The Growth page strips the designer out of the primorial tower: keep one structural demand and one blind rule — always the cheapest legal move that meets it — and a ring grows itself. Run cold, the three natural demands have three fates (proved there): demand independence — every new piece shares nothing with the ring so far — and growth builds exactly the primorial tower; demand novelty — every move must create dynamics the ring has never seen — and growth falls down a single prime's well forever; demand safety — capacity without new dynamics — and growth dies at a wall. This page turns the choice into chance and heats it. What the demand wrote survives at every temperature; what only the greed wrote melts — and the melting has laws of its own: what a grown world can ever know of its genesis, how a history is hidden on purpose, what the same knob does in the mirror worlds of polynomials and number rings, where forgetting snaps — and, at the end, what kind of machine the whole process is.
Soften the greed: pick among legal moves at random, cheaper more likely, with a temperature knob whose coldest setting is the greedy rule (the chance of a move of size m falls off as m−β, with β above 1; smaller β is hotter, β = ∞ is the greedy). What the demand wrote survives at every temperature (proved): safety still dies at exactly its wall; independence still gets every prime, with probability one. What only the greed wrote melts: the increasing itinerary becomes one order among many, and — growing from 1 — a perfectly square-free world becomes a matter of probability, exactly 1/ζ(β), with ζ the Riemann zeta function — each prime's depth comes out geometric with ratio p−β, independent across primes, square-free meaning every depth stopped at one. The zeta function is the heated tower's partition function.
The novelty demand melts most spectacularly. Its single column turns out to be a zero-temperature artifact: at every temperature short of the frozen greedy, the basins all merge and the growth opens every window and deepens each without bound, with probability one (proved), from any seed — heat carries the choosiest demand to the largest completion the construction has. The independence demand, by contrast, spans a one-parameter family under the same knob: perfect crystal at zero — this exact tower — and a nearly uniform cloud at the hot edge. The primorial tower is the ground state of its own thermal family.
Stand inside a grown world, holding only its present shape, and ask what it can know of its own genesis. The answer splits by fate. A world grown wide — every window once, arithmetic like ours — is amnesiac (proved): all the evidence it can ever assemble that it was grown cold rather than warm is capped forever at ζ(β)-to-1 odds — against a moderately warm origin, β = 2, the cap is ζ(2) = π2/6 ≈ 1.64 : 1 — and a majority of worlds grown at that warmth come out perfectly square-free anyway, handing their inhabitants a false “crystal” reading. Cold genesis is refutable, never confirmable. A world grown deep remembers (proved): every deepening adds a fixed increment of evidence, without bound — at that same moderate warmth, one deepening out-earns the wide world's whole eternity.
The memory was never really in the world. What a snapshot of the state loses is exactly the information carried by the history the state forgot (proved as an identity), and an observer who watches the moves as they happen accumulates temperature information linearly, move after move — the thermal laws are exponential families in β with sufficient statistic −log m, so each watched move carries Var(log m) of information, bounded below across the tested range: past a certain world size, one watched move tells more about the temperature than the entire infinite state (computed: at β = 2 the crossover sits between the 33,000th and the 100,000th window; at β = 3, from every tested size). A grown world's memory lives in whoever watched it grow.
An inhabitant with a hand can do more than watch. In a deep world, one injected move — raising the 2-window's depth to max(3, v2(λ)+2), where v2(λ) counts the 2's in the current period λ — re-locks the growth onto the 2-column (proved), and one composite move steers it to any chosen prime (verified on every tested target): the destiny map is not just decidable, it is programmable. In a wide world pushes cannot move the destination — healing absorbs them — but every push leaves a permanent scar, and scars forge (computed): even a single planted defect makes a cold-grown world read as warm.
And in a mortal world grown from the bare seed 1, the cheapest life support — inject the smallest legal piece, 2, at every death — grows exactly the Fermat primes: 3, 5, 17, 257, 65,537, each opening at its own revival while most revivals open nothing, and it opens another window if and only if another Fermat prime exists (an exact criterion); a different seed programs its own frozen window list (proved: the windows are exactly the primes p with p−1 dividing 2t·D, where D is the odd part of the seed's period and t counts the revivals). Whether the seed-1 phoenix ever opens a sixth window is one of the oldest open questions about primes.
