EXACT HEISENBERG FINITE-DETERMINANT LIMIT FOR ODD PRIME TOWERS
===========================================================
Written: 2026-10-02 12:22:26 UTC. Target AIM-GEOMETRIC_GROUP_THEORY-0134.
Complete auxiliary theorem, NOT original-source closure. The fixed element
is ONLY f=1-2a+2b. No broader coefficient claim is made. Novelty remains
undetermined: Kammeyer's2026 example and Boschheidgen's representation/
zero-eigenmass work are prior art; source lane owns the bounded novelty audit.

THEOREM
--------
Let p be an odd prime, q=p^i, Q=H_3(Z/q), and M_q be right multiplication
by f on C[Q]. Write D_q=|det_C M_q|. Then

  lim_(i->infinity) log(D_q)/q^3=log(2)/(p+1),
  lim_(i->infinity) det_FK(M_q)=2^(1/(p+1)).

Here det_FK(M_q)=D_q^(1/q^3), since M_q is invertible. The exact formula
and rigorous tail proof below do not use floating eigenvalues or numerical
extrapolation. For p=3 the limit is2^(1/4), strictly below the credited
infinite Fuglede-Kadison determinant2. This sharpens the published bound
5^(1/3) for that same example; no first-result claim is made.

1. DEFINITIONS AND JOINT CENTRAL/POWER EIGENSPACES
------------------------------------------------
Use H and its coordinate convention from the companion subgroup proof.
The finite generators a,b,c satisfy a^q=b^q=c^q=1 and ab=bac, with c
central. Right-regular operator composition may invert the commutation
root, but its order, eigenvalue set and all determinants below are
unchanged. Avoid identifying source parameters with a scalar power without
checking them: OUR alpha and beta are defined intrinsically as eigenvalues
of a^r and b^r on the indicated joint eigenspace.

Fix a central eigenvalue zeta of EXACT order r=p^s, 0<=s<=i, and put
m=q/r. Its central eigenspace V_zeta has dimension q^2: restriction of the
regular representation to the central cyclic subgroup gives q^2 regular
copies. On V_zeta, a^r and b^r commute with a and b and with one another,
because zeta^r=1. Their eigenvalues alpha,beta satisfy alpha^m=beta^m=1.

Each joint space W_(zeta,alpha,beta) has dimension EXACTLY r^2. This can be
proved without a full irreducible classification. Let

  U_r=<a^r,b^r,c>.

In coordinates its first two entries are multiples of r and its central
entry arbitrary, so |U_r|=m^2*q and [Q:U_r]=r^2. The assignments
a^r->alpha, b^r->beta, c->zeta define a one-dimensional character of U_r:
its commutator c^(r^2) is killed since zeta^r=1, and the generator orders
are respected. In each coset copy of the regular U_r action the weighted
character vector spans precisely one such eigenline. Thus the full joint
space has one line per coset, giving dimension r^2. This also proves that
every pair alpha,beta of m-th roots occurs, with the stated multiplicity.

2. r WEYL BLOCKS INSIDE EACH r-SQUARED JOINT SPACE
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Write A,B for the actions of a,b on one joint space. They are unitary,
A^r=alpha*identity, B^r=beta*identity, and AB=zeta^(+-1)BA. The r roots
lambda of lambda^r=beta are distinct. A cyclically permutes the B-eigenspaces
through all these r roots. They have equal dimension; because the total
dimension is r^2, each has dimension r.

Choose a basis of one B-eigenspace and take its A-iterates. Each basis
vector produces an invariant r-dimensional weighted-cycle block, and the
r resulting blocks form a direct sum. In each block, A is a cyclic shift
with wrap weight alpha, while B is diagonal with entries lambda*zeta^j
(or the inverse ordering), 0<=j<r. Therefore

  det(identity-2A+2B)
    =product_j(1+2lambda*zeta^j)-2^r*alpha
    =1+2^r*beta-2^r*alpha.

For r>=2, only the full diagonal and full cyclic-shift permutations can
contribute to this determinant. For r=1 it is the same scalar formula.
The root product follows from product_j(1-t*zeta^j)=1-t^r and ODD r.
There is no missing power: there are r such blocks per joint space.

