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, the squarefree limit, 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, the state with its age, 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 (the menus never changing along the run) 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 mutual-information 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

Amnesia survives the removal of temperature. Freeze the breadth world at zero temperature, where the law is deterministic and the route is the increasing one, and the state still fails to determine its own past. The pairs (seed, age) — the state a world started from and the number of moves since — whose greedy trajectory passes through N are exactly the subsets of D(N) = {p : p divides N exactly once, and every prime below p divides N}: the subset is the set of primes the run picked, the seed is N divided by their product, and every such pair really is a run. So a state supports 2^|D(N)| pasts — N = 210 has 16 — at the one temperature where the law itself leaves nothing to chance. The ambiguity is over where the run began and not over the order it went in — seed and age fixed, the increasing route is the only one, which is why the count is of starting points. What buys uniqueness is the genesis convention, seed 1, and nothing the world holds: initial condition and history are separable from inside only by decree.

Scope. Proved, and checked exhaustively at every state N ≤ 2000. The count is of zero-temperature breadth preimages; at β < ∞ every ordered coprime factorization is a candidate history and the posterior above is what sorts them.

verifier: explore_observer_view.py

Can a demand offering many moves at every state — 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): 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 both routes to a two-route endpoint equiprobable 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, its primes reused forever) 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 — the ledger identity's conditional term read on the temperature, I(β; path | dated) > 0, the route still carrying temperature information the snapshot lost — 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; 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

Every move splits into what it spends of the percept and what it opens:

A modulus divides its own wall and the wall never falls, so the wall's gain at each prime covers whatever the move fails to pay out of V — which is what makes the opening factor G′, the premium, an integer, and what makes it a law that a move can never destroy more headroom than it spends. The burn is gcd(m, V): truncated subtraction on the exponents, the primitive lossy counter machines are built from, and the opening runs that same primitive upward on λ's exponents plus one term the spending side has no counterpart for — the windows that were not there before, which is the whole reason V can rise. It rises exactly when G′ > gcd(m, V); a move that also burns transparency comes out behind (N = 3 under m = 16 sends V from 8 to 5).

Neither half is a property of the move alone. The wall gain is a ratio of endpoint values, so it telescopes along a split m = ab where the unpaid part does not — and since V(Nab) is one state either way, an inflated opening half arrives with an equally inflated spending half, so the gross flows are path-dependent and only their quotient is not. The correction relating a split's chained premium to the direct one is a unit fraction every time, which makes the premium submultiplicative along every factorisation: a split never undercounts what the move opened. Spending and opening are properties of the move together with the granularity it is watched at, and the wall gain is the only one of the three that is a state function of the endpoints. What the move opens closes in the same style: the state enters through λ and one coordinate of the percept per prime of the move, and through nothing else — but the move's primes interact, since a window opens on a divisibility condition reading the whole of λ. At λ = 1 the move 35 opens the window 13, which neither 5 nor 7 opens alone: 13 − 1 = 4·3 draws its 4 from λ(5) and its 3 from λ(7). So a move's premium is not assembled from its factors'.

The wall map itself is a closure operator: N ↦ W(λ(N)) is monotone, inflationary and idempotent, so a transparency episode is a walk inside the interval [N, W(λ(N))] — the divisor lattice of V — and its fixed points are V = 1, the closed states, those with no transparent move left. Those are the mortal tombstones, and above 2 every one of them is divisible by 24, for an elementary reason with no Bernoulli numbers in it: λ is even from 3 up, so the wall carries 2^v₂+2 ≥ 8 and (3−1) | λ admits the 3. The state 2 escapes by the property that distinguishes it — the only closed state with an odd λ.

Scope. The ledger and the premium's closed form are proved prime by prime for all N and m, and confirmed over 23084 moves (N < 400, m < 60), 13134 of them with more than one prime in the move. The closure operator is proved outright and checked on its own battery — idempotence and inflation below 3000, monotonicity below 400 against every divisor. Submultiplicativity is proved prime by prime over 52536 splits and is exactly trivial on 97.0% of them. The exponent identity is a statement about the ODD windows, which are the machine's counters — the prime 2 carries its own formula — and it is exact at an open window, parting company at a shut one where the counter is unseated.

verifiers: explore_headroom.py, explore_premium.py, explore_composite_move.py

Two states with the same headroom are one state to this probe, and a transparent move keeps them so with certainty — the wall gain is 1 exactly when the multiplier divides V. Blindness is entirely a cross-λ phenomenon (λ and V determine N, so the probe is injective on a λ-fibre), and what shatters it is the premium: d tells a blind pair apart iff the two wall gains differ, for every d, with no coprimality hypothesis. Since every such d — every resolver — is a non-divisor of V,

the least non-divisor of V — always a prime power, since a least non-divisor p^am with m > 1 coprime to p has both parts strictly smaller, so both divide V, so their coprime product does too. That bound is a function of the reading alone. A probe that cannot tell which state it is in computes a sharp lower bound on its own cheapest way of finding out, knowing neither state: V = 24 gives 5, and the pair 10, 11 resolves at exactly 5. It is usually the whole answer — 90 of the 96 pairs of the standard pool (the pairs below 1500 with multipliers under 60) attain it — and the residue is exactly what the reading cannot supply, namely which blind partner is on the other side: 34 of the 52 classes with three or more members carry more than one least resolver across their pairs.

