e_010303

This module is chunk 1 of a 256-cell kernel certification that the folded M2 numerator equals eight times an explicit integer table on all of $(\mathrm{Fin},4)^6$. The setting is the 4D Regge midpoint analysis of the M2–TT identity in the gravity stack.

The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: start at $0$ and add each term's contribution at the six indices. The explicit kernel $Z^{\mathrm{expl}}$ is a pattern-matched integer table on the same six $\mathrm{Fin},4$ arguments (sample clauses include $4$ on diagonal-type pairs and $-2$ on mixed pairs).

The global claim is that these two agree up to the constant factor $8$ at every multi-index. Each chunk theorem discharges one concrete cell so the assembly can recombine them without re-running the full fold.

One-line computational proof: decide. Both sides reduce to concrete integers once the six indices are fixed at $(0,1,0,3,0,3)$, so the kernel decision procedure checks equality of the evaluated fold against $8$ times the matched table entry. No lemmas beyond the definitions of the numerator fold and the explicit table are required.

Feeds the assembly theorem $m_2^{\mathrm{num}}=8,Z^{\mathrm{expl}}$ for all six indices in ReggeExactMidpointM2TTIdentity4DM2NumAssemble, which introduces the six variables and splits by fin_cases on each. That global identity is the certified numerator half of the 4D Regge midpoint M2–TT kernel comparison used in the gravity analysis path.

Within Recognition Science gravity work, these kernel cells underwrite exact discrete curvature bookkeeping (Regge-type edge and deficit combinatorics) rather than continuum curvature postulates. The chunking into 256 decides keeps each certificate tiny and machine-checkable while the assemble step restores the universal statement. No forcing-chain landmark (T5–T8) is touched directly; the link is through the discrete gravity layer that consumes the identity.