The Tower page reads the primorial tower as a finished object: one residue window per prime, taken in the order of the primes — the Eratosthenes sieve in algebraic form. This page grows it instead. First with the designer in view: climb the rungs and abilities switch on, each birthday priced. Then with the designer removed: keep one structural demand and a blind rule — always the cheapest legal move — and let a ring grow itself. What follows is that story: three demands with three fates, what heat melts and what it cannot, what a grown world remembers of its own genesis and what it can be made to forget, what changes in other number worlds and at what price, and what kind of machine the whole process is.
Climb the rungs and abilities switch on. Channels only accumulate — a rung keeps every channel below it — so an ability's whole lifetime is readable from its logical shape (the fate rule, proved): an ability that needs some channel with a given property is immortal once born, and an ability that needs every channel to have it dies at the first failing prime and never returns — 2 and 3 are permanent vetoes. Zero divisors are born at rung 2. The square root of −1 dies at rung 2, while “an element that turns a quarter turn” — order 4, the rotation half of i — is born at rung 3, immortal: the tower keeps the turn and discards the square root. Error correction is born at rung 4.
The staircase has a top step. Of the abilities charted (computed per rung, every row's direction proved), every mortal one is dead by rung 6, and the last qualitative birth — more data than parity in the error-correcting split — lands at rung 7: Z/510,510, this site's demo rung. Past it, births are quantitative — bigger capacity, longer orbits — and each wait is priced by primes in arithmetic progressions: an order-m element is born when every prime-power part of m first divides the ring's period, so the birthday is set by least primes in progressions and Linnik's theorem prices the wait. The staircase describes the one trajectory where the primes arrive in increasing order — and that order was a choice. The rest of this story removes the chooser.
Strip the designer out. Keep one structural demand and one blind rule — always extend by the cheapest piece that meets the demand — and let the ring grow itself. No demand mentions primes. Three natural demands turn out to have three different fates.
Demand independence and you grow this tower. If every extension must share nothing with the ring so far — the new piece splits off as its own channel — the cheapest legal move is provably always the next missing prime: anything smaller is built from primes the ring already carries. Nobody demanded prime fields or a square-free modulus; they come free. From 1 the trajectory is exactly the primorial tower, and from any starting ring the process picks up precisely its missing primes, in increasing order (the healing rule, proved). The tower is the attractor of one sentence: grow by the least piece that shares nothing with what you are.
Demand novelty and you fall down one prime's well. If every move must instead create new dynamics — an orbit period the ring has never seen — growth abandons breadth and tunnels: it commits to a single prime and deepens that one window forever (from 1: the column 3, 9, 27, …). Which prime it commits to is decided at the first committing move — at most a seed-priced number of cheap “ghost” moves precede it, one more than the count of distinct odd primes in the seed's period — and the map from starting ring to destiny is decidable (proved). Every prime is somebody's destiny.
Demand safety and you die. If every move must add capacity without new dynamics — transparent growth: the ring gets bigger, the period stays frozen — growth halts (proved): only finitely many rings keep the period frozen, and the greedy dies at the largest of them, a wall fixed by the seed. The walls — 24, 240, 504, … — are old celebrities: topology knows the same integers as the image-of-J orders of stable homotopy (that reading is textbook). Exactly: the wall at frozen period L is the denominator of the Bernoulli quotient BL/2L — verified here from scratch for even L ≤ 12.
Two riders. Growth versus choice (the tie-or-beat proved, the overpayment computed): a designer who knows the target always ties or beats a grower whose every move must show progress — on the tested targets with several parts to buy, the impatient grower overpaid on 126 of 139, mean factor 4.2 — because the best move often buys several needs at once and looks like no progress to each one. Who signs what (proved): the demand alone, readable through the windows with no notion of size, fixes the destination — which windows eventually exist. “Cheapest,” the only place size enters, authors the route: the entry order, the schedule, the plateaus. Windows fix where growth ends; size picks the order it travels.
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.
The demand that commits — novelty, one prime's well — is the sharpest probe of what a number world is, so run it cold in the mirror worlds. Over polynomials it never commits (proved over the two-element field's polynomials, under any tie-break): fresh windows keep opening forever — from the empty seed, exactly one new window per degree, each shutting out its same-degree siblings for good, with deepenings squeezed in as the depth crosses each power of two — so depth and breadth, exclusive fates in the integers, co-occur. In the integers the demand always commits (proved, the well above). In both number rings tested it commits too, in both readings (computed, every seed in both censuses). The well is not a fact about greed; it is a fact about which world the greed runs in.
