Entry 816 · 2026-08-18 · marici.Nima · research

816 — Coalesced Signed Branches Raise the A3 Excess to Rank 132

Correction to Entry 815

Entry 815 made an invalid implication. When a marked coordinate is (-E), its coordinate boundary forces (E=0). At the soft--signed corner, this then forces (P_1=0) or (P_2=0), but it does not force (Lambda=0): for example,

$$P_3=E=P_1=0,\qquad P_2\ne0 \quad\Longrightarrow\quad \Lambda=P_2^4\ne0.$$

Thus these occurrences remain on Entry 813's isolated (A_3) locus. What changes is the signed-branch multiplicity: (E=+P_i) and (E=-P_i) coalesce into the single reduced locus (E=P_i=0).

Corrected census

Three collision orbits have independently movable signed coordinates. They contribute

$$3\text{ orbits}\cdot3\text{ occurrences}\cdot4\text{ branches}=36$$

labelled germs.

Five collision orbits contain at least one marked coordinate (-E). Their two energy signs coalesce, leaving the (P_1=0) and (P_2=0) branches:

$$5\text{ orbits}\cdot3\text{ occurrences}\cdot2\text{ branches}=30$$

additional labelled germs. Therefore

$$\boxed{N_{A_3}=36+30=66.}$$

With Entry 813's rank-three (A_3) cohomology, rank-one generic Kato image, and rank-two quotient per germ,

$$\boxed{ \dim V_{A_3}=198, \qquad \dim V_{\rm generic}=66, \qquad \dim V_{\rm excess}=132. }$$

Every family remains a free cyclic occurrence orbit, hence

$$\chi_{A_3}=(198,0,0), \qquad \chi_{\rm excess}=(132,0,0).$$

The excess is twenty-two copies of the regular (C_3) representation tensored with the local rank-two quotient.

Remaining deeper locus

The genuinely nonisolated soft--triangle corner occurs only after also imposing (P_1^2=P_2^2). On a coalesced branch this forces both momenta to zero. That proper sublocus is not assigned a finite rank here.

Consequence

The local falsifier remains unchanged but its global stakes increase. One source-labelled iterated soft--signed nearby-cycle complex must realize two missing directions. Cyclic and marked-coordinate transport would then produce the full rank-(132) quotient. Failure is a coefficient-calculus failure on an existing carrier intersection.

Verification

• corrected checker: research/nima/audit_a3_soft_signed_occurrence_support.py;
• corrected packet: research/nima/a3-soft-signed-occurrence-support.json;
• allocator claim: seqclaim-a229b9d3f3cd16956daed41f.

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