Recognizing a filler does not yet maintain its current answer

13 October 2026. Dedicated comparison of two primary proofs. This review identifies failures of literal matcher-based implementations. It makes no new native theorem, general lower bound or priority claim.

The next learned-filler construction needs three paid mechanisms: recognition, activation updates and access to the selected current answer. Classical pattern matching supplies strong recognition mechanisms. It does not by itself bound the number of indexes changed by a literal owner activation. Construction 82 succeeds because every activation updates just three shared roots, independently of the number of learned namespace views.

Primary proof 1: one pattern and its borders

Donald E. Knuth, James H. Morris, Jr. and Vaughan R. Pratt, Fast Pattern Matching in Strings, SIAM Journal on Computing 6(2), 323–350, 1977. DOI. Proof scope: sections 1–5, including preprocessing, the scan invariant and the complete per-character delay proof; section 7 was checked for the binary-alphabet and real-time qualifications. The palindrome application is not a transferred result here.

The scan retains a matched pattern prefix. After a mismatch, border information selects the longest remaining candidate without moving backward in the text. Preprocessing builds these links from shorter prefixes. Forward text progress and decreasing candidate length bound total work linearly in pattern plus text length. The optimized next link skips candidates that would repeat a known failed comparison.

Section 5 proves a logarithmic upper bound on consecutive next-link steps at one character. Two successive overlap shifts imply a Fibonacci lower bound on the prefix length needed for a long sequence; Fibonacci patterns witness sharpness for that loop. This is a delay statement about that implementation. Section 7 explicitly records a binary-alphabet improvement and a real-time conversion. It would be incorrect to infer that pattern matching inherently requires logarithmic delay.

For this project, separately running a matcher for every learned filler can multiply work by the number of fillers. A fixed compiled machine for each pattern also does not supply one fixed native controller with input-paid preparation for every pattern.

Primary proof 2: shared recognition and inherited outputs

Alfred V. Aho and Margaret J. Corasick, Efficient String Matching: An Aid to Bibliographic Search, Communications of the ACM 18(6), 333–340, 1975. DOI. Proof scope: the complete paper, particularly Algorithms 2–4, Lemmas 1–3, Theorem 2 and the separate output-cost discussion.

The keyword trie shares prefixes. Breadth-first construction assigns failure links to longest proper suffixes that are trie prefixes and inherits the associated outputs. The suffix invariant proves exact matching. Trie depth increases by at most one on a successful symbol transition and decreases on a failure, so Theorem 2 bounds total recognition transitions by fewer than twice the text length. Section 6 completes transitions in advance, giving one recognition transition per symbol; a fixed alphabet bounds table width per state.

Section 5 separates recognition from output emission. Shared output lists economize construction of inherited sets; they do not make enumeration of all reports constant cost. The construction is batch processing of an already built trie, using a queue and subsequent table assignments. These proofs do not give an input-paced immutable old-top producer or incremental persistent maintenance under new keyword insertions.

Three independent costs, three explicit failures

All examples use the binary alphabet. Let $k\ge1$.

These are written implementation counterexamples, not imported lower bounds. Their proofs are direct. Before the final 1, every suffix consists only of zeros. In the first machine each proper-prefix failure drops one zero until reaching the root. In the second dictionary every keyword requires a final 1, and all are suffixes of the completed text. For insertion, no old prefix ends in 1; afterward the longest available suffix ending in 1 is exactly the new keyword 1. Thus the last example changes $k+1$ explicit entries after a one-symbol keyword insertion.

For the second route, suppose each report releases a separate unit-cost index update only when that match is reported. With at most $B$ jobs per symbol and $c$ remaining ordinary symbols, including the reporting symbol, only $Bc$ such jobs can finish. Choosing $k>Bc$ defeats this route. The actual final-symbol contract has constant $c$, not a padded tail. Total input work may still be linear; the failure concerns release time. Speculation before reporting, compressed output sets, deduplication and shared aggregate updates are outside this failed route and remain possible.

A related output-size control uses keywords $0,00,\ldots,0^k$ and text $0^n$, $n\ge k$. Explicitly reporting every occurrence requires

$$\sum_{j=1}^{n}\min(j,k)=kn-\frac{k(k-1)}2$$

reports. This counts the requested reports, not the information needed for a late Boolean query. It cannot establish a native language lower bound.

Connection to persistent composition and reuse

Construction 82 uses injective fixed-width normalizers and three current activation roots. Its query producer locates the appropriate normalized owner under the ordinary query deadline. The number of namespace descriptors does not multiply literal activation work. Replacing those roots with one index per matched filler forfeits precisely this property unless reports share a bounded update representation.

The two proofs therefore sharpen, without solving, the charged factorization target:

1. Build the learned-pattern representation from the actual ordinary records. Count validation, allocation and name routing; accept no supplied machine or ready handles.
2. Give a bounded shared activation summary. A literal owner update must not scan the growing filler family or enumerate all matching views.
3. Produce the selected normalized current predicate in time. A shared recognition state or an any-match bit is insufficient unless it preserves the view-specific answer and its local access route.
4. Supply the fixed-degree, fixed-radius, finite-label old-top implementation. Completed automaton edges are abstract transitions until their representation and access are paid. A per-instance transition table in finite control would be a nonuniform family.

Persistent descriptors, indirection and shared output aggregates might avoid explicit table rewrites and per-report dispatch. The insertion example does not exclude them. Their decisive obligation is simultaneous update locality and timely selected readout, including negative observations. Preparation, retained storage, old-storage observations, deferred work and output must fit the actual encoded length and every deadline. No EOF work is available. Input orientation remains material: a native recognizer yields a PEG for the reversed language only under the established correspondence.

Evidence and handoff

The finite checker tests all four families for $k=1,\ldots,64$. It also compares completed-state outputs with direct suffix tests for every nonempty subset of binary words of lengths 1 and 2, on every length-5 binary text. These are finite semantic controls, not a native implementation. An independent reviewer read the complete Aho–Corasick proof and construction 82, checked the first two families independently through $k=32$, and identified the insertion witness.

Contribution assessment: the matching bounds are established imported results. The burst and insertion examples are direct consequences of the classical representations, with high confidence; their contribution here is to identify exactly which transfer obligations construction 82 must preserve. This is a mechanism comparison, not a new pattern-matching result.

The failed route is literal matcher completion followed by independent match dispatch or explicit table repair. Reopen it with a charged shared representation that avoids those obligations. No root claim is admitted; no existing theorem, SAT/scaffold result or open frontier is retired. Primary PDFs and extracted text remain outside public source.

Next decisive test: replace common zero filler by independently learned filler words using a bounded shared normalization basis. The candidate must exhibit its shared activation record and its paid selector-to-current-answer operation before relying on a pattern-recognition time bound. Charge discovery and literal activation updates without scanning the growing family. Preserve arbitrary whole-suffix permutations, arbitrary-name routing, cyclic sharing and heterogeneous service as open.

Accounting: scheduled 144 to 145 once; mathematical 145, window five, last reassessment 140 and native 82 unchanged. Assessment 240 is the dedicated primary review; assessment 239 remains the last editorial pass. Next scheduled 146 is research; scheduled 150 requires both primary review and editorial work. The mathematical target survives unchanged, with the three matcher costs now separated.