Universal refinement receipts and the bounded-diversity escape

Assessment142, 8 October2026. Conditional transfer using the established OMv framework, alongside the new written native construction in reader27. No separate novelty claim, root admission or mathematical count for this note.

Assessment146 continuation: the exact degree-free gate

Reader29 now applies the snapshot mechanism to the precise extension of reader28 with unrestricted normalized row degree. Identity normalization makes per-key width one automatic, so arbitrary matrices satisfy the gate, unlike the global bounded-diversity restriction discussed below. Equal-length names also meet span zero. The native-to-OMv implication is unconditional; the resulting nonrecognition is conditional. A one-key independent-join control explains why reader28’s serial access is not a constant-cost join of completed independent stores. The earlier argument and its global-diversity qualification are retained.

A stronger receipt contract is conditionally too strong

The next step after assessment141 was to allow arbitrary target-correlated preparation instead of exposing only old masks and counts. Consider the following uniform deterministic contract. Ordinary predecessor-first right incidences prepare arbitrary finite target sets X_i in O(N) work and storage. A later monotone anchor update selects a row subset B, with bounded local work/allocation per actual input character. After a constant number of historical partition snapshots, an ordinary target name must determine whether X_i intersects B, at its final delimiter. Full, partial and empty intersection classification is at least this strong. Fixed degree/radius, finite labels/control, old-top semantics and fixed initialization remain; all preparation, observations, allocation and routing are charged. The interface is required to answer on arbitrary prepared target families, not only globally validated two-type inputs.

Conditional obstruction. Such a native receipt would give deterministic online Boolean matrix-vector multiplication in O(n^2 log(n+2)) word-RAM time and words, including all preparation and persistent forks. Thus it contradicts the OMv conjecture. The implication to the fast algorithm is unconditional; nonexistence depends on the conjecture. It is not an entry129 port bound.

Here is the actual two-anchor encoding. Given an n-by-n Boolean matrix M, create row names x_1,…,x_n and two marker rows a,y. First anchor A contains all x_j and a; second anchor B initially contains only y. For each i create actual target t_i with predecessor set

X_i = {a} union {x_j : M_ij=1}.

The marker makes even an all-zero matrix row an actual target. It never belongs to B and contributes nothing to the queried intersection. These are ordinary charged edges, not padding or idle work symbols. The first two targets are A,B; all row names are already present before the first t_i, as required by the historical frozen-universe contract.

Choose L=ceil(log2(n+2)) and distinct L-bit suffix names for all rows and distinct L-bit target identifiers for A,B,t_i. Encode target identifier q as paired bits 0b, followed by 11 and the common key z=0^L. The fixed subsequential normalizer discards paired identifier bits through the aligned 11 and outputs z. Every target has length 3L+2; every row and normalized key has length L. Thus normalized degree is one and dyadic row/key span zero. The growing target alphabet has been encoded into a fixed binary alphabet with all names charged.

Save the prepared native configuration. For each arriving Boolean vector v, fork from that saved configuration and add (x_j,B) for precisely its one bits. Finally add (y,A). The first new B incidence, if any, triggers one historical snapshot; the final new A incidence triggers another. Including initial closure, there are at most three snapshots. Each row changes anchor profile at most once after preparation. The current B class is a union of at most a fixed number of historical classes. A receipt supporting all such class intersections can OR those finitely many answers.

For each i, fork the vector configuration, read ordinary target name t_i and output the intersection bit. It equals OR_j(M_ij AND v_j). Output all n bits before receiving the next vector. The matrix preparation has O(n^2) records, each O(log(n+2)) characters. Across n vectors there are at most n^2 activation records, n closing-anchor records and n^2 target queries. Total simulated characters are O(n^2 log(n+2)); vector input, name generation and output fit the same bound. The native simulator charges constant word-RAM work per character for a fixed machine’s bounded observation and allocation. All abandoned branches are retained and counted. Its addresses have O(log(n+2)) bits, and the literal bit-space bound is O(n^2 log^2(n+2)).

Forking is a simulator consequence of immutable old records, not an added input operation. Store the top pointer and finite control; each branch reproduces its own ordinary sequential continuation. Native rules cannot observe the simulator’s address gaps, identity, timing or global allocation count. This is the same all-representation snapshot argument already proved in the Lab’s OMv boundary. No general inverse reduction or same-orientation PEG transfer is assumed.

The primary assumption is Henzinger, Krinninger, Nanongkai and Saranurak, Unifying and Strengthening Hardness for Dynamic Problems via the Online Matrix-Vector Multiplication Conjecture, arXiv:1511.06773v1, Conjecture1.1, printed page3. Its online protocol and conjecture were checked against the primary source on8 October2026. This was a targeted source check, not a scheduled multi-paper review. Their stronger bounded-error conjecture implies the deterministic restriction used here. No primary PDF or extracted text is committed. Source: https://arxiv.org/pdf/1511.06773 .

Why this does not settle bounded final diversity

The prepared X_i are arbitrary matrix rows and may have n distinct predecessor sets. A recognizer of a language that checks at most two final target types may reject such a prefix/final instance, without implementing an intersection oracle on it. It is invalid to obtain answers by ignoring that gate. The conditional obstruction therefore retires the universal receipt specification, not the actual bounded-diversity question. A persistent implementation may exploit a joint structural check instead of evaluating arbitrary partial sets.

The successful escape in reader27 uses a bound on simultaneously distinct predicates after every right record. It stores all K current representatives and their pairwise exact Hamming distances. One insertion changes one predicate by one element; equality with a current class is detected by a distance-one counter and the named row’s bit. Slot populations decide whether to replace a singleton class, split off a target or merge it into another class. No extensional set scan or broadcast occurs. Class labels can be reused safely because empty slots have no current aliases and old configurations retain their old headers.

This is a charged deterministic discovery mechanism, unlike a semantic quotient alone. Global K roots are available before the predecessor name starts. The key-first corollary selects one local K-slot header during the actual key prefix, before the remaining row suffix. It allows unboundedly many interleaved groups, with no supplied class labels or relative name-length bound. The stronger arbitrary-family intersection interface is never exposed.

Final diversity is weaker than prefix diversity: diagonal incidences followed by completion to the all-ones relation have unbounded prefix diversity and final diversity one. Likewise, late target-side key discovery is weaker than key-first routing. Neither extension follows from the positive theorem or this negative transfer. These are the remaining frontiers, not evidence of universal native nonrecognition.