factorizedSlotTerm
plain-language theorem explainer
Single-slot contribution to the factorized type-(1,1) Bloch fold of the committed flat Regge Hessian: the product of two phased class dots (area covariance times full star kernel) when the oriented hinge is orbit type (1,1), else zero. Gravity analysts cite it as the atomic summand of the cell fold and of the zero-momentum consistency gate. Pure definitional gate-and-product; no proof content.
Claim. Given a $4\times 4$ metric perturbation $H$, a wave vector $m\in\mathbb{R}^4$, and oriented hinge indices $(s,t)$, the factorized slot term equals the product of the phased class-dot of the type-$(1,1)$ area covariance with the phased class-dot of the full star class kernel, both evaluated at the hinge base point, whenever the hinge has orbit type $(1,1)$; otherwise the term is $0$.
background
The module carries out the exact phase-decorated Bloch fold of the committed true-weight flat Hessian for type-$(1,1)$ triangle hinges inside one Kuhn cell, using the midpoint plane-wave convention of the 4D edge stencil. Scope is the $(1,1)$ orbit only (72 oriented slots per cell).
A hinge base is the coordinate mask of the triangle's vertex support. The predicate isT11 holds exactly when the hinge orbit classifier returns type $(1,1)$. The phased class-dot contracts a fixed 15-component class vector $v$ against the plane-wave class perturbation of $H$ at momentum $m$ and base point $x$. The area-covariance vector for type $(1,1)$ is the sparse load $(1/4,1/4,0,\ldots)$; the full star class kernel is the committed deficit kernel on the same 15-class space.
The shifted cost $H(x)=J(x)+1$ from the Recognition cost algebra is ambient infrastructure here; the local objects are purely the Regge class loads and plane-wave decorations.
proof idea
Definitional, not a proof. On the decidable branch where the hinge is type $(1,1)$, return the ordinary real product of two phased class-dots (area covariance times full star kernel) at the same hinge base; otherwise return $0$. No lemmas are applied.
why it matters
Atomic summand of the factorized $(1,1)$ cell fold: summing this term over all $24\times 10$ oriented slots defines the factorized Bloch fold used throughout the module. Downstream, the zero-momentum consistency gate shows that fold recovers the committed orbit quadratic at $m=0$, and the all-orbit symbol layer identifies the generic orbit fold on type $(1,1)$ with this specialization.
In the QG campaign this is the factorized building block for the finite-momentum $(1,1)$ symbol before continuum comparison. Module tags are explicit: it does not evaluate the $m^2$ Taylor coefficient against the Einstein–Hilbert/TT continuum symbol, does not prove RS action convergence to EH in 4D, and does not flip the gap-action-recovery flag. It sits inside the discrete-gravity analysis lane that ultimately supports Recognition Science continuum recovery, not inside the T0–T8 forcing chain itself.
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