One law decides both sides, and prices the habit (proved; instances verified at the integers, at the class-number-3 ring of Q(√−23), and locally over fields of size 2, 3, and 9): the cost of a habit is the size of the room it writes into. A deepening writes new dynamics into the window's module of 1-units — the units reading 1 to the current depth — and everything follows from that room's rank. In a number ring the rank is finite — e·f, ramification times residue-field degree — so the period ticks linearly, one tick per e depths, at an eventually constant price of pef per tick: p to the rank of the room, a habit you can afford. Over polynomials the rank is infinite: the period ticks only as the depth crosses p, p², p³, … (at depth a it is p⌈logp a⌉), the gaps between ticks grow without bound, and no habit is affordable forever.
And affordability is commitment (proved, growing by ideals — growing by elements is the next section): a window deepened infinitely often at bounded cost absorbs the whole tail of the growth, two such windows cannot coexist, so a bounded-price habit and a lock are the same thing — at exactly the recurring price p to the rank.
A rider (computed): the greedy's decisiveness is a rational accident. In the integers, candidate moves are priced at powers of distinct primes — distinct integers, never a tie. Over the two-element field's polynomials, price sees only degree, and ties are everywhere. In Z[√−5] every tie in the census — 21 in the ideal world, 16 in the element world — is a pair of windows swapped by the field's own conjugation symmetry — same norm, same price — and the growth respects that symmetry move by move (verified): to proceed at all, the greedy must spontaneously break its world's symmetry, starting with its very first move from the empty seed. Only over the rationals does “cheapest” need no tie-break clause.
Open, and named: whether every trajectory in every number ring commits. The only escape is an infinite ladder of primes of the special form m·pv+1+1 landing inside narrow exponential windows at every stage — heuristically impossible, but past the reach of current results on primes in progressions. The integers' case is proved; the tested rings are censused wall to wall.
In a ring with class number above one, the commitment story runs once per reading, and the two readings disagree in geography. Read in ideals, both tested rings reproduce the integers' map (computed): from the empty seed each falls down the well of a norm-3 window at 3 per move — now as a broken tie between conjugates — and every censused seed locks some window at exactly the module-law price (the censused destinies: 2, 3, 7, 23 in Z[√−5]; 2, 3, 5, 13, 29 in Q(√−23), where locks onto 2 are common because 2 splits there).
Read in elements, the world pays its class group in price, and the basin map becomes a norm race among the vehicles a trajectory can afford. A window whose ideal is not principal cannot be deepened alone — no element does only that — so its cheapest vehicle is the free-rider door, the window bundled with a flat partner at the product norm: in Z[√−5], a norm-6 move deepens the 3-window while carrying the 2-window along one depth, and every such ride lifts the passenger until the 2-window wakes and absorbs the trajectory. There, that is the whole story (computed): every censused seed ends in the (2)-well at 4 per move.
It looked like a law of element worlds. The class-number-3 ring of Q(√−23) refuted it (computed): 38 censused seeds ride to the 2-well at 4 per move — but three lock the inert 5 at 25 per move, a pure principal column at the untaxed ideal price, because its very opening already carries period 24, swallowing the novelty of every cheaper move, and a trajectory that never rides a bundle never lifts the 2-passenger. The passenger lift is real but conditional: it forces the 2-tail on exactly the trajectories that ride.
The price a committed element tail pays is one comparison (computed, quadratic fields, all four censused tail species): its cheapest one-move vehicle is either the window's class-order power (norm pfm, m the window's order in the class group) or the plain rational prime (norm p² in a quadratic field), whichever is smaller — pmin(fm, 2) per move.
| world | tail | vehicle | per move |
|---|---|---|---|
| Z | the p-column | p itself | p |
| Z[√−5] | 2, ramified | the square is the bundle | 4 |
| Q(√−23) | 2, split | the rational 2 | 4 |
| Q(√−23) | 5, inert | 5 itself | 25 |
The tax over the ideal price exists only when fm > 2, and it is pure myopia: in Q(√−23) a cube of the 2-window sits at norm 8 forever — three depths per purchase, 2 per tick, the untaxed price — and the per-move greedy, paying 4 for one depth, never once picks it. Greed that prices moves cannot see prices per tick.
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 monopoly 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 cost law above. 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 diverges logarithmically (its gap law measured: slope −1.499 vs the predicted −1.496, R² = 0.99997). 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 cold 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.
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.