The 9-Dimensional Binary Hypercube
512 vertices — the 9-bit binary state space F29 that underlies the model
State Space: F29
The ASH Model is built on the binary vector space F29 — all 9-bit binary strings. This yields exactly 29 = 512 vertices, each a distinct 9-bit state (a vertex of the hypercube Q9). In the model’s interpretive layer these states are read as candidate “realms,” but that reading is a research hypothesis, not an established cosmological result.
Of the 512 states, exactly 256 satisfy the integrity relation x9 = x1 ⊕ … ⊕ x8 — the even-parity hyperplane E, an 8-dimensional linear subspace the model treats as its admissible states.
The 512 states are organized into 10 Hamming shells (weights 0 through 9), where shell k contains all vertices with exactly k ones, with binomial degeneracy C(9,k). These counts are proved finite mathematics.
Why 9 Bits?
The 9-bit width is a versioned modeling choice, not a result derived from physics. Eight coordinates carry deterministic image/video measurements and the ninth is a recomputed XOR-parity (integrity) bit, giving the F29 state space. The repository explicitly does not claim a physical derivation of nine dimensions, and the finite-math results below stand independently of any such motivation.
- Coordinates 1–7Deterministic image/video measurements — mean luminance, RMS contrast, edge / texture / chroma energy, horizontal and vertical gradients — each thresholded with fixed hysteresis
- Coordinate 8Temporal-change bit; invariant under every code (XOR) translation and zero in every codeword
- Coordinate 9Recomputed XOR-parity of coordinates 1–8 — the integrity bit that defines the parity-valid states
Hamming Shell Distribution
Under controlled symmetric bit-flip noise the state occupancy converges to uniform, whose Hamming-weight marginal is exactly Binomial(9, ½) (mean 4.5, variance 2.25) — a generic finite-hypercube baseline that non-ASH controls also reproduce, not an ASH-specific result. With XOR transforms and no noise the dynamics stay confined to a 16-state code orbit (weights {0,4,8}) and do not converge (TV ≈ 0.73 from the binomial).
The [9,4,4] Transform Code
The canonical transform code C is a rank-4, doubly-even binary linear code with parameters [9,4,4]: length 9, dimension 4, minimum distance 4 — hence exactly 16 codewords, with weight enumerator {weight 0: 1, weight 4: 14, weight 8: 1}. It acts on the state space by XOR translation (x → x ⊕ c), partitioning it into code orbits of 16 states each.
The canonical [9,4,4] transform code is doubly-even and self-orthogonal (16 codewords, minimum distance 4) — not self-dual in nine dimensions (dim C = 4, dim C⊥ = 5). Self-duality holds only for the punctured [8,4,4] code obtained by removing the code-invariant coordinate 8.
Graph Structure
The 9-cube Q9 = (V, E) has 512 vertices connected whenever they differ in exactly one bit (Hamming distance 1). Every vertex has degree 9, yielding 2,304 edges total. These are proved finite-mathematics counts; the graph’s spectra and spectral gaps are finite combinatorial invariants, not physical light cones, diffusion constants, or expansion rates.
Scientific status
ASH is an exploratory finite-mathematics and computational-ontology framework (reference implementation v1.1.0). Its claims fall into three tiers: proved finite mathematics, specified deterministic computation, and interpretive research hypotheses. It is not an empirically validated theory of physics, cosmology, or consciousness — it does not derive a Friedmann equation, spacetime metric, dark energy, or the CMB, does not establish that its branching realizes quantum measurement, and has no confirmed observational predictions. The five Axioms of Existence are interpretive postulates, not established laws.