Pure Time Theory - Chapter VII - Deterministic 3SAT decision in polynomial time via structural amortization and multi-modular closure

We present a minimal deterministic framework for deciding 3SAT, based on: (i) a global order T (stratification of choices); (ii) a local perceived time tau defined by d(tau) = dt / g(rho), with an information density rho satisfying structural properties (P1–P5) and an increasing slowing function g; and (iii) a weighted discrete potential Phi* that measures only reducible heads (unit clauses, binaries handled via Rbeta–BinClose, triplet motifs) and linear defects defect_p for p in {2,3,5}. The multi-mod scheme relies on deterministic syntactic detection of linear gadgets mod p and Gaussian elimination mod p over extracted subsystems.

We establish two structural lemmas: (L1) deterministic fairness in tau, and (L2) amortization of Phi* (each valid operation ensures Delta(Phi*) per unit of normalized time is ≤ −1 with g(rho) = (1 + rho)^alpha, alpha in (1/2, 1]). We further use a Combined Lyapunov Lambda := Phi* + epsilon * M to make the episode bound scheduler-independent. A key lemma, No-False-Silence under T, formalizes that every active stratum carries either a local operation or an interface flux triggering one at the next step; the only allowed “silence” is a certified one (Phi_k = 0 and J_k = 0).

Results:
(T1) Unconditional SAT completeness: the dynamics reaches a window where the local readout stabilizes, then Rbeta–BinClose + unit propagation completes the assignment (witness verifiable in O(m)).
(T2) UNSAT ⇒ unconditional certificate: Boolean conflict or 0 = 1 mod p for at least one p in {2,3,5}, with a verifiable Gaussian trace.
(T3) Unconditional polynomial bound: with Phi*_init ≤ poly(n, m) and the minimal decrease above, the number of episodes is poly(n, m); each episode has polynomial cost (at most O(n^3) for a linear block).

The framework remains strictly discrete and deterministic (no tuning) with three global constants fixed a priori: alpha in g(rho) = (1 + rho)^alpha, the set P = {2,3,5}, and a lexicographic tie-break order.

After R7.5 (unique external injection and constant threshold), the local cell is canonically “ripple + fold-back”; this yields a purely structural reading of q (no state simulation). We establish the bijective local classification (the only admissible pattern under span ≤ 1 and external degree 1 is ripple), complete normalization of linear constraints (XOR/counters/equalities), and the SPC bifurcation (Small-Prime Completeness): non-saturated ⇒ multi-mod conflict {2,3,5}; saturated ⇒ controlled fall into the scalar regime and empty clause. The bit-complexity satisfies log q = O(|F|) and the cost of R8_bin(q) is O(K log q). It follows that 3SAT ∈ P, hence P = NP.

All per-step cost bounds (UP, Rbeta–BinClose, LinClose^+(p), R8_bin(q), detection) are polynomial and summarized in the complexity statements of the paper.