Forgetting

A certificate for X at Φ is a witness at one of four grades, increasing in strength. R, readable: X is a function of Φ, and there is no certificate. P, possibilistic: every fiber meets at least two X-classes, so X is not recoverable as a value; the strongest witness for it is a state-side factoring, a presented split of the history space into X and a remainder with Φ constant in the X slot, which needs no weight at all. T, tuned-flat: at one named temperature the posterior of X is uniform on every fiber's consistent values. S, robust-flat: uniform at every weight in the family. Flatness is fiber-uniformity — maximum entropy behind what the living state logically pins, and never posterior = prior, since a grower's state is its present and so always informs its past possibilistically. It upgrades to posterior = prior for every prior exactly where the split carries the WEIGHTS with it, which is the inheritance route below.

Grade S is bought two ways, and the two forgetting specimens here sit there by different routes — one lemma apart, not one construction apart. Inheritance: where the history space splits as X against a remainder, with X's prior weight-free and Φ reading only the remainder, the posterior of X equals its prior at every state and every weight, with no tuning. Destination universality is grade S by that route: the adapted reader's state is a function of its metabolism and never of its data, which is the factoring. Weight-side symmetry: where the multiset of interior normalizers agrees along every route of a fiber, the route posterior is uniform at every β — route weight is proportional to the product of the reciprocals of those normalizers, and equal multisets have equal products. The depth column is grade S by that route, every normalizer in it being the same polynomial. A third lemma collapses the obvious alternative to a witness: a quotient through which Φ factors, each of whose classes meets every X-class, exists if and only if every Φ-fiber meets every X-class. So “factors through a quotient killing X” is extensionally the FULL-SPREAD case of the possibilistic grade — every fiber meeting every class rather than merely two — and a split witness adds provenance rather than extension.

Scope. The three lemmas proved, scope-free. The grading is an observation exhaustive at the census scope — 406 fibers over the three worlds read to ages 2, 5 and 3 across β ∈ {1, 2, 3}, exact in rationals throughout — and the four grades land whole, with zero fibers they cannot express. The tables read R 3 and T 2 in the tuned world, R 10 and S 10 in the depth column, R 18 and P 363 in breadth. The destination-universality row enters by inheritance together with the factoring measured on the reader side, which holds under every loss that reads only what the reader has committed to and fails outside that family.

verifier: explore_forgetting_certificate.py

A certificate carries a structural clause — is there a witness? — and a measure clause — is the posterior flat? Neither implies the other, and both crossings print. Structural kill without flatness: at all 363 of breadth's multi-route fibers the state factors through the quotient that remembers which moves were made and not the order they came in, so the ORDER is possibilistically dead at every one of them — and the posterior still reads that order through the normalizers at β = 1. Flatness without any structural witness: the tuned world's endpoint 6 at age 2 is flat at β = 1 with the witness absent, which is the signature of tuning — it buys one temperature and no more, and the census separates tuned from structural flatness by inspection. What a world forgets is also fiber-local: that same tuned world hides the ORDER at endpoint 6 and hides the move SET at endpoint 30 ({2, 15} against {3, 10}, each admitting exactly one order).

Scope. Exhaustive at the census scope above. The crossing in the remaining direction — robust flatness with no witness — occurs at no fiber here, and whether it can occur at all is the obstruction the coprimality dichotomy settles.

verifier: explore_forgetting_certificate.py

A certificate for X does not descend to functions of X. Take the depth column, which is route-uniform at every temperature with the weight-side witness present — perfect symmetry, the strongest grade there is. The posterior of the FIRST MOVE at the endpoint 2^a3^b is nonetheless exactly (a/(a+b), b/(a+b)) — independent of β, and a leak. Route-uniformity does not coarsen to feature-uniformity, because a uniform posterior counted over unequal coarse classes is not uniform. Two leak channels now stand named: the WEIGHT leak, where the measure itself is asymmetric, and the COUNT leak, where only the geometry of the fiber is. Every coarsening needs its own flatness clause.

Scope. The a/(a+b) law proved for the two-move constant menu — equal route weights with binomial fiber counting — and verified at every mixed fiber to age 5; other menus were not scanned. The naming of the two channels is an observation.

verifier: explore_forgetting_certificate.py

Two-class conservation. On a g-fiber carrying exactly two X-classes, of masses m₁ and m₂, every spread block holds both classes and flatness forces their two masses equal inside each block; summing over blocks gives m₁ = m₂. So a two-class fiber is curable only where Φ = g is already flat: refinement never cures a two-class leak. The count leak is therefore not merely undescended but UNREMOVABLE while the dated endpoint is retained — deleting a record's attribute while keeping the record runs into a parity-style obstruction that no state design crosses.

