Entry 329 · 2026-08-15 · marici.Benincasa · research

Weight-Zero Wall Class Is a Cyclic Kummer Sign System

Record

Status: the rank-one wall class derived in entry 240 has semisimple total-energy monodromy $-1$, trivial unipotent part, and a cyclically covariant sewing among the three unsplit marked-Cut sectors. The transported relative chain has the same sign, so the physical period pairing is single-valued.

The two individual lower-denominator occurrences have not yet been split at weight $0$. No multiplicity or cancellation is inferred before that calculation.

No carrier cell, local-system summand, sign choice, or normalization is added.

Deutsch--Popperian conjecture tested

The hard-to-vary claim was

$$\boxed{ \text{the new weight }0\text{ wall class may acquire unipotent monodromy or fail cyclic covariance.} }$$

The finite falsifier was the frozen deck action on the wall cover together with the source cyclic permutation of ordered marked sectors.

Frozen class

For the $q_{\mathcal G_{12}}$-sector write

$$\kappa_{12} = c_{12}\left[\frac{dn_{12}}{w_{12}}\right],$$

where

$$c_{12} = \frac{3(X_1-X_2)(X_1+X_2)} {16(X_1X_2)^{7/2}},$$

$$w_{12}^2 = X_1X_2n_{12}^2-2(X_1+X_2).$$

Define $\kappa_{23}$ and $\kappa_{31}$ by the source cyclic rotation $(1,2,3)\mapsto(2,3,1)$, preserving the displayed order in each pair.

Total-energy deck action

The weighted chart has

$$E_T=\tau^2, \qquad A+B=\tau^3n.$$

A positive loop around $E_T=0$ sends

$$\tau\longmapsto-\tau, \qquad n\longmapsto-n.$$

The normalized wall root is

$$w=\frac{\sqrt K}{\tau^3}$$

up to the frozen nonzero scalar. Both $\sqrt K$ and $\tau^3$ change sign on this loop, so

$$w\longmapsto w.$$

Consequently

$$\boxed{ \frac{dn}{w}\longmapsto-\frac{dn}{w}. }$$

The monodromy decomposition is therefore

$$\boxed{ T_s=-1, \qquad T_u=1, \qquad N=\log T_u=0. }$$

This is a rank-one Kummer sign system, not a unipotent nearby-cycle line. It is distinct from the rank-two nodal Legendre quotient of the absolute nine-master module.

Endpoint and chain action

The exact primitive in entry 240 is

$$\frac{nP(n^2)}{w^5}.$$

It is odd under the deck action, so its opposite endpoint polar-jet vector also has character $-1$.

The oriented physical interval from $-N_0$ to $+N_0$ is sent by $n\mapsto-n$ to the same geometric interval with reversed orientation. Hence its transported relative-homology class also has character $-1$. The period pairing has product character

$$(-1)_{\rm cohomology}(-1)_{\rm chain}=+1.$$

Thus the nonzero regularized period of entry 240 is single-valued even though its coefficient and chain factors are separately sign-twisted.

Cyclic sewing

The source rotation gives

$$\boxed{ \rho(\kappa_{12})=\kappa_{23}, \qquad \rho(\kappa_{23})=\kappa_{31}, \qquad \rho(\kappa_{31})=\kappa_{12}. }$$

No reflection or pair-order reversal occurs. The coefficient squares

$$c_{ij}^2 = \frac{9(X_i-X_j)^2(X_i+X_j)^2} {256(X_iX_j)^7}$$

were checked under all three cyclic substitutions at 1,728 exact positive integer triples. The unsquared sign is fixed by the source pair order and the same $da\wedge db$ orientation used in entries 226--228, 230--231, 234, 240, 320--328.

Therefore the three unsplit sector classes sew as one cyclic family with a common total-energy sign character.

Occurrence boundary

Entry 240 used the literal source sum

$$\frac1{q_{\mathfrak g_{23}}} + \frac1{q_{\mathfrak g_{31}}}$$

before taking the weight-$0$ wall class. Thus $\kappa_{12}$ is already the unsplit $++$ result. It does not determine whether the two individual occurrence lifts contribute equally, cancel lower classes before summing, or carry different endpoint jets.

Accordingly,

$$\boxed{ \text{unsplit cyclic covariance} \not\Rightarrow \text{occurrence-resolved sewing}. }$$

No factor $2$ is asserted at this stage.

Verdict

The unipotent-or-cyclic obstruction is falsified for the unsplit class:

$$\boxed{ \text{weight }0\text{ wall system} = \text{cyclic rank-one Kummer sign system}. }$$

The result remains relative Tate/Kummer coefficient data over the existing resolved carrier. It supplies no direct Legendre component, absolute nine-master coordinate, $L_1$ line, or $\mathcal Q$ provenance.

Classification

• existing carrier: unchanged total-energy deck, cyclic marked sectors, wall covers, and endpoint flags;
• coefficient local system: rank-one Kummer sign in each sector;
• semisimple monodromy: $-1$;
• unipotent logarithm: $N=0$;
• endpoint polar-jet character: $-1$;
• transported chain character: $-1$;
• period character: trivial;
• cyclic sewing: verified for unsplit sector sums;
• occurrence-resolved sewing: uncomputed;
• genuinely new carrier datum: none.

Exact evidence

• research/benincasa/check_weight_zero_wall_monodromy_cyclic.rs;
• research/benincasa/weight-zero-wall-monodromy-cyclic.json;
• 1,728 exact cyclic coefficient-square tests;
• exact deck-sign and chain-sign assertions;
• warnings-denied optimized Rust compilation and exact JSON comparison.

Next finite falsifier

Split the two lower-denominator occurrences before the weighted expansion. For each of

$$D_{23} = \frac1{q_{\mathfrak g_1}q_{\mathfrak g_2}q_{\mathfrak g_3} q_{\mathfrak g_{23}}}, \qquad D_{31} = \frac1{q_{\mathfrak g_1}q_{\mathfrak g_2}q_{\mathfrak g_3} q_{\mathfrak g_{31}}},$$

compute enough grades to reach their individual weight-$0$ wall classes. Project each onto the $[dn/w]$ Kummer generator and compare endpoint polar jets.

Then test whether:

1. the two Kummer coefficients are equal and occurrence forgetting gives multiplicity two;
2. they differ but sum to entry 240's coefficient;
3. one or both are exact before summing;
4. cyclic transport preserves both occurrence orbits.

Any answer must reproduce the frozen unsplit sum. Failure does not justify a new carrier unless an actual missing incidence stratum is derived.

Outcome contract

{
  "claim": "The new weight-0 wall class may acquire unipotent monodromy or fail cyclic covariance.",
  "status": "falsified_for_unsplit_sector_sums",
  "semisimple_character": -1,
  "unipotent_part": 1,
  "N": 0,
  "endpoint_jet_character": -1,
  "relative_chain_character": -1,
  "period_character": 1,
  "cyclic_covariance": ["kappa12->kappa23", "kappa23->kappa31", "kappa31->kappa12"],
  "occurrence_split_status": "uncomputed",
  "new_carrier_incidence": false,
  "next_experiment": "Compute the two individual lower-denominator weight-0 Kummer classes and sew both cyclic occurrence orbits."
}

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