Explanation of weight_polynomial_decay_summable
(1) In plain English, the theorem states that if prime weights λ_p satisfy a polynomial decay bound |λ_p| ≤ C / p^(1+ε) for some constant C > 0 and exponent ε ≥ 0, then the infinite sum of λ_p² over all primes is finite (i.e., the sequence is square-summable).
(2) In Recognition Science this matters because the cost operator T_J is built from these weights in the NumberTheory layer; square-summability is a structural precondition for the operator to be well-defined as a spectral object on the recognition Hilbert space, feeding into the regularity hypotheses that support discrete spectrum and trace-class heat kernels.
(3) The formal statement reads: for a function lamP : Nat.Primes → ℝ and real ε, given the assumptions (hε : 0 ≤ ε) and (h : WeightDecayPolynomial lamP ε), the conclusion is WeightSquareSummable lamP. WeightDecayPolynomial asserts existence of C > 0 such that ∀ p, |lamP p| ≤ C / (p.val : ℝ)^(1+ε); WeightSquareSummable asserts Summable (λ p, (lamP p)^2).
(4) Visible dependencies in the supplied source are the definitions WeightDecayPolynomial and WeightSquareSummable in the same module, plus the bundling theorem cost_operator_regularity_certificate that lists this implication among the module's structural facts. The proof is complete (no sorry).
(5) The declaration does not prove the three regularity sub-conjectures themselves; EssentialSelfAdjointness, CompactResolvent and TraceClassHeatKernel remain hypothesis structures whose analytic content is placeholder True. It likewise does not establish any specific RS weight values or downstream physical predictions.