SpectralTuringBridgeHypothesis
plain-language theorem explainer
SpectralTuringBridgeHypothesis packages four quantified bounds that translate a positive spectral gap on the J-cost landscape into an Ω(n²) Turing-machine time lower bound for SAT. Researchers closing the P versus NP argument inside Recognition Science would cite the structure to supply the missing simulation-overhead step. It is introduced as a bare structure definition whose fields are later instantiated in conditional separation theorems.
Claim. A hypothesis structure asserting that for every natural number $n$ there exists a real gap with $0 < gap ≤ 1$ (spectral gap of the SAT J-cost Laplacian); the recognition convergence time satisfies $T_R ≤ n+1$; each global gradient step requires at least $n/2$ Turing-machine tape operations; and the resulting TM time lower bound satisfies $T_{TM} ≥ n · (n/2)$.
background
The CircuitLedger module models Boolean circuits as feed-forward sub-ledgers lacking global J-cost coupling on the Z³ lattice. The J-cost is the derived cost of a multiplicative recognizer comparator; UNSAT formulas induce a defect moat of width at least 1 because every assignment has J-cost ≥ 1. Upstream results include the local step operation on cellular-automaton tapes and the non-negative cost of recognition events from observer forcing.
proof idea
The declaration is a structure definition that introduces its four fields directly. No lemmas or tactics are applied; the body is empty and the structure functions as a hypothesis interface whose fields are supplied to downstream conditional theorems.
why it matters
It supplies the open gap identified in the module documentation for obtaining P ≠ NP from circuit separation. The structure is used by CircuitLedgerCert and by the conditional_separation theorem, which assumes the hypothesis to show TM time exceeds recognition steps for n ≥ 4. In the Recognition Science framework it links the phi-lattice eigenvalue gap (≈ 1 − O(1/n)) to the D = 3 spatial dimensions of the underlying lattice and the RCL composition law. The remaining open question is the Cheeger-inequality proof of the gap lower bound.
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