The inside view

The breadth world is the amnesiac fate: its total possible evidence for cold genesis against temperature β is capped at ζ(β) : 1 odds forever (1.64 : 1 at β = 2) — one deep column refutes the crystal outright, so cold genesis is falsifiable but never confirmable — and a majority of β = 2 worlds come out squarefree anyway, handing their observers a false crystal reading. The depth world remembers: its cold-genesis log-odds grow linearly in depth (0.69 nats per deepening at β = 2), so one deepening out-earns the breadth world's capped eternity. The split is a property of states, not of watching — the dated state being a function of the path, I(G; path) = I(G; dated) + I(G; path | dated) exactly (G the generator: law and temperature), so what a snapshot loses about its generator is precisely what the forgotten route carried. The fossil paradox closes the ledger: the column keeps every generator bit while its route posterior sits at the uniform maximum of route entropy, and necessarily so — zero menu drift forces the route posterior uniform, which proves the fossil and maximizes the amnesia at once. Memory for the law and memory for the past are different resources. The chart is a Bayes ceiling and the exact posterior attains it (a smoothed plug-in pays +200%), so grown worlds are induction benchmarks that know their own optimum, scoring a compression-driven inducer against the literal Bayes value; the test is lopsided — the whole generator is log₂ 3 bits forever while the unrecoverable route entropy crosses it by age 3 and grows ≈ 1 bit per move.

Scope. The bounded-evidence law, the unbounded-memory law, the MI identity, and the fossil paradox are proved; the ceiling chart is truncated-menu exact (twelve moves, ages ≤ 4), with the mnemonic crossover measured — young breadth worlds out-inform young columns, depth wins from age 3.

verifiers: explore_observer_view.py, explore_induction_ceiling.py

Can a many-window demand — every admissible move coprime to the state — hide exponentially many histories? A dated endpoint's route posterior is uniform iff the interior-normalizer product is route-invariant (the balance criterion), so the design target is normalizer balance, and the answer is a trichotomy. At every temperature, impossible (the quarantine theorem, at every age and with no exchange hypothesis): in any coprime demand the leading move is quarantined from every tail menu, the normalizer product's support pins both routes to one prime set, and the valuations then force route identity — so β-free perfect amnesia forces a single route. The escape is PRIME RECYCLING, and that is the exact boundary: a world free to reuse its own primes can hold a two-route fibre flat at every β with menus that differ (the rogue world), and coprimality is precisely what forbids the reuse — the constant menu ({2, 3} at every state, old windows re-ground) is the extreme case, hiding exponentially many routes at every β, which is the depth column's own privilege and the fossil paradox's other face. At the world's own temperature, a construction (the tuned amnesiac): the balance criterion is a finite system of mass equations, solved exactly by Egyptian-fraction junk menus — at k = 2 both two-route endpoints are exactly uniform at β₀ = 1, detuning to 117:70 odds at β = 2; at k = 3 all 3! routes of one endpoint are exactly equiprobable — and the tuned world keeps nonzero menu drift and a positive witness gap while its routes sit uniform at β₀: it forgets its past without becoming a fossil (the fossil split). And plain breadth hides nearly for free at height (the footprint law): the posterior's only leak is the used primes' menu-mass footprint, the singleton-footprint model capturing all but ≤ 0.04% of it from support {11, 13, 17, 19} up. History-hiding is thermally tuned: a grown world can be built unreadable exactly at the temperature it was grown.

Scope. The balance criterion is immediate; the quarantine theorem is proved at every age for coprime demands, subsuming the age-2 and exchange-closed scope of the earlier exchange-rigidity argument; the amnesiacs and the rogue world are exact constructions (k = 2, 3; general k argued); the footprint law is measured across tested supports and temperatures.

