Thermal and density

The greedy argmin softened into a Gibbs selection — and the asymptotic underneath it that temperature never touches.

A growth law is a structural demand plus a greedy move: extend the modulus N by the least m ≥ 2 meeting the demand. Each demand realizes one fate — breadth (independence) seats every prime, depth (λ must grow) collapses onto one prime's column, mortality (capacity with λ frozen) halts. Soften every law's argmin into a thermal selection — pick admissible m with probability proportional to m^−β, β > 1; the greedy is β = ∞ — and the blueprint's properties grade by how much of the selection each needs. Mortality's wall needs none: every admissible trajectory absorbs at exactly W(λ(seed)), a deterministic bound, at every temperature. Breadth's destination needs only full support: healing is almost sure at every β. What only the argmin enforced — the route, squarefree-ness, the lock — melts.

The hot-limit theorem rule

The dynamics demand's admissible set is upward-closed in divisibility — λ never falls, so admissibility is inherited by multiples — hence every prime rides the multiples of every admissible move, P(p | pick) ≥ p^−β uniformly in the state, and hot dynamics-greed reaches Ẑ itself almost surely: every window opens and deepens without bound, for every seed and every β > 1. The single column, the lock, and the whole basin geography are zero-temperature artifacts; the discontinuity sits at β = ∞ exactly. The proof needs only the closure: any thermal growth on the divisibility order with upward-closed nonempty admissible sets reaches Ẑ almost surely (the upward-closure law, tower-free).

verifier: explore_hot_limit.py

The zeta measure rule

The hot-breadth limit is a random supernatural number with independent geometric depths (ratio p^−β) — squarefree, i.e. the exact blueprint, with probability

\prod_p \bigl(1 - p^{-\beta}\bigr) \;=\; \frac{1}{\zeta(\beta)},

so the Riemann zeta function is the thermal tower's partition function. At the pole β → 1⁺ the depth profile converges to Haar measure on Ẑ conditioned coordinate-wise on full support. The primorial tower is the ground state of a one-parameter Gibbs family running from the crystal (β = ∞) to the Haar shadow (β → 1): windows write the invariants, the softened deleted place writes a measure over itineraries.

verifier: explore_thermal_growth.py

The two hot limits are separated by the Jacobson radical: ∏_p F_p = Ẑ/J(Ẑ), so cold independence grows the semisimple quotient (from a squarefree seed — a general seed's excess powers persist) and hot dynamics the full completion from any seed — temperature's gift to the dynamics demand is exactly the radical. Over F₂[x], where cold growth already runs one column's radical, the gift is breadth instead (the function-field melt).

The density theorem: transparency → 1

The tower alternates between complexity rungs, where λ jumps, and capacity rungs, where a transparent prime adds space on the same dynamical skeleton. In the limit the balance tips entirely to capacity — long the program's central open problem, now a theorem; what remains open is the rate's constant.

Transparency density approaches 1 theorem

The fraction of transparent primes among the first k approaches 1 — equivalently, log lcm{p−1 : p ≤ x} = o(x). The proof is elementary and unconditional. Repeated prime powers cost O(√x); primes q ≤ x/log x enter the lcm once each, at most θ(x/log x) ~ x/log x in total; and a prime q > x/log x divides some p − 1 only with p − 1 = mq, m < log x — a prime pair, and the sieve upper bound (Halberstam–Richert) caps such p at O(x log log x/(log x)²), each contributing less than log x. Total: O(x log log x/log x) = o(x). The mechanism, measured: a prime is transparent when p − 1 is smooth against the accumulated λ (transparent primes have mean P⁺(p−1)/p = 0.056 vs 0.286 for non-transparent); the fraction rises 47% at k = 50, 65% at k = 200, 74% at k = 1000, 80% at k = 10⁴ — log-speed in range, not a power law — and the rate ratio L(x) log x/(x log log x) sits flat at 0.76–0.79 to x = 10⁷, so the proved upper bound looks tight in range. The matching lower bound — sharpness, and the rate's constant — stays open and is Elliott–Halberstam-hard: its mass sits at q = x^1−o(1), needing primes in progressions to moduli near x. Safe primes (p = 2q+1, q prime) are reliably non-transparent and conjectured infinite, so non-transparent primes are expected never to stop appearing — only to become rarer, which is all the theorem forces.

Scope. The vanishing (density → 1) is proved — elementary sieve tools, general in x, no computation at specific k. The trend is computed k = 2..10⁴ (to 200, to 2000, and to 10⁴ — the three density scripts in order), the rate charted to x = 10⁷; the lower bound and the constant are open (EH-hard).

verifiers: explore_lcm_shifted_primes.py, explore_shifted_prime_density.py, explore_asymptotic_density.py, explore_density_extended.py, explore_complexity_ledger.py

The complexity ledger rule

λ grows only at non-transparent steps, and every λ-jump raises exactly one prime power — a new-prime jump forces P⁺(p−1) itself, 100% in range — so log λ(p_k#) is the accumulated log-size of the prime powers the rough shifted primes force into λ. Each raise is smaller than log p_k, so with N_nt(k) the number of non-transparent primes among the first k, log λ(p_k#) < N_nt(k) log p_k: transparency-density → 1 forces the complexity ratio α = log λ / log φ → 0. Measured, α drifts monotonically 0.30 (k = 50) → 0.165 (k = 10⁴).

Scope. The jump structure is verified k ≤ 10⁴; the domination is exact; the α drift is an observation.

verifier: explore_complexity_ledger.py

The collision equivalence theorem

With N(q^a) = #{p ≤ p_k : q^a | p−1}, exactly:

\begin{aligned} \log\varphi &= \sum_{q^a} N(q^a)\log q, \\ \log\lambda &= \sum_{q^a} \bigl[N(q^a) \ge 1\bigr]\log q, \end{aligned}

so α = 1 − collision/log φ, the collision mass being the gap Σ (N−1)⁺ log q — prime powers dividing several shifted primes. The equivalence: N_nt(k)/k − α(k) → 0, at rate O(log log x/log x) with x = p_k — so α → 0 iff transparency density → 1, with no extra hypothesis, and limsup α = limsup N_nt/k. Forward is the ledger's domination; the converse is the write-once bound

N_{\mathrm{nt}} \;\le\; \sum_{q \le y} \Bigl\lfloor \tfrac{\log x}{\log q} \Bigr\rfloor \;+\; \frac{\log\lambda}{\log y}

for any y strictly between 1 and x: a prime power is raised at most once ever, so raises by primes q ≤ y are rationed by the prime-counting function π(y) and every other raise carries ≥ log y; y = x/log³ x gives the rate. Equivalently, density → 1 states: log lcm{p−1 : p ≤ x} = o(x) — proved (the density theorem above), so lim α = 0 unconditionally. What stays open is the rate's constant: the sharpness direction needs the shifted-prime distribution circle (Elliott–Halberstam).

Scope. The identity is exact at every k; the equivalence's proof is elementary and general (Chebyshev, Mertens, partial summation — no computation at specific k); the in-range gap 0.22 → 0.034, monotone; lim α = 0 via the density theorem, its rate constant the open residual.

verifiers: explore_collision_equivalence.py, explore_ledger_threshold.py

One statement, read twice: the transparency criterion is the self-growing tower's window-opening rule verbatim — a shifted prime dividing an accumulator — so density → 1 is also the breadth trajectory's asymptotic absorption rate. What a DESIGNED economy makes of the same shifted primes — reserves whose solvency thresholds realize the analytic-prime-constant zoo, with the density above sitting at the degenerate end of that zoo — is Constants.