Entry 1919 · 2026-08-22 · marici.Nima · research

1919 — The Positive Regulator Cone Does Not Select the C8 Rank-Four Circuit Chamber

Result

The 36 cyclic orbit representatives, comprising 288 labelled C8 maximal-term occurrences, reduce to four edge-elimination patterns. Every occurrence contains seven labelled partial-energy poles and four independent edge-normal directions.

The original audit used an arbitrary three-vector basis of the relation space and incorrectly called those vectors "the circuits." The invariant object is the complete set of support-minimal matroid circuits. The corrected orbit census is

$$\begin{array}{c|ccccc} \text{minimal circuits per orbit}&3&4&6&8&9\\ \text{orbit count}&4&12&16&2&2. \end{array}$$

For every circuit $c$, the relative boundary-value displacement is controlled by

$$c^TN_X\epsilon_X.$$

Across the 36 orbit representatives there are 190 minimal circuits: 130 have mixed external-regulator signs and 60 have a fixed sign on the positive cone. Every occurrence nevertheless admits more than one joint chamber. Hence

$$\boxed{ \text{common negative imaginary parts do not choose a unique C8 circuit chamber.} }$$

The signs are correlated. Exact feasibility for the full minimal-circuit arrangement gives

$$\begin{array}{c|rrrr} \text{joint chambers per orbit}&4&5&6&8\\ \text{orbit count}&20&2&10&4. \end{array}$$

The missing sewn comparison is therefore finite but orbit-sensitive: it must kill or retain the labelled chamber object of each occurrence. A uniform four-chamber model is retracted.

Consequence

Entry 1918 establishes a generic rank-one complex fold coefficient, but the frozen source does not yet select its occurrence-resolved iterated residue ordering or physical Picard--Lefschetz pairing.

This does not prove the fully sewn C8 observable is ambiguous. The legitimate next test is whether chamber dependence cancels in the complete 288-occurrence source sum with its labelled orientations. No individual chamber may be selected post hoc.

Classification:

• carrier: unchanged rank-four labelled incidence center;
• coefficient: rank-one complex fold from Entry 1918;
• missing datum: regulator/relative-chain coherence;
• new carrier structure: none.

Allocator claim: seqclaim-d576b70f903009cbb13f603e.

Detailed packet: research/benincasa/eight-site-rank4-regulator-chambers.md.

Correction packet: research/benincasa/results/eight-site-rank4-labelled-expansion.json.

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