rawCosineFoldAtScale_zero
plain-language theorem explainer
At vanishing scale the raw cosine fold of a 3×3 edge-weight matrix is identically zero, for every real mode direction. Continuum-limit arguments for Regge TT cite this to strip the constant term before dividing by q². The proof rewrites the scale-zero fold as the assembled constant block and invokes the hinge-aware zero-mode cancellation.
Claim. For every real mode direction $x \in \mathbb{R}^3$ and every edge-weight matrix $E : \{0,1,2\}^2 \to \mathbb{R}$, the raw cosine fold evaluated at scale $q = 0$ equals zero: $\mathrm{rawCosineFoldAtScale}(0,x,E) = 0$.
background
This module is the C-DAG2 continuum stage for Regge TT. Exact Bloch orthogonality has already removed the cell sum, so the remaining work is a local cosine two-jet limit of the finite raw bucket fold. The reusable continuous-variable object is a free real scale $q$ whose phase is $q \sum_i x_i (u_i/2)$, preserving the doubled-midpoint convention.
The raw cosine fold aggregates signed stencil weights over geometry-derived buckets. Each bucket key records the doubled midpoint displacement between left and right slots of a tetrahedron/slot triple; the bucket amplitude is the honest fiber sum of those signed weights. At scale zero the cosine evaluator collapses to the constant $1$, so the fold becomes the assembled constant block of the stencil.
Upstream, the assembly layer supplies the identity of the fold with a triple sum over the $6^3$ raw triples, and the hinge-aware assembled zero-mode theorem asserts that constant block vanishes. That cancellation is geometric (hinge structure), not a bare stencil identity.
proof idea
Rewrite the scale-zero fold as the Bloch fold of the $N=1$ raw cosine evaluator on the zero integer mode, via unfolding and a pointwise congruence on buckets. Apply the assembly identity that equates that fold to the raw triple sum. On every triple the zero-mode evaluator is $1$ by direct normalization of the commensurate momentum. The weighted triple sum therefore collapses (after product-type summation and one_mul) to the assembled constant block. Finish by the hinge-aware theorem that the assembled constant block is zero.
why it matters
This is the constant-term removal step for the continuum two-jet. The reusable headline rawCosineFold_scale_tendsto divides the fold by $q^2$ in a punctured neighborhood of zero and needs the value at zero to vanish so the quotient is well-defined in the limit; the module doc states explicitly that the constant term is removed by the assembled zero-mode theorem. That tendsto result then feeds the P1.1a headline: for every fixed nonzero integer mode, the finite reduced Regge symbol over momentum-norm-squared converges to the raw Regge TT moment at the normalized real mode direction.
In the Seven Gaps ledger the same fact is cited when certifying the Pillar 2 cutoff-limit blocker: zero phase fails oscillatory-tail cancellation, while the continuum half of the gap obligations relies on genuine scale-dependent cancellation of the kind established here. Within RS gravity this sits on the continuum side of the Regge TT analysis, after Bloch assembly and before mesh refinement $q_N = 2\pi/N$.
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