IndisputableMonolith.Gravity.Analysis.ReggeTTContinuumLimit
Continuum-limit stage for Regge transverse-traceless Bloch modes. It packages midpoint-displacement phases, scaled cosine folds, and the cosine two-jet limit that turn finite-cell evaluators into continuum symbols. Downstream M2-tendsto and algebraic-closer modules cite these limits. Argument is analytic: Taylor/jet control of cos and scale tendsto on normalized real modes.
claimDefines the midpoint-displacement phase $\sum_i x_i(u_i/2)$, raw cosine evaluators and folds at lattice side-scale, normalized real modes, and proves $(\cos\theta-1)/\theta^2\to -1/2$ as $\theta\to 0$, plus the matching scaled fold limit for Regge TT symbols.
background
This module sits in the Regge TT gravity analysis pipeline, immediately after the finite Bloch assembly stage. That upstream stage builds a C-DAG1 finite-cell cosine evaluator from a bucket integer phase key, independent of any quadratic moment evaluator, for side lengths and commensurate integer wave vectors whose doubled frequency is non-aliased in one coordinate.
Here the continuum step begins. The midpoint-displacement phase is the literal linear form $\sum_i x_i(u_i/2)$. From it one builds raw cosine evaluators and folds at a lattice side-scale, together with real-mode norms and normalized real modes. The analytic core is the two-jet of cosine near zero and the corresponding scale-tendsto for the fold, which convert discrete Bloch data into continuum symbols used by later closers.
Notation is lattice-first: side scale, integer wave vectors, and fold-along-direction symbols, not continuum metric fields.
proof idea
Definition layer first: linear and quadratic raw phases, cosine evaluator and fold at scale, real-mode norm-squared and normalization, and side-scale bookkeeping, including the zero-fold identity.
Analytic layer: an iterated-derivative identity for $a\mapsto\cos(a\cdot)$ supplies the two-jet. The key lemma is the standard limit $(\cos\theta-1)/\theta^2\to -1/2$. That jet is then transferred to the scaled raw cosine fold, yielding the fold-scale tendsto used by continuum closers.
No algebraic TT isotropy identities live here; those are deferred to the algebraic closer.
why it matters in Recognition Science
Production stage C-DAG2 in the panel-locked D-dag order ReggeTTBlochAssembly → ReggeTTContinuumLimit → ReggeTTAlgebraicCloser → ReggeTTContinuumCloser (QG full-theory campaign, Paper C / Pillar 1).
ReggeBlochM2Tendsto4D imports the proved cosine two-jet to close the punctured Tendsto along symbolDir for axis TT and pure gauge, using also zero-momentum vanishing of the deficit kernel on those sectors. ReggeTTAlgebraicCloser is the sole production importer of the committed algebraic certificate spike and takes this continuum limit as its immediate predecessor before emitting the C8 closed form $(1/2)x^T\mathrm{adj}(E)x$ and the TT isotropy value $-1/4$.
Without the cosine two-jet and fold-scale tendsto, the finite Bloch assembly cannot pass to continuum TT symbols.
scope and limits
- Does not prove full 4D continuum Einstein or Regge gravity reconstruction.
- Does not establish TT isotropy value $-1/4$ or the C8 algebraic closed form.
- Does not treat aliased or non-commensurate wave vectors beyond the upstream non-aliasing hypothesis.
- Does not close FoldAlongM2Tendsto for sectors other than those handled downstream.
- Does not import or certify the algebraic certificate spike (Gate B bridge).
used by (2)
depends on (1)
declarations in this module (18)
-
def
rawPhaseLinear -
def
rawPhaseQuadratic -
def
rawCosineEvaluatorAtScale -
def
rawCosineFoldAtScale -
def
realModeNormSq -
def
normalizedRealMode -
def
sideScale -
theorem
rawCosineFoldAtScale_zero -
theorem
iteratedDeriv_two_cos_mul -
theorem
of -
theorem
cos_sub_one_div_sq_tendsto -
theorem
rawCosineFold_scale_tendsto -
theorem
rawPhaseQuadratic_normalized -
theorem
reggeTTMoment_normalized -
theorem
realModeNormSq_intCast_pos -
theorem
rawCosineEvaluator_eq_scale -
theorem
momentumNormSq_eq_scale_sq -
theorem
canonicalFiniteH_div_momentumNormSq_tendsto