Later-partition adoption: conservation of earlier incidences

Assessment 141, 8 October 2026. Written certificate argument and restricted observation obstruction. No new native recognizer, root admission or novelty claim. Reader 26 remains the maintained native result.

Result and payoff

A target can transfer every complete old partition class to a refinement without revisiting its members. For incomplete classes, resetting the target’s mask and coverage deficit loses information: exact adoption must also account for the old incidences that have not been reclassified. A separate nonnegative counter makes that loss explicit and prevents false acceptance.

The resulting certificate is exact under paid incidence re-observation, but it does not repair the original unrepeated-input problem. Even a split into two large classes has a linear observation requirement for the isolated-summary contract below. This identifies what a stronger persistent operation must supply: a target-correlated certificate for the intersection with a newly discovered class, prepared before the adoption deadline. Global populations, old masks and cardinalities alone do not supply it.

1. Fixed partitions and the certificate invariant

Fix a finite row universe V and a bounded chain of refining partitions P_0,…,P_b. These are mathematical parameters of the certificate argument, not input-supplied decompositions of a native language. Reader 26 constructs particular bounded historical partitions from actual anchor incidences. For a target w, let E be its set of distinct predecessors observed so far. Repeated clauses do not enlarge E.

At its currently selected partition P_i, retain:

• a set A of credited predecessors, with A contained in E;
• a finite count c_C=|A intersect C| for each class C of P_i;
• a debt rho=|E minus A|;
• a coverage deficit q_C=|C|-c_C for each selected class c_C>0.

A and E are sets in the correctness proof. An implementation need not enumerate them. Exact-pair records distinguish first occurrences from duplicates, while finite generation tags record whether a pair has already been credited in the current adoption generation. Class populations and counts have immutable unit-stack representations; deficits and sums can use reader 20’s batch stacks. No arithmetic-valued labels are assumed.

Certificate lemma. The two conditions rho=0 and q_C=0 for every selected class hold exactly when all historical incidences have been credited and E is the union of the selected P_i classes. In that case the class mask answers membership in E exactly, hence also answers every intersection test S intersect E nonempty by a finite disjunction over the selected classes.

Proof. rho=0 and A contained in E give A=E. Each selected class is full exactly when its count equals its population. Empty classes in the mask contribute nothing. Conversely, complete credited coverage satisfies both equalities. The additional rho gate cannot be inferred from the q_C gates.

2. Transfer complete cells; conserve the incomplete part

Allow adoption from P_i to a later refinement P_j only when rho=0. Coverage may still be incomplete. Let F be the selected old classes whose deficits are zero. Every member of every C in F is already in E. Every P_j class D contained in such a C can therefore be inherited as complete without examining any pair. Set

A_new = union of all C in F,

rho_new = sum of c_C over selected old classes C not in F.

The new selected mask initially consists of the descendants of F. Their counts are the new class populations; every other count is initially zero. These assignments establish the invariant because the old credited set was exactly E. Only the earlier edges in incomplete old classes require further evidence. This is stronger than discarding all earlier credits and replaying the whole target, and avoids an unnecessary scan of already complete blocks.

On a subsequent ordinary incidence (v,w), use its actual name to retrieve its old-pair flag, current generation tag and current class D.

1. If D is inherited complete, the pair is already known to belong to E and A. No debt or coverage debit is allowed.
2. If the pair is already tagged with the current generation, do nothing.
3. Otherwise, if the pair was in E, debit rho once. If it is a genuinely new pair, enlarge E and do not debit rho.
4. Credit v, update the generation tag and increment c_D once. When D first becomes selected, install its population as a coverage batch; then debit that coverage batch for v.

A new edge cannot discharge an old edge’s debt. A duplicate cannot discharge the same debt twice. Inherited full classes cannot be used as replay credit. The inclusion A contained in E and exact count identities follow inductively. No count becomes negative. The universe and the chosen historical classes are frozen, so later anchor changes do not invalidate a credited membership.

A later adoption is permitted only after rho returns to zero. This condition is essential to the stated transfer rule: if prior debt remains, c_C describes only credited edges, and summing partial-class counts would forget still older edges. One may stay at an earlier stage while debt is outstanding, then adopt any available later stage. The number of generations is bounded by b+1; tags are finite labels. No counter reset overwrites an old version.

3. What is charged, and what this does not prove

For a fixed partition budget, the transfer is a fixed-size table operation. Copy prepared population roots for inherited descendants, add the finitely many partial-cell count roots as debt batches, and discard the old current certificate by changing its root. Each actual record performs at most one new-pair increment, one debt debit, one coverage debit and one tag update. Per-target coverage is necessary here; the aggregate historical cell deficit of reader 26 cannot simply delete one target’s contribution by a pointer reset. Keeping local counters and a global count of invalid targets supplies a constant-size boundary test once the named target state has been accessed.

