Two suffix maxima with delayed classification

10 October 2026. Assessment 194. Written native theorem 62, extending the existing persistent constructions. No root admission or external novelty claim.

Current continuation: Ordered suffix reuse extends this construction to every fixed number of final maxima. The two-maximum proof and its dated next test are retained below; the checkpoint records the open frontier.

Result and example

Written theorem 62. In the ordinary right-first Horn language of the encoding contract, require every nonempty final right row to contain exactly all binary words with lengths in an interval $[b_v,H_v]$, where $1\le b_v\le H_v$. Require at most two distinct final maxima $H_v$. One fixed deterministic native scaffolding automaton checks these restrictions and recognizes the pinned SAT language with bounded work and allocation per actual input symbol, including its last symbol.

The maxima and their class labels are not supplied. Names, lower endpoints, intermediate maximum diversity, row interleaving, duplicate input, selected populations and retained versions are unrestricted. This removes the common-maximum condition from the sparse-age suffix construction for two final maxima. It does not establish arbitrary maximum diversity or a join of independently stored graphs. Native recognition gives a PEG for the reversal of this checked encoded language.

For example, $[1,3]$ and $[8,12]$ must retain both pieces of coverage. Keeping only the smallest lower endpoint loses lengths 8 through 12; keeping only the largest maximum can lose length 1. The new construction maintains an all-row minimum immediately, and a second minimum behind a paid maximum-classification worker.

Let $L$ be the smallest final right maximum, including rows not selected by the current source. Through length $L$, every row’s upper test is true. Therefore the all-row lower minimum answers those queries. Only beyond $L$ does the distinction between maximum $L$ and the possible larger maximum $U$ matter. This common initial range pays for classification without delaying earlier answers.

1. Export the smaller maximum during actual right input

Retain the literal complete-level validator, occupied-run interval gate and prepared lower/upper unary rulers and binary numerals from the suffix producer. Retain the current unary maximum histogram from input-dominated class routing, section 2. Its payload at depth $j$ counts rows of current maximum $j$; its occupied-bucket counter checks final diversity at most two. Intermediate diversity is not bounded.

Here is a paid extension that exports the minimum occupied histogram depth. During each actual target word of length $n$, walk the current histogram from its root, one level per target symbol. In parallel consume the current row’s old maximum ruler of length $h$ and build the usual current-position unary ruler and numeral. At depth $h$, if the histogram population is a singleton, treat that payload as zero for this speculative walk. At every other depth, treat its actual positive population as occupied. Capture the ruler and numeral at the first occupied depth encountered, using a finite seen flag. Inspecting zero/singleton populations is local.

At the delimiter:

• If $n\le h$, the row’s maximum did not change. Discard this speculative export and retain the old global minimum.
• If $n>h$, perform the existing two-terminal histogram update, removing this row at $h$ and adding it at $n$. The captured first occupied depth is the new global minimum if one was found; otherwise the new minimum is $n$ itself.

For a new row, $h=0$ and there is no departure population. A fresh maximum is necessarily a fresh literal pair. The position roots captured while spelling the word are already prepared; no numeral conversion happens at the delimiter.

Proof. When $n>h$, the walk observes exactly the old histogram with the departing row removed, through depth $n$. The new histogram adds a row at $n$, so its minimum cannot exceed $n$. Thus the first surviving occupied depth at most $n$, if present, is its minimum; otherwise $n$ is the minimum. Populations beyond $n$ cannot change this conclusion. When $n\le h$, there is no histogram change. Induction covers arbitrary insertion order, aliases, duplicate targets, minimum-class removal and arbitrarily distant successors.

This walk shares the already paid target-driven histogram traversal. A fixed number of extra roots captures its first hit; it does not search after the word ends or follow a stale class handle. The final right gate checks interval shape, literal completeness and at most two occupied maximum buckets. Empty right input is a separate finite case. On nonempty valid input the frozen export $L$ and its binary numeral are ready at this boundary. The export does not enumerate or relabel rows.

2. A receipt with three fixed service lanes

For each left source retain the existing selected-name trie, suppressing duplicate nonempty row releases. Its receipt has:

1. The sparse-age minimum worker $A$ for every selected lower endpoint.
2. A strict persistent FIFO and one digit comparator classifying selected upper endpoints against the frozen $L$.
3. The sparse-age minimum worker $B$ for rows whose maximum differs from $L$, plus the first such row’s upper ruler $U$.

