e_310303
plain-language theorem explainer
For the six-index tuple (3,1,0,3,0,3) in (Fin 4)^6, the midpoint M2 numerator equals eight times the explicit Z coupling. Gravity analysts assembling the 4D Regge midpoint identity cite this as one of 256 kernel cases. The proof is a single kernel decide on concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(3,1,0,3,0,3)$ with each index in $\{0,1,2,3\}$, the folded M2 numerator equals $8$ times the explicit integer coupling $Z$ at those indices.
background
This module is chunk 13 of a 256-case kernel that certifies the algebraic identity between two integer-valued tensors on $(\mathrm{Fin},4)^6$ arising in the 4D Regge exact-midpoint analysis.
The numerator side is defined by folding a fixed coupling list: each term contributes an integer contrib and the fold starts at zero, yielding an Int. The right-hand side is an explicitly tabulated coupling explicitZ, a pattern-matched function $\mathrm{Fin},4^6\to\mathbb{Z}$ that returns small integers such as $4$, $-2$, and so on on the support of the kernel.
The local claim is the pointwise equality of these two at one concrete multi-index. The surrounding certificate module supplies both definitions; this file only discharges individual decide goals.
proof idea
One-line computational proof: decide evaluates both sides as concrete integers (the fold of m2Num at $(3,1,0,3,0,3)$ and eight times the matched explicitZ clause) and closes the equality by kernel reduction. No lemmas are invoked beyond the definitions of the two sides.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every six-tuple in $(\mathrm{Fin},4)^6$ by exhaustive fin_cases and dispatches each leaf to a chunk theorem of this form. That global equality is the numeric core of the 4D Regge exact-midpoint M2/TT identity certificate in the Gravity analysis stack. It is pure discrete linear algebra on the coupling kernel, not a continuum GR derivation; within Recognition Science it supports the discrete geometric side of the gravity sector rather than the T0–T8 forcing chain directly.
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