Everything the inside view can and cannot know fits one exact chart (computed exactly at the tested truncation: twelve moves, ages up to four): read a grown world as bare state, dated state, or watched path. The two unknowns live apart — the demand is written in the state, the temperature in the clock, the timing of the moves. The chart is a ceiling and the exact Bayes posterior attains it (computed; a smoothed plug-in pays +200%), so grown worlds are induction benchmarks that know their own optimum: a compression-driven inducer — minimum description length, a language model — is scored against the best possible value, not other solvers. The test is lopsided: the whole grower is log2 3 bits forever; the unrecoverable history crosses it by age 3 and grows about a bit per move.
One row is a paradox (proved). The deep world's bare state keeps every bit about its law that the whole watched history carries, yet its past is maximally unreadable: every history that could have produced it, exactly equally likely. The wide world is the reverse, and the lock is an identity: what a snapshot loses about the law is precisely what the forgotten history carried. Memory for the law and memory for the past are different resources; a perfect fossil of the one is a perfect amnesiac of the other.
The paradox leaves a design question: the column hides its past perfectly but has little to hide; a wide world has exponentially many histories. Can it hide them all — every route exactly equally likely? At every temperature at once, no (proved for two-move history pairs and exchange-closed route sets): such uniformity forces parallel histories' menus to coincide as sets, and coprimality forbids exactly that while two routes remain — temperature-free hiding collapses to a single route. The escape belongs to depth: a constant menu ({2, 3} forever, old windows deepened again) hides exponentially many routes at every temperature; independence can never hold a menu constant.
At one chosen temperature, yes, by construction: “all routes equally likely” is a finite system of balance equations on menu masses, solved exactly by Egyptian-fraction menus of junk moves — built at two and three windows, every route exactly equiprobable at the design temperature (all 3! routes at three windows; the two-window build detunes to 117 : 70 at β = 2), the general case argued. A tuned world forgets its past without becoming a fossil — the paradox's two notions split. And at height hiding is free (computed): a plain wide world's only leak is its used primes' menu-mass footprint — a first-order model captures all but at most 0.04% of it from support {11, 13, 17, 19} up — while small-prime worlds stay readable by cooling. History-hiding is thermally tuned: impossible at every temperature at once, designable at one, free at height.
None of this is secretly about the integers. The fates stratify by bare structure (proved floor by floor): to die at a wall needs only order — any demand that confines growth to a finite region absorbs under every policy; to reach everything under heat needs only multiplication with cancellation; to be arithmetic — independent windows, the square-free crystal — needs unique factorization, and necessarily so: in the monoid generated by x2 and x3 — pieces multiply, factorization is not unique — the independence demand dies by its second move even though everything in that world remains reachable under the laxer demands. And the zeta in the crystal law is the zeta of the world: for polynomials over the two-element field the partition function is rational and the crystal probability is 1 − 21−β — exactly one half at β = 2. The primorial tower is what growth does on the ground floor of unique-factorization worlds.
One floor up sits the sharpest mirror: a number ring with a nontrivial class group — the textbook specimen is Z[√−5], where 6 = 2 · 3 = (1+√−5)(1−√−5) — runs both readings at once: its ideals factor uniquely while its elements do not. The ideal world's partition function is the Dedekind zeta — the family is three deep: Riemann for the integers, rational for polynomials, Dedekind for number rings — and the gap between the two worlds' partition functions is exactly the class-group L-function (the identity is classical genus theory; the growth reading is the point).
Grow that world by elements instead of ideals and the crystal is not rare but dead — the chance of a square-free outcome is exactly zero (proved in the finite world; the infinite limit argued) — and heating both readings makes their mass ratio converge to one over the class number (computed: to one-half in the two-class specimen): temperature reads the class number. Temperature, which above read a world's distance from its coldest self, here reads a world's distance from its own arithmetic.
Turn the same temperature knob over the two-element field's polynomials and the melt comes back with its constants in closed form (proved). The partition function — the Riemann zeta function for the integers — is here plainly rational: the 2d moves of degree d weigh in at rd total, with r = 21−β, and the whole sum is r/(1 − r). A perfectly square-free world is again a matter of probability — now exactly 1 − 21−β: one half at β = 2, where the integers answer 1/ζ(2) = 6/π² ≈ 0.608. The transcendental content of the integers' melt evaporates.