3. EXACT FACTORIZATION AND ITS INTEGRAL INTERPRETATION
-----------------------------------------------------
Define, for odd m and even positive integer t,

  B_m(t)=product_(alpha^m=beta^m=1)(1-t*alpha+t*beta).

It is the ordinary determinant of identity-t*R_x+t*R_y on the abelian
group Z/m x Z/m, hence an integer congruent to1 modulo2. It is positive:
complex-conjugate root pairs contribute positive squared moduli, and the
only conjugation-fixed pair for odd m is alpha=beta=1, giving factor1.
Invertibility follows independently from the matrix being identity mod2.

There are phi(r) central roots of exact order r. Section2 gives

  D_(p^i)=product_(s=0..i) B_(p^(i-s))(2^(p^s))^(p^s*phi(p^s)),
  phi(1)=1, phi(p^s)=(p-1)*p^(s-1) for s>=1.

The dimension checksum is q^2*sum_(s=0..i)phi(p^s)=q^3. In particular the
primitive central-order q summand has m=1, B_1(2^q)=1, so contributes no
logarithmic determinant. This recovers Kammeyer's unipotent-block mechanism
without assuming it covers nonprimitive central roots.

4. A RIGOROUS NONNEGATIVE TAIL BOUND
-----------------------------------
Let E_s be the direct sum of the central eigenspaces with exact order p^s,
and let D_s be the absolute determinant of M_q on E_s. This is a rational
subspace: it is the kernel of the integral cyclotomic polynomial
Phi_(p^s)(R_c). Its intersection with Z[Q] is a full-rank saturated lattice.
T=R_a-R_b preserves both the subspace and this lattice. Relative to an
integer lattice basis, M_q=identity-2T has an odd integer determinant.
Consequently D_s>=1, which proves NONNEGATIVITY of every logarithmic term.

E_s is a reducing subspace for M_q, since the central unitary commutes with
M_q and its adjoint. Its dimension is phi(p^s)*q^2. The operator M_q is a
sum of five unitary summands, so its norm is at most5. Every singular value
of its restriction is at most5; thus

  0<=log(D_s)/q^3<=phi(p^s)/q * log5.

No bound based on the eigenvalue moduli of a nonnormal matrix is silently
used in place of singular values here. For j=i-s>=J,

  sum_(s=0..i-J) phi(p^s)/q =p^(i-J)/p^i=p^(-J).

Hence the entire tail with j>=J is bounded uniformly in i by p^(-J)*log5.
This is the crucial justification for the order of the two limits below.
It avoids an unjustified uniform asymptotic in growing m.

5. EACH FIXED j CONTRIBUTION
----------------------------
Fix j>=0, m=p^j, and let i tend to infinity with r=p^(i-j). Then m is fixed
and r tends to infinity. In B_m(2^r), the m diagonal pairs alpha=beta
contribute exactly1. Each of the m^2-m off-diagonal pairs has

  log|1+2^r(beta-alpha)|
     =r*log2+log|beta-alpha|+o(1).

For complete control, when m>1 the finite minimum separation is
d_m=2sin(pi/m)>0. Once 2^r*d_m>=2, the error for each pair is at most
2/(2^r*d_m), by |log|1+z||<=2|z| for |z|<=1/2. For m=1 the product is1.
For each fixed alpha, the derivative of X^m-1 gives
product_(beta!=alpha)|beta-alpha|=m. Thus the total constant term is
m*log m, and

  log B_m(2^r)=(m^2-m)*r*log2+m*log m+o(1).

The s=i-j logarithmic determinant has multiplicity r*phi(r). Divide by
q^3=(rm)^3 and use phi(r)/r=(p-1)/p for all sufficiently large i:

  lim_i log(D_(i-j))/q^3
    =((p-1)/p)*(1/m-1/m^2)*log2
    =((p-1)/p)*(p^(-j)-p^(-2j))*log2.

The constant/error term vanishes after the normalization, since m is fixed
and its coefficient is O(1/r). The j=0 contribution is exactly0.

