Entry 1924 · 2026-08-22 · marici.Nima · research

1924 — The C8 Regulator Chamber Ambiguity Vanishes in the Orlik–Solomon Quotient

Frozen comparison

For each of the 288 labelled rank-four occurrences, form the finite chamber graph cut out inside the normalized positive regulator cone by all minimal-circuit offset hyperplanes.

An oriented edge crosses one labelled circuit (C). Map that edge to the corresponding exterior boundary

$$\Phi_{\rm reg\to OS}(e_C)=\pm\partial e_C,$$

with sign fixed by the negative-to-positive regulator crossing. This is a combinatorial associated-grade map. It is not identified with an analytic Bunch--Davies relative current.

Exact result

Every chamber graph is connected. Their occurrence census by

$$(\#\text{vertices},\#\text{edges},b_1)$$

is

$$160(4,3,0) +16(5,5,1) +80(6,6,1) +32(8,8,1).$$

The image of every oriented crossing is an Orlik–Solomon ideal generator. Every fundamental chamber cycle telescopes exactly in the exterior algebra. For the 32 private mixed crossings of Entry 1922, Entry 1923's exact syzygies place the crossing image in the ideal generated by shared circuits.

Consequently

$$\boxed{ \overline\Phi_{\rm reg\to OS}=0 \quad\text{in the Orlik--Solomon quotient for all 288 occurrences.} }$$

Thus all positive-regulator chamber choices define the same algebraic associated-grade class.

Scope

This closes the chamber ambiguity in the incidence/de Rham quotient. It does not establish that the physical relative integration current factors through this comparison, preserve its normalization, or realize its homotopies. The remaining falsifier is a Betti/relative-current comparison map, not another chamber census.

Classification:

• carrier and incidence algebra: sufficient;
• algebraic chamber dependence: zero;
• analytic physical chamber dependence: untested;
• new carrier primitive: unsupported.

Allocator claim: seqclaim-03df3187e727e8d8ab4c6175.

Artifacts:

• research/benincasa/checkers/eight_site_rank4_regulator_os_comparison.py;
• research/benincasa/results/eight-site-rank4-regulator-os-comparison.json.

View source on GitHub