I Dyadic full characters and the odd critical endpoint
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Checkpoint: Paper_XI_Dyadic167. Parent: final Paper_XI_Attachment166. Author scope: this file only; no Block166 file is modified.
Evidence: ordinary mathematical proof using the actual Artin–Schreier matrix tower and the retained Bushnell–Kutzko character and transfer theorems. No Lean verification is claimed. This note proves no dyadic quotient-capacity, quadratic-phase, packet-attachment, or H5 result.
I.1 1. Exact theorem
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Let F=F_2((t)), let psi_F be an additive character of conductor t O_F (trivial there and nontrivial on O_F), and let
0<h_1<...<h_d, each h_i odd,
q_i=3h_i, N=2^d.Use the dyadic Artin–Schreier recursion specified in Section2. It gives the actual maximal simple parameter beta_d with its native arithmetic defining sequence. There is an actual full simple character theta of its maximal simple stratum such that
theta restricted to U_F^1 is trivial. (1)The construction retains the actual canonical beta extension and can be completed with any prescribed central quasicharacter omega_F trivial on U_F^1. Because \(k_F^\times={\mathbb F}_2^\times\) is trivial, such an omega_F is unramified. Thus this covers every tamely ramified central quasicharacter in this particular residue-field setting.
The corresponding compact induction is an actual irreducible admissible supercuspidal of rank N, by the retained maximal-simple-type theorem. No claim is made here about its Heisenberg quotient size or about realization of a supplied dyadic finite packet.
The trace-zero argument from166 does not apply verbatim: at p=2 the primitive Artin–Schreier parameter has trace one. The correct invariant is integral trace, not necessarily zero trace. Since psi_F kills t O_F, integral correction trace is sufficient on scalar principal units. The recursion below preserves the stronger exact value one.
I.2 2. Actual dyadic recursion and exact native traces
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At a recursive step let L be a characteristic-two local field with residue field F_2, uniformizer t_L, and a positive odd q. Define
E=L(alpha), alpha^2-alpha=t_L^(-q).The right side is not an Artin–Schreier coboundary: a negative valuation of x^2-x is even, whereas -q is odd. The degree-two polynomial is therefore irreducible and separable. Its valuation equation gives total ramification and v_E(alpha)=-q. Put
a=(q+1)/2, z=t_L^a alpha.Then z is an actual uniformizer and its Eisenstein polynomial is
f(Z)=Z^2+t_L^a Z+t_L. (2)Its two conjugate alpha values are alpha and alpha+1, so
Tr_(E/L)(alpha)=1. (3)Use the coefficient basis 1,z, the trace corestriction pr, and the same normalized corestriction convention as166:
kappa=-f'(z)/z = t_L^a/z = alpha^(-1),
s=kappa pr.Minus and plus agree in characteristic two. If P0 extracts the constant coefficient on every E-coordinate, then
u_0=s(P0)=t_L/z^2 is a unit,
Lift_E(C)=u_0^(-1) C P0,
s(Lift_E(C))=C.The trace-dual calculation in166 gives, as an identity of L-matrix traces for every matrix C over E,
tr_L Lift_E(C)
=Tr_(E/L)(kappa^(-1) tr_E(C))
=Tr_(E/L)(alpha tr_E(C)). (4)The lift preserves the complete-chain filtration valuation, by expansion in 1,z. Thus it is the actual filtered right inverse used in the field constructor, not an arbitrary inverse on vector spaces.
Define B(L,t_L;h_1,…,h_s) recursively. At s=1 use the L-matrix of multiplication by alpha with q=q_1. At s>=2 use (2) with q=q_s, m=2^(s-1), and
C=B(E,z;h_1,...,h_(s-1)),
B(L,t_L;h_1,...,h_s)=alpha I_m+Lift_E(C). (5)The base trace is one by (3). If tr_E(C)=1, then (4) gives lift trace one. Since m is even, the trace of alpha I_m as an L-matrix is m Tr_(E/L)(alpha)=0. Consequently
tr_L B(L,t_L;h_1,...,h_s)=1 (6)for every native recursion, by induction. There is no division by the rank and no confusion between a native trace and its amplified trace.
The actual field proof from166 applies with this degree-two extension and normalized lift. At a nontrivial step its hypotheses are n=2^(s-1)q_s, R=2^(s-2)q_(s-1), s(Lift(C))=C, and
n>2R,
-2R>-n-q_1.The first strict inequality follows from h_s> h_(s-1). These are the conditions used in BK Remark2.2.7. The displayed beta itself generates the totally ramified degree-2^s field normalizing the complete chain, and its final critical exponent is -q_1. This statement does not identify the final field with its auxiliary derived field.