The cold polynomial monopoly — from the empty seed, one window opening per degree, its same-degree siblings shut out for good — melts the way the well did. At every temperature short of the frozen greedy, the growth opens every window and deepens each without bound, with probability one (proved) — the cold world's one-column geography is a zero-temperature artifact, exactly as the integers' single column was above. And temperature's gift trades places: over the integers, heat bought depth, the thing cold breadth never buys; over the polynomials, where cold growth already runs one column ever deeper while shutting its siblings out, heat buys breadth — the degree-one sibling starved forever in the cold opens in 96 to 98 of every 100 heated runs (computed). Heat even supplies the decisiveness the cold polynomial greedy lacked: equal-degree ties split democratically, the missing tie-break paid in randomness rather than in an imported notion of size.
What no temperature touches is the cold story's cost law — the cost of a habit is the size of the room it writes into (the Growth page's one law: a deepening writes into a module whose rank is finite in a number ring — a linear pump, one period tick per e depths — and infinite over polynomials). Watch a single column's deviation line — the chance, depth by depth, that the growth's next move leaves the column (computed at β = 2). The integers' 2-column holds a near-constant band, leave probability between 0.312 and 0.374 at every depth: the linear pump prices the next deepening at 2 forever. The polynomial x-column breathes: exactly one-half sticky at each power-of-two depth (a closed form in the deep-window limit; 0.500 to 0.501 measured), and molten everywhere between — leave probability 0.9989 and up measured one step past the depth-32 frontier. A finite-rank room prices its door at one constant number; an infinite-rank room's door recedes, and its price surges, at every power-of-two frontier. The cost law is visible in a thermometer.
Bound the alphabet and the split turns mortal. Allow only moves of degree at most D — a world that can spell only so much per move — and every heated polynomial trajectory halts (proved): each move at least doubles the universal period; fresh odd factors are rationed — the degree-d door contributes the Mersenne number 2d − 1 (degree one contributes nothing: 21 − 1 = 1, leaving D − 1 doors), opens at most once per degree, and opening a degree shuts every unopened degree dividing it — so a linear budget chases exponentially receding power-of-two frontiers and loses. The ledger is exact (proved): every legal move either raises the world's clock — the power of two in its universal period — or spends one of its D − 1 doors, so no trajectory outlives D − 1 moves plus the clock's total rise.
The death carries a clock reading. From a light seed it concentrates at ⌊log2 D⌋ + 2 — the bit-length of the alphabet bound, plus one — because a frontier gap of 2j is affordable exactly when 2j ≤ D (the modal reading is exact at all seven alphabet bounds tested, D = 2 to 12). Only that typical reading is temperature-proof. The tail above it is priced by the dyadic phase of D — how far D sits below the next power of two — and rises with heat: at D = 6 the mode is exceeded in 3.25% of runs at β = 3, 12.5% at β = 2, 18.75% at β = 1.25, the modal reading 4 at all three. Heat can even strand the clock below the mode, on a column too expensive to cross. And the integers, under the same bounded alphabet, keep an immortal witness (proved): along the pure 2-column the move 2 stays admissible forever — a linear pump never outruns a bounded purse. Mortality is not a fact about heat; it is the room's rank, read as a lifespan.
Here, how much of a genesis is beyond recovery from its endpoint is an exact quantity — the route probabilities are closed-form — making an old informal idea measurable: computational irreducibility, the thesis that some processes are their own shortest description. It gains an order parameter. Per move the unrecoverable entropy is analytic in the temperature knob (proved): a smooth crossover, no transition anywhere. What snaps is the geometry (computed): a deep world has a critical temperature — exactly where the smooth entropy peaks at one bit per move — where the origin's uncertainty reorganizes into the infinitely deep tail (its occupation jumps from zero to one) and the depth to which the origin stays uncertain climbs logarithmically (its gap law measured: slope −1.499 vs the predicted −1.496, R² = 0.99997; whether the climb is unbounded is itself a question about primes — below). A phase transition invisible to the thermodynamics — geometric, not energetic.
The critical point knows its carrier (the structural law, both fates verified): an interior critical temperature exists exactly when the fate's menu keeps nonzero weight at unbounded depth — the deep fate's does; the wide fate's clocks pile up only at β = 1, the pole of its partition function ζ. And every prime q carries its own clock (computed across the family): its column's critical temperature is rooted in exactly the primes it grows through transparently, and that set is q's residue-and-power data (2 for every q; 5 exactly when q ≡ 1 mod 4; a prime is uniquely blind to its own column). The clocks all land between 1 and 1.7287, the root of ζ(β) = 2; richer transparent sets pull the clock lower, and the top of the spectrum is the 2-column at 1.6045 (the maximum over the tested family): cancelling its own powers, it sees only the Fermat primes 3, 5, 17, 257, 65,537 — the poorest menu, the highest clock.
