Thermal and density

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 — which seeds reach which limits — 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).

Scope. Proved; the runs stand as verification.

verifier: explore_hot_limit.py

The upward-closure law asks a demand for one thing only — the shape of its admissible set in the divisibility order — so the question is how many shapes there are and how much each buys. The three demands have exactly three, and the fate-power is graded. Dynamics is upward-closed and co-finite: its complement is finite and downward-closed, being exactly the divisors of W(λ(N))/N — the moves that leave λ where it stood — so dynamics and mortality partition the moves at every state, the reach-everything demand and the finite-death demand complementary. Upward closure forces the top by itself, which is the law above. Independence is support-avoiding, a move admissible iff coprime to the state, and that shape alone forces entry-once and surely finite depths — a window's depth its exponent: once p divides N no later move touches p, and entering at all is the weight's full support. Mortality is downward-closed and finite, and its shape alone forces nothing — the constant demand admitting only the move 2 is downward-closed, finite and forever nonempty, and grows the 2-adic column instead of dying. Absorption additionally uses mortality's budget structure: its admissible set is the divisor set of a cofactor that every move strictly consumes. Two fates are shape-authored, the third is shape plus budget, and every one of the three arguments runs on the divisibility order alone.

Scope. Proved, and tower-free. Stated for the weights m^−β, though the argument needs only w(pm) = c_pw(m) with c_p > 0 and w summable; wider weight families are untested. The closure and the partition are swept exhaustively over 2000 states × 199 moves.

verifier: explore_hot_limit.py

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

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 — that squarefree limit — at β = ∞ to the Haar shadow at β → 1: windows write the invariants, the softened deleted place writes a measure over itineraries.

Scope. The squarefree probability is exact, and it is the partition-function identity. The β → 1⁺ statement is coordinate-wise convergence conditioned on full support — the global full-support event is Haar-null.

verifier: explore_thermal_growth.py

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 is smoothness of p − 1 against the accumulated λ, which is the criterion itself; smoothness against p is the measurable proxy for it, and it is a weaker one than its class means suggest. Transparent primes do have mean P⁺(p−1)/p = 0.056 against 0.286 for non-transparent — but read as a classifier at the classical P⁺ ≤ √p boundary it scores 67.3% where calling every prime transparent already scores 64.8%, and 39.5% of transparent primes are rough by it. What does separate them is P⁺ ≤ (log p)², at 75.9% — a polylogarithmic threshold, not either power of p tested. 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 always non-transparent — for q > 2 no prime below p is ≡ 1 (mod q), so the factor q first appears at p itself, and p = 5 forces the 2² bump — 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 class means and the classifier scores are computed over k = 2..200 only, and are an observation at that range: they say how much of the criterion the p-relative proxy carries, and nothing about the limit. 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

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 at the milestones k = 50 … 10⁴, α drifts 0.30 → 0.165, monotone across those milestones only: rung by rung it is a sawtooth, rising only where λ moves — at non-transparent rungs — and falling at every transparent one.

Scope. The jump structure is verified k ≤ 10⁴, and its one-prime-power clause much further, for every p ≤ 10¹⁶ (the serial ledger); the domination is exact; the α drift is an observation.

verifier: explore_complexity_ledger.py

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

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

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, falling across the milestones; lim α = 0 via the density theorem, its rate constant the open residual.

verifiers: explore_collision_equivalence.py, explore_ledger_threshold.py