C Native-rank full characters and tame central completion
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Parent checkpoint: Paper_XI_Base_Realization165. New arithmetic input: attachment166/tower/TRACE_ZERO_AS_TOWER166.md, together with its RECURSIVE_TOWER166.md defining-sequence proof. Input hashes are recorded in TRACE_ZERO_CHARACTER_PINS166.json.
Evidence: ordinary proof using the explicitly constructed arithmetic parameters and the named Bushnell–Kutzko character and transfer theorems. No arithmetic-induction/descent constructor or inverse correspondence is used. No Lean verification is claimed.
C.1 1. Statement
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Let p be odd, F=F_p((t)), and
0<h_1<...<h_d, p not dividing any h_i,
q_i=(p+1)h_i, N=p^d.Let beta_d be the actual trace-zero Artin–Schreier matrix constructor in the named tower note. There exists an actual full simple character theta for its maximal simple stratum such that
theta restricted to U_F^1 is the trivial character. (1)Every step constructing theta is specified below: a finite abelian character extension, the standard simple-character transfer, or the explicit additive correction from the displayed defining sequence. The field, order, and parameter are the actual tower ones, not an unspecified arithmetic realization.
The actual canonical beta extension can then be completed to a maximal extended type and supercuspidal pi with any prescribed smooth central quasicharacter omega_F trivial on U_F^1. Its rank, Swan conductor, torsion number and self-pair conductor remain those proved for the same tower. The residue and noncompact extension choices must be free or already compatible with omega_F.
For p=3, this gives full scalar-pushout attachment and same-candidate sector recovery for any number d of the supplied cubic source planes, with declared stabilizer C={1} and the one-vertex contract. All original full source character values are retained in the attachment. A general nontrivial stabilizer, transport diagram, or H5 conclusion is not part of this statement.
C.2 2. The exact arithmetic inputs
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For 1<=j<=d, put
b_j=B(F,t;h_(d-j+1),...,h_d) in M_(p^j)(F),where B is the trace-zero constructor. The tower proof gives:
K_j=F[b_j] is a totally ramified maximal field of degree p^j, normalizing the native complete period-p^j chain A_j.
For \(1\le j<d\), in rank p^(j+1), gamma_j=b_j tensor I_p is a simple approximation to b_(j+1), at the critical level r_j=q_(d-j)>0 of b_(j+1). They are equivalent as strata at that level.
For \(1\le j<d\), delta_j=b_(j+1)-gamma_j has grade -r_j and F-matrix trace zero.
At native rank p, b_1 is multiplication by alpha in the explicit Artin–Schreier field
E=F(alpha), alpha^p-alpha=t^(-q_d).Its actual field trace is zero, since p is odd.
For completeness, the tower's exact trace identity is
tr_L Lift_E(C)=Tr_(E/L)(kappa^(-1) tr_E(C)),where s=kappa*pr is its normalized trace corestriction and Lift_E(C)=s(P0)^(-1) C P0. The recursive inner matrix has trace zero, so this proves zero trace at every native rank. It is not merely the automatic zero obtained by amplifying any matrix p times in characteristic p. The literal compatibility of the truncations is proved in Section6 of the tower note.
C.3 3. A finite character-extension fact
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Let K/F be a finite totally ramified local extension. Suppose chi is a smooth character of U_K^a, and chi is trivial on U_K^a intersect U_F^1. Then chi extends to a smooth character of U_K^1 trivial on U_F^1.
Indeed, on U_K^a U_F^1 define chi_0(ku)=chi(k). Agreement on the intersection makes this a character. Choose M>=a such that chi is trivial on U_K^M. The character chi_0 descends to a subgroup of the finite abelian group \(U_K^1/U_K^M\). It extends to that whole group by successively adjoining generators and selecting complex roots. Pulling it back proves the assertion. This also describes an actual finite procedure for making the extension choices.
This fact will be used only when the parameter field is maximal in the current native matrix rank. Direct use in a nonmaximal centralizer would require checking its determinant image and would not follow from the scalar intersection alone.
C.4 4. Base character at native rank p
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Let E=K_1, and let n_1=q_d be the native level of alpha. Since n_1 is even, set a=n_1/2+1. On U_E^a the forced field-unit character is
phi(v)=psi_F(Tr_(E/F)(alpha(v-1))).It is trivial on U_E^a intersect U_F^1 because Tr_(E/F)(alpha)=0. Section3 gives a full character chi_E of U_E^1 extending phi and trivial on U_F^1. No additive-character normalization is needed for this implication: the trace is exactly zero.