6. COMPLETE LIMIT AND NORMALIZATION
-----------------------------------
First fix J. The finitely many j<J terms converge by Section5. The remaining
terms are nonnegative with total at most p^(-J)*log5 by Section4. Now let J
tend to infinity. This proves convergence of the FULL normalized logarithm,
not only a limsup bound. Its value is the convergent geometric sum

  ((p-1)/p)*[sum_(j>=0)p^(-j)-sum_(j>=0)p^(-2j)]*log2
   =((p-1)/p)*[p/(p-1)-p^2/(p^2-1)]*log2
   =log2/(p+1).

Finite M_q is invertible, so normalized finite Fuglede-Kadison determinant
is exp(log D_q/q^3). Continuity of exp gives2^(1/(p+1)). The infinite
Fuglede-Kadison value2, and its injectivity/determinant-class hypotheses,
are the credited Kammeyer/Deninger primary result, not a conclusion from
these finite computations.

7. ACTUAL FINITE CHECKS, BOUNDARY FAILURE AND SOURCE CREDIT
---------------------------------------------------------
check_heisenberg_factorization.py independently constructs and eliminates
the FULL regular matrices at q=3 (27x27) and q=5 (125x125), obtaining

  D_3=1729, D_5=3277440001,

and matches the derived central-order factors. At q=9 it EVALUATES the
factorization, rather than claiming an independent729x729 elimination:

  B_9(2)=1836032914900504688395761745921,
  B_3(8)=7077889, B_1(512)=1,
  D_9=B_9(2)*B_3(8)^6
     =230835684981286860588733237627265586640390748412914623442454481337057281.

A first hand-resultant check erroneously compared the r=3,m=3 block to
B_3(512) instead of B_3(8). It failed closed and is preserved in
factorization_control_failure_20261002T121947Z.txt; the matrix/factorization
calculation was already correct. The corrected actual run passed828,902
explicit checks. No assert or floating logarithm is used.

The odd-order hypothesis is REAL: at q=2 the actual full regular determinant
is -735, while the incorrect odd-order block formula would give -15.
For even r the root product changes to1-2^r*beta, rather than1+2^r*beta.
We make no claim for a p=2 limit. This counter-control is not an AIM target
counterexample; it only rejects enlargement of the odd-prime calculation.

Prior-art context, credited without claiming novelty:
  Holger Kammeyer, arXiv:2609.15567v1, Theorem2 and Section2: this fixed
  Heisenberg element, congruence tower, finite bound and infinite value.
  Jan Boschheidgen, arXiv:2210.15240v3, Section4: finite representation
  matrices/multiplicities and zero-eigenmass p/(p+1), matched by source lane.

The present proof uses intrinsic power eigenvalues and derives its block
determinant directly. It does not import a scalar-power formula with a
possibly omitted multiplicity exponent. Source lane has flagged precisely
that notation hazard in the retained Boschheidgen version. No general
prior-art absence is proved by this bounded audit.

8. CONSEQUENCE FOR THE PARENT'S FINITE-CW CONSTRUCTION ONLY
---------------------------------------------------------
For the finite three-dimensional CW complex Y constructed from the genuine
nilmanifold N wedge S^2 by attaching one3-cell along f, the companion proof
gives rho^Z(Y_i)=log D_q-log q and rho^(2)(tildeY)=log2. The exact limit is
therefore

  lim_i rho^Z(Y_i)/q^3=log2/(p+1).

This is tower-dependent for the SAME space Y as p varies over odd primes.
It disproves the corresponding finite-CW integral-torsion approximation
statement, subject to its explicitly matched definition/hypotheses.
It does NOT answer the original closed-aspherical-manifold question:
tildeY has nonzero ordinary H_2=coker f, and Y is not a closed manifold.
No canonical source-resolution, paper-ready or publication flag is changed.

REPLAY AND HONEST CHECKER LIMITS
--------------------------------
  python3 -I -B check_heisenberg_factorization.py
  python3 -I -B -O check_heisenberg_factorization.py

The standalone standard-library checker reads/writes no files, checks the
stated finite algebra and exact geometric-series/tail weights, and leaves
the representation proof and infinite-limit argument to this analytic
packet. A passing finite executable is not their substitute.
