e_032301
plain-language theorem explainer
Pointwise identity: the folded Regge midpoint numerator at multi-index (0,3,2,3,0,1) equals eight times the explicit integer table at that same point. Gravity analysts cite it as one cell of the 256-case kernel that assembles the global m2Num = 8·explicitZ theorem. The proof is a single kernel decide on concrete integers.
Claim. At the multi-index $(a,b,c,d,i,j)=(0,3,2,3,0,1)\in(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals eight times the explicit closed-form integer: $m_2^{\mathrm{num}}(0,3,2,3,0,1)=8\,Z_{\mathrm{expl}}(0,3,2,3,0,1)$.
background
This module is chunk 3 of a 256-cell kernel certifying that the folded Regge midpoint numerator equals eight times an explicit integer table on every sextuple of indices in $\mathrm{Fin},4$. The local slogan is $m_2^{\mathrm{num}}=8\cdot Z_{\mathrm{expl}}$.
The numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ is defined by folding a fixed coupling list: it sums a contribution functional over that list and returns an integer. The companion $Z_{\mathrm{expl}}$ is a total function $(\mathrm{Fin},4)^6\to\mathbb{Z}$ given by an exhaustive pattern match (sample values include $4$, $-2$, and so on on the diagonal and near-diagonal patterns).
Both definitions live in the kernel certificate module imported here. The present theorem fixes one concrete sextuple and asserts the scalar identity at that point only.
proof idea
One-line proof by decide. Both sides reduce to concrete integers once the six $\mathrm{Fin},4$ arguments are substituted: the fold that defines the numerator evaluates fully, and the pattern match that defines the explicit table fires on a single clause. Lean’s kernel decision procedure checks integer equality; no lemmas are invoked.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states $\forall a,b,c,d,i,j,; m_2^{\mathrm{num}}=8,Z_{\mathrm{expl}}$ and discharges the universal quantifier by nested fin_cases over all six indices. Each chunk theorem such as this one is a leaf of that case tree (chunk 3 covers a block of the 256 cells).
In the broader Gravity analysis, the identity is the algebraic backbone of the exact midpoint M2/TT relation in 4D Regge calculus: once the numerator is replaced by the closed table, subsequent curvature and deficit identities can quote a pure integer expression rather than a fold. It does not itself touch the RS forcing chain (T5–T8) or the J-cost; it is infrastructure inside the discrete gravity layer.
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