The ordinary source name opens its current capsule. During the following ordinary intermediate name, service the already released receipt. At its delimiter, frozen right lookup supplies the row’s prepared roots. A first nonempty selection releases its lower endpoint into $A$ and its row into the classification FIFO. No service before that delimiter is credited to the new job. Missing and repeated selections release no work. Source suspension and reconstruction use the inherited serial dictionaries.

The classifier compares the binary numerals of $L$ and $H_v$ from their low-order ends. A differing digit or unequal termination certifies inequality; simultaneous termination without a difference certifies equality. It never converts a numeral to an integer. Since every valid final $H_v\ge L$, classification needs at most

$$C(L)=\ell(L)+2,\qquad \ell(x)=\lfloor\log_2 x\rfloor+1,$$

normalized steps: one setup step, at most $\ell(L)$ digit steps, and a termination/release step. Constant record work is included in a normalized step. An inequality releases the prepared lower endpoint into $B$ and, if needed, captures that row’s upper ruler. All unequal rows have the same $U$ because the checked final diversity is at most two. Equal rows release nothing into $B$.

A service macro performs one bounded step of each of these three lanes. Idle lanes do nothing. A FIFO completion and its new $B$ release are a bounded record update, not a drain. Neither a large endpoint in $A$ nor a large endpoint in $B$ blocks classification. The first two buckets suffice for each sparse worker exactly as in theorem 61.

The receipt is fully immutable. Every new root is built from current records and bounded local observations. The classifier holds its own operand cursors; it never later replaces an old receipt root. The two minimum workers preserve all their interruption invariants. Old versions retain their own queues, comparisons, worker roots and upper export.

3. Pay downstream work at the original release

Delayed classification can release many minimum jobs. Counting only the FIFO debt would leave this downstream work unpaid. Reserve both stages at the original ordinary-name release.

Fix a lower-numeral cutoff $k$. For each original row release reserve at most $C(L)$ classifier credits and $2k$ credits for that row’s possible $B$ work through ages $0,\ldots,k-1$. The latter is the screening-plus-comparison bound from theorem 61. Equal classification discards its reserved minimum credits. Unequal classification transfers them to $B$ without creating new credit. Promotion beyond the cutoff, dominated-tail deletion and unused comparison budgets discard credit. Suspensions do not restart comparisons.

While this combined debt is positive, each macro removes at least one credit: an unfinished classifier takes its step; if no classification remains, positive minimum debt is serviced at the head of $B$. A higher-age $B$ step cannot interfere with the classifier’s separate lane. Charge a minimum comparison to the entering candidate, as in the sparse-worker proof. Unspent reservations are an analysis device, not a stored unbounded counter.

Run $q=64$ macros per ordinary intermediate-name bit. The final busy-period argument therefore applies to the combined per-release envelope

$$D_k=C(L)+2k, \qquad W_k\le D_k+\sum_{d\ge1}2^d(D_k-qd)_+ \le D_k+4q\,2^{D_k/q}.$$

There are at most $2^d$ distinct nonempty names of length $d$. The first release in a busy period contributes at most $D_k$; each later release contributes at most $(D_k-qd)_+$. Repeated names add service but no job. The bound uses only this source’s actual selected-name subsequence, so unrelated-source input is not free credit. Each macro is one fixed batch even though it services three lanes.

Let $b$ be the minimum lower endpoint among selected unequal rows, if any. Put

$$k=\max(4,\ell(L),\ell(b)),$$

omitting $\ell(b)$ when that set is empty. Then $C(L)\le k+2\le2k$, so

$$W_k\le4k+256\,2^{k/16}<1024\,2^{k-4}.$$

For the final inequality, after division by $2^{k-4}$ the first term is at most 16 and the second at most 512 for every $k\ge4$. Both bounds decrease after their initial cases. These constants are deliberately conservative.

Draining this debt finishes all classification. If there is no unequal row, $B$ is empty. Otherwise it also makes $B$ completely idle: a shortest lower numeral has at most $k$ bits and its completion discards all higher-age buckets. Thus quiescence is locally recognizable by an empty FIFO, no active classifier and an idle minimum head. The machine does not use $k$, a cutoff selector or an integer-order oracle.

4. Every query prefix has its exact answer

The positive pin retrieves a receipt through the inherited ordinary-name dictionary. During each bit of the negative target pin, run $Q=1024$ service macros and the fixed reader/repair lanes. Run the original $L$ ruler from the first bit. Forks of old receipts use the same schedule.

Through query length $L$, answer with $A$’s lower test. All selected maxima are at least $L$, so this is exact. Theorem 61’s minimum discovery and four-step lag repair apply with the larger service rates used here. Its proof did not use the common upper endpoint to construct the minimum worker; that endpoint only bounded the final membership test.

