e_033232
plain-language theorem explainer
For Fin-4 indices (0,3,3,2,3,2), the folded M2 numerator equals eight times the tabulated explicit Z-kernel entry. Gravity analysts assembling the 4D Regge exact-midpoint TT identity cite this as one atomic case among 256 kernel checks. The proof is a single kernel decision that evaluates both closed integer sides.
Claim. For indices $a=0,b=3,c=3,d=2,i=3,j=2$ in $\mathrm{Fin}\,4$, the folded M2 numerator equals eight times the explicit integer kernel: $m_2^{\mathrm{num}}(0,3,3,2,3,2)=8\,Z_{\mathrm{exp}}(0,3,3,2,3,2)$.
background
This module is chunk 3 of a 256-case kernel certification that the folded M2 numerator equals eight times an explicit integer table on six Fin-4 indices. The local setting is 4D Regge exact-midpoint analysis for a TT-sector identity.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a contribution map over a fixed coupling list, starting from 0. The explicit kernel $Z_{\mathrm{exp}}$ is a piecewise integer function on $(\mathrm{Fin},4)^6$, with tabulated values such as $4$, $-2$, and so on for distinguished index patterns.
Upstream, both maps live in the kernel-cert module; this chunk only instantiates one concrete six-tuple and checks the scalar identity.
proof idea
One-line kernel proof: decide. With all six indices literal, both sides reduce to closed Int expressions (fold of concrete contributions versus a lookup in the explicit table, scaled by 8). The kernel evaluates the equality and closes the goal; no lemmas or rewriting are required.
why it matters
The parent theorem is the universal statement that $m_2^{\mathrm{num}}=8,Z_{\mathrm{exp}}$ for every six-tuple in $(\mathrm{Fin},4)^6$. That assembler runs nested fin_cases on all six indices and dispatches each leaf to a chunk theorem of this form.
In the Recognition gravity stack, the identity is bookkeeping for the exact-midpoint Regge M2 TT sector in four dimensions. It does not itself invoke the forcing chain (T5–T8), RCL, or the phi-ladder mass formula; it is pure discrete-kernel arithmetic supporting the continuum-facing gravity analysis.
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