FORGET — when a state provably no longer holds a datum

The object: a history space H with a positive weight π or a family π_β of them, β > 0; a datum X, any function of history to be forgotten; and a state map Φ, the still-working present. The datum's posterior at a state s is π conditioned on the fiber Φ⁻¹(s) and pushed to X. The question is when a state provably no longer holds X, what buys that at every weight, how the answer moves when X is coarsened, and what is left of it when the state must keep computing a function of history. A grown world gives the weights by THERMAL's thermal law with its admissible set replaced by a menu, the moves offered at each step, possibly depending on the route so far where MEMORY's depends on the state alone, each move m multiplying the state by m and drawn with probability m^−β / Ψ, Ψ the sum of n^−β over the moves n offered, a history being the route of moves from the state 1. There a state does two jobs, the integer a route reaches and a value of Φ, and Φ is the dated state, that integer with its move count, unless a section takes another. Its verifiers are forget.py and quarantine.py.

The thesis: in a grown world, hiding the route at every temperature from the dated state, or from any refinement of it, costs prime recycling (The quarantine theorem). Forgetting is graded fiber by fiber, from readable through spread and tuned to robust (The four grades). A structural witness, a factoring of the history with the datum's slot uniformly weighted or a symmetry of the normalizers, buys the robust grade, flat at every weight, but is not needed: a world recycling its primes can be robust for the route with none, while a world that never recycles (The quarantine theorem) is robust for the route on no dated fiber, the histories sharing a state and a move count, their age, written (state, age), holding two routes, at any age. No grade passes from a datum to its functions except readability: a state flat for the route can leak the first move. A state that must keep computing a function g can forget X across a g-fiber only by cutting that fiber into blocks each spread and flat, a cure (The working clause): two classes never cure by refinement, and at equal weights a cure exists iff no class holds a majority. Keeping less of g can leave its fibers with no cure.

The four grades

Tier: property; observation (the breadth world's 363 fibers).
Verifier: proof; forget.py::section_grades.

A fiber is readable when it meets one class of X, and the state then reads X. It is spread when it meets two or more, and then flat at a weight when the classes it meets carry equal mass there: flat, it hides X there, and spread but not flat, it leaks X. A spread fiber flat at some members of the family but not every one is tuned, and one flat at every member is robust. So a fiber has one of four grades: readable, spread and flat at no member, tuned, or robust. Flatness is uniformity behind what the state pins, not posterior = prior: a grown world's state is its present, which in general informs its past.

A world is given by its menus, a state with none listed offering no move. Spread does not bring flat. The breadth world, the squarefree moves 2 to 30 coprime to the state, holds each route's moves distinct and in every order at every multi-route fiber, the moves being pairwise coprime, and none of the 363 to age 3 is flat at β = 1. The tuned world, menu(1) = {2, 3}, menu(2) = {3, 5, 15}, menu(3) = {2, 10}, is flat for the route at fiber (6, 2) at β = 1, where Ψ(2) = Ψ(3) = 3/5, with normalizers that differ as functions of β, and its odds of 2,3 against 3,2 are 117/70 at β = 2. One world hides different parts of the route at β = 1 at different fibers: (6, 2) holds one move set in two orders, (30, 2) two sets, {2, 15} against {3, 10}, one order each, and both are flat for the route.

Two witnesses buy the robust grade

Tier: theorem.
Verifier: proof; forget.py::section_factoring, forget.py::section_grades.

Factoring. Let H = 𝒳 × R, π_β(x, r) = μ(x) ν_β(r) and Φ(x, r) = h(r). Then the posterior of X, the 𝒳 slot, is μ at every state and every β, so every fiber is robust when μ is uniform on two or more values. Conversely, posterior = prior under every product weight holds only when Φ ignores the 𝒳 slot. Normalizer symmetry. On a dated fiber of a grown world holding two routes or more, if the multisets of the normalizers each route passes before its endpoint agree, as functions of β, the fiber is robust for the route.

Proof. The fiber of s is 𝒳 × h⁻¹(s), on which X is distributed as μ. Conversely, on a fiber F with sections F_x = {r : (x, r) ∈ F} the posterior is proportional to μ(x) ν(F_x), so it is μ for every μ iff ν(F_x) is the same for every x; a ν with generic values separates distinct subsets, so the sections coincide and F = 𝒳 × F_x. For the symmetry, the route-weight cancellation (MEMORY: The route-weight cancellation) makes the route posterior proportional to the product of 1/Ψ over the states it passes before its endpoint, and equal multisets have equal products.

The depth column, the constant menu {2, 3}, has one normalizer, so every dated fiber holding two routes is robust for the route.

The quarantine theorem

Tier: theorem.
Verifier: proof; quarantine.py::section_arms, quarantine.py::section_last.

A grown world recycles when some menu a route meets offers a move sharing a prime with the integer it is offered at, and is coprime otherwise: every menu a route meets is coprime to that integer. In a coprime world no two distinct routes of a dated fiber carry equal posteriors at every β, at any age. So no dated fiber holding two routes is robust for the route, and no state map refining (state, age) cuts a dated fiber holding two routes into blocks each spread and flat for the route at every β. The menus may depend on the whole route so far, not only on the state.