That census is an old one — the ways to measure a rational number are one residue window per prime plus size (Ostrowski's theorem), the census the tower is built from — and the clock spectrum is the census read thermodynamically: one critical temperature per window. The reading survives a change of field (computed; structural identities verified in range). Over the two-element field's polynomials — the mirror world above — the clock equation clears to a polynomial in 2−β (the degree-1 clock solves 2−β = 1 − 1/√2, exactly) and same-degree windows share a clock exactly. In Z[√−5] the clock splits by the class group — element clock 1.3527 below ideal clock 1.5492, the ideal/element split above read thermally — and near the pole the pair's ratio reads the class number. One law holds in every field tested: the highest clock sits at the window whose residue field has two elements — 2 for the integers, degree 1 for the polynomials, the prime above 2 for Z[√−5], its Fermat menu gated out.
Beneath the census sits one gear train (a rule, verified by exhaustive census across fourteen local fields and two power-series rings, 1.13 million checks): at every window, the level at which a unit's repeated p-th powers sit follows a single two-branch map — multiply the level by p, or add the window's ramification e, whichever is smaller — and only one seat, level e/(p−1), can cancel instead. A window ramified e ≥ 2 deep therefore breathes: past a finite transient, the levels where its universal period climbs recur with period exactly e.
And the breathing factors (rule in range, 4,440 checks). The critical temperature reads only the window's limit menu — which primes the field's splitting law admits into the column, and their depth caps — and is provably blind to everything the seat does along the way: two worlds sharing one tick chain can carry different temperatures (adjoin √2 or √−2 to the 2-adic window: same chain, clocks 1.486 vs 1.632, split by a mod-8 gate on which Fermat primes enter), and the two quadratic worlds ramified over the 3-adic window — adjoin √3, or a cube root of unity; twins by every coarse invariant — split 1.447 vs 1.334 by menu alone, a chain transplant moving neither. What the temperature drops, finer layers keep: the set of values the breath cycles through counts e, the phase of the breath reads the torsion, and only the finite-time transient remembers how each unit arrived.
Even that last memory is classical (rule in range; two censuses, 551 and 170,364 checks, worlds over p = 2, 3, 5): where an arriving unit lands reads how the p-power roots of unity sit over the window — the ramification of the cyclotomic tower, read dynamically. Over the 2-adic window the landing spells a full three-letter alphabet: across all six quadratic worlds ramified over it, exhaustively, the arrival lands at level 5, at 6, or at 7 and beyond, exactly as adjoining i ramifies, stays unramified, or is unnecessary — i already inside. At odd windows the tower's first step shows only two of the letters: wherever the seat is a whole level, adjoining a p-th root of unity is a tame twist, never ramified — a first step spelling the full alphabet is a privilege of the wild prime 2.
And the transient reads deeper than letters: it recites digits. Divide the window's root out of the prime 2 itself — e times, leaving a unit — and expand that unit digit by digit in powers of the root: where a deep arrival lands is a staircase that is a function of the leading digits alone (rule in range; three censuses, 164,327, 189,483 and 225,983 checks; the stop is the same whichever root is chosen, verified rung by rung with alternative roots). At a window ramified four deep the staircase has six rungs and reads the first five digits; at eight deep, twelve rungs and the first eleven — every rung realized by some world.
The count obeys one law: an arrival class watching an intermediate storey of the 2-power roots of unity reads 2m − 1 digits (the class of the 2m+1-th roots; a count intrinsic to the storey, the same at every censused window), while the window's own top storey reads 3e/2 − 1 — half again the window's depth, a bonus exclusive to depths that are powers of two: a depth-12 probe's odd-indexed top class got its unit freedom at the fourth digit already. And the law named its own test in advance: written down before any sixteen-deep world had been censused, it said the four classes there must read 1, 3, 7 and 23 digits — and the third census returned that row exactly: a staircase of twenty-four rungs reading the first twenty-three digits, every rung hit. The rungs the random sample missed were reached by reading the law backwards — sparse worlds designed to stop on a chosen rung, the readout as a construction kit. Before freedom takes over, the deepest arrivals recite the wild prime's own digits.
