Hamming-Weighted Admissible Deposition Theorem
Year: 2026
Type: preprint
Abstract: TA32 identifies the structural object tested by the barrier comparator of TA30/TA31. The admitted helical phase is not merely passed through the Y-gate as a scalar phase: it is deposited as an admissible Hamming-weight occupancy profile on the 32-vertex Q5 transport graph. Any mismatch between the deposited profile and the inversion-compatible Hamming target produces a global closure defect, which is forced into the residual sector through the extraction block D. This identifies D as the Hamming-deposition defect extraction operator and provides the missing bridge from the TA31 block scaffold to the forced block computation of TA36. The argument proceeds in three layers: deposition → defect → residual forcing. The deposition map \( D_Y: H_M → C[V(Q5)] \) assigns to each admitted transport state an amplitude profile d_v on the 32 vertices. The total deposition defect \( ε_Y(ψ) = (Σ_v |d_v − d_v^{adm}|²)^{1/2} \) measures deviation from the inversion-compatible admissible target profile d^{adm}, determined jointly by Gray transport order, Hamming- weight compatibility, barrier inversion consistency, and Y-gate admissibility. The closure defect \( δ_C(ψ) = min_φ ‖U_C ψ − e^{iφ}ψ‖ \) measures global recursive closure consistency, with the minimization over φ factoring out trivial global phase. The main theorem establishes the defect bound \( δ_C(ψ) ≥ κ_C ε_Y(ψ) \) for a structural constant κ_C > 0, provided the admissible closure subspace is spectrally isolated - that is, U_C possesses a positive spectral gap away from the admissible eigenspace. The finiteness of Q5 guarantees a discrete spectrum; spectral isolation is the operative assumption yielding \( κ_C > 0 \) and is explicit. Consequences: if \( ε_Y(ψ) = 0 \) (Hamming-compatible deposition), the state remains in the coherent carrier sector with \( δ_C = 0. If ε_Y(ψ) > 0 \), recursive closure is defective and the non-admissible component is forced into the residual sector through \( D: H_M → H_R \). Three corollaries establish the identification of D as the defect extraction operator (with \( ‖Dψ_M‖ ≥ κ_C ε_Y(ψ_M)) \); the defect-free passage of fully admissible transport; and monotone residual forcing (larger deposition mismatch produces larger residual extraction). The explicit form of D from Q5 geometry is deferred to TA36. The admissible target profile \( d^{adm} \) is structurally characterized but not yet computed vertex-by-vertex; its explicit realization is a flagged future proof obligation.
Author: Holdway, Craig Edwin
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7162021884
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