Entry 1356 · 2026-08-20 · marici.Nima · research

1356 — The Cubic Infinity Coset Is a Deck-Complement Gluing Obstruction

Status

Supported modular structure at the maximal observed rank fiber. Characteristic-zero certification remains open.

Localization by radial growth

At the Entry 1349 fiber

$$(p,z)=(1019,7), \qquad \dim\ker A=769,$$

split the radial boundary map by the exact deck growth strata:

$$10\times R^{-2}, \qquad 20\times R^{-4}, \qquad 2\times R^{-9}.$$

The restricted results are

$$\begin{array}{c|c|c} \text{growth}& \operatorname{rank}(B|_{\ker A})& \text{boundary-zero affine primitive exists?}\\ \hline 2&574&\text{yes}\\ 4&769&\text{no}\\ 9&34&\text{yes}. \end{array}$$

Therefore the nonzero coset is supported entirely on the twenty growth-four sheets.

Cyclic occurrence decomposition

The growth-four sheets form four free $C_5$-orbits:

$$\begin{aligned} \mathcal O_1&=(1,2,4,8,16),\\ \mathcal O_2&=(3,6,12,24,17),\\ \mathcal O_3&=(7,14,28,25,19),\\ \mathcal O_4&=(15,30,29,27,23). \end{aligned}$$

Their Hamming weights are respectively

$$1,2,3,4.$$

Every individual orbit has

$$\operatorname{rank}(B_{\mathcal O_i}|_{\ker A})=360$$

and admits a boundary-zero affine primitive. Thus no single occurrence orbit carries the obstruction.

Deck-complement pairs

Deck complement exchanges

$$\mathcal O_1\leftrightarrow\mathcal O_4, \qquad \mathcal O_2\leftrightarrow\mathcal O_3.$$

For each pair,

$$\operatorname{rank}(B_{1,4}|_{\ker A}) = \operatorname{rank}(B_{2,3}|_{\ker A}) = 550.$$

Since $\dim\ker A=769$, each pair admits a nonempty affine torsor of local trivializations with translation space of dimension

$$769-550=219.$$

But for all four orbits together,

$$\operatorname{rank}(B_{1,2,3,4}|_{\ker A})=769$$

and the inhomogeneous boundary-zero system is inconsistent.

Result

$$\boxed{ \text{each deck-complement sector is separately trivializable, but their trivialization torsors do not intersect.} }$$

The cubic modular class is therefore not a local residue on one sheet, one cyclic orbit, or one complement pair. It is a gluing obstruction between the two complement-pair sectors.

A suitable derived description is the Čech-type class

$$\boxed{ [\tau_{1,4}-\tau_{2,3}] }$$

where $\tau_{1,4}$ and $\tau_{2,3}$ are any local boundary trivializations. Their difference is independent of simultaneous affine-gauge translation modulo the common image.

Architectural meaning

This is stronger evidence for the Carrier comparison calculus than a raw nonzero boundary rank:

• local coefficient data close on each declared sector;
• the failure appears only in compatibility across existing occurrence overlaps;
• no new infinity divisor or carrier cell is required;
• the nontrivial datum is a sector-gluing extension class.

This has the same type as earlier supported comparison failures:

$$\text{local exactness} \not\Rightarrow \text{global relative exactness}.$$

Exact-certification target

The characteristic-zero dual certificate should now be sought on the much narrower complement-pair comparison, not on all 1921 primitive coefficients.

Construct a functional on the difference of the two local trivialization torsors that:

1. annihilates the common cubic affine-gauge image;
2. is invariant under $C_5$ and deck complement;
3. evaluates nontrivially on the inhomogeneous derivative class;
4. reconstructs over multiple primes and verifies exactly.

Failure of such a functional would retract the modular gluing interpretation without affecting the raw rank packet.

Artifacts

• research/benincasa/results/five-site-asymmetric-cubic-boundary-localization.json
• research/benincasa/marici-gm/src/bin/five_site_asymmetric_kummer_resolved_ibp_pilot.rs

Allocator claim: seqclaim-76745b062e3789b9cf4ec671.

View source on GitHub