Sparse content reuse along learned spines

10 October 2026. Assessment 204. Written native construction, theorem 67, subject to the explicit checked contract below. Root admission and external priority remain separate.

Statement and payoff

Long shared prefixes need not be copied when selected rows are combined. Separate the content of a path from the depths at which its sibling pairs occur. The input itself can learn a fixed number of paths, while unboundedly many selected rows contribute depth profiles along them. The existing mixed-region scheduler can then merge those profiles, and a query checks path content as its own bits arrive.

For example, many rows may place pairs along two paths that agree for a million bits and then diverge. Their sets of terminal depths can differ. A selected union is represented by two pending depth computations and the two learned paths, rather than by an eagerly copied million-node common prefix. The bound is on global path diversity, not on the number of rows selected by a source.

Use the ordinary right-first Horn syntax of reader section 12. Every right target is a nonempty binary word. Fix integers $K\ge1$ and $r\ge0$ for the machine. The machine checks the following conditions on the actual right block.

1. Every row is sibling-closed: $p0$ occurs exactly when $p1$ occurs. Repetitions and arbitrary row interleaving are allowed.
2. The set of observed pair parents $p$ has at most $K$ maximal elements in prefix order. The greedy, stable-slot procedure below learns this cover; no guide strings, partition or handles are supplied.
3. For every row $v$ and learned slot $\ell$, its deterministically assigned depth set $D_{v,\ell}$ obeys the mixed budget below.

Set $I_j=[2^{j-1},2^j-1]$ and $J_t=[2^{t-1},2^t-1]$. Let $C_{v,\ell,j}$ be the canonical EMPTY/FULL/BRANCH tree for $D_{v,\ell}\cap I_j$. Count unfolded mixed-coordinate occurrences by $M$, as in theorem 66, and require

$$\sum_{j\in J_t}M(C_{v,\ell,j})\le r2^{t-1}.$$

Written theorem 67. One fixed deterministic native scaffolding automaton for each fixed $K,r$ recognizes exactly the well-formed encodings passing these three gates that are satisfiable. It has bounded work and allocation per actual symbol, including the final symbol. Ordinary source/intermediate names, target lengths, row populations, left fanout, duplicates, row interleaving and retained versions are unrestricted within the gates. There is no name/target-length compatibility condition.

The third gate refers to the stable assignment produced by the input order, not to the existence of an arbitrary favorable partition. This is a checked language of serialized formulas. Equivalent reorderings can have different profile gates. No invariance under arbitrary clause reordering or renaming is asserted. For $K=1$, every pair uses the unique slot, so that assignment ambiguity disappears.

The exact encoded length is unchanged:

$$N=\sum_{(v,w)\text{ records}}(9+|v|+|w|) +\sum_{(u,v)\text{ records}}(10+|u|+|v|)+11+|a|+|b|.$$

The sums count duplicate records. The language consequence is a PEG for the reversal of this checked language. Independent-store composition and unrestricted Horn SAT remain open.

1. Learn the content cover without an endpoint comparison

Maintain at most $K$ used guide slots in creation order, initially none. Each guide is a strict persistent FIFO of binary labels, with an explicit used flag so an unused slot differs from a used empty guide. Guides are pair parents, not complete targets.

While reading a target $w=pb$, keep its most recent bit buffered. On each subsequent target bit, append the previous bit to a fresh FIFO and feed it to one comparison cursor for each used guide. Thus at the target delimiter the FIFO contains exactly $p$. No reverse or deletion is needed there. Each cursor has a mismatch flag and an extension flag. If the old guide has a next bit, compare it and advance by one queue view. If it has ended, set the extension flag and continue without a comparison. Mismatch remains latched. All $K$ cursors are initialized from current guide roots before the target starts.

At the delimiter choose the least-index slot with no mismatch. If its extension flag is true, replace that slot’s root with the just-built FIFO; otherwise retain the old guide. If none is compatible, create the next slot with that FIFO, or latch rejection if $K$ slots already exist. Selecting among $K$ flags and copying roots is fixed work. A duplicate target is still compatible with its existing cover. Parsing may continue in a bounded rejecting state after overflow.

