e_310130
plain-language theorem explainer
One kernel case of the 4D Regge midpoint identity: the folded mass-squared numerator at index tuple (3,1,0,1,3,0) equals eight times the explicit integer coupling Z at those same indices. Gravity analysts cite it only as a brick in the full six-index identity. The proof is a single kernel decide on concrete Fin 4 values.
Claim. For indices $(a,b,c,d,i,j)=(3,1,0,1,3,0)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer kernel $Z(a,b,c,d,i,j)$.
background
This module is chunk 13 of a 256-case kernel certification that the midpoint Regge mass-squared numerator equals eight times an explicit integer table on $(\mathrm{Fin},4)^6$. The local slogan is $m_2^{\mathrm{num}}=8\cdot Z$.
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add each contribution term evaluated at the six indices. The explicit kernel $Z$ is a closed-form integer lookup on those same six $\mathrm{Fin},4$ arguments (sample values include $4$, $-2$, and other small integers on the listed patterns).
The ambient setting is 4D discrete gravity analysis: certifying an exact algebraic identity for the midpoint Regge TT sector before any continuum or continuum-limit argument.
proof idea
One-line computational proof: decide evaluates both sides on the concrete six-tuple $(3,1,0,1,3,0)$ and checks integer equality. No lemmas are invoked beyond the definitions of the folded numerator and the explicit $Z$ table; the kernel is small enough for Lean’s decision procedure.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every six-tuple in $(\mathrm{Fin},4)^6$ by exhaustive fin_cases and discharges each leaf with a chunk equality such as this one. Without the full set of 256 decides, the global numerator-versus-explicit-$Z$ certificate does not close.
In the Recognition gravity stack this is pure discrete bookkeeping: it locks the algebraic skeleton of the 4D midpoint Regge TT identity so later continuum or phenomenological layers can quote a proved integer relation rather than a schematic coupling table. It does not itself touch the forcing chain (T0–T8), $\phi$-ladder masses, or $\alpha$; it is infrastructure under the gravity analysis path.
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