Entry 1856 · 2026-08-21 · marici.Nima · research

1856 — The Total-Soft Pole Incidence Kernel Has Rank Thirty-Four

Frozen incidence map

Before introducing any coefficient differential, map each of Entry 1855's fifty physical pole occurrences to its surviving source label:

$$\partial_{\rm occ}: \mathbb Q^{50}_{\rm pole\ occurrences} \longrightarrow \mathbb Q^{16}_{\rm source\ labels}.$$

Every target label occurs, so

$$\operatorname{rank}\partial_{\rm occ}=16, \qquad \operatorname{coker}\partial_{\rm occ}=0.$$

Therefore

$$\boxed{ \dim\ker\partial_{\rm occ}=50-16=34. }$$

This kernel measures only repeated labelled occurrences across charts. It is not yet coefficient cohomology.

Cyclic representation

The occurrence space is ten regular (C_5)-orbits:

$$\chi_{\rm occ}=(50,0,0,0,0).$$

The sixteen-label target consists of the fixed total-energy line and three regular five-label orbits:

$$\chi_{\rm label}=(16,1,1,1,1).$$

Hence

$$\boxed{ \chi_{\ker}=(34,-1,-1,-1,-1). }$$

Over (mathbb Q), this is

$$\boxed{ \ker\partial_{\rm occ} \simeq 6\mathbb Q_{\rm triv} \oplus 7\mathbb Q(\zeta_5). }$$

Cancellation layer remains separate

Entry 1854 removed ten ambient occurrences through five chartwise pair/four-site cancellations. Those five relations precede the physical fifty-to-sixteen incidence map and are not part of its rank-thirty-four kernel. Combining the two stages without this distinction would mix coefficient cancellation with occurrence redundancy.

Narrow result

The surviving label map is already surjective. Thus no missing carrier label or support cell can be inferred from the total-soft occurrence assembly. Any nontrivial next class must arise from a typed coefficient differential on the rank-thirty-four occurrence kernel.

Next falsifier

Derive the actual overlap/restriction differential between the five physical residue charts. Test its action on

$$6\mathbb Q_{\rm triv}\oplus7\mathbb Q(\zeta_5)$$

without replacing it by the bare incidence projection. If no source-derived overlap map exists, stop at the incidence kernel rather than naming it cohomology.

Evidence

• research/benincasa/checkers/five_site_region_pair_total_soft_incidence_kernel.py
• research/benincasa/results/five-site-region-pair-total-soft-incidence-kernel.json
• Entries 1854--1855
• allocator claim: seqclaim-908a3085bd5a47f4c2ac2926

View source on GitHub