Inductively, used guides are exactly the maximal observed parents, in stable slots. They are pairwise prefix-incomparable. A new parent that extends an old guide cannot extend two incomparable guides. A parent that is a prefix of an existing guide adds no maximal element. A parent incomparable with all guides creates one new maximal element. The empty parent is handled by the used flag and is extended in its existing slot. The number of maximal parents cannot decrease: no later string can extend two incomparable strings. Consequently an early overflow cannot be repaired by later input.

Most importantly, a guide in a used slot is only extended, never replaced by an incompatible string or moved to another slot. Every parent assigned to that slot remains a prefix of its final guide. This monotone extension is the content-coherence invariant; it replaces an uncharged longest-common-prefix operation.

2. Assign each row pair once and check its completion

Retain the native outer-name and inner-target capsule dictionaries from section 12.2. At each inner trie node store both its own target-terminal bit and a finite slot field for the sibling pair immediately below that node, initially unassigned. These fields have different meanings. In particular, a target can be a parent of longer targets without losing its terminal bit.

On the first member of a row’s pair $\{p0,p1\}$, save the slot chosen by the cover procedure at parent $p$. On its second member or a duplicate, retain that original assignment. An older common-prefix pair can be compatible with several later guides, but its first slot remains the least compatible one: earlier incompatible slots stay incompatible under extension, its chosen slot stays compatible, and new slots have larger indices. Saving the assignment makes that invariant explicit without repeating classification for the row pair.

The target traversal saves the parent view, the path FIFO before the last frame, and the selected child’s endpoint view. At the delimiter create a child tip with terminal true and its previous children and slot field; create a parent tip preserving its own terminal and untouched child, while retaining or setting its pair-slot field. Publish the capsule with the saved ancestor FIFO and that parent tip. This is a two-level explicit-tip update, not an upward path copy. For a length-one target the parent is the root and the ancestor FIFO is empty. The suspended outer-name capsule then installs the current row header in bounded work.

As in sparse completion, maintain one global unary count of half-present row pairs. A first member increments it; a new second member decrements it; a duplicate changes nothing. Use the children’s terminal bits, not child existence. Its zero test at the right boundary is exactly sibling closure.

A first member of length $n$ releases one profile point job $(\ell,n)$ for its row, using the saved finite slot and the unary length root prepared during the actual word. Because all parents assigned to one slot are prefixes of its final guide, there is at most one such parent of any given length in that slot. Thus a row/slot/width releases at most one job. Different slots can have pairs at the same width; they remain different jobs with different witnesses.

On a passing right block every released job has a later occurrence of the other member of that exact row pair. These are distinct witness words for distinct jobs. The witness’s whole target is later than the release. Repeated records are not counted as new jobs or required distinct witnesses.

3. Sparse input pays the complete profile producer

Each row owns one serial FIFO worker for all its slots. Its state includes a fixed tuple of $K$ profile directories and at most one active point job. The worker uses the point insertion, route conversion, signed block budget and current-version ownership of theorem 66, sections 2-4. A job focuses the indicated slot’s current directory and inserts depth $n$ into its canonical band tree. It never installs a captured old row header. Other slots and the foreground pair dictionary remain current.

The inherited point operation costs at most $a_{K,r}(n+1)$ fixed local actions, including traversal of dense band/block directories, capacity preparation, frames, canonical collapse, unary budget adjustment, allocation and commit. It does not require complete binary target levels. Completeness was used in theorem 66 to supply later witnesses; the sibling gate supplies those witnesses here instead.

Give this one row worker $4a_{K,r}$ service steps on every later target bit of that row. If work survived a passing right boundary, consider its last uninterrupted busy period. Every released job in the period has its distinct later witness inside that period. If $T$ is its number of target bits, then

$$\sum_i n_i\le T,\qquad \sum_i a_{K,r}(n_i+1)\le2a_{K,r}T<4a_{K,r}T.$$

All service while the worker is busy is used, contradicting unfinished work. The word causing a release is not prepaid service for its own job. Interleaving other rows preserves the argument on each row’s subsequence. A word that completes the final missing pair supplies service before its delimiter changes the gate. There is no boundary drain.