The floor's prime-power shape does not transfer to the price itself: the pair 2¹¹, 2¹² — blind at V = 65535 — survives every multiplier up to 27 and splits first at 28 = 2²·7, where 12289 = 3·2¹²+1 — a door of the deeper state's λ, a prime p with p−1 dividing it, which is exactly what its wall carries — opens at the deeper state only. On adjacent powers of two the whole question is primality at exact depths: a move resolves exactly when a prime u·2^k+1 (a Proth prime; Fermat when u = 1) sits at the deeper state's depth with u dividing the odd part of the move's λ, so the least resolver is composite exactly when the cheapest such prime needs both a deepened seat and an odd route — a move 2^x·m with x ≥ 1 and m > 1 — which happens twice in the family's eleven blind pairs. And the threshold is unbounded: at every even L the blind-spot theorem's lift to every even λ puts infinitely many primes at the reading W(L), so every such reading hosts blind pairs, while along L = lcm(1, …, n) every prime up to n+1 is a door of L and each door's own term in the floor exceeds n times that door, so floor(W(L)) exceeds n+1. What is measured is the price against the floor: every member of the class of reading W(L₀) has λ divisible by an explicit base, the base-multiple enumeration reaches λ in the billions, and readings designed to carry floors 47 and 81 both host blind pairs — resolving at exactly their floor, on states of up to 453 digits.

No pair is blind forever. Given a blind pair, a prime q ≡ 1 mod lcm(λ₁, λ₂) drives both states to λ = q−1, making the walls agree while the two readings cannot — so the resolving set is never empty, on Dirichlet and nothing stronger. That buys certainty, not economy: the constructed q runs to 111827 over the pool while every pair is in fact resolved by some multiplier under 60. Which is why the harder position is the watcher's, who takes the move it is given and cannot go looking. A watcher wins on m iff some divisor of m resolves, so every blind pair survives at least τ(V) − 1 moves with certainty — one per divisor of the reading above 1, τ the divisor count — and a chooser never picks one. Choosing when to read and choosing what to play are two faculties the word observer bundles into one.

One blind spot is provably infinite and needs no conjecture: V = 1 is the class of closed states, W(L) is closed for every L, and W(2^k) ≥ 2^k+2 is unbounded. Every wall reads exactly like every other wall, and there are infinitely many walls.

Scope. The resolution criterion, the floor, its prime-power form, the non-existence of permanent blindness and the threshold's unboundedness are proved — the last on that lift, so on Bombieri–Vinogradov and nothing conjectural; the pool figures are measured over the pairs below 1500 with multipliers under 60, nothing censored under a search to 200; the witness pair, the Proth-window table and the deep enumeration are exact finite checks, and that a designed reading's pairs resolve at exactly the floor is measured through floor 81. The least resolver is not always a prime power (the 2¹¹, 2¹² witness), so the prime-power shape belongs to the reading's floor, never to the price. The floor is not a benchmark ceiling and does not follow from one: a ceiling is an expectation over an ensemble, computed by a designer standing outside the ignorance being priced, where this is a per-instance bound computed by the ignorant party from a strictly poorer percept.

verifiers: explore_resolver_threshold.py, explore_proth_window.py, explore_silent_set.py, explore_composite_move.py

Acting is fate-graded opposite to memory: where depth sells the hand a destination and mortality sells it time, the breadth world — the amnesiac fate — sells inscription. Healing absorbs every push: the demand goes on seating new primes and the destination is push-proof. But multiplication never removes, so every scar — the extra depth a push writes into the state — is permanent (200 of 200 thermal continuations keep an injected 2²). In the amnesiac fate the only fossils are therefore artifacts. And artifacts forge: a crystal dressed in scars with geometrically distributed depths is state-indistinguishable from thermal genesis by construction, and a single 2-scar — one squared prime on an otherwise squarefree state — moves the crystal's temperature reading from infinity to β̂ = 1.88. False memories are implantable. The deniability split sorts pushes by who can call them: a gcd-scar, a push sharing a prime with the state, is a probability-zero move under the thermal law — signed, certainly artificial to any watcher of the path — yet the state it leaves is thermally reachable, so it is deniable in the state; a coprime scar, a push the law itself could have made, is a positive-probability thermal event, deniable even to the watcher. States can always be forged; only watched paths can be signed, and only gcd-scars sign.

Scope. Permanence is proved (the healing rule's dark half) and held over 200 thermal continuations; the forgery readings and the single-scar estimate are measured at β = 2 through the maximum-likelihood temperature read from the state alone; the deniability split's two specimens are exact routes.

verifier: explore_interactive_observer.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 (push p itself), depth (push l^e+1 for each prime power l^e exactly dividing odd(p−1)), 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-genome and spectrum laws are proved at every genome — each death lands on the wall of a λ whose odd part a push of 2 never moves, so the windows are the wall's own doors at its own exponents — the seed-1 window list exact, the criterion the spectrum law read at D = 1; the bill doubling is a rule and the price-list census a rule in range (primes < 50000); exact-edit feasibility is the Proth/Sierpiński question, open. The steering law is proved, and instantiated at 16/16 targets.

verifiers: explore_interactive_observer.py, explore_phoenix_bill.py