For the final length d write the native shorter matrices as
b_j=B(F,t;h_(d-j+1),...,h_d), 1<=j<=d.They are the compatible native truncations of (5), because the outer extension, uniformizer, and lift depend on the same retained largest h at each stage. For \(1\le j<d\), in rank 2^(j+1), put
gamma_j=b_j tensor I_2,
delta_j=b_(j+1)-gamma_j.The defining-sequence calculation of166 gives gamma_j as a simple approximation at the critical level r_j=q_(d-j) of b_(j+1), and delta_j has grade -r_j. Equation (6) gives the exact scalar identity
tr_F(gamma_j)=0,
tr_F(delta_j)=1. (7)Thus for every scalar u in U_F^1 on which its additive correction is defined,
psi_(delta_j)(u I)=psi_F((u-1)tr_F(delta_j))
=psi_F(u-1)=1. (8)It is (8), not a false trace-zero assertion, that drives the full character construction.
I.3 3. The odd-critical-level rounding
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The critical levels r=q_i are odd. Write r=2s+1, and set
m=floor(r/2)=s, m+1=(r+1)/2.The relevant BK statements use the floor, with no assumption that r or the residue characteristic is odd or even:
BK3.1.9(ii) identifies H^k(beta) and H^k(gamma) for k>=floor(r/2)+1.
BK3.3.18 gives the correction bijection at floor(r/2)<=m<=r-1.
BK3.2.5 extends the resulting character at floor(r/2) to a full character, its fibers being the indicated determinant characters.
At our exact endpoint m=s, the correction character is legitimate. For x,y in the radical power P^(s+1),
delta*x*y belongs to P^(2(s+1)-r)=P^1.The level-one additive matrix character is trivial on P^1, so the multiplicative cross term contributes one. Therefore psi_delta is a character on U^(s+1), and hence on the recursive subgroup used below. This checks the borderline odd-level rounding directly. Replacing m by r/2 as if it were an integer would be incorrect.
The same check applies at the native minimal base level q, which is odd: its forced additive character begins at floor(q/2)+1=(q+1)/2, and 2*(q+1)/2-q=1.
I.4 4. Finite character extension at maximal rank
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We reuse the elementary extension procedure with its compatibility condition explicit. If K/F is a finite totally ramified field, and chi is a smooth character of U_K^a trivial on U_K^a intersect U_F^1, define a character on U_K^a U_F^1 by
chi_0(ku)=chi(k).It is well defined by the intersection condition. Choose M>=a with chi| U_K^M=1. Then chi_0 is a character of a subgroup of the finite abelian group \(U_K^1/U_K^M\). Extend it by adjoining generators and choosing the required complex roots, and pull it back. The resulting chi_K extends chi and is trivial on U_F^1.
All freely chosen character extensions below take place at maximal native rank, where the centralizer is K itself. We do not apply this scalar-intersection argument directly inside a nonmaximal matrix centralizer, which would require a separate determinant-image check.
I.5 5. Base full character with trace one
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At native rank two, let E=F[alpha], with q=q_d and a=floor(q/2)+1. The forced character on U_E^a is
phi(v)=psi_F(Tr_(E/F)(alpha(v-1))).For scalar u in U_E^a intersect U_F^1, equation (3) gives
phi(u)=psi_F((u-1)Tr_(E/F)(alpha))=psi_F(u-1)=1.Section4 constructs chi_E on U_E^1 extending phi and trivial on U_F^1. Define theta_1 on \(H^1=U_E^1\) U_(A_1)^a by chi_E on the first factor and psi_alpha on the second. The restrictions agree on the intersection. BK3.2.1 gives the actual full simple character, with theta_1| U_F^1=1.
This uses the true trace value one and only the stated conductor condition on psi_F. It does not require an artificial zero-trace primitive parameter in a quadratic extension.
I.6 6. Native-rank induction via full beta-determinant transfer
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Assume theta_j is a full simple character for native b_j in rank 2^j, with theta_j| U_F^1=1. Let K_j=F[b_j]. Since K_j is maximal, the character chi_j=theta_j| U_(K_j)^1 is its full beta-determinant character.
Transfer theta_j to the multiplicity-two realization gamma_j=b_j tensor I_2 in rank 2^(j+1), using BK3.6.14. Denote the full transferred character by Theta_j. BK3.6.13(ii) preserves the entire beta-determinant character on U_(K_j)^1 at level m0=0. Hence for every literal scalar u in U_F^1,
Theta_j(u I)=chi_j(det_(K_j)(u I_2))
=chi_j(u^2)=theta_j(u^2)=1. (9)This statement includes nonminimal b_j. The determinant exponent two is essential and has not been suppressed.