Maintain the current global count of over-budget row/slot/index-block records as in theorem 66. Active budget overrides are authoritative until commit; other rows update the current global root rather than restoring an old one. At a passing sibling boundary all workers are already quiet, so the bad-block zero test checks the exact final profile gate. Intermediate profiles may violate and later repair their budget. A nonpassing snapshot is not exported as an exact completed profile merely because its worker happens to be empty.

Let $g_\ell$ be the final guide in slot $\ell$, and let $\operatorname{pref}_d(g)$ denote its length-$d$ prefix when that prefix exists. The constructed profiles now satisfy exactly

$$S_v=\bigcup_{\ell=1}^{K}\ \bigcup_{n\in D_{v,\ell}} \{\operatorname{pref}_{n-1}(g_\ell)0, \operatorname{pref}_{n-1}(g_\ell)1\}.$$

Unused slots have empty profiles. The prefix exists for every depth in a nonempty profile by monotone guide extension. Each assigned pair occurs literally, and closure supplies both members. This proves both containments, including overlapping pair meanings across slots.

4. Combine unboundedly many selected profiles

Freeze the guide tuple and right-row dictionary at the right/left phase boundary. For each source, retain the inherited ordinary-name Seen dictionary and $K$ instances of theorem 66’s persistent band consumer. A previously unseen intermediate is looked up while its actual name arrives. At its delimiter, capture its fixed tuple of completed profile directories and release at most one cursor per slot. Missing rows contribute empty profiles. Duplicate selections release nothing. All copying and slot dispatch is bounded for fixed $K$.

For each slot, the checked block budget gives

$$\sum_{j\le k}M(C_{v,\ell,j})\le2rk.$$

Theorem 66’s one-time mixed-coordinate coupling therefore bounds the increase in that slot’s truncated residual work by $c_r k$, including its new merge, the disturbance of all old waiting jobs, restored directory prefix and promotion work. An active insertion keeps its frozen base until it completes. The $K$ consumers run fixed parallel logical batches. One original intermediate name pays their combined $K$-dependent constant; it does not supply a separate uncharged name for each component or pair.

Each consumer receives $64c_r$ steps per subsequent intermediate-name bit, $1024c_r$ worker steps per query bit and 1024 enumerator steps per query bit. These are the conservative rates from the charged band criterion. Their sum over $K$ slots is still one fixed batch. Distinct-name scheduling, ordered sealing, the local EMPTY-tail certificate and persistent FIFO output give the exact profile bit

$$A_{u,\ell}(n)=\bigvee_{v:(u,v)\in R}[n\in D_{v,\ell}]$$

by query position $n$, even when huge later bands remain pending. This uses theorem 66’s consumer proof, not its former complete-level producer gate. Neither the rows’ populations nor the number of selected intermediates enters finite control.

5. Charge prefix comparisons and negative readout

The source pin’s actual bits locate the receipt. At the start of the target pin initialize one ordinary queue cursor for each frozen guide, with a mismatch flag false. Buffer the current target bit. At position $n\ge2$, compare the previous target bit with the next guide bit, advance that cursor, and latch failure if they differ or the guide ended. At position one the empty-prefix comparison succeeds. The current last bit is deliberately not compared yet.

After the fixed service batch at position $n$, let $E_\ell(n)$ be the resulting finite match flag. It is exactly the assertion that the target’s first $n-1$ bits equal the first $n-1$ guide bits and that this guide prefix exists. Thus the reachability answer is

$$[b\in(RS)_u]=\bigvee_{\ell=1}^{K} \bigl(E_\ell(|b|)\land A_{u,\ell}(|b|)\bigr).$$

The sibling pair permits either final bit; delaying its comparison is necessary. A prefix that matched at length $n$ may correctly fail at length $n+1$. Empty or absent profiles contribute false. A guide may end exactly at depth $n-1$, which is valid, but cannot support a longer parent. The parser separately rejects an empty target. At the required final semicolon, negate the finite OR for pinned Horn SAT. No comparison, union, mask expansion or queue drain remains due after the input.

Retained receipts share the frozen guide tuple and retain their own immutable worker/deque roots. Each query fork pays its own fixed service and comparison batches. No unbounded version selection or independent-store handle is free.

