e_301322
plain-language theorem explainer
For the six-index pattern (3,0,1,3,2,2) on Fin 4, the folded midpoint numerator m2Num equals eight times the closed-form kernel value explicitZ. Gravity analysts assembling the 4D Regge midpoint M2TT identity cite this as one of the 256 atomic cases. The proof is a single kernel decide on integer equality.
Claim. For indices $a{=}3,\,b{=}0,\,c{=}1,\,d{=}3,\,i{=}2,\,j{=}2$ in $\mathrm{Fin}\,4$, the folded coupling numerator $\mathrm{m2Num}(a,b,c,d,i,j)$ equals $8$ times the explicit kernel integer $\mathrm{explicitZ}(a,b,c,d,i,j)$.
background
This module is one chunk of the 4D Regge exact-midpoint M2TT identity certification. The local goal, stated in the module header, is to show m2Num = 8·explicitZ by discharging 256 kernel decides, of which this declaration is case 12 in its chunk.
The numerator m2Num(a,b,c,d,i,j) is the fold of contrib over couplingZList: an integer accumulated from discrete coupling terms at the six Fin-4 indices. The comparison value explicitZ is a closed-form Int-valued kernel on the same six indices, tabulated by pattern (e.g. (0,0,1,1,2,2) ↦ 4, (0,0,1,2,1,2) ↦ −2).
The ambient setting is discrete gravity analysis: verifying that the midpoint second-moment numerator factors exactly as eight times this kernel, index by index, before the universal quantifier is reassembled.
proof idea
One-line tactic proof: by decide. Both sides are closed integer expressions once the six Fin-4 literals are substituted into m2Num and explicitZ, so the kernel evaluates the equality and closes the goal with no lemmas or rewriting.
why it matters
Feeds the assembler m2Num_eq_eight_explicitZ, which states ∀ (a b c d i j : Fin 4), m2Num a b c d i j = 8 * explicitZ a b c d i j and proves it by exhaustive fin_cases on all six indices. Each atomic e_* theorem is one leaf of that case split.
In the Recognition gravity stack this identity is infrastructure for the exact midpoint M2TT relation in 4D Regge calculus: once the numerator is identically eight times the explicit kernel, downstream curvature and mass-ladder comparisons can quote a single closed form rather than a fold. It does not itself touch T0–T8 or the RCL; it is a certified arithmetic brick inside the gravity analysis layer.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.