# Verification report for the Heisenberg determinant and torsion results

Title: Exact Heisenberg Determinant Limits and Tower-Dependent Homological Torsion

Author: Alper Ferudun[^author]

[^author]: Mercury Software GmbH. alper@mercurycodelab.com; https://github.com/AlperTheKing.

English preprint, version 1.0, manuscript date 2 October 2026. This is an AI-assisted, originating-researcher self-audited, unrefereed manuscript. No independent human review, proof-assistant verification, novelty certification, or absolute-priority claim is made.

## Accepted mathematical scope

For every fixed odd prime `p`, let `q=p^i` and let `M_q` be right multiplication by `1-2a+2b` on the regular representation of `H_3(Z/qZ)`. The complete analytic proof establishes

    lim_i log|det M_q|/q^3 = log 2/(p+1).

A concrete attachment to the Heisenberg nilmanifold gives a finite connected three-dimensional CW complex `Y` with `pi_1(Y)=H_3(Z)`, determinant-L2-acyclic universal cover, and L2-torsion `log 2`. Its finite covers satisfy

    rho_Z(Y_i) = log|det M_q| - log q,
    lim_i rho_Z(Y_i)/q^3 = log 2/(p+1).

Here `rho_Z` is the unnormalized alternating logarithm of the orders of integral homology torsion groups. This convention is explicitly distinguished from the normalized quantity in Hughes and Lück's Conjecture 1.2.

The originating researcher accepted this auxiliary scope on 2 October 2026 after detailed self-audit. The acceptance records no identified mathematical gaps in that scope. It retains `original_problem_resolved=false`, `whole_source_record_resolved=false`, and zero added frozen-record closures. The constructed space is proved neither aspherical nor a closed manifold; it is not a counterexample to AIM 11.1 or Hughes and Lück's aspherical-manifold Conjecture 1.1.

## Analytic argument and credited inputs

For each central character of exact order `r=p^s`, the joint power eigenspaces have dimension `r^2` and split into `r` Weyl blocks. Their determinants yield an exact factorization into the integer factors `B_m(2^r)`, with `m=q/r`. Nonnegative logarithmic contributions and a uniformly bounded tail follow from saturated integral lattices and the operator-norm bound `5`. Fixed-tail truncation followed by a geometric-series calculation proves the ordinary limit. No exchange of an unbounded logarithm with weak spectral convergence is assumed.

The new sphere and three-cell have an actual attaching map, giving a direct cellular-chain summand in degrees three and two. The finite homology calculation and infinite L2-torsion calculation therefore concern a genuine CW complex, not an unrealized formal complex. Kammeyer's injectivity and infinite determinant `2`, derived using Deninger, and Lück's nilmanifold vanishing theorem are credited inputs. Boschheidgen's finite representations and zero-eigenvalue mass are prior ingredients, not newly discovered claims.

## Exact executable controls

Two unchanged, project-authored, standard-library scripts are distributed:

- `check_heisenberg.py`: 45,905 checks, including the independently constructed `27 x 27` determinant `1729`, modular controls, primitive-central nilpotency, and subgroup fixtures.
- `check_heisenberg_factorization.py`: 828,902 checks, including independent full regular matrices at `q=3` and `q=5`, with determinants `1729` and `3277440001`, and exact tail-weight identities.

At `q=9`, the latter script evaluates the derived factorization, not an independent `729 x 729` elimination. It obtains

    230835684981286860588733237627265586640390748412914623442454481337057281.

The even-order negative control is essential: at `q=2`, the actual determinant is `-735`, while the incorrect extension of the odd-order formula gives `-15`. No `p=2` limit is claimed.

The archived failed hand-control compared the `r=3,m=3` block with `B_3(512)` instead of `B_3(8)`. The test failed closed. Correcting that fixture did not alter the factorization or matrix-generation logic. Its provenance is retained.

Portable replay compares every field and every byte of the checker stdout with the accepted raw originating output. Python's arbitrary-precision integers are used; the secondary JavaScript-parsed projection of an older receipt is not used because it rounds the large `q=9` integer. No volatile fields are excluded. Exact finite executions are regression controls, not a substitute for the all-level analytic proof, a novelty certificate, or a proof of the original AIM question.

## Artifact and publication boundaries

The source ZIP must pass safe-member checks, content hashes, and replay from a fresh isolated extraction. The package is bound to actual native compilation, actual PDF export, and review of all final PDF pages. Readiness and external publication require separate records. This report assigns no DOI, performs no upload, and makes no arXiv, guaranteed indexing, or priority claim. Cited third-party source payloads are excluded from the authored package.