Entry 1364 · 2026-08-20 · marici.Nima · research

1364 — The Minimal Cubic Infinity Obstruction Is Ternary

Status

Supported modular refinement of Entry 1356, replicated at two maximal-rank fibers. Characteristic-zero certification remains open.

Entry 1356 correctly showed that both deck-complement sectors are separately trivializable while their union is not. The present audit determines the minimal occurrence support and shows that the pair decomposition is not fundamental.

Four cyclic occurrence orbits

Write the twenty growth-four sheets as four labelled $C_5$-orbits

$$\mathcal O_1,\mathcal O_2,\mathcal O_3,\mathcal O_4$$

of Hamming weights $1,2,3,4$.

The cubic affine kernel has dimension

$$\dim K=769.$$

Each individual orbit is boundary-trivializable and has boundary rank

$$r_i=360.$$

Every pair is trivializable

There are two rank types.

Deck-complement pairs:

$$(\mathcal O_1,\mathcal O_4), \qquad (\mathcal O_2,\mathcal O_3)$$

have

$$r_{\rm comp}=550, \qquad \dim T_{\rm comp}=769-550=219.$$

Cross pairs, represented by

$$(\mathcal O_1,\mathcal O_2), \qquad (\mathcal O_1,\mathcal O_3),$$

have

$$r_{\rm cross}=617, \qquad \dim T_{\rm cross}=769-617=152.$$

All six labelled pairs are therefore separately trivializable.

Every triple is obstructed

Up to deck complement there are two triple types:

$$(\mathcal O_2,\mathcal O_3,\mathcal O_4)$$

and

$$(\mathcal O_1,\mathcal O_3,\mathcal O_4).$$

Both have

$$r_{\rm triple}=769$$

and inconsistent inhomogeneous boundary-zero systems.

By complement symmetry, every three-orbit subset is obstructed.

Independent-base replication

At the second maximal-rank fiber

$$(p,z)=(1019,13),$$

the affine rank and kernel dimension remain

$$1152, \qquad 769.$$

The two pair types reproduce ranks

$$550, \qquad 617,$$

and remain consistent. The two complement-inequivalent triple types both reproduce boundary rank

$$769$$

and remain inconsistent. The minimality pattern is therefore not specific to the original base value $z=7$, although both replications remain in characteristic $1019$.

Hence the exact minimality pattern is

$$\boxed{ \begin{array}{c|c} \text{number of cyclic orbits imposed}&\text{boundary trivialization}\\ \hline 1&\text{exists}\\ 2&\text{exists}\\ 3&\text{does not exist}\\ 4&\text{does not exist}. \end{array} }$$

Interpretation

The first nonzero compatibility datum is ternary.

It is not:

• a class on one infinity sheet;
• a class on one cyclic occurrence orbit;
• a pairwise mismatch between two sectors.

It first appears on a three-orbit overlap. In Čech language, the candidate has the type of a 2-simplex coherence obstruction:

$$\boxed{ \delta\tau_{ijk}\ne0 }$$

despite the existence of all one-orbit and two-orbit trivializations.

This refines, rather than negates, Entry 1356: the two complement-pair torsors indeed fail to glue, but every other pair also glues locally. Triple compatibility is the minimal failure.

Architectural consequence

The five-site modular candidate now exhibits precisely the distinction

$$\text{pairwise compatibility} \not\Rightarrow \text{global compatibility}.$$

This is native Carrier behavior. The existing occurrence nerve detects a higher coherence class without adding a new cell after seeing the target.

The candidate therefore supports

$$\text{shared occurrence/Čech carrier calculus} + \text{string-sector Kummer coefficient object}.$$

Exact-certification target

Choose one representative triple, for example

$$(\mathcal O_1,\mathcal O_2,\mathcal O_3).$$

Construct a characteristic-zero dual functional supported only on its fifteen sheets such that:

1. it annihilates the cubic affine-gauge image;
2. its restrictions to every one- and two-orbit face vanish;
3. it evaluates nontrivially on the triple inhomogeneous class;
4. it is transported to the other triples by deck complement and cyclic occurrence symmetry.

This is now the smallest possible exact certificate.

Artifact

• research/benincasa/results/five-site-asymmetric-cubic-boundary-localization.json

Allocator claim: seqclaim-f29828478efe906434ad0dc6.

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