On a breathing window the reorganization above climbs a staircase (rule in range; the step ordering proved): the deep tail's occupation jumps not once but in e geometric steps, one temperature per breath phase, strictly ordered — in the eighth-roots-of-unity world, where the 2-adic window ramifies four deep, the four steps sit at 1.2295 < 1.2426 < 1.2775 < 1.4002 and the occupation climbs 0, 1/4, 1/2, 3/4, 1. Every step temperature is menu-only, blind like the parent clock — but which depths strike is not: between steps the struck depths form residue stripes mod e whose phases read the tick chain. The 3-adic twins' stripes are complementary, and transplanting one twin's chain onto the other flips the stripe while every step temperature holds to a part in a billion: the geography of forgetting reads the torsion that its temperatures provably cannot. Breathing also taxes the entropy: the per-move rate stays smooth through every step, but its peak sits strictly below one bit per move on every breathing window (short by half a percent at the √3 twin, up to 1.4% at the eighth roots) — perfect per-move irreducibility belongs to breathless windows alone.
And each step carries a correlation length — the depth to which the origin stays uncertain as the temperature closes on the step, the quantity that climbed logarithmically above — and it reads the window's transparent family (computed; the threshold walk exact in range). At the √3 twin, whose family is provably finite and fully entered at depth one, the onset is constant: a sharp jump, bounded unconditionally. At the 2-adic windows the onset walks the Fermat primes' entry thresholds one at a time (4, 6, 10, 18 over the integers) and saturates when the known family runs out — whether it stays bounded past 65,537 is the question whether the Fermat primes are finitely many. At the cube-roots twin the onset walks its own family 2·3i + 1 exactly, diverging if and only if that family is infinite. So the transition's sharpness is pinned, window by window, to the finiteness of a prime family — settled at √3, open at the others — and the standard counting heuristics put the two open cases on opposite sides: sharp at the 2-adic seat, long-ranged at the cube roots. The thermodynamics of forgetting, at these windows, is an open problem about primes.
Read the whole growth process as a machine. Its state is the depth vector — how deep each window runs — and it has two primitives: raise a window's depth by one (an increment), and test whether a window is still at zero (a zero-test), the check that decides whether a fresh channel may open. A general-purpose computer needs two ingredients (Minsky's counter machines): that zero-test, and a borrow — a move that lowers a count while reading its value on the way down. The growth owns the zero-test natively and is missing exactly the borrow: every move it has — grow, deepen, scar, revive — multiplies the modulus, so depths only ever rise and nothing subtracts (the no-decrement law, proved exhaustively). It can only add.
That one absence decides everything. A machine that can increment and zero-test but never borrow always has decidable halting: the depths only climb, each window's zero-test flips at most once, and the whole run is settled on a finite quotient — a decision procedure for questions whose naive simulation would never end. The three fates are its three long-run behaviors: converge to the crystal, lock to a single column, die at a wall. Put the borrow back and the machine turns universal, its halting undecidable (Minsky). So the growth sits one primitive below a general-purpose computer — and that primitive is one the tower deletes by construction. A borrow reads a count's magnitude as it descends, and magnitude — size, carry, overflow — is the single window the tower removes.
The edge is sharp on both sides. A forgetful borrow — one that wipes a whole window without reading it — is harmless: the machine stays finite-state (proved). Only a borrow that reads the value it destroys buys universality, and it pays undecidability for the privilege. The danger is exactly the reading, not the destruction.
What the tower keeps in exchange is a clean ledger. Its native arithmetic — carry-free addition, multiplication by a unit — is reversible, a bijection of the ring that erases nothing (verified). Every test it runs natively — unit, quadratic residue, error-correction syndrome — reads only the residue tuple, so its whole native vocabulary is one decidable class: the purely periodic tests with a squarefree period (proved). And the missing borrow pays a dividend: the tower has canonical forms by construction — the CRT tuple is an element's unique normal form — so equality of two tower computations is decidable, and a result carries its own check, the ECC syndrome, verified without redoing the work.
This is the opposite trade from the one a general-purpose computer makes, and it is exact. A universal machine buys the freedom to run anything and pays with undecidability — no algorithm decides whether two programs compute the same function (Rice's theorem). Give the borrow up and you buy the reverse: canonical forms, decidable equality, results that certify themselves. The tower keeps every residue window and deletes the one that carries size; read as computation, that deletion is what swaps the power to run anything for the power to check what ran.