Learning

With no budget in force the deficit is a perfect loss over the committing policies: monotone strict descent converges from every one of them to the exact global optimum — greedy in both kinds, and route-free — and the sole trap these rows carry is total refusal, which sits on a flat plateau of infinite loss where every single-coordinate neighbour prices infinite too (designed streams and short counted windows carry more — the stall census). The corner is a real trap and not a shallow basin: a stalled reader's shortfall grows quadratically in the horizon, the per-step deficit growing like the step times the stream's scale rate. Two cures built from the window's own prices address it, both working at the original space and separating when the patience axis grows. The signal cure refines the comparator lexicographically — finite deficit beats infinite, and two infinite policies compare by their committed-scale shortfall, the finite part of the infinite loss — and converges from every start at both widths, for a reason that does not depend on width: that shortfall orders the plateau however far it extends. The move cure keeps the plain comparator and widens the move set with the patience diagonals, moves that lower or raise both patiences at once; it is radius-bounded, stalling exactly where the whole neighbourhood is infinite, since the blind region grows with the axis while this cure's own reach does not — one notch on each patience, however far the axis extends. Signal cures scale; a move cure built to a fixed reach is sized to the plateau it was designed against.

Scope. Exhaustive at both spaces — 100 policies on the patience axis (0, 1, 2, 3, ∞) and 324 on the extended axis (0…7, ∞) — over eight rows whose streams are quadratic irrationals, so their expansions are eventually periodic, including the wall row (x², √2), where the map sends the stream onto a vertex: a reader confined to the tree cells freezes there at rank 1 forever, and this row on its own is enough to make the corner a flat infinite plateau, the corner and both its single-refusal neighbour families all pricing infinite on it together. The straddle chains cure the wall itself (the wall criterion): a greedy reader on them commits exactly rather than boundedly, its deficit falling to zero, below the generic commitment bound. Horizon 120, counted from step 8. Toy scale.

verifiers: explore_ratchet_learner.py, explore_bootstrap_cures.py

Fix a bounded nondegenerate interval. The cover cells containing it form a sub-poset with an inclusion minimum, and greedy multi-commit — taking commit moves within one input step for as long as any remains available — reaches that minimum in any preference order: the containing cells are finitely many, every commit move strictly raises rank, and local confluence holds in three cases, so Newman's lemma applies. Two corollaries follow. Confluence: at greedy patience the committed cell sequence is preference-free, which is why the funnel's optimum is route-free. Pointwise global optimality: greedy patience dominates every policy at every step, in any policy space of this family — so the optimum with no budget in force carries nothing stream-specific, and nothing about it can be learned from data at all. That minimum has a closed form, read off the interval with no policy in it, and greedy multi-commit's committed cell is it at every step (the floor theorem). The hypothesis that the two kinds read one shared reference is tight: at mixed patience the cell the multi-commit settles at can depend on preference. In the partition cover — the tree cells alone, where the containing cells form one nested chain — loss is instead pointwise monotone in patience, strictly where the compared losses are finite, with total refusal worst. Global here means best in this policy family, not best conceivable; what the best conceivable solver could score, and whether a task family can be designed to state that number in closed form, is an eval's ceiling.

Scope. The lemma and the partition valley proved for this cover and move set; the engine checks the lemma's statement at every counted step (169,082 containments) and the tightness witness is a policy at the same slate and horizon.

verifier: explore_bootstrap_cures.py

Pointwise optimality consumes unbounded multi-commit, so a rank budget breaks its proof and prices the coordinates that proof made free. The start-delay law: a patience-p reader opens about B·p ranks behind greedy, and under a binding budget that gap is never erased — what used to erase it, in the unthrottled reader, was exactly the unbounded multi-commit the budget removes. So patience costs a permanent lag rather than a transient one, and the patience valley is strict wherever a row lags. The catch-up threshold law: greedy at budget B matches the unthrottled loss exactly once B reaches a per-row threshold — the ceiling of mean demand where fluctuation is bounded, one rank higher on a row whose large leading digit demands more rank than the budget's first steps can supply, leaving a warm-up debt standing — and never on exponential demand, the wall row's per-step deficit growing linearly in the step so that no finite budget catches up. A threshold read off a coarse budget grid is an interval and not a value: on one row the grid of 1, 2, 4, 8 records 4 where the banking verifier's grid, which includes 3, measures 3. Banking moves exactly one threshold on the slate, and by warm-up funding rather than by smoothing — a row whose greedy reader waits while the first digits arrive banks that step's income and later repays the debt that warm-up left, taking that row's threshold from 2 to 1. Off the optimum an early cheap commit can lock a route: 83 strict wins for a withholding schedule over spend-all, each steering a preference-fixed route past a better later cell. Every one of them is strictly off the optimum, where every drawing schedule ties.

Scope. Exhaustive at budgets 1, 2, 4, 8 and unthrottled, bank caps 0, 2, 4, 8, over the eight quadratic rows, horizon 120; the banking verifier re-runs budgets 1 through 4, the finer grid. Toy scale.

verifiers: explore_throttled_reader.py, explore_banking_reader.py

Each row's argmin class is the set of behaviours minimizing that row's loss. At every setting one universal policy sits in every row's argmin class, and transfer gaps — the excess a policy picked on one ensemble of rows pays on another — are exactly zero: under the deficit on the eight quadratic rows at every budget and cap, and under the clock loss, taken with the deficit as tiebreak and taken alone, on all nine rows, the one aperiodic non-quadratic row included. No loss tried that reads only the committed-cell trace lets training data pick the destination — and both of these do. What the optimum does depend on is the resource environment: at exactly one setting, (B, W) = (4, 2), the route order inverts and the universal set shifts — verbatim under both losses. That the shift is identical under both losses tests two layers for stream dependence at once — the allocation layer (route, patience, drawdown schedule) and the loss layer (scale units against whole steps) — and finds it in neither. A reader adapts to its metabolism, never to its data; what descent learns here is how to spend, not what the stream says. Read as forgetting, that independence is a certificate at the strongest of the four grades — flat at every weighting of the data rather than tuned to one — inherited from the destination being a function of metabolism alone (the four grades).