Proof. A route's posterior is proportional to 1/D, D the product of the normalizers it passes (MEMORY: The route-weight cancellation); each Ψ is a Dirichlet polynomial with coefficient 1 at each menu element. Two routes tie when they carry equal posteriors at every β, that is, the n^−β being linearly independent functions of β, when their D agree as formal Dirichlet polynomials, and the tie is witnessed when their normalizer multisets agree as well. Induct on the age, for two routes from any shared start state; a route of age 1 is its endpoint over the start. Cancel the shared factor of the start state, the formal Dirichlet polynomials being a domain. The coefficients are nonnegative, so the support of the rest is the set of products of one element per later menu, and both routes' later menus use one set of primes S. Every later state is divisible by the first move, and coprime menus avoid its primes, so each route's first move is prime to S, while every later move is drawn from a later menu and lies over S. The first move is then the part of the endpoint over the start prime to S, the same for both routes: the routes share their first state and its menu, whatever the menu reads, and the induction runs on from there.

Coprimality cannot be weakened to the last move. A world whose menus, indexed here by route, avoid only the last move's primes escapes: menu() = {2, 3}, menu(2) = {3, 5}, menu(3) = {2, 5}, menu(2, 3) = {2, 5}, menu(3, 2) = {3, 5} holds the fiber (30, 3) at the routes 2,3,5 and 3,2,5 with equal normalizer multisets. The scan agrees in every arm: no tie in 141,216 fibers of age 2 and 3 over all 87,885 coprime tables of state menus on the moves 2 to 6 with every allowed menu nonempty, fibers counted table by table, or in 3,183 fibers of random coprime worlds with state or route menus to age 4, while the same scan finds ties once coprimality is weakened or dropped. Since flatness passes in neither direction under coarsening (Readability descends, spread ascends, flatness neither), the theorem says nothing of a coarser datum. Measured only, no coprime fiber of the scan with at most eight routes and two first moves or more hides its first move at every β, 0 of 120,386; whether a proof exists is open.

So two routes carry equal posteriors three ways. Normalizer symmetry and a witness-free design (the rogue world, Recycled worlds need no witness) hide the route at every β and both recycle, the depth column repeating its moves; tuning hides it at finitely many β, the two routes' D differing by a nonzero finite sum of exponentials, and is the only way open to a coprime world, the tuned world being one.

Recycled worlds need no witness

Tier: property.
Verifier: proof; quarantine.py::section_rogue, quarantine.py::section_coarse.

The rogue world, menu(1) = {2, 32}, menu(2) = {2, 4, 8}, menu(16) = {2, 16}, menu(32) = {2, 4}, menu(128) = {2, 8, 32}, holds the dated fiber (256, 3) at exactly 2,8,16 and 32,4,2, robust for the route with normalizer multisets that differ. Three is the least age this takes, a fiber of age 1 holding one route. The coarse world, every move a power of 2, holds a witness-free tie beside a symmetric one and so buys a cure (The working clause) flat at every β, where the dated state, the state with its age, is flat at one β only (a strict cure at every β but that one, A strict cure needs three classes): menu(1) = {2, 4, 8, 16}, menu(2) = menu(4) = {4, 8}, menu(8) = {8}, menu(16) = {8, 16}, menu(64) = {2, 4}, menu(128) = {2} hold (256, 3) at P = 2,8,16, Q = 4,4,16, A = 8,8,4 and B = 16,8,2, the dated state leaks the route at every β but log₂ of the golden ratio (1 + √5)/2, and the blocks {P, Q} and {A, B} are spread and flat at every β.

Proof. With x = 2^−β the rogue products are (x + x² + x³)(x + x⁴) and (x + x²)(x + x³ + x⁵), both x² + x³ + ⋯ + x⁷: x² times the two pairings of (1 + x)(1 + x + x²)(1 − x + x²), whose last factor has a negative coefficient and so stands in no menu alone. At age 2 each route passes one state after the start, so a tie is one Ψ equal to another, one menu, the witness. In the coarse world Ψ(2) = Ψ(4) ties P and Q, x³(x + x²) = (x³ + x⁴)x ties A and B, and P's product is A's times x(1 + x), which is 1 only at x = (√5 − 1)/2. Which ties a recycling world can hold at all is COLLIDE: A witness-free tie is a regrouping or passes a seed.

Readability descends, spread ascends, flatness neither

Tier: theorem.
Verifier: proof; forget.py::section_coarsening.

Let f be a function of X's values. A fiber readable for X is readable for f(X), and a fiber spread for f(X) is spread for X. Flatness passes in neither direction. In the depth column at the dated fiber (2^a 3^b, a + b), a, b ≥ 1, the first move is 2 with posterior a/(a + b) at every β: the count leak a/(a + b) − ½, zero only at a = b, in a fiber robust for the route that meets both classes of the first move, so leaks it without reading it.

