e_033230
plain-language theorem explainer
Pointwise identity: the folded midpoint numerator at index sextuple (0,3,3,2,3,0) equals eight times the explicit integer table at those indices. Gravity analysts cite it as one cell of the 256-way kernel that certifies the closed-form Regge midpoint mass term. The proof is a single kernel `decide` on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(0,3,3,2,3,0)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator equals $8$ times the explicit integer table entry at those indices.
background
In the Regge exact-midpoint 4D analysis, the numerator of the discrete mass term is assembled by folding a fixed coupling list: each triple contributes an integer contrib, and the fold yields an Int-valued six-index array on Fin 4. Parallel to that fold sits an explicit closed-form table on the same domain, with sparse nonzero pattern (e.g. value 4 on diagonal-type slots, -2 on selected off-diagonal slots).
This module is chunk 3 of the 256 kernel decides that discharge the pointwise claim numerator = 8 · explicit table. The local setting is pure finite enumeration: every sextuple is a concrete Fin 4 tuple, so equality of two integers is decidable.
Upstream, the fold definition and the pattern-matched table are the only inputs; no continuum limit or continuum curvature identity is invoked here.
proof idea
One-line kernel proof: by decide. Both sides reduce to concrete Int values (left via the fold over the coupling list, right via the explicit table match), and Lean’s decidable equality on integers closes the goal with no lemmas or rewriting.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every sextuple in (Fin 4)^6 by exhaustive fin_cases and invokes each chunk cell such as this one. That global equality is the algebraic certificate that the folded Regge midpoint numerator collapses to a sparse closed form, a step in the discrete gravity analysis chain inside Recognition Science.
It does not itself touch the forcing chain (T0–T8), the J-cost, or the phi ladder; it is bookkeeping infrastructure for the 4D midpoint mass identity. Closing all 256 cells removes scaffolding from the assemble step.
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