e_012033
plain-language theorem explainer
For the single index sextuple (0,1,2,0,3,3) on Fin 4, the folded Regge midpoint numerator equals eight times the explicit integer kernel. Gravity analysts cite it when discharging one cell of the 4D m2Num = 8·explicitZ identity. The proof is a pure kernel decide on the two closed definitions.
Claim. With indices in $\{0,1,2,3\}$, the midpoint numerator satisfies $m_2^{\mathrm{num}}(0,1,2,0,3,3)=8\,Z_{\mathrm{explicit}}(0,1,2,0,3,3)$, where $m_2^{\mathrm{num}}$ is the fold of coupling contributions and $Z_{\mathrm{explicit}}$ is the closed integer kernel.
background
This module is one chunk of the 4D Regge exact-midpoint certification: the claim that the folded numerator m2Num equals eight times a hand-written integer table explicitZ on every sextuple of Fin-4 indices. The module doc states the local goal as "m2Num = 8·explicitZ, chunk 1 (256 kernel decides)."
m2Num a b c d i j is defined by folding contrib over couplingZList, starting from 0. explicitZ is a pattern-matched Int-valued kernel on six Fin-4 arguments (sample clauses give 4, −2, etc.). Both live in the KernelCert import; this file only decides individual cells.
The ambient setting is discrete gravity analysis: verifying an exact algebraic identity between the summed midpoint coupling numerator and an explicit sparse table, index by index, before assembling the universal statement.
proof idea
One-line computational proof: by decide. Lean reduces both sides on the concrete sextuple (0,1,2,0,3,3) from the closed definitions of m2Num (fold of contrib over couplingZList) and explicitZ (pattern match), then checks integer equality. No lemmas are invoked beyond the two upstream defs.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states ∀(a b c d i j : Fin 4), m2Num = 8·explicitZ and discharges the quantifier by nested fin_cases, invoking one cell theorem per sextuple. This cell is the (0,1,2,0,3,3) branch of that case split.
In the Recognition gravity stack the identity certifies that the 4D Regge midpoint numerator collapses exactly onto the explicit integer kernel, a bookkeeping step toward continuum or continuum-limit comparisons. It is pure discrete algebra: no continuum limit, no forcing-chain landmark (T0–T8), and no open physics claim beyond the certified equality on this cell.
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