Sparse content reuse along learned spines
10 October 2026. Assessment 204. Written native theorem 67 for a checked serial-reuse subclass; no root admission.
Later developments: review 205 isolates the selected-incidence obstacle; theorem 68 gives a separate shared-context construction. These positive classes are not claimed to subsume one another.
Contents
Result and mechanism
The complete proof separates path content from terminal-depth coverage. Ordinary right input learns at most a fixed number of global spines. A row places sibling pairs at selected depths along those spines. Its depth sets satisfy the same blockwise mixed-coordinate budget used by theorem 66. Unboundedly many selected rows can then contribute to each pending depth union, while actual query bits pay for spine comparisons and exact positive or negative readout.
This is a content-sensitive extension beyond complete binary levels. The bound is on global spine diversity, not on selected row count. Long shared prefixes remain shared; the receipt need not eagerly materialize each selected union. Sparse sibling closure supplies distinct later words that pay every released point-insertion job. One serial worker per row handles all fixed slots. The prior consumer proof applies separately to each slot, with the total fixed batch charged to the original input symbols.
The contract retains right-first ordinary Horn syntax, fixed degree/radius/finite control, old-top updates, explicit initialization, all input and name routing, allocation, retained observations, pending work, per-symbol service and the final-symbol answer. The checked gates are sibling closure, fixed maximal-parent count and the deterministic assignment’s mixed-profile budget. There is no name/target-length compatibility gate. One machine is fixed for each fixed pair of parameters; growing spine diversity is not covered. The PEG consequence is for reversal of the checked language.
Independent review and correction
Independent adversarial review checked the online antichain cover, first pair assignment, one-bit-delayed parent and query comparisons, two-level capsule tip, distinct completion witnesses across all slots, serial current-version ownership, mixed-budget check, and the inherited pending-consumer deadlines. No native gap was found relative to the maintained theorem 64/66 primitives. It also produced a separately designed semantic audit without importing the primary or inherited model.
The review corrected an overstatement in the first draft. With creation-order slots and the least-compatible choice, an old parent always retains the least compatible slot as guides extend. Earlier incompatible guides stay incompatible, the chosen guide stays compatible, and later slots have larger indices. Arbitrary repartitioning would incur updates, but this deterministic procedure does not move old pairs within one history.
Across different input orders, the checked budget really can change. For fixed parameters two and one, row a has sibling pairs below parents 000, 00000 and 000000; row b has a pair below 000001. If a’s depth-seven pair arrives first, its mixed profile count is two and passes capacity two. If b’s pair arrives first, a’s depths four and six use the older slot and have mixed count three, so that ordering fails. All pairs are completed in either ordering. This exact limitation appears in the theorem statement and the independent audit.
The primary model initially requested a guide-extension query beyond its allocated finite oracle answer array. Enlarging that array to cover both stored word lengths and guide-test lengths fixed the test harness. It did not change the mathematical algorithm or service schedule.
Evidence and comparison
Primary transition model: 180 histories, 8,620 records, 79,474 non-idle producer service steps, 1,957 passing sibling boundaries and 8,620 retained versions. Of these histories, 163 pass the mixed budget. Their 814 distinct consumer releases produced 752,242 query-prefix checks, including 572,327 negative answers. A guide-overflow rejection control also passed. The model services one row worker only during actual target bits and does no delimiter drain; immutable producer and consumer transitions are reused from the maintained theorem 66 model. Query experiments deliberately omit pre-query consumer service. Semantic Python containers and finite constants are not claims of native unit-cost implementation.
Independent audit: 2,801 exhaustive cover histories, 10,738 cover prefixes, 720 record interleavings and 240 seeded variable-depth histories. It checks 7,938 record prefixes and immutable snapshots, 2,173 passing sibling boundaries and 1,978,956 query prefixes, including 1,530,508 negative answers. A same-formula/different-order witness changes the global mixed-budget gate. String/set/integer operations are explicitly semantic oracles. The written proof, not these finite counts, establishes the all-length native resource claim.
The method extends strict persistent sequences, the Lab’s serial capsule dictionaries, theorem 66’s mixed-band producer/consumer and assessment 202’s sibling-witness certificate. Its additional argument is the ordinary-input stable content cover and its charged product with unbounded selected depth profiles. This is classified as a written proof extension, not an external priority finding. No additional imported theorem was needed; the next dedicated primary-proof review remains due at 110.
The direct controls were reader sections 12 and 34, mixed-budget theorem 66 and the sparse preparation note. Section 12 already supports a bounded number of arbitrary selected predicates; this result instead permits unbounded selections on each fixed spine. Section 34’s shared mixed-prefix/name gate is different: one pair at every depth of a distant full depth band can have zero depth-profile mixed count and a short intermediate name while failing that name compatibility gate. Theorem 66’s radial complete-level condition does not cover these sparse word rows.
Replay forks each pay their own query service. Affine-fragment joins provide no independent-store conversion here. Persistent causal algebra preserves history but does not pay access; the explicit schedule does. Unary coherent cubical algebra supplies no automatic growing-branch sewing closure. Native SAT controls retain ordinary late endpoint pins, complement readout and the reversal qualification. Entry 129’s growing-alphabet quotient and conditional state/port bounds are preserved and are not asserted as native-language lower bounds.
Failure ledger and next decisive test
| Route | Exact limitation | Reopen gate |
|---|---|---|
| Infer a formula-invariant mixed gate | Different clause orders can assign shared parents to different slots | Prove an input-paid canonical assignment or retain the explicit serialized language |
| Replace content by occupied depths alone | Equal depths can contain different target words | Retain learned guides and pay each query comparison |
| Copy the union before late pins | The earlier eager-output residual-node bound still applies | Share compressed content or retain paid pending readout, as here |
| Let the spine count grow in this implementation | Classification and simultaneous query comparison cease to be fixed batches | Construct local content-prefix routing and a matching disturbance bound |
Next: a growing spine forest with locally routed, query-relevant depth profiles. First test long shared prefixes that branch at staggered depths, sparse unequal terminal-depth sets and late short-name selections. Charge the construction and access of the routing structure and the effect of every selection on all pending work. A supplied decomposition, free longest-common-prefix oracle, hidden per-spine scan or EOF work does not close the gap. Primary-proof review 110 should compare compressed persistent trie routing and colored or marked ancestor access. Unrestricted radial profiles and independent-store joins remain separate.
Integration and publication scope
- Custody: one maintained GitLab proof and two audit sources; no primary PDFs, extracted sources or scratch exported.
- Claims: unregistered native theorem 67 in the existing SAT program; root CLAIMS.yaml and admission status unchanged.
- Evidence: written resource proof, reconciled independent review and both finite models above.
- Failures: input-order gate and fixed-comparison-family boundary retained with explicit reopen conditions.
- Frontier: growing spine routing, unrestricted radial profiles and independent-store joins remain distinct.
- Reader synthesis: update the existing foundations homepage entry and program overview; no new paper or significance promotion.
- Site: generate styled HTML/KaTeX readings and syntax-highlighted code with line anchors and byte-exact raw downloads from the maintained source. Full-suite, build, rendered-link checks and exact deployment/source-parity receipts remain in integration state. Browser viewport inspection is unavailable under the supported workflow; responsive layout and equation clipping are unverified.
Scheduled 108 to 109 and mathematical 129 to 130 once for this combined construction. Mathematical window two; reassessment 128 retained. Next 110 is a substantive primary-proof review; next editorial 114. Legacy reverse-writer retirement remains a separate unverified follow-up.