrix subgroup. Such an embedding can fail even for a genuine scalar tame source, as Section4 shows.
E.2 2. Construction and proof
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Let D=k×, and let Teich:D -> O_F× be the canonical multiplicative lift. For p=2 this is still the prime-to-p root-of-unity subgroup. When q=2, D is trivial and the argument below forces chi trivial as well.
Local reciprocity identifies the determinant image of inertia with
det rho(I_F)=omega(O_F×).The unit group decomposes as
O_F× = Teich(D) times U_F¹.The smooth character image omega(U_F¹) is a finite p-group, while omega(Teich(D)) has order prime to p. Therefore the prime-to-p elements of omega(O_F×) are exactly omega(Teich(D)).
For every c in C, the actual-source hypothesis implies
chi(c)^n=det(Omega_src(c)) in det rho(I_F).This value has order prime to p, and hence lies in omega(Teich(D)). Since n is a p-power and chi(C) has order prime to p, the map z -> z^n permutes chi(C). Consequently
chi(C) is contained in omega(Teich(D)). (1)In particular every source phase has order dividing Q. This conclusion does not require selecting a splitting of the tame quotient or choosing Frobenius representatives.
Choose an integer k_0 with n k_0 congruent to1 modulo Q. When Q=1 take k_0=1. Define
delta(u)=omega(Teich(u))^(k_0), u in D.Because exponentiation by k_0 is an automorphism on every group of roots of unity of order dividing Q, delta(D)=omega(Teich(D)). Define the finite fiber product
C_tilde={(c,u) in C times D : chi(c)=delta(u)}. (2)It is a subgroup, and projection pr:C_tilde -> C is surjective by (1). Define the actual central matrix homomorphism
g(c,u)=Teich(u^(k_0)) I_n. (3)Its action on Lambda and Pi is the central-character scalar
Lambda(g(c,u))=omega(Teich(u^(k_0)))I
=delta(u)I=chi(c)I. (4)If (1,u) belongs to ker pr, (4) is the identity. Hence the represented action factors uniquely through C. It is exactly the prescribed source action. Although selecting a particular matrix representative for c may require a choice of u, the represented operator is independent of that choice. Changing k_0 by a multiple of Q leaves (3) unchanged.
All matrices (3) are central. Their conjugation on actual groups, characters, and scalar pushouts is therefore the identity, and their Lambda operators are scalar. Thus they preserve every e and every sector E. The source extension action is also the identity by hypothesis, so any attachment of those scalar extensions is equivariant. Both honest carrier actions are chi(c)I, so every Heisenberg intertwiner intertwines C without a new phase choice.
For the recovery step, put P={a in N:Lambda(a)eLambda(a)^(-1)=e}. Then F× is contained in P. The retained maps
Phi:Ind_P^N E -> Lambda,
j:Lambda -> Ind_P^N Eare N-equivariant. They therefore commute with all actual matrices (3), and ker pr also acts trivially on Ind_P^N E: its matrices are central elements of P acting trivially on E, hence trivially on every induced translate. Thus the equality descends through pr to C. The recovering idempotent, and then compact induction to GL_n(F), preserve this equivariance. They recover the same fixed Lambda and Pi. QED.
E.3 3. What this closes and what it does not
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For a source packet whose marked tensor carrier is the entire rank-n source, this proves the stabilizer step whenever its genuine tame operators are scalar and an actual host with the correct central character and full-character attachment is already present. There is no remaining scalar-phase equation to solve in that situation: the determinant and central character supply it.
If only the source symplectic C action is known to be trivial, the preceding STABILIZER_CONTRACT166.md proves triviality of its full scalar-extension action for prime-to-p C. Schur then gives a scalar packet character. To apply the present theorem one must still identify that character with the action of actual inertia on the full source. That identification holds when the supplied honest packet carrier is the whole source carrier. It is not automatic for a proper multiplicity-space factor, whose omitted factor can contribute to the determinant.
For several factors the total packet character is the product of the supplied factor characters. The theorem matches that product on the whole attached packet. It does not represent independent factor actions by distinct central matrices unless those additional marked actions are separately specified and verified.
The result uses the fixed full inducing representation Lambda. If Lambda| J=kappa_can tensor sigma, then kappa_can and sigma are both unchanged. Equation(4) concerns Lambda, and does not assert that kappa_can alone has the same scalar character. A contract prescribing an action specifically on the canonical beta extension requires that distinction to remain explicit.
No nontrivial source symplectic action is realized here. No new base candidate, new H5 comparison, dyadic quadratic refinement, or general rank theorem is claimed. Existing quadratic refinements are preserved because the actual central conjugation action is the identity.
E.4 4. Why a direct map C into residue units is too strong
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Take q=5, n=5^r, and let omega on k×=F5× be the quadratic character. Let C=C2 and let chi be its nontrivial character. Such scalar tame behavior is obtained by twisting a tame-trivial rank-5^r source by the quadratic tame character, whenever such a source is supplied; the retained odd-prime two-factor family supplies rank25 examples.
A group homomorphism a:C2 -> F5× must send its generator to1 or -1. Both have quadratic character1. Consequently
omega(a(c)) = chi(c)^nis impossible at the nontrivial c, since n is odd. The proposed direct section is not forced by genuine scalar tame source data. Reciprocity supplies a quotient of residue units; that quotient need not split.
The construction above still works. Here k_0=1 modulo4, C_tilde is the cyclic group of order4, and its central matrices act by the quadratic character. The kernel of C_tilde -> C2 acts trivially on the fixed candidate, so the actual operator action descends correctly. This example explains why finite-cover descent is necessary rather than a cosmetic reformulation.
E.5 Dependencies and review status
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External input: rank-one local class field theory, with the same fixed normalization used for omega=det rho; the existing actual candidate's central character identity; the retained sector recovery theorem. The finite-cover construction and its descent are proved above and use no local Langlands candidate selection. No computation is needed. This version has undergone author self-review; a parent audit is requested. No Lean run is claimed.