When the classifier and $B$ become quiescent, an empty $B$ certifies that no selected row extends above $L$. Otherwise capture its final lower ruler $b$ and the retained upper ruler $U$. Start both reader cursors with the current query-prefix unary stack as their replay debt. Run four repair steps per target bit for each ruler, including their birth batch; after alignment each advances once per bit. These are two fixed lanes, regardless of selected population.

Let $M=\max(L,b)$ when $B$ is nonempty and $M=L$ otherwise. The preceding bound makes quiescence occur by target time $t\le2^{k-4}$. If $k>4$, then $M\ge2^{k-1}$, and both reader repairs finish by

$$t+\lceil t/3\rceil<M.$$

If $k=4$, quiescence and the one-symbol replay finish in the first query-symbol batch. Until alignment, use false for the high-range branch. If $b>L$, this is correct before $b=M$; if $b\le L$, alignment finishes before the first length above $L$ can require it. The upper ruler is at least $M$, so it is ready before it can impose a negative answer. If there is no high row, quiescence by the same bound certifies absence before leaving the low range, although the default false answer already suffices.

The exact membership predicate is therefore

$$(n\le L\ \wedge\ \min_{v\text{ selected}}b_v\le n) \quad\vee\quad (n>L\ \wedge\ B\ne\varnothing\ \wedge\ b\le n\le U).$$

Positive lower endpoints give false membership at an empty target. Complement the relation-membership bit for pinned Horn SAT. Syntax and gates are checked at their existing boundaries. Every possible final target prefix has its answer by the final symbol; there is no EOF work, supplied target length or pause for a queue drain.

5. Resource contract and structural meaning

Initialization contains empty dictionaries, queues, worker roots and finite flags. Literal row preparation, speculative histogram walks, numeral/ruler exports, both name lookups, duplicate detection, all three service lanes, delayed releases, retained old records, query forks, reader replay and output are charged to actual encoded length $N$.

The new producer adds one target-driven walk and fixed capture roots. A receipt adds a fixed FIFO, digit comparator, two already implemented sparse workers and fixed ruler readers. Every normalized operation observes a bounded number of old pointer fields and finite labels. Fixed $q,Q$ batches unroll to a finite record plan. The inherited simultaneous-record compiler gives one fixed degree, radius, finite control and old-top update rule, independent of $L,U,b$, names and history. Storage is $O(N)$ logical records, with the inherited $N+1$ physical scaffold-node convention and $O(N\log(N+2))$ explicit pointer bits. No address equality, numerical pointer, metadata, timing or control-flow side channel encodes the maximum class.

This is a proof extension of the existing serial dictionaries, name-funded queues and sparse-age workers. Its additional mechanism is a common initial range that defers classification, together with credits reserved for downstream work at original release. A semantic two-interval summary alone would not supply either mechanism. Entry 129’s imported full-relation, growing-alphabet and matching/positive-path hypotheses remain unchanged; no native lower bound or all-relation representation follows.

The finite audit uses explicit linked-digit classification and the maintained immutable minimum worker. A separate set-of-intervals oracle checks query prefixes and retained forks. Histogram maps, tuple FIFO operations and unary reader lag arithmetic model inherited primitives; their Python running time is not native evidence. The all-length schedule and producer argument above carry those costs.

6. Failed routes and the next test

Discarding every row above the global minimum loses the two-interval example. Classifying at each short name delimiter requires an unbounded digit scan. Paying only classification and releasing all minimum jobs for free omits downstream debt. Reclassifying every old row when the global right minimum changes uses stale aliases and an uncharged scan. The construction respectively keeps a separate high worker, defers classification through $L$, reserves the joint envelope, and freezes the producer export only at the right boundary.

For three final maxima, the all-row minimum remains valid through the smallest one, but the rows above it no longer share an upper endpoint. A second deferred classification threshold is needed. The next test is to export the ordered first fixed $K$ occupied maximum depths during actual right input and compose a fixed-depth classification/minimum pipeline, while charging speculative exports, downstream credits and reader handoffs. First test removal of a sole low bucket when all successors lie beyond the changing word. Do not assume a free sorted palette or bulk relabeling. Arbitrary maximum diversity and independent-store joins remain separate.

Custody: one maintained proof and checker. Claim mapping: written native theorem 62, no root-ledger change. Evidence: the general construction and finite audit, with inherited primitive scope. Failure ledger and reopen gates: this section. Reader synthesis: the common initial range and paid pipeline. Site representation: generated styled HTML and highlighted exact-download source, with deployment receipts in integration state. Existing replay, affine, causal-algebra and SAT controls remain intact.