verifiers: explore_one_way.py, explore_rogue_world.py

How much of a genesis is beyond recovery from its endpoint is an exact quantity here — computational irreducibility with closed-form route posteriors — and it has a critical point the thermodynamics cannot see. The per-move route-entropy rate is analytic in β, a smooth crossover peaking at log 2 at the critical clock β_col (the root of the depth column's thermal normalizer Z_col(β) = 1; ≈ 1.496 for the 3-column). What reorganizes there is the genesis mode: the unbounded-depth tail's occupation jumps 0 → 1, and the depth to which the origin stays uncertain — a correlation length — climbs as Θ(ln 1/(β−β_col)), walking the column's transparent prime family one entry at a time (the clock gap is governed by the next unincluded family prime: measured slope −1.499 against −β_col = −1.496, R² = 0.99997), so it diverges exactly iff that family is infinite. The structural law: an interior critical point exists iff the fate's normalizer keeps a nonzero unbounded-depth limit — depth's does; breadth's clocks pile up only at the ζ pole. Every finite place carries its own clock: β_col at lock prime q is rooted in that column's transparent set, which is q's residue-and-power data (2 transparent for every q, 5 iff q ≡ 1 mod 4, a prime uniquely blind to its own column), so the finite places of Q map into the window (1, β*), β* = 1.7287 the root of ζ(β) = 2, richer transparent sets pulling the clock lower. The top is the 2-adic seat: the q = 2 clock, 1.6045, cancels its own powers and sees only the Fermat primes — the poorest menu, the highest clock — so the divergence question at the seat, its transparent family being the Fermat primes, is the Fermat-prime question. The reading survives a change of field: over F₂[x] the clock equation clears to a polynomial in 2^−β (the degree-1 clock solves 2^−β = 1 − 1/√2 exactly; same-degree places share a clock), over Z[√−5] the clock splits by the class-group characters (the 3-column's element clock 1.3527 below its ideal 1.5492, the pair's ratio reading h near the pole), and in every tested field the highest clock sits at the place whose residue field has two elements — Ostrowski's census of the places, read thermodynamically.

Scope. The rate's analyticity and the structural law are exact statements about the closed-form normalizers; the spectrum values, orderings, and the seat law are computed across the tested families; the correlation length's divergence is pinned, column by column, to the finiteness of the column's transparent prime family (the walk is measured at the 3-column; the per-column pinning rides the mode staircase's step ordering) — settled where the family is provably finite, open at the 2-adic seat and the 3-column.

verifiers: explore_irreducibility_order.py, explore_irreducibility_places.py, explore_irreducibility_crossfield.py

Over Z, minimal life support — inject the least move, 2, at each death — freezes the genome D = odd(λ(seed)), the odd part of the seed's period, and opens exactly the windows {p : p − 1 | 2^t·D} at frozen depths (t counts the revivals). From seed 1 those windows are the Fermat primes, opened at their own exponents (3, 5, 17, 257, 65537 at λ = 2, 4, 16, 256, 65536), and whether the phoenix ever opens a sixth window is the Fermat-prime question: the cheapest escape from death grows the Fermat primes. Above the minimum the push writes the genome — windows {p : odd(p−1) | D} at depths v_p(D)+1, the exponents compounding with the state rather than with the push, since the realized genome reads the accumulated product and is blind to the order the pushes arrived in — and opening a chosen prime has a three-branch price list: fiat, depth, or cover (push the least prime q* with odd(p−1) dividing odd(q*−1)). Cover routinely undercuts fiat — 97 opens via a push of 7 at 2.8 bits against 6.6, and across primes < 50000 a third cost strictly less. The floor is the exactness obstruction: an exact genome edit (add l^e with no odd side effect) needs a Proth prime l^e·2^a + 1, whose price is unbounded — and nonexistent for a Sierpiński l^e. Over F₂[x], where a bounded alphabet makes every thermal world mortal (the mortality split), life support buys exactly one native move per revival and the bill doubles: bill_k+1 = 2·bill_k + D (2, 10, 26, 58, 122 at D = 6) — the module law's diverging cost read as a price the operator pays, against Z's rank-1 column at bill 0.

Scope. The frozen-DNA and spectrum laws and the bill doubling are rules, the seed-1 window list exact — the criterion is the spectrum law read at D = 1; the generalized spectrum and the price-list census are rules in range (primes < 50000); exact-edit feasibility is the Proth/Sierpiński question, open. The rudder is proved, the steering instantiated at 16/16 targets.

verifiers: explore_interactive_observer.py, explore_phoenix_bill.py