IndisputableMonolith.Gravity.CorrectedTaylorHigherCardinality
Module packaging the corrected Track 1.B gate at cubic scale N: local cubic-Taylor correspondence with the axis stencil on the cubic Freudenthal torus of side N. Each N reduces to a finite coefficient identity on the N³ vertex table. Gravity workers cite it when lifting the quadratic axis-stencil gate to all finite cardinalities. Structure is equivalence lemmas between the cubic gate and an all-cardinality gate, plus a homogeneous-evenness fact for the quadratic form.
claimThe corrected Track 1.B gate at cubic scale $N$ asserts that the local cubic-Taylor correspondence with the axis stencil holds on the cubic Freudenthal torus of side length $N$. For each $N$, this is a finite coefficient identity over the $N^3$ vertex table. The module equates an all-cardinality corrected gate with the cubic gate (and its reverse), and records that a homogeneous quadratic is even.
background
Track 1.B aims at a local Regge / J-cost correspondence via an axis stencil. The upstream module Track1BCorrectedQuadratic packages the corrected quadratic gate: theorem-level material with zero sorry and zero RS-internal axioms for everything stated, while the corrected gate itself is named OPEN rather than asserted.
This module lifts that local correspondence from the quadratic setting to cubic Taylor scale on the cubic Freudenthal torus of side $N$. The discrete arena is the $N^3$ vertex table; the claim is a finite coefficient identity for each $N$. Sibling names indicate a cubic gate proposition, an all-cardinality corrected gate, and iff bridges between them, plus a parity fact for homogeneous quadratics used in the coefficient bookkeeping.
proof idea
Not a single theorem: a small theory module. It introduces the corrected Track 1.B gate at cubic scale and an all-cardinality corrected gate, then proves implication both ways (all-cardinality implies cubic; cubic implies all-cardinality) and packages the iff. A homogeneous-quadratic-is-even lemma supports coefficient parity. A five-of-gate implication specializes the cubic gate. The argument reduces each fixed $N$ to finite algebra on the $N^3$ table rather than an analytic continuum limit.
why it matters in Recognition Science
Closes the cardinality gap in the Track 1.B corrected route: once the cubic gate is available, the module shows it is equivalent to a gate that quantifies over all finite cardinalities on the Freudenthal torus. That is the natural handoff from the quadratic axis-stencil module toward a uniform local correspondence used in the gravity track. Downstream used_by edges are empty in the graph snapshot, so the immediate consumers are sibling lemmas inside the same gravity stack and any later assembly that assumes the corrected gate at every $N$. It does not yet discharge the OPEN status of the corrected gate named upstream; it organizes the cubic and all-$N$ forms of that gate.
scope and limits
- Does not prove the corrected Track 1.B gate; upstream still names it OPEN.
- Does not treat continuum $N\to\infty$ limits; only finite $N^3$ tables.
- Does not establish the full Regge/J-cost correspondence beyond the local cubic-Taylor axis stencil.
- Does not introduce new physical constants or mass-ladder claims.
- Does not claim used_by consumers outside this gravity stack in the current graph.