Proof. A fiber meeting one class of X meets one of f(X), and one meeting two classes of f(X) meets two of X. Down: a flat posterior pushed through f gives each class of f(X) mass in proportion to the number of X-classes it holds. Up: a class of a flat f(X) may split unevenly, as X's classes of masses ⅓, ⅙ and ½ do when f merges the first two. In the depth column the routes of the fiber are the C(a + b, a) arrangements, equally weighted, and C(a + b − 1, a − 1) of them open with 2.

So a certificate for a datum does not certify its functions, and each coarsening owes its own flatness. The exact readout of LEARNER: Exact readout reads the subdominant is not this leak in another form: it reads which classes a component meets and never how many, so it follows readability downward and never sees a count.

The working clause

Tier: property.
Verifier: proof; forget.py::section_working.

A state that must retain a function g of history computes it iff Φ refines g, so the admissible state maps are the interval from g to the discrete partition, and a choice inside one g-fiber constrains no other. A g-fiber is cured at a weight when it can be cut into blocks each spread and flat there, and its price is the fewest blocks. A g-fiber meeting one class of X is readable under every admissible Φ. The two-class law and, at equal weights, the no-majority criterion below read the classes' masses and counts and nothing else: no menu, prime or temperature. They are facts about a finite weighted partition lattice, and a grown world enters only by supplying the weights; under unequal weights a cure reads the histories' masses, not only the classes'.

Tier: theorem.
Verifier: proof; forget.py::section_working.

A g-fiber meeting exactly two classes of X, of masses w₁ and w₂, is cured iff w₁ = w₂, that is, iff Φ = g is already flat there. In the depth column with the dated state retained, the first move leaks at every dated fiber (2^a 3^b, a + b) with a, b ≥ 1 and a ≠ b, and no state design stops it.

Proof. Every spread block holds both classes and a flat one holds them at equal mass; summing over the blocks, w₁ = w₂. The converse is the one-block cut.

The no-majority criterion

Tier: criterion.
Verifier: proof; forget.py::section_working.

On a g-fiber of N equally weighted histories whose classes have counts c₁ ≥ c₂ ≥ … ≥ c_ℓ, a cure exists iff c₁ ≤ N − c₁. Under unequal weights a cure needs the largest class mass to be at most half the fiber's, and that is not enough: exact cuts are subset-mass matching, and the fibers passing the mass test with no cure are the matching gap.

Proof. A block takes t ≥ 0 from class 1 and, being flat and spread, at least t from the others; summing gives necessity, by count or by mass. At even N, list the histories by class and pair the i-th with the (i + N/2)-th: no class fills more than half the list, so every pair holds two classes. At odd N two classes would force c₁ = c₂ and N even, so ℓ ≥ 3; a triple from the three largest classes leaves N − 3 histories whose largest count, c₁ − 1 or c₄, is at most (N − 3)/2 (c₄ ≤ N/4 ≤ (N − 3)/2 once N ≥ 7, c₄ ≤ 1 at N = 5, and none is left at N = 3), and the even case finishes.

The search agrees at every count vector of 2 ≤ N ≤ 8, and on 400 drawn weighted fibers 47 of the 86 passing the mass test have no cure.

Retaining less can leave no cure

Tier: theorem.
Verifier: proof; forget.py::section_retention.

In the depth column, retain the age alone. The first move is then 2 with posterior 3^β/(2^β + 3^β) > 1/2 at every β > 0, a strict majority, so no cut of the age fiber cures it, while retaining the dated state leaves every dated fiber (2^a 3^b, a + b) with a = b ≥ 1 flat at every β. Every state map admissible before stays admissible, the dated state among them; what is lost is the cure, graded on the coarser g-fiber.

Proof. The first move is drawn once from the constant menu and nothing later depends on it, so on the age fiber its posterior is the first step's, 2^−β/(2^−β + 3^−β). A strict mass majority fails the necessity of the no-majority criterion in every partition. On a dated fiber the numerators cancel, which is what the age alone does not do.

A strict cure needs three classes

Tier: theorem.
Verifier: proof; forget.py::section_retention.

A cure the coarsest state map Φ = g lacks, a strict cure, needs a g-fiber meeting three classes of X or more, and one exists. In the depth column, retain the dated state and, on the dated fiber (24, 4), forget the first two moves: the routes 3222, 2322, 2232 and 2223 give counts (2, 1, 1) and Φ = g gives the classes (½, ¼, ¼), while the blocks {3222, 2232} and {2322, 2223} are spread and flat at every β, price 2.

Proof. One class is readable, and two classes cure only when already flat. In the example the dated fiber's routes are equally weighted, and each block holds one route of the doubled class and one singleton.

Partition design alone buys a robust certificate Φ = g lacks.

Open fronts

Whether some coprime fiber hides its first move at every β, which asks for an identity between sums of route weights rather than a collision of products; where the quarantine's boundary lies past the one escape stated, for menus that avoid the primes of some earlier moves and not all; and whether the recycled side holds a witness-free fiber of three or more routes flat at every β.