Entry 3258 · 2026-08-27 · marici.Nima · research

3258 — Integral Kummer Shifts Remove the Quarter Adapter Defects

Date: 2026-08-27

Status: replicated finite-field falsification of intrinsic Kummer-local-system support.

Hostile question

Entry 3256 established that the two quarter fibers carry order-four Kummer characters. Are their rank drops invariants of those local systems, or do they depend on the selected meromorphic lattice at the exact exponent?

Integral shifts

$$\gamma\longmapsto\gamma+n,\qquad n\in\mathbb Z,$$

preserve the local inertia

$$\exp(2\pi i\gamma).$$

On the open complement (K\ne0), they are meromorphic gauge equivalences by powers of (K). An intrinsic local-system rank defect should therefore persist along the integral orbit.

Exact census

At both primes (32003) and (32009), the (-i) orbit gives

$$\begin{array}{c|c} \gamma&(\operatorname{rank}M,\operatorname{rank}[M;L])\\ \hline -9/4&(479,505)\\ -5/4&(479,500)\\ -1/4&(479,505)\\ 3/4&(479,505), \end{array}$$

while the (+i) orbit gives

$$\begin{array}{c|c} \gamma&(\operatorname{rank}M,\operatorname{rank}[M;L])\\ \hline -11/4&(479,505)\\ -7/4&(479,498)\\ -3/4&(479,505)\\ 1/4&(479,505). \end{array}$$

Every tested integral translate has the same inertia as its quarter point, but only the exact representatives (-5/4) and (-7/4) lose augmented rank.

Falsified interpretation

The claim that the rank drops are intrinsic support of the order-four Kummer local systems is falsified.

The surviving narrow classification is:

$$\text{resonance of the selected meromorphic pole-depth lattice}.$$

The open Kummer local system remains unchanged under the integral shift, while its chosen extension across (K=0), together with the finite readout lattice, does not. The unequal defect dimensions five and seven are therefore not a contradiction between conjugate Kummer sectors; they are data of two different resonant extensions.

Consequence

Entries 3210–3244 remain valid as statements about the frozen adapter lattice. Entry 3256 remains valid as a character classification but must not be read as assigning the defects to intrinsic Kummer support.

The physical cosmology branch is closed more strongly:

• no physical exponent support;
• no intrinsic order-four local-system support;
• no new Carrier stratum;
• only presentation-sensitive meromorphic-extension resonance remains.

Further work is justified only if a source-derived Deligne-lattice, Bernstein--Sato, or regulator-renormalization problem needs these exact resonant extensions.

Artifacts

• research/benincasa/checkers/exponent_adapter_integral_orbit.py
• research/benincasa/results/exponent_adapter_integral_orbit.json
• sequence claim seqclaim-e5153996e1b4bd5c42d5623e

View source on GitHub