IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.MultiDistinctionGeometry
Geometry of n independent binary distinctions: each channel sits on one of two sides, so configurations are vertices of the n-cube. The module builds vertices, edges, 0- and 1-chains, and boundary maps with d^{2} = 0. Downstream cubical and delta-native modules import it as the discrete substrate. Structure is definitional plus elementary commutation and nilpotence lemmas.
claimFor $n \in \mathbb{N}$, a configuration is an assignment of each of $n$ independent binary channels to a side. The configuration space is the vertex set of the $n$-cube; edges flip one channel. Chain groups $C_0$, $C_1$ and boundaries $d_1$, $d_2$ satisfy $d^2 = 0$.
background
In the Primitive Recognition Calculus, the elementary act is a binary distinction: a channel is on one side or the other. A multi-distinction configuration packages $n$ such independent channels. The natural geometry is cubical: vertices are full side-assignments; an edge is a single-channel flip.
The module names that geometry. Config is the space of $n$-channel side-assignments. Vtx and Edge are the 0- and 1-skeleton. Formal chains $C_0$ and $C_1$ carry integer (or ring) coefficients on vertices and edges. Boundary operators $d_1$ and $d_2$ implement the usual incidence rule (oriented sum of faces).
Sibling facts record that channel flips commute (diff_comm, diff_self_comm) and that the boundary squares to zero (boundary_squared_zero). A vertex indicator supplies the standard basis of $C_0$. Only Mathlib is imported; no deeper RS constants appear here.
proof idea
Primarily a definition module: Config, Vtx, Edge, C0, C1, d1, d2, and vertexIndicator are introduced by construction on the product of two-point sets (the n-cube). Commutation of single-channel flips is by case analysis on coordinates. Nilpotence of the boundary is the standard cubical identity: each 2-face contributes opposite orientations twice and cancels. No deep analytic or forcing argument; the content is discrete linear algebra on the cube.
why it matters in Recognition Science
This is the discrete geometric substrate for the Primitive Recognition Calculus. Three modules import it directly: CubicalChainComplex (lifts the skeleton to a full cubical complex), DeltaNativeAnalysis, and DeltaNativeStrongClosure (analytic and closure results for the native delta on this geometry).
In the broader Recognition framework the n-cube of distinctions is the combinatorial precursor to the eight-tick octave (period $2^3$) and to dimension forcing: once distinctions are organized cubically, chain-level identities become available for later forcing steps. The module itself does not force $D=3$ or $\varphi$; it only supplies the multi-distinction complex those arguments sit on.
scope and limits
- Does not define higher cubical chains $C_k$ for $k \ge 2$ beyond the $d_2$ piece needed for $d^2=0$.
- Does not prove topological invariants (homology ranks, homotopy type) of the n-cube.
- Does not introduce J-cost, $\varphi$, mass ladders, or any RS physical constant.
- Does not force spatial dimension or the eight-tick period; those live upstream/downstream in the forcing chain.
- Does not treat continuous limits or embeddings into manifolds.