Constants

R(k) = θ(p_k)·[ρ − S(k)/θ] with S = Σ w, so the reserve thrives (R → +∞) iff ρ exceeds ρ_c(w) = lim Σ w(p_i)/θ(p_k) — the weight's Cesàro log-average, when it converges. Calibration weights land exactly: w = log p realizes ρ_c = 1.000000 — sinking at ρ = 0.9, thriving at 1.1 — and w = ½ log p realizes 0.500000. Every nonnegative value is therefore trivially reachable (w = c·log p gives ρ_c = c), and the realizer re-expresses a prime log-average, never derives it. The content is which structured weights land on which named constants — the two families below.

Scope. Elementary (the drift calculus); calibrations exact in range.

verifier: explore_reserve_zoo.py

Weight w_j(p) = log P_j(p−1), the j-th largest prime factor of p − 1 with multiplicity (P⁺ denotes P₁, the largest). The factors partition log(p−1) exactly — Σ_j log P_j(p−1) = log(p−1) at every prime — so the thresholds sum to 1 in the limit: Σ_j ρ_c^(j) = log φ/θ → 1, in range 0.999971. The spectrum is ordered, positive, decreasing: ρ_c^(1..3) = 0.5791, 0.1931, 0.0980, tail (ranks ≥ 4) 0.1298. In the limit, under Elliott–Halberstam, it is the Poisson–Dirichlet(1) / Shepp–Lloyd spectrum of the Dickman ordered factorization (0.6243, 0.2096, 0.0883, …); the identification is open unconditionally. The integer positive control — the same machinery on unshifted n ≤ x — gives [0.6523, 0.2006, 0.0789], inside the PD bands and above the shifted value at j = 1: the straddle. Finite range confirms the ordered spectrum and the conservation, not the limit values.

Scope. The partition and the per-k conservation are exact, Σ → 1 elementary; the in-range values are observations. The PD identification is a theorem under Elliott–Halberstam — level of distribution 1 gives the full joint PD(1) law (Bharadwaj–Rodgers, arXiv:2402.11884; the j = 1 Dickman case Pomerance's conjecture, EH-proof Granville–Wang) — and open unconditionally: shifted primes hold level ½ (Bombieri–Vinogradov), buying restricted-support PD correlations only.

verifiers: explore_reserve_zoo.py, explore_ledger_threshold.py

Weight w(p) = log p · 1[E(p)] for an event E of prime density δ realizes ρ_c = δ: the threshold is the density itself (the log-weighted density equals the natural one). E = “p ≡ 1 mod 4” gives 0.4987 → ½ (Dirichlet). E = “2 is a primitive root mod p” gives 0.3749 — Artin's constant 0.3740 under GRH (Hooley), an infinite Kummer intersection rather than a single Chebotarev class. A density constant is a different kind than the rank family's Dickman size-averages; the one lever w reads both kinds.

Scope. Threshold = density is a rule; the in-range values are observations. The Artin value is Hooley's theorem under GRH; the ½ anchor is Dirichlet's theorem.

verifier: explore_reserve_zoo.py

log φ counts every hit of every prime power in the shifted primes with multiplicity; log λ counts each hit power once. Their gap is the collision mass — prime powers dividing several shifted primes — and α = log λ/log φ = 1 − collision/log φ, exactly. In range the collision mass is 0.8349 of capacity (α = 0.1651), and nearly all of log λ is distinct large primes: the mass from primes above √p_k is 0.9749, while the Linnik-guaranteed range — prime powers up to p_k^1/5 — carries 4.5·10⁻⁴. Elementary bounds sandwich α only between the Linnik floor 7.5·10⁻⁵ and 1, and α → 0 is equivalent to the transparency-density conjecture (the collision equivalence, elementary), so forcing it amounts to a lower bound on how much large shifted-prime factors repeat — the shifted-prime distribution circle. The hinge: the count reserve (lcm, distinct) loses the collision mass — its threshold is 0 iff transparency density → 1, the open conjecture itself; the size reserve (product, multiplicity) keeps every occurrence — its threshold is positive and stable in range (λ_GD under EH). One distinction, both fates.

Scope. The identity is exact (three computations agree to 10⁻⁶, k ≤ 10⁴); the mass-locus is an observation; the equivalence is a theorem — it settles the question's shape, not its value.

verifiers: explore_ledger_threshold.py, explore_complexity_ledger.py, explore_collision_equivalence.py

A genuinely different solvency mechanism — the least s at which D(s) = Σ_p 1/P⁺(p−1)^s converges, the abscissa σ_c, rather than an average — realizes no constant, and the reason is an inversion. The Cesàro log-weight reads the bulk of the P⁺ distribution (0.7456 of its mass on the rough set P⁺ > √(p−1)); the reciprocal weight reads the smooth tail (0.0461 rough — the band P⁺ ≤ 11 alone carries 0.7638 of D(2)). That tail is carried by smooth shifted primes — P⁺ = 2 exactly the Fermat primes, frozen at 5 in range; P⁺ = 3 the Pierpont primes — and D_k(2) climbs 2.25 → 12.66 with a non-vanishing tail. Divergence is a theorem at every computed exponent: friable-shifted-prime counts force D(s) = +∞ for every s < 1/0.2844 ≈ 3.52, so σ_c > 3.51 and only the endpoint σ_c = +∞ stays open — where Pierpont infinitude is a sufficient certificate, not an equivalent. The integer control isolates the shift at that endpoint: unshifted smooth counts are unconditionally infinite (3-smooth 100 vs the shifted 31 in range), so the integer endpoint is provable while the shifted one waits.

Scope. The inversion and the climb are observations, k ≤ 10⁴. The fixed-s divergence is Lichtman's theorem (arXiv:2211.09641, Thm 1.1, won by extending the Bombieri–Friedlander–Iwaniec mean-value range; Baker–Harman 1998 before it); the endpoint is open.

verifier: explore_abscissa_reserve.py