For the native minimal stratum, \(H^1=U_E^1\) U_(A_1)^(a). Define theta_1 by chi_E on U_E^1 and by psi_alpha on U_(A_1)^a. They agree on the intersection, so BK3.2.1 supplies the actual full simple character. It satisfies theta_1| U_F^1=1.
C.5 5. Transfer the full field character before adding the next layer
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Suppose theta_j has been constructed on the native rank-p^j stratum of b_j, with theta_j| U_F^1=1. Its restriction to U_(K_j)^1 is the full beta-determinant character chi_j, since its native centralizer is K_j itself.
Use the standard simple-character transfer to the multiplicity-p realization gamma_j=b_j tensor I_p in rank p^(j+1). Denote the transferred full character by Theta_j. The exact primary result is BK3.6.13(ii), with existence in Theorem3.6.14: transfer preserves the full beta-determinant character on U_(K_j)^1. Consequently its restriction to the centralizer order is chi_j composed with det_(K_j).
For every literal scalar u in U_F^1 this gives
Theta_j(u I)=chi_j(u^p)=theta_j(u^p)=1. (2)The determinant exponent p is essential. We do not replace (2) by an unsupported pullback along a scalar matrix embedding. This transfer also deals with nonminimal b_j: beta-determinant preservation is the general theorem, not just the minimal-character formula.
C.6 6. Add the actual correction and extend at maximal rank
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Let r=r_j=q_(d-j), m=floor(r/2)=r/2, beta=b_(j+1), gamma=gamma_j, and delta=beta-gamma. The actual defining sequence gives
D=H^(m+1)(beta)=H^(m+1)(gamma).By BK3.3.18, the character
vartheta=Theta_j restricted to D times psi_deltais an actual simple character at level m for beta. For every scalar u in D intersect U_F^1, equation (2) and the exact trace identity give
vartheta(u)=Theta_j(u)
psi_F((u-1)tr_F(delta))=1. (3)The new parameter field K=K_(j+1) is maximal and totally ramified, and the native chain has K-period one. Thus
U_K^1 intersect D=U_K^(m+1),
U_F^1 intersect D=U_F^ceil((m+1)/p^(j+1)).In particular (3) supplies the exact compatibility condition for extending vartheta| U_K^(m+1) while imposing triviality on all U_F^1. Apply Section3 to obtain such a character chi_K of U_K^1.
BK3.2.5, including its explicit extension formula, now constructs
theta_(j+1)(kd)=chi_K(k) vartheta(d),
k in U_K^1, d in D.The proposition supplies normalization of vartheta by the field units, well-definedness, and the full simple-character condition. This theta_(j+1) has the claimed trivial wild center. Induction proves (1) at the actual final rank p^d.
The proof deliberately performs every arbitrary character extension at maximal native rank. Extending directly along amplified intermediate centralizers M_(p^(d-j))(K_j) would introduce determinant-image compatibility conditions. Merely knowing that their ambient matrix trace is zero would not discharge those conditions. Transfer first, followed by maximal-rank extension, removes that ambiguity rigorously.
C.7 7. Complete this same type with the prescribed tame center
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Keep the actual canonical beta extension kappa of the resulting theta. The final K/F is totally ramified of degree N, with residue rank one. The compact type group satisfies
J/J^1 = k_K^times = F_p^times.On U_F^1 the beta extension acts by theta, hence trivially. For a prescribed omega_F trivial on U_F^1, choose the residue character
sigma=omega_F|F_p^times divided by omega_kappa|F_p^times.Its inflation gives lambda=kappa tensor sigma with the right center on O_F^times. Complete lambda to an actual extension Lambda_0 on the inducing group K^times J using the retained maximal-type completion theorem. If its center is omega_0, choose s in C^times with
s^N=omega_F(t)/omega_0(t).Multiply Lambda_0 by the character of K^times J trivial on J and taking a K-uniformizer to s. The quotient (K^times J)/J is Z, and t has image N; this proves exact agreement of the resulting central character with omega_F on all F^times. Canonical kappa is unchanged.
