Adinkra Error Correction
Doubly-even codes provide structural stability and guaranteed single-bit error correction (via an explicit radius-one decoder) across the hypercube
Error Correction Bound
A linear code with minimum distance d corrects all error patterns of weight t when:
If codeword c receives error e (weight t) giving r = c ⊕ e, then c is the unique nearest codeword to r. Nearest-codeword decoding succeeds.
For the ASH canonical [9,4,4] transform code the minimum distance is d = 4, so the guaranteed correction radius is t = ⌊(d−1)/2⌋ = 1. Single-bit errors are corrected uniquely; two-bit errors are detected and rejected, never silently corrected. Exhaustive decoding of all 512 states of F29 gives 16 exact codewords, 144 single-bit corrections, and 352 uncorrectable states.
Decoder Verification
Error correction is a property of an explicit radius-1 nearest-codeword decoder, not of the simulations: when invoked it corrects every single-bit corruption (144 across all 512 states) and rejects every two-bit corruption (576 rejected, never silently healed). The Markov simulations apply noise and code XORs but never decode.
For known affine-orbit anchors all single-bit recoveries also succeed (256 × 16 × 9 = 36,864 corrections). Guaranteed correction remains radius-1 only; multi-bit corruption is rejected by policy, not repaired.
Adinkra Graph Origins
Adinkra graphs (Faux & Gates 2005) encode supersymmetry multiplet relationships. The doubly-even property emerges from the algebraic requirement that SUSY transformations square to translations, constraining codeword weights to multiples of 4.
In ASH this connection is realized as a finite, verified construction rather than a physics claim: the self-dual doubly-even [8,4,4] code (the [9,4,4] transform code punctured at the code-invariant coordinate 8) and its 16-vertex Q8/C8 Adinkra quotient with exact N=8 Garden matrices (64 identities, integer residual 0). It is a representation-theoretic reference layer, not a claim of observed physical supersymmetry.
Scientific status: This page describes interpretive and computational ASH content, not empirically established physics. ASH (v1.1.0) proves specific finite mathematics and specifies deterministic computations; it is not a validated theory of cosmology or consciousness, and the Axioms of Existence are interpretive postulates, not laws.