Stalls

Where an exact descent stops short of the bottom, including where no measurement tie sits anywhere near it: which stall species exist, what cures each, and how far an escape is when one exists. What decides whether a lowered patience improves anything at all is the geometry underneath them — nesting.

The thing being descended is a commitment policy over a cover of intervals (descent in reader space), and what follows states what it needs of one. A stream is the continued-fraction expansion of a real number, arriving digit by digit, and a map is a function applied to it. After n digits the value is pinned to a cylinder, and what the reader is certain of about the output is the image interval, that cylinder's picture under the map; these nest and shrink as digits arrive. A row is one (map, stream) pair.

What a reader commits to are cells drawn from one fixed family, the cover: the tree cells, the Stern–Brocot cells, each the set of positive reals whose expansion opens with a given run of digits, and the straddle chains standing at their vertices, a cell's vertex being the mediant of its two endpoints — the fraction whose numerator and denominator are their sums. Both kinds are graded by rank, which steps by one along every child relation (the redundant cover). A commitment is a ratchet: never undone, and permanently sound, since a cell containing the current image contains every later one. What a policy chooses is the route past a vertex — the tree child or the chain — and how long to wait before committing: four coordinates, a route preference at each of the two cell kinds and a patience for each. A kind at patience p commits a candidate only once it has contained the image for p+1 consecutive steps; patience 0 is greedy, committing on sight, and patience ∞ refuses that kind outright. A kind's reference is the image interval its patience makes a candidate contain, the one from p steps back at patience p, so a policy with equal patience on both kinds reads one reference at both.

Two exact rulers price a policy over a counted window, a fixed span of input steps, the same for every policy compared; its length is the horizon. The deficit at a step is the scale the reader is short — writing a cell's scale for ln(1/length), the image interval's scale less the committed cell's — and the clock ruler is that same lag counted in whole steps. Both compare by cross-multiplying big integers, so no float enters any decision. Descent is the matching finite-graph notion: a move is taken only when it strictly lowers the loss — the descent fixes one of the two rulers and runs against it alone — and a stall is a policy with no strictly better neighbour. Descent runs on the behavioural quotient — policies identified when their committed-cell traces over the window agree, since two policies with identical traces cannot be ordered by any loss reading that window; the resulting classes are what descent moves between. Resources, where they are in force, enter as settings rather than as coordinates: a rank budget B of cover-rank units per input step, and a bank of capacity W holding what a step does not spend, against which the policy gains a fifth coordinate for how much of the bank a single step may draw. What they are spent against is the row's demand, the rank per input step that row asks for. A setting with neither in force is unresourced, and there the four coordinates are all of it.

A stall is a fact about a move set before it is a fact about a landscape, and the move set here is the unit step. Neighbouring policies differ by one notch in one coordinate, or by both route preferences flipped together, and nothing below is a claim about a reader who may do more. Let a reader move a patience two notches at a time and the corpus's stall count over its ten landscapes — a landscape being one loss surface, a row and a setting fixed — falls from 10 to 1; at three notches it is zero — not at the ten specimens only but over every one of the 601 off-bottom classes those landscapes hold — and a reader free to set patience outright adds nothing beyond that (explore_move_set.py). So every stall named on this page is a reader who may change patience ONLY one notch at a time, and every distance below is counted in those notches.

The stall census pattern

Every stall found at the scanned scopes, with the cure the species takes. The cures name three instruments: a refinement of the comparator, the quotient the descent already runs on, and lookahead — accepting a short sequence of moves whose net effect improves where no single move does.