Scope. Exact and exhaustive at the stated scope and nowhere beyond it — this policy space, these nine rows, horizon 120, these two losses. Toy scale, and the statement is about this family of exact readers: outside it universality can fail, and the two laws of when it fails are the data door and the substrate door.

verifiers: explore_throttled_reader.py, explore_banking_reader.py, explore_scale_clock.py

Destination universality is a property of (space, loss) jointly, and the loss is the axis that breaks it. Relax exactly what the loss reads — a decision-valued prediction score, where the committed state alone guesses which side of its cell's vertex the stream will land on, and is charged for a miss — and destination universality dies. The rows' argmin classes stop intersecting under both orders, the score alone and the score with the deficit beneath it as tiebreak; greedy leaves the argmin on six of nine rows, and eight rows are perfectly predictable in-family but by different policies. Two hand lemmas locate the door. Collapse: any local proper score — one a truthful prediction minimizes — read against the flat belief on the committed cell telescopes back into the family of losses that see only the committed-cell trace, so leaving that family takes a discrete decision. Coarsening: the ratchet state is coarser than the public past, so the honest score is a function of the state alone. The door has no width. Each counted step carries one future bit — which side the stream in fact took — and each of the 112 of them, taken alone as the top order with the full deficit beneath it, already empties the intersection, while zero bits restore it: a step at zero rather than a threshold. So the price of data is not conserved across bandwidth: thinning the scored bits collapses it, and at a single scored bit a row's specialist ties the pooled winner's miss total outright, losing only the deficit tiebreak. “Reads the stream” is not the law, and it fails from both sides. Losses consulting future bits only in their values keep universality whenever the per-row orderings they induce never move, and at this scope they never do: 467 of the 475 within-row pairs that tie in value are identical in trace, so a tiebreak sitting below the deficit has nothing to act on. Meanwhile 146 losses that read nothing of the stream at all — indicators of a band of deficit values, row-uniform functions of the deficit alone — break universality outright. What does break is every non-degenerate loss tried whose top order consults future bits, the perverse direction included: no universal anti-predictor exists either.

Scope. The hand lemmas proved. Everything else exact and exhaustive on one frozen arena — the behavioural quotient with no budget in force, 100 policies in 38 classes, nine rows, horizon 120 — and over the loss species tried, which is what makes the surviving criterion statements patterns over the species tried rather than rules.

verifiers: explore_prediction_door.py, explore_bandwidth_dial.py, explore_score_criterion.py

An empty intersection turns out to be the generic case, so the thing needing a law is the keep. Order classes by containment of their committed cells, step by step and row by row; a loss whose optimum is pulled monotonically toward one extreme of that order keeps universality, and both extremes exist. The greedy class is pointwise-tightest — its committed cell contained in every class's cell at every counted step on every row — a short corollary of the bottom lemma, confirmed against a cell family rich in overlaps: 76,058 sites where two classes' cells overlap without either containing the other, so no chain argument reaches it — and the refuser is pointwise-coarsest by construction, holding the root cell throughout. All four monotone loss species tried keep, each through the extreme class its direction points at. For the losses that simply reward landing in a band of deficit values, the criterion is exact and proved. Pool the deficit values of every (class, row) pair into one sorted order — 117 distinct ranks at scope — and a class's hull is the interval its rank in that order spans across the rows; a band is clean when on every row it catches some class and not all, and a clean band keeps if and only if it contains some class's hull — the class it then hands every row at once. That hull spectrum is bimodal, and the gap is the mechanism behind generic emptiness: the refuser's hull has width 0 and its near-refuser sibling's width 2, where every adapted class's hull spans at least 85 of the 117 ranks, so adaptation makes a class's standing violently row-dependent and only refusal is row-stable. The law is sufficiency-only, and a band pair shows it: 213 keepers built from two disjoint bands exist, and in 201 of them the universal optimum set contains neither extreme — a keep handed out by where the values happen to fall, not by monotone structure.

What holding data costs is read off the same arena. Within a row, the committed geometry exhausts the visible: no loss tried separates any of the 475 within-row pairs of classes whose committed cells agree in length, step for step, on that row — the same 475 pairs the data door counts as value ties: on this arena the two descriptions pick out one set. So data content is a cross-row quantity, witnessed only where two rows collide in geometry — canonically at the golden near-twins, the two rows on which φ² = φ + 1 acts as a translation of the cover tree and which therefore agree in geometry on 32 of the 38 classes. The price is then structural rather than statistical: the optimum set becomes the row's fingerprint, an advantage on one row costs loss on the pooled slate wherever it is not free (measured at full bandwidth: 4 to 15 pooled misses per miss of advantage on the row), and no adapted optimum keeps its cross-row standing.

Scope. The coarse extreme, the hull criterion and the two-band construction proved; the tight extreme proved from the bottom lemma and engine-confirmed on the frozen arena the data door names; the loss-species results exhaustive there. Toy scale.

verifiers: explore_keep_law.py, explore_score_criterion.py, explore_prediction_door.py