Coherence thresholds for optimal-rate stabilizer purity testing
Scott Schubr, Luminary Oddities, 2026

The paper introduces the name Schubr's Prime-Characteristic Coherence Law
for its optimal-rate coherence theorem.

The article proves that, for every odd prime p and every fixed number of
p-level registers, a p-copy block is exactly the minimum coherence
permitting optimal epsilon^-1 log(1/delta) exact one-sided purity testing.
Smaller fixed blocks have a quadratic-contact epsilon^-2 rate obstruction,
even with classical adaptation. Exact integer-budget survival, higher-contact
regimes and rank-sensitive code bounds refine the quantitative conclusions. A discriminant-phase measurement and product
MUB conditioning cover arbitrarily entangled inputs. In dimensions five and
seven, a smaller family uses one fixed local circuit; independent support
obstructions also rule out every shorter universal exact test there.

Contents
  main.pdf       Complete article
  main.tex       Editable LaTeX source, including bibliography
  CITATION.cff   Machine-readable citation metadata
  citation.bib   BibTeX citation for this manuscript
  code/          Exact finite certificates and reproduction runners
  MANIFEST.json  Explicit inventory, sizes and SHA-256 digests

Build
  tectonic main.tex
A standard LaTeX installation supporting the packages in main.tex also works.
The bibliography is embedded; no separate bibliography processor is needed.

Python certificates (Python 3.10 or later; standard library only)
  python code/reproduce.py
The runner executes six specialized proof certificates and one general
construction control in normal and optimized modes, failing on any failed
check. The six certificates cover the 41,056 characteristic-five
affine supports, the 3,125 characteristic-five MUB tuples, both exact odd-parity
projections, characteristic-seven odd/even MUB minors, and the eight-plane
Macaulay matrix (rank 105 in degree 13). The latter is a coefficient-rank
certificate, not finite-field sampling of projective points. The general
control checks exact modular tangent ranks and quadratic-image covers at
p=3,5,7,11,13; the all-prime proof is analytic and cites its support input.

Characteristic-seven support and MUB certificates (.NET 8 SDK or later)
  dotnet run --project code/seven/SevenCertificate.csproj --configuration Release
Alternatively, with PowerShell 7.2 or later, in a fresh process:
  pwsh -NoProfile -File code/seven/run.ps1
The support check covers 54,899,600 affine supports by analytic filtering and
17,587,080 explicit coset checks. The MUB check covers all 823,543 label tuples.
Both runners require full completion and exact expected coverage/digest values.
A slower machine may need a larger 180000 millisecond limit in its runner;
partial output is never a completed certificate. PowerShell uses its bundled
compiler/runtime; the dotnet command requires the SDK.

The article explains how exact modular computations certify characteristic-zero
claims and supplies the remaining analytic proofs. The manifest covers every
package file other than itself.

Copyright 2026 Scott Schubr.