specieswhat stallscure
the plateau (space-born) a blind region of unbounded width — total refusal, where every neighbour prices infinite and the comparator has no signal a SIGNAL: lexicographic refinement by the loss's own finite part; radius-free, so it scales with the plateau
the degeneracy (space-born) a flat in the quotient — behaviourally identical policies tied together, manufactured by resource abundance (a full bank smooths two patiences into one trace) the QUOTIENT: descend counted-window behaviours rather than coordinates, on which the species cannot form
the ruler disagreement (loss-born) two exact losses over one window ordering an edge oppositely — the coarse ruler dams the fine one's gradient, so the deficit's own exit is clock-blocked and the class stalls under the clock order alone a two-move lookahead route, MEASURED; or descending the finer ruler across the coarse ruler's plateaus
the burst trap (resource-born) a strict local minimum with no tie in it — under a binding budget a demand burst makes one patience pay a little early to win a lot late, a paired trade every single move breaks a two-move lookahead route, MEASURED, which at one of the six is the quotient's walk and costs a reader three moves; neither refining the ruler nor quotienting the space touches any of them
the horizon-cut stall (window-born) a strict local minimum with no budget anywhere in it — the counted window ends inside a trade's transient stretch (two traces part, the early loser collects late, they re-merge), freezing the late win in place at the cut a two-move lookahead route, MEASURED — or the count itself: the census specimens dissolve once the window outgrows the trade

The three lookahead rows say MEASURED because two-move lookahead also has a proved cure attached to it, and that cure's hypothesis holds at none of the ten finite-loss stalls these three rows hold between them. What reaches them is an existence claim carrying no such hypothesis, plus a route someone counted: the escape radius below.

One more species once held a seat and lost it: coordination blindness, the trap a binding budget seemed to open under the unthrottled optimum itself, cured by a diagonal move — on the quotient the trapped class merges with its behavioural twin one route flip from the bottom, so the diagonal cure was two ordinary moves through a twin-flat and the space, rightly quotiented, never had the disease.

What the survivors share was once an open conjecture — every stall is a tie artifact, a measurement tie being neighbouring policies whose exact losses read equal, so that the coincidence dissolves under a refined ruler or the quotient — and it settles as a dichotomy. The spine is the policies whose two kinds share one reference: equal patience on both, unresourced, every patience shorter than the counted window's start. On the spine the conjecture is a theorem: the cover cells containing an interval have an inclusion minimum, that minimum is monotone in its interval, so committed cells nest pointwise along the patience diagonal and lexicographic deficit descent on the behavioural quotient stalls nowhere off the bottom, which is greedy — at any nested strictly shrinking stream and any horizon. Off the spine the counterexamples are genuine hills with no measurement tie anywhere in their neighbourhoods, so the instrument pair does not suffice in general: the burst trap under a binding budget, and the horizon-cut stall with no budget at all — the latter three specimens: two found by census in plain period-2 digit streams under the squaring map at horizons 11 and 12, one built from three digit bursts at horizon 16, where the same census over 177 generic streams finds none. The census-found pair dissolves once the window outgrows its trade, so a stall-free landscape is a property of the battery it was drawn from, generic streams and long counts and slow maps, never of the unresourced space; what says where nesting still holds there is the decision lemma.

Scope. A census over the settings scanned plus a designed battery, not a classification of exact reader spaces, and every row of it read at the unit patience step. The two space-born species and their cures are rules at the scopes the funnel and the resource laws state; the ruler disagreement is a single specimen, at (B, W) = (2, 0), and whether it is knife-edge or a family is open; the burst trap is six designed specimens (spike digit streams under the doubling map at budget 2); the horizon-cut stall is three specimens in exact arithmetic, one hand-verified at each horizon; the spine theorem is proved for this cover and move set.

verifiers: explore_bootstrap_cures.py, explore_scale_clock.py, explore_stall_tie.py, explore_stall_assembly.py, explore_move_set.py

