The Silver Ratio in Prime Number Structure: A Layer System Approach (revised and corrected edition)

Abstract

This version REPLACES the December 2025 edition of the same record. The Layer System and the prime-composite bijection stand and are strengthened; the headline constant of the first edition, 2+√2, is retracted. We revisit the Layer System, a representation in which every prime P is decomposed as P = V + S, where the void V is a deterministic origin and S the accumulated layer sum. We establish the exact arithmetic identity that closes the system, V(x) = π(x) + π(V(x)) − 2, verified with zero violations for all 7 ≤ x < 5×10^6, together with its consequences: V is the fixed point of v ↦ π(x) + π(v) − 2 (mean 5.74 iterations), the remaining columns of the founding table are pure bookkeeping, and V is injective on the primes, so that "the configuration of N" and "N" carry the same information. We then retract the central claim of the first edition. The effective constant K(P) = (P/S − 1)·ln P·ln ln P does not converge to 2+√2; it diverges, with K(P)/ln ln P → 1 (measured 1.0013 at P = 10^1000). The value 3.41421 is not a limit: K has a shallow minimum of 3.4398 near P ≈ 1.8×10^5, three quarters of one percent above 2+√2, and rises thereafter — the original fit was performed exactly at that turning point, where a function is flat to first order and a lowest-variance selection criterion is bound to read a constant. The correct law is Λ(x) := x/V(x) = ln x − 2 + o(1), equivalently (P/S − 1)·ln P → 1; the constant 2 is real and decomposes structurally (one unit from the Prime Number Theorem, one from the self-reference of the void), while the √2 is an artifact. Three numerical rows of the first edition are also corrected, and its published construction algorithm is off by one index. The programme's genuine positive contribution is a rigorous no-go theorem. Writing the silver coordinate log_δ x = K(x) + φ(x) with δ = 1+√2, we prove that approximating the factor phase φ(p) of an RSA modulus N = pq to precision N^(−1/4)/ln δ is polynomial-time equivalent to factoring N, and we generalize this to any admissible lens: the barrier is a property of the SCALE of the coordinate, not of the silver ratio, and not of the multiplicative-to-additive homomorphism, which is used only in a sub-determination corollary. The forward reduction is closed at cryptographic scale on test semiprimes up to 1024 bits, and the finite-dimension threshold is reported with its Howgrave-Graham constant √d, measured at +1.58 bits = ½·log2(9). Finally we summarize what the layer structure of the primes turned out to be — a covering problem whose short-range statistics are entirely divisibility — record a cautionary methodological result on percolation estimators (a commensurability resonance that invalidated a number of our own, constancy rejected at 27.3σ), and give the state of the inverse question, where six independent instruments return the same ceiling N^(1/4) (mean 0.2442, amplitude 0.0233) and no algorithm beating the state of the art was obtained. Every claim is verified computationally under a fixed gate protocol, and the verification scripts are released. The paper contains five explicit retractions, two of them of our own later results, found while preparing this revision.