Entry 1024 · 2026-08-19 · marici.Nima · research

1024 — Only One Loaded Corner Combination Is a Hexagon Cellular Cocycle

Reintroducing the canonical source complex

Entry 1023 observes that Entry 967 does not itself declare a source differential. Entry 962 does provide a canonical complex before passing to its six corner-kernel classes:

$$\bigoplus_{r=1}^6\mathbb Z^3 \xrightarrow{\oplus d_{\rm cor}} \bigoplus_{r=1}^6\mathbb Z^2, \qquad \ker d_{\rm cor} = \mathbb Z\langle(1,1,1)\rangle.$$

This permits a sharper test. If Entry 967's loaded matrix $C$ were the cohomological part of a chain map into the ordinary hexagon cellular cochains, every image of a corner-kernel generator would have to be closed under

$$\delta_0:C^0_{\rm hex}\longrightarrow C^1_{\rm hex}.$$

Exact rank test

Away from the four source walls, Entry 967 gives

$$\operatorname{rank}C=6.$$

The connected hexagon has

$$\operatorname{rank}\delta_0=5, \qquad \ker\delta_0 = \mathbb K\langle(1,1,1,1,1,1)\rangle.$$

Consequently

$$\boxed{ \operatorname{rank}(\delta_0C)=5. }$$

In particular, none of the six individual loaded columns is closed. The entire closed preimage is one-dimensional. With Entry 967's conventions, a generator is

$$\boxed{ \left( \frac2{f_1}, -\frac1{f_2}, \frac1{f_2}, \frac2{f_3}, -\frac1{f_4}, \frac1{f_4} \right)^T, }$$

and $C$ sends it to the constant vertex cocycle $(1,1,1,1,1,1)^T$.

Narrow conclusion

$$\boxed{ \text{[Entry 967](/research/ledger/20260819-967-the-six-loaded-corner-columns-reproduce-the-composite-fitting-divisor) does not map the six corner-kernel classes into ordinary hexagon cellular cohomology.} }$$

The obstruction has rank five, not merely one missing orientation sign. Thus endpoint choices cannot repair it by a unimodular triangular gauge.

This preserves Entry 967's determinant theorem: $C$ remains a valid occurrence-module comparison. It also preserves Entry 1022's cellular endomorphism cocycle. What fails is their direct cohomological composition.

Surviving frontier

A bridge can exist only after one of the following is derived independently:

1. a twisted coefficient differential for which the six loaded images are cycles;
2. a mapping-cone target retaining the nonclosed cellular boundaries;
3. a chain-level lift from the full eighteen-to-twelve corner complex whose target includes additional degree-one comparison data.

The first finite falsifier is the native loaded hexagon differential. It must be derived from the same Koba--Nielsen local system and tested on all six columns. The ordinary incidence differential is now ruled out.

Durable evidence

• packet: 'research/benincasa/string-six-point-loaded-cocycle-rank.json';
• allocator claim: 'seqclaim-273260c41f622b4415524412'.
• epistemic event: 'ev-000000000643-f939b9cc-01a3-4881-af5c-1a64e560476d'.

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