e_031333
plain-language theorem explainer
Kernel case identity: the midpoint mass-squared numerator at multi-index (0,3,1,3,3,3) equals eight times the explicit Z-coupling table entry. Gravity analysts assembling the 4D Regge midpoint M2TT identity cite these chunk lemmas. Proof is a single kernel decide on concrete integer arithmetic.
Claim. For the multi-index $(a,b,c,d,i,j)=(0,3,1,3,3,3)$ with each coordinate in $\mathbb{F}_4$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer table value $Z(a,b,c,d,i,j)$.
background
In the 4D Regge midpoint analysis, two integer-valued maps on six $\mathrm{Fin},4$ indices are compared. The numerator $m_2^{\mathrm{num}}$ is the fold of a fixed coupling list: it sums a local contribution at each list entry for the given multi-index. The companion map $Z$ is an explicit pattern-matched table of small integers (entries such as $4$, $-2$, and so on).
The module is chunk 3 of a 256-case kernel-decide partition of the identity $m_2^{\mathrm{num}}=8Z$. The local setting is pure finite arithmetic on the discrete index cube; no continuum limit or physical units enter at this layer.
proof idea
One-line proof by decide. Both sides reduce to concrete integers once the six indices are fixed: the left-hand side evaluates the fold that defines $m_2^{\mathrm{num}}$, the right-hand side looks up $8Z$ in the explicit table. The kernel checks integer equality; no lemmas beyond the two definitions are invoked.
why it matters
Feeds the universal assembly theorem $m_2^{\mathrm{num}}=8Z$ for all six $\mathrm{Fin},4$ indices, which is proved by exhaustive case split and consumes these chunk identities. That identity is a certified algebraic step inside the 4D Regge midpoint M2TT analysis in the Gravity domain: it replaces a folded coupling sum by a sparse explicit table, scaled by eight.
Within Recognition Science this sits in the discrete gravity bookkeeping that supports continuum limits and mass-ladder comparisons; it does not itself invoke the forcing chain (T5–T8), RCL, or $\varphi$-ladder constants. It closes one concrete cell of the kernel partition so the parent forall can discharge without sorry.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.