The slow-map refusal and the one-cell wall observation

Of the battery's three legs — generic streams, long counts, slow maps — the slow map is the one an adversary without a budget cannot buy back — the burst trap already buys the doubling map back with one. Widen the digit range to 256 and take horizons 9–20: the identity and doubling maps refuse across some 2,800 distinct censused landscapes and 70,000 adversarial search evaluations, under the same detector that reproduces every squaring-map specimen. A large digit buys one step of shrink under any map, but only the squaring map compounds it along the window, so the best slow-map stall margin — the signed gap, in the scale's log units, by which a class's best neighbour fails to improve on it, positive exactly at a stall — approaches zero inversely in the digit cap and never crosses. Behind the approach stands a wall. The two frontier records — doubling at digit cap 64, identity at 256 — each equal −ln(1 + 1/R) exactly: the price of one further step down the cover, from a cell to a child of it, at the binding cell's own lopsidedness R, the ratio of its endpoint denominators, each record's R just under the bound its digit cap sets. The doubling record traces to its cells — a single mediant step, the other counted step cancelling — and the identity record is a single-counted-step ratio by construction. That one-cell wall is proved for any reader class holding a one-move neighbour whose committed cells sit nested-or-equal at every counted step. That domination condition holds for all 9,401 classes measured under the identity map, without exception; the doubling map dents it in 632 classes, of which 540 keep a neighbour shorter step for step without nesting and 92 still lose only on aggregate; the squaring map's stalls break it outright — every neighbour of a stall concedes somewhere inside the window.

Scope. The refusal is an adversarial search at the stated digit caps and horizons, so it is an observation and not a law about slow maps. The one-cell wall is proved for this cover, conditional on the domination condition; that condition is exhaustive at scope under the identity map (9,401 off-bottom classes at horizons 9, 10, 12 and 16, zero exceptions) and measured, not derived, under the others. Toy scale.

verifiers: explore_stall_maprate.py, explore_stall_domination.py

The escape radius rule

A stall's escape radius is the length of the shortest improving move sequence out of it, and the answer depends on who counts. Through the cure graph — whose edge exists as soon as some policy in a class carries the move — all ten finite-loss stalls above have radius exactly 2. Counted as one reader's own moves, nine of the ten agree and the tenth needs three. At the burst trap on the spiked stream under doubling, the one class supporting a two-edge walk out of the stall is entered at three policies and left at two others, and those sets are disjoint, so no reader takes the walk in two — nor does changing identity inside that class help: one change plus two moves is three, and three moves reach a better class outright.

Both figures are measured at these ten rather than inherited from the proved two-move cure, because that cure does not reach them. Lowering both patiences by one is an exact delay and improves strictly wherever it changes the class, on any map at once (the nesting neighbour), but its hypothesis is a policy with both patiences finite and positive whose counted window still refines, and no stall in the table above holds one. Every one of the ten has a patience pinned to the current image — a patience of 0, whose reference is the image itself — and no species straddles the split: the three horizon-cut stalls sit pinned at the chain, the ruler disagreement and all six burst traps at the tree. So the proved cure is a theorem about the classes that do NOT stall. What reaches the stalls carries no patience hypothesis and no map fence, and it is an existence claim rather than a route — that a strictly better class exists from every class off the bottom at finite loss, the floor theorem, proved at the three stalls carrying no budget and measured at the seven that do. How far away it is, is what the radius adds.

Scope. Measurements over the stalls the census holds, exhaustive at those ten under each reading of the count and at the unit patience step, the pinning anatomy likewise; a rule at that scope and not a theorem. The route behind each radius-2 figure, and the nesting that picks it out, are the reference geometry's. Toy scale.

verifiers: explore_charged_radius.py, explore_scale_clock.py

Nesting

What decides whether a lowered patience improves anything at all is one question: do the two readers' committed cells nest? Where no resource cap is in force, a greedy reader commits the floor of the current image — the inclusion minimum of the cover cells containing it, a closed form that reads the image and runs no policy — and that floor sits inside every cell any policy commits, which is why a strictly better class exists from every class off the bottom at finite loss, with no map fence and no patience hypothesis on it. Which single moves nest is measured, and the answer is narrow: patience-down moves and nothing else. One corner of it is proved — a run preferring the straddle chain at both kinds, under the identity map — and that proof's hypothesis turns out to be two inequalities on the indices of the references a run reads at a single step, the step where it can leave a chain for the cell the chain sits in. Off the spine, nesting dies only where the commit loop had a route preference to consult, though not at every such place: nesting.