e_010133
plain-language theorem explainer
Pointwise identity: the Regge midpoint mass-squared numerator at multi-index (0,1,0,1,3,3) equals eight times the explicit Z coupling there. Gravity analysts cite it as one kernel cell in the 4D midpoint M2–TT identity. The proof is a single kernel decide on the concrete integer values of both sides.
Claim. At the multi-index $(a,b,c,d,i,j)=(0,1,0,1,3,3)$ with each index in $\{0,1,2,3\}$, the folded mass-squared numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer coupling $Z(a,b,c,d,i,j)$.
background
This module is one chunk of the 4D Regge exact-midpoint analysis that certifies $m_2^{\mathrm{num}}=8\cdot Z$ on the full $4^6$ index space by splitting into decide-sized blocks (module doc: "m2Num = 8·explicitZ, chunk 1 (256 kernel decides)").
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start from $0$ and add each local contribution contrib at the six Fin-4 indices. The comparison target explicitZ is a closed-form integer table on those same indices (sample clauses include values $4$, $-2$, and so on for distinguished patterns).
The local setting is pure integer arithmetic on a finite discrete index set; no continuum limit or variational argument is invoked at this layer.
proof idea
One-line computational proof: decide evaluates both sides at the concrete sextuple $(0,1,0,1,3,3)$ and checks integer equality. No lemmas beyond the definitions of m2Num (the fold) and explicitZ (the table) are required; the kernel reduces the fold and the matching table clause to numerals and compares them.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every $(a,b,c,d,i,j):(\mathrm{Fin},4)^6$ and discharges the universal quantifier by exhaustive fin_cases. Each chunk cell such as this one supplies a decide-closed instance so the global equality is not a single giant kernel goal.
In the Recognition gravity stack this is bookkeeping for the exact midpoint Regge mass-squared / TT identity in 4D, not a forcing-chain (T0–T8) step. It closes a finite combinatorial obligation that later continuum or phenomenological layers can quote as an already-certified algebraic identity.
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