The resulting pi=c-Ind Lambda is irreducible, admissible and supercuspidal. All these choices are made before H5. If a residue character or noncompact extension has been immutably supplied, it must already satisfy this compatibility; an incompatible datum is not silently replaced.
By the exact same parameter and defining-sequence calculations,
rank(pi)=N=p^d,
torsion(pi)=1,
Swan(pi)=p^(d-1)(p+1)h_d,
a(pi^vee x pi)=p^(2d)-1
+ sum_i (p^(2i)-p^(2i-2))h_i.No determinant twist changes the depth. These are the proved arithmetic tower profile formulas, not a proof of H5.
C.8 8. Full packet attachment in arbitrary cubic length
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For p=3, retain d original supplied full characters theta_i and their actual character groups H_i^1 normal in J_i^1, with whole symplectic quotients V_i that are the d labelled planes. Assume the declared stabilizer is C={1} and the diagram is one vertex. The source balanced scalar extension has quotient V_src of dimension 2d and Heisenberg carrier dimension 3^d, the original rank.
The actual final host has total quotient dimension N-1. Its leading nondegenerate symplectic layer has dimension
(p-1)p^(d-1)=2*3^(d-1) >= 2d.This follows either from the explicit defining-sequence quotient calculation or the independently proved leading-layer lemma. Choose a labelled symplectic embedding of the supplied V_src in that layer. Write W=V_src perpendicular-sum V_perp. Choose a maximal isotropic subspace L in V_perp. Its associated eigenspace in the actual Heisenberg carrier has dimension 3^d, and L-perp/L is V_src.
Form the actual reduced scalar extension using the full host character and its extension to the inverse image of L. Form each original full pushout
P_i=(C^times x J_i^1)/
{(theta_i(h)^(-1),h): h in H_i^1}.The center-fixing symplectic-extension isomorphism, constructed by matched Darboux lifts and complex roots, maps their balanced product to that actual reduced scalar extension. In particular each original h in H_i^1 maps to the scalar theta_i(h). Every prescribed shallow and deep source character value is retained. Source character attachment does not claim that the original field units are the same literal scalar matrices in the new host.
Finite Stone–von Neumann supplies the full carrier intertwiner. The retained sector induction and recovery idempotent then recover this same Lambda and this same pi from the source-sized sector. The canonical beta extension, residue factor, central character, and noncompact scalar are those actually selected in Section7. No source-independent alternate candidate is substituted in recovery.
For the retained independent tensor family with d>=2, each factor's wild determinant has order dividing p, because the abelianization of its extraspecial order-p^3 image is elementary abelian. The tensor determinant raises each factor determinant to p^(d-1), so the full source determinant is trivial on wild inertia. Thus the source's prescribed central quasicharacter lies in the class covered by Section7. At d=1, matching this central conclusion requires the source determinant itself to be trivial on wild inertia; it is not inferred merely by taking a tensor power of length one.
For p>3, the same arithmetic host and character theorem hold, and the same attachment works for any actually supplied valid packet whose whole quotient and markings meet the stated symplectic contract. A proposed source plane is not automatically a whole rank-p type quotient: that latter quotient has dimension p-1. No admission of all such plane sources into original H2 is inferred.
C.9 9. Boundary and primary-source check
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This provides the previously missing full-character and actual wild-center construction for the explicit odd-prime arithmetic tower in every prime-power rank. The proof uses no classical automorphic-induction/descent candidate. It retains the supplied finite characters through the declared scalar-pushout attachments.
It does not preserve old summand matrix conjugacy classes, produce a prescribed nontrivial stabilizer action, preserve arbitrary numerical depth labels, handle arbitrary transport graphs, settle the dyadic case, or prove original same-candidate H5. It therefore does not close unrestricted claim66.
The retained primary BK text, SHA256 d69db0827256707a34cd2562af68b9de11fc889ab3f43535ec47b620d58ca9c8, was directly checked at:
Definition3.2.1 and Proposition3.2.5, printed98–100: minimal character condition and explicit full-character extension.
Corollary3.3.18, printed112–113: the actual additive correction takes the approximation character to the next simple character.
Condition3.6.13(ii) and Theorem3.6.14, printed136–137: transfer preserves the full beta-determinant character, including at m0=0.
The actual field and defining-sequence assertions are the separately proved and independently reviewed trace-zero tower inputs, with BK2.2.7 and the retained simple-stratum theorems. No newly downloaded source or new novelty claim is involved.