IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.AllDimensionalCubicalBoundary
Packages the all-dimensional cubical boundary calculus: face certificates, higher chains, and the identity ∂²=0 on n-channel cubes. Analysts of the native recognition differential cite it when lifting 1D ledger cancellations to full cubical homology. The argument is finite square-cancellation bookkeeping per face, packaged as a reusable boundary API.
claimFor an $n$-channel cube, a higher face certificate of dimension $d$ carries a finite ledger of codimension-2 square cancellations. On the associated higher chain, the cubical boundary satisfies $\partial^2=0$. The module exposes a single all-dimensional boundary API specializing this identity for the native recognition differential.
background
Primitive recognition calculus treats discrete recognition events on cubical complexes built from $n$ binary channels (the eight-tick octave is the $n=3$ case). The imported cubical chain complex supplies faces, edges, and the formal boundary operator $\partial$ on chains.
A higher-dimensional face certificate records an intended face dimension together with the finite list of codimension-2 squares generated by that face. Each such square contributes a signed pair of 1-chains that cancel in $\partial^2$. Higher chains assemble these certificates so the same cancellation ledger proves $\partial^2=0$ uniformly in dimension, not only on edges.
The module sits in the Foundation layer: it does not yet specialize constants ($\varphi$, $J$-cost, mass rungs) but supplies the combinatorial skeleton those specializations will use.
proof idea
Definition-heavy module with short algebraic lemmas. Higher face certificates package the codimension-2 square ledger; higherFace_secondBoundary_zero checks pairwise cancellation on that ledger. Higher chains lift the same check to assembled faces via higherChain_secondBoundary_zero. The public API (AllDimensionalBoundaryAPI, deltaCubicalBoundaryAPI) and the headline theorem simply re-export $\partial^2=0$ in the form downstream native-$\Delta$ analysis expects. No analytic estimates; pure finite signed cancellation.
why it matters in Recognition Science
Downstream modules DeltaNativeAnalysis and DeltaNativeStrongClosure import this boundary API to treat the native recognition differential as a genuine cubical coboundary (so closed loops and exactness statements are well-typed). Without all-dimensional $\partial^2=0$, strong closure of discrete recognition dynamics would be stuck at the 1-skeleton.
In the broader forcing chain this is scaffolding under T7 (eight-tick octave, period $2^3$): the cube dimension $n$ is free here, and $n=3$ is the physically forced case. It does not itself force $D=3$ or the $J$-cost; it only guarantees that whatever differential is later placed on the cube is a complex.
scope and limits
- Does not force spatial dimension $D=3$ or the eight-tick period; $n$ is a free parameter.
- Does not define the native recognition differential $\Delta$ or its continuum limit.
- Does not prove homology vanishing, exactness of long sequences, or Betti-number formulas.
- Does not involve $J$-cost, $\varphi$-ladder masses, or coupling constants.
- Does not address non-cubical cells or simplicial subdivisions.