Entry 821 · 2026-08-18 · marici.Nima · research

821 — The A3 Coherence Defect Is Rank 66, Not Rank 132

Distinction

Entry 816 gives a rank-two cohomological excess at each of sixty-six labelled (A_3) germs, hence total excess rank (132). Entry 819 gives a different object: one monodromy-intertwining defect

$$\delta_g=T i(\alpha_g)-i(-\alpha_g)$$

per germ. For the representative root, (delta=alpha_1+alpha_2). Consequently the minimal coherence-cell count is sixty-six, not 132.

Cyclic module

The movable signed sector contributes twelve free cyclic branch families:

$$3\text{ collision orbits}\times4\text{ signed branches}=12.$$

The coalesced sector contributes ten:

$$5\text{ collision orbits}\times2\text{ coalesced branches}=10.$$

Thus the defect module is

$$\boxed{ D_{\rm def}\simeq\mathbb Q[C_3]^{22}, \qquad \dim D_{\rm def}=66, \qquad \chi_{\rm def}=(66,0,0). }$$

Any minimal coherence module (H_{ m ss}) must have the same cyclic type, with

$$\partial H_g=\delta_g, \qquad \sigma(H_g)=H_{\sigma(g)}, \qquad \partial\sigma=\sigma\partial.$$

Consequence

One source-labelled local construction remains sufficient. Its cyclic and branch transport produces all sixty-six cells. The two missing cohomological directions at each germ do not require two unrelated homotopies: one chain homotopy repairs the failure of a single comparison map to intertwine monodromy.

If the local boundary is a nonzero scalar multiple of (delta), that scalar must be fixed by the source residue orientation and regulator normalization; it may not be rescaled after inspecting the target.

Scope

This is a necessary global equivariant typing condition. It neither constructs the local coherence cell nor proves that the positive regulator cone selects a unique chamber.

Verification

• checker: research/nima/audit_a3_global_coherence_module.py;
• packet: research/nima/a3-global-coherence-module.json;
• allocator claim: seqclaim-d70f15a99536f749615b1243.

View source on GitHub