Entry 815 · 2026-08-18 · marici.Nima · research

815 — The Generic Soft-Signed A3 Excess Occupies Three Collision Orbits

Superseded by Entry 816. This entry incorrectly treated the five occurrences with a marked coordinate (-E) as automatically lying on (Lambda=0). They instead retain isolated (A_3) germs with coalesced signed branches when the other momentum is nonzero. Its (36/108/72) census is withdrawn.

Question

Entry 813 finds a local (A_3) singularity of rank three at a double-coordinate, soft--signed corner, with only one direction supplied by the generic Kato line. The eight collision orbits of Entry 803 do not all meet this locus generically: the marked-coordinate equations must be imposed before cyclic assembly.

Support gate

The three collision representatives

$$(g_1,g_2),\qquad(g_1,g_3),\qquad(g_2,g_3)$$

have two independently movable marked coordinates. Their simultaneous coordinate boundary can therefore meet

$$P_3=0,\qquad E^2=P_1^2\quad\text{or}\quad E^2=P_2^2$$

at a generic (A_3) point.

For each of the other five representatives, at least one marked coordinate is exactly (-E). Its double boundary forces (E=0); the signed condition then forces an additional soft momentum. Those occurrences lie on Entry 813's deeper nonisolated soft--triangle corner and must not be counted as generic (A_3) germs.

Cyclic count

Each of the three eligible collision orbits has three cyclic occurrences. Each occurrence has four signed branches: two choices of (P_1) versus (P_2), and two energy signs. Hence

$$N_{A_3}=3\cdot3\cdot4=36$$

labelled generic germs.

Entry 813 gives rank three, generic Kato rank one, and excess rank two per germ. Therefore

$$\boxed{ \dim V_{A_3}=108, \qquad \dim V_{\rm generic}=36, \qquad \dim V_{\rm excess}=72. }$$

The cyclic action is free on every transported family, so

$$\chi_{A_3}=(108,0,0), \qquad \chi_{\rm excess}=(72,0,0).$$

Equivalently, the excess is twelve copies of the regular (C_3) representation tensored with the rank-two local coefficient quotient.

Meaning for H2

The support gate prevents a spurious factor-of-eight census. The genuine frontier is smaller but nonzero: a rank-two local extension on three free collision orbits. This is coefficient complexity on already declared soft--signed intersections, not a new carrier divisor.

Only one source-labelled iterated soft--signed nearby-cycle complex now needs to be constructed. If it realizes the two missing local directions, cyclic naturality transports it across all thirty-six germs. If it does not, the current coefficient calculus fails H2 on this corner.

Scope

The other five collision orbits belong to the deeper nonisolated soft--triangle audit. No finite rank is assigned to them here.

Verification

• checker: research/nima/audit_a3_soft_signed_occurrence_support.py;
• packet: research/nima/a3-soft-signed-occurrence-support.json;
• allocator claim: seqclaim-08bc2c6d4aa68947010c4ece.

View source on GitHub