Now set beta=b_(j+1), gamma=gamma_j, delta=delta_j, r=r_j, m=floor(r/2), and
D=H^(m+1)(beta)=H^(m+1)(gamma).The equality and endpoint m were verified in Section3. BK3.3.18 constructs the actual recursive simple character
vartheta=(Theta_j restricted to D)*psi_delta.For u in D intersect U_F^1, equations (8) and (9) prove
vartheta(u)=1. (10)The new field K=F[beta] is maximal and totally ramified in native rank 2^(j+1). Its chain has K-period one, so
D intersect U_K^1=U_K^(m+1),
D intersect U_F^1=U_F^ceil((m+1)/2^(j+1)).Thus (10) is exactly the compatibility required by Section4. Extend vartheta| U_K^(m+1) to chi_K on U_K^1 trivial on U_F^1. BK3.2.5 constructs the full simple character by
theta_(j+1)(kd)=chi_K(k)*vartheta(d),
k in U_K^1, d in D.That proposition supplies normalization and well-definedness; its proof is a character-extension argument and imposes no exclusion of residue characteristic two. The new theta_(j+1) is trivial on U_F^1. Induction proves (1) in every native rank 2^d.
More generally the character argument needs only integral base field trace and integral traces for the native corrections. The dyadic tower establishes their exact value one, while the odd-prime tower of166 establishes zero.
I.7 7. Canonical beta extension and prescribed tame center
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Let K/F be the final totally ramified degree-N parameter field and let eta be the Heisenberg representation for the constructed theta. The compact type group is
J=O_K^times J^1.Here \({\mathcal O}_K^\times=U_K^1\) because the residue field is F_2. Since U_K^1 is contained in J^1, we have
J=J^1, J/J^1=F_2^times={1}. (11)Thus the actual canonical beta extension on J equals eta; there is no residue-character choice and no change of that extension. It acts trivially on the central U_F^1 by (1). Equation (11) does not identify any externally prescribed dyadic quadratic phases.
Let omega_F be any smooth quasicharacter trivial on U_F^1. Since \({\mathcal O}_F^\times=U_F^1\), it is determined by its value omega_F(t). The actual maximal type extends to K^times J. Concretely, (K^times J)/J is infinite cyclic, generated by a K-uniformizer. An intertwiner for its conjugation action on eta gives an extension Lambda_0. Denote its central character by omega_0.
Choose a complex scalar s with
s^N=omega_F(t)/omega_0(t).Multiply Lambda_0 by the character of K^times J trivial on J and taking the K-uniformizer to s. Because t has valuation N in K, this changes its scalar by s^N and gives the requested central quasicharacter on all F^times. The value at every element of J, including the canonical beta extension, is unchanged. This does not assume that a chosen K-uniformizer raised to N equals t.
The retained maximal-simple-type theorem now gives the actual irreducible admissible supercuspidal pi=c-Ind_(K^times J)^GL_N(F) Lambda. The noncompact scalar is fixed before any H5 comparison. An incompatible immutably prescribed extension is not overwritten; as in166, its value must be free or already compatible.
I.8 8. What this closes and what it does not
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This closes the full-character and actual tame-central-character construction for the explicit dyadic arithmetic tower. The failure of trace zero at p=2 is repaired by the correct integral trace invariant. Both the odd-level floor and the multiplicity-two determinant transfer are accounted for explicitly.
This note does not compute \(J^1/H^1\) or prove that it contains a particular dyadic source packet. It does not establish a quadratic refinement, canonical phase matching, nontrivial stabilizer action, general transport graph, source admission, or H5. In particular, an actual supercuspidal host with the required central character is not asserted to be a completed realization of the original dyadic packet without those separate proofs.
I.9 9. Exact source checks and input pins
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The following final166 files were read and reused without editing:
attachment166/central/TRACE_ZERO_CHARACTER_INDUCTION166.mdSHA25699cb6e80c419e0bf7325e2b2b3fbaba1391903448a3b6a0c4cd9f3944ed406a1.attachment166/tower/TRACE_ZERO_AS_TOWER166.mdSHA25649a18fb733f0209b3e38118f5132839a060961edd0bdc0211e1f7b441a4e5653.attachment166/tower/RECURSIVE_TOWER166.mdSHA2566f39a077df18eb03f9fab2b6e426529dbe14cf4f7e1cfce95ddb0fcd187d7b33.attachment166/audit/TOWER_CENTRAL_REVIEW166.mdSHA2564bc71dd050ddb321a253105c8a4ee7a86185af255532d5605caa7a9de383e26b.
Primary retained source: Bushnell–Kutzko, The admissible dual of GL(N) via compact open subgroups, text SHA256 d69db0827256707a34cd2562af68b9de11fc889ab3f43535ec47b620d58ca9c8. The current dyadic check reread definitions3.1.7–8 and Proposition3.1.9(ii) (printed90–91), Definition3.2.1 and Proposition3.2.5 (printed98–100), and Corollary3.3.18 (printed112–113). Their level thresholds are floor(r/2)+1 or floor(r/2), as used above, and their statements do not exclude p=2. The transfer's preservation of the full beta-determinant character is BK3.6.13(ii), with existence Theorem3.6.14 (printed136–137), already directly checked in166 and reread for this application.
The character-extension and correction proofs use neither division by two nor an odd-order averaging argument. The exact dyadic trace calculation and the borderline correction-character check are given in Sections2–3. No new publication lookup, external arithmetic-induction/descent constructor, or inverse LLC is used.