6. Native resource contract and comparison

Fix $K,r$, the strict persistent FIFO/deque implementations, capsule dictionaries, counters and simultaneous-record compiler. A record has fixed arity and finite tags; slot indices range over the fixed set of $K$ slots. Guide comparisons use queue labels and emptiness, not identity or addresses. Point jobs and merges use the same bounded local views and continuations as theorem 66. The two-level capsule tip and fixed guide tuple add only a bounded batch and fixed observation radius to the inherited old-top semantics.

Initialization has empty dictionaries, queues and directories, unused slot flags and finite parser state. Actual target bits pay the guide candidates, their $K$ comparisons, literal dictionary traversal, unary length preparation and serial profile worker. Names pay ordinary routing and the consumer service; target pins pay content comparisons, pending profile completion, enumeration and output. Discarded candidates, superseded guides, suspended paths, signed counters, old-storage observations, all allocation and retained forks are included. Logical work and retained storage are $O_{K,r}(N)$; packing yields $N+1$ physical scaffold nodes and $O_{K,r}(N\log(N+2))$ explicit pointer bits. No supplied decomposition, padding, advice, free handles, external preprocessing, EOF work or metadata/timing/control-flow side channel is assumed. This is one fixed automaton for fixed $K,r$, not a uniform claim for growing $K$.

This is a content-sensitive composition class with unbounded left fanout. It is not the fixed-number-of-selected-predicates construction: each slot profile can union arbitrarily many independently named rows. For a power of two $H\ge4$, a short-named row may contain only one sibling pair at each depth in $[H,2H-1]$ along a long learned guide. Its sole occupied band is FULL as a depth profile, so its mixed count is zero; the actual word row is far from a complete binary level. Multiple distinct guides and distinct depth intervals are also permitted when the checked budget holds. This sparse family violates theorem 66’s radial hypothesis and, for sufficiently large $H$ relative to a fixed short intermediate name, the name/target compatibility gates in reader sections 33-34. It does not contradict their conditional results.

The construction adapts known strict persistent sequences, the Lab’s serial dictionaries, the sibling-witness proof and the mixed-band consumer. Its additional content is the ordinary-input stable cover and the charged coupling of content comparison to depth-profile reuse. This is a written proof extension, not an established external priority claim. Entry 129’s relation quotient, growing alphabet and conditional state/port bounds remain unchanged.

7. Failed extensions and next decisive test

Arbitrarily repartitioning old pairs among compatible guides would require charged profile updates. The deterministic least-compatible procedure does not itself reclassify old pairs as guides extend. Its trace dependence is across different input orders. For example, take $K=2,r=1$: row $a$ has parents 000, 00000 and 000000 (depths 4, 6 and 7), and row $b$ has parent 000001 (depth 7). If $a$’s depth-seven pair arrives first, its single-slot profile has mixed count 2 and passes the relevant capacity 2. If $b$’s pair arrives first, $a$’s depths 4 and 6 go to its slot, with mixed count 3, while its depth 7 goes to another slot. The same literal rows then fail the checked gate. All pairs are completed in both orders. This is a limitation of the stated serialized language, not an invariant claim about the underlying formula. Merely retaining a fixed number of arbitrary predicates does not handle unbounded selected populations. Dropping $K$ makes guide classification and simultaneous query comparison unbounded; the present proof does not cover that extension. The eager ordinary-trie obstruction remains valid: these receipts deliberately retain shared guides and pending profile computations rather than expose a materialized union before pins.

The next target is a growing spine forest with locally routed, query-relevant depth profiles. Replace the fixed $K$ comparison family by a paid content-prefix routing contract, and bound a newly selected row’s disturbance of all pending work. First test many long spines sharing a prefix and branching at staggered depths, with sparse unequal terminal-depth sets and late short names. A free longest-common-prefix oracle, a supplied spine decomposition or a hidden per-spine scan is not a solution. Primary-proof review 110 should compare compressed persistent trie routing and colored or marked ancestor access before claiming this stronger transfer. Unrestricted radial profiles and independent-store joins stay separate.

Evidence and review are recorded in assessment 204. The primary model and independent audit separate finite semantic checks from the native all-length argument.