Assessment153: prefix-profile access boundary
8 October2026. Scheduled research completion59; no new native endpoint or substantive mathematical-result increment.
Contents
Result and payoff
The maintained argument is docs/research-programs/prefix-depth-profile-access.md in the repository root. It gives a written sharp quadratic scan bound for independent unary sibling profiles, including maximal suffix sharing, and realizes those profiles in ordinary finite rows with linear mixed frontiers at each exact width. It separates three attempted routes: independent cursors, delayed conjunctions, and OR of partial-level markers. None supplies the charged joint operation.
The root cardinality counter from152 bypasses the scan on the witness family. That dependent control is essential: the family has exponentially much explicit input, and a different prepared certificate can be cheap to use. The missing target is a joint prefix/width index with paid routing and union, not another state-count or information-size argument.
Actual-draft audit
The writing agent checked the final argument and code. No independent review is claimed.
- Last-one distance and first-one distance distinguish every required unary suffix in the all-one instance. Tail sharing after the first marker cannot enter a two-marker initial path.
- Each omitted distinguished bit can flip with all pointers and other labels fixed. The resulting representation is valid even if not minimally shared. Canonicalization certificates and identity observations are not granted.
- Free roots and free advance knowledge of n strengthen the client. The lower bound is on successor-only access; joint metadata and jump pointers escape its premises and must be charged separately.
- Sibling cylinders are prefix-incomparable. Anchor level n differs from every variable residual level n-d-1, including d=n-1 and its empty suffix.
- The actual global anchor widths are n+d+1. For powers n, span1 suffices and exact-width mixed frontiers are at most n. This does not mean aggregate class frontiers satisfy reader34.
- The ordinary right encoding is Theta(n^2*2^n), so the fresh query bound cannot be relabeled a superlinear-in-input preparation lower bound.
- The target root fullness is an auxiliary profile operation. Every native SAT recognizer is not forced to compute it through this interface.
- A single width counter bypasses that auxiliary query. Its weighted count reaches2^n-1 exactly when all variable regions are present, then insertion reaches Full. Counter input and distinctness remain charged.
- Fixed-depth fullness and cardinality agree on the two one-hole witnesses, but singleton union distinguishes them. All-prefix terminal information remains semantically sufficient; its operational access is the question.
- Width-first replay has linear total work per isolated record, but a following short record or pin gives no automatic completion deadline. No free EOF work or pending-queue readout is assumed.
Finite experiment
tools/research/check_prefix_profile_access.py passed508 literal row instances, 29688 sibling-profile bits and4092 ancestor cutoff flags. It separately built 63 maximally suffix-shared unary DAGs for n=2 through64, checked45759 distinct required initial-path nodes and2079 fixed-topology label-flip controls. Six one-hole witnesses checked the failure of fixed-prefix fullness plus counts. The existing linked counter bypassed the scan in508 cases;17 exact-width frontier checks on powers n=2,4,8 confirmed the simple-slice control.
Literal sets and exhaustive binary suffixes are independent semantic oracles. Hash-consing IDs are finite-check bookkeeping, not operations exposed to the query client. The experiment does not execute a native machine. The general restricted bound follows from the written two-distance argument.
Seven-surface disposition
| Surface | Disposition |
|---|---|
| Custody | Original note and finite checker in GitLab; no primary sources or private scratch added |
| Claims | Supporting restricted proposition and exact witnesses; CLAIMS.yaml unchanged, no novelty assertion |
| Evidence | Written proof, explicit encoding, dependent counter escape and finite model; no new imported theorem |
| Failure | Independent unary scans and deferred forcing fail the charged access test; coarse OR loses overlap; reopen with joint preprocessing and paid direct pending queries |
| Frontier | Joint prefix/width preparation and complementary-coverage union remain open; width-first replay is a candidate with unresolved deadlines |
| Synthesis | Program checkpoint and agendas point to the precise operation and preserve152’s positive theorem |
| Site | No reader theorem or homepage advance; public545 retained, continuation metadata waits for a later warranted projection |
The imported entry129 theorems and prior native controls are unchanged. Replay, affine joins, causal algebra and unary cubical coherence do not by themselves provide the new joint index. Independent-store and conditional OMv claims keep their original quantifiers. No generalized native nonrecognition or PEG claim is made.
Next assessment and accounting
Next60 is a dedicated multi-paper primary-proof review, at least two complete primary arguments. Compare persistent indexed sequences, shared batched access and worst-case deferred computation against the diagonal workload and the two-stage width-first replay candidate. Seek a charged joint interface or an input-checkable structural gate, including union of complementary partial coverage and strict final-symbol readout. Preserve arbitrary name-linear rows, final-only diversity and unknown coordinate-order frontiers.
Scheduled58->59 once; next60 dedicated, last completed dedicated55/run149. Mathematical103/window5/last reassessment98 unchanged. The next substantive result104 still requires the six-result reassessment. Supporting bounds and control applications in this failed-route assessment do not count separately. Validation and conditional-commit receipts belong to integration state.