The numerator effect. Retention is not monotone the way admissibility is. Conditioning on a dated endpoint cancels the move weights in the numerator, the moves' product being the endpoint itself; conditioning on the AGE alone surfaces them. In the depth column the posterior of first move 2 at an age fiber is exactly 3^β/(2^β + 3^β) — 3/5 at β = 1 — a strict majority at every temperature, and a mass majority is partition-free, so every age fiber leaks whatever state map is built over it. A coarser retained function admits MORE state maps and certifies FEWER: retaining less exposes more.

Scope. Two-class conservation proved. The numerator effect's mechanism is proved and its instances measured at scope: in the depth column with the dated endpoint retained and X the first move, every fiber whose two exponents differ leaks (8 of 20), the equal-exponent ones cure at price 1 and the pure powers are readable, while coarsening to the age alone converts route-uniform fibers into majority-readable ones — the same reversal holding in the tuned world. Where the majority test passes and the individual weight ties a cure would need do exist, no cure assembles at ages 2 or 3 either, so the effect is not a majority artifact.

verifier: explore_working_amnesiac.py

On a uniform-weight g-fiber whose X-class counts are c₁ ≥ ⋯ ≥ c_r totalling N, a spread flat partition exists if and only if c₁ ≤ N − c₁ — if and only if no class holds a strict majority. Necessity survives the weighted case as a MASS-majority test, which is partition-free and therefore runs at any fiber size, however far past exhaustive search. Sufficiency is exact subset-mass matching and is NOT claimed weighted, and the distance between the two is where the real price sits: with unequal weights, all 118 searchable non-majority breadth fibers fail to cure. In uniform-weight fibers necessity IS sufficiency; the gap between them is exactly the exact-subset-sum structure of the weight family.

Scope. Proved at uniform-weight scope, and cross-validated against exhaustive partition search — capped at 8 routes per fiber — on all 49 uniform fibers inside that cap, with zero mismatches. The matching-gap count is an observation at the census scope above.

verifier: explore_working_amnesiac.py

What the three laws price is curability; what they leave open is whether refinement ever STRICTLY beats the coarsest working state. It does, rarely, and cheaply. In the depth column with the dated endpoint retained and X the first TWO moves, five fibers cure STRICTLY: the coarsest working state Φ = g leaks, and a refinement of it is spread and flat at every temperature. The specimen is class counts (2, 1, 1) with baseline posterior (½, ¼, ¼), cured at price 2 by pairing the two singletons. Partition design alone buys a robust certificate the coarse state does not have. But it is confined to worlds carrying exact weight ties, and plain breadth is a TIE DESERT: across all 363 of its multi-route fibers the interior-normalizer product is INJECTIVE — not one pair of routes carries equal products at β = 1, so no pair can tie across all temperatures at once, and no working state hides the route anywhere in it. For X the first move, of the 291 fibers inside the search cap 173 die by mass majority and 118 pass necessity but admit no exact-tie partition, and none cures. Partition design never manufactures the exact ties working amnesia needs; a symmetry or a tuning has to supply them.

Scope. Exhaustive at the census scope above, exact in rationals, the partition search capped at 8 routes per fiber. The 72 breadth fibers above that cap carry no all-temperature mass majority — the partition-free test runs at any size — so they stay genuinely open rather than decidably leaking. The tie desert is a scope fact and not a theorem: no argument yet says that breadth normalizer products can never tie.

verifier: explore_working_amnesiac.py

Read a menu as a Dirichlet polynomial, the sum of m^−β over its members. A route's weight is then set by a product of such polynomials, and two routes collide when two products agree identically in β. Collisions are generic rather than exotic: 4849 colliding pairs, falling into 4487 classes by the product they share. Measured directly, one scaling family accounts for 4369 of them: writing Z_A for the polynomial of a menu A, and cA for A with every member multiplied by c, they are the instances of Z_A·Z_cB = Z_cA·Z_B. Of the 480 non-scaling pairs, 6 involve a single prime, where every menu's polynomial is a polynomial in one variable and factorization already fails to be unique — a factor carrying a negative coefficient cannot itself be a menu, so the remaining factors regroup around it two ways — and 474 involve at least two primes, such as {2}, {3, 6} against {3}, {2, 4}, both products 6^−β + 12^−β. Non-unique factorization is the generic state of this semiring, not a cyclotomic curiosity. What limits the door is therefore not the algebra but REALIZATION — whether menus carrying those polynomials assemble into an actual world — and realization's teeth are exactly one degeneracy: the unrealized collisions coincide precisely with the all-singleton ones, 44 of 44, where the four singleton menus of such a pair force equal move products, hence equal leading moves, hence a conflict at the first state. Everything with a non-singleton menu realizes, 4805 of 4849.

Scope. Exhaustive at the collision census's own scope — menus drawn from {2..16} ∪ {32} of size at most 3, with realization seeds up to 8. The singleton direction is proved, over all orders. The same scan cross-checks the coprime half of the dichotomy from outside its proof: zero coprime realizations across all 4849 collisions, against 4805 realizations for worlds free to recycle.

verifier: explore_rogue_world.py