These are operations at already routed records, not a free-handle native representation theorem. Applying them natively also requires the actual ordinary-name dictionaries, simultaneous old-top update schedule, finite labels/control, fixed degree/radius, fixed initialization, and the finite lazy row-code readout of reader 26. All preparation, old-storage observations, allocation, retained versions, pointer routing, speculation and final output must remain charged to the actual encoded input length N. A lookup available only after a target name cannot pay for an earlier scan of all its predecessors. No timing conclusion is drawn from the Python model.

Under any realization of those access obligations, this certificate accepts only streams in which necessary re-observations actually occur. It does not insert repeated clauses, provide a supplied replay block, or convert an unreplayed instance into an accepted one. If a stage leaves r old partial-cell edges uncredited, exactly r distinct old-edge re-observations are necessary for this rule to discharge the debt; fresh edges and any number of duplicates of already credited edges contribute zero to it. Their actual name spellings, not merely r, determine the encoded work available.

An offline expansion that appends copies of old records after a refinement may make the certificate succeed, but it supplies extra input time. Bounded total expansion, semantic equivalence of duplicate clauses and a small number of stages do not establish a deadline-preserving native inverse translation. The Lab’s completed replay-slice results already warn that repetition can alter native recognizability. This run makes no new closure or PEG claim. Were a native checked-stream construction completed, its immediate PEG consequence would concern reversal of that complete checked language.

4. The late split test, including the tempting false zero

For the smallest retained split control, rows v,x initially share a class and y has a separate class. An early target E={v,y} is incomplete on {v,x}. A later partition separates v and x. Adoption inherits {y} and leaves debt one for v. Seeing y again does not pay it; seeing v again pays it and gives exact coverage. Without that additional occurrence, the target still fails. Thus the proposed adoption mechanism does not resolve the original late-split input.

For a direct unsoundness witness, take an old class {a,b,c}, old target E={a,b}, and refine to singleton classes. Reset the mask and process a again. The newly selected class {a} has zero coverage deficit, but b is forgotten. The retained debt is one and correctly prevents acceptance. Testing only the new coverage deficit would produce an incorrect future answer on source {b}.

Complete classes really can be retained: start with old classes {0,1} and {2,3,4}, target {0,1,2}, then refine to {0},{1},{2},{3,4}. The first two singleton classes transfer at no incidence cost. Only edge2 is debt. A new edge3 cannot pay it; replay2 pays it but coverage of {3,4} remains incomplete until edge4. These controls distinguish transfer, debt and coverage as three different facts.

5. A charged obstruction for isolated-summary adoption

Here is a precise restricted observation contract, not a lower bound for all persistent representations. At adoption the target store exposes only its old coarse class mask and exact per-class cardinalities, plus fixed metadata independent of which row names belong to the target. The shared new partition and all its populations may be available for free, making the task easier. To learn further target-correlated information, the algorithm may ask membership probes v in E. A probe returns only that Boolean answer. There are no target-correlated receipts, structural side channels, exposed numeric addresses, root-identity tests, data-dependent metadata or timing observations. Control flow can depend on previous probe answers. Unlimited internal computation and adaptive choice of named probes are allowed.

Restricted exact-coverage lower bound. Let an old class C split as C_0 disjoint-union C_1, with both new classes of size m>=2. The store reports |E intersect C|=m. On the valid instance E intersect C=C_0, any deterministic exact verifier of complete-new-class coverage must make at least m membership probes in C in the worst case, even with negative answers and all populations known.

Proof. Follow the verifier’s transcript on C_0. If fewer than m distinct rows have been probed, choose unprobed x in C_0 and y in C_1. Replace E intersect C by (C_0 minus {x}) union {y}. Its cardinality and all exposed initial summaries are unchanged. Every positive and negative answer on the actual transcript is unchanged; by induction adaptive choices and control flow are unchanged too. The original is a complete class. The replacement is neither C_0 nor C_1 and is not a union of new classes, since m>=2. The verifier must give different answers, a contradiction. Repeated probes only increase cost. This is an ordinary indistinguishability proof, not an information-volume argument.

The split budget stays constant. To realize the obstruction inside the retained two-anchor pattern, take rows C_0,C_1,marker a and row y. Initially anchor A contains C_0 union C_1 union {a}, and anchor B contains only y. An early target contains C_0 union {a,y}. Next B gains a, then every row in C_0; finally A gains y. The first late B edge creates one snapshot, and the last A edge creates the next. Final historical cells distinguish C_0,C_1,a,y with fixed k=2. The target is now a union of cells. The good final relation has exactly two predecessor types: A is universal and B equals the early target. Swapping unprobed x,y’ with x in C_0 and y’ in C_1 changes only that target’s old incidences, preserves its old coarse counts and mask, but yields incomplete new cells and a third final type. Earlier detailed target-correlated state would distinguish these instances; the restricted contract deliberately excludes it. The comparison is for a recognizer that checks its structural language conditions, including the final type bound. It is not a lower bound for a promise algorithm that may behave arbitrarily on the three-type replacement.

For names of length O(log m), strict input and late-pin deadlines provide only O(log m) local steps per such name, not m independent observations. Use equal-length row names and a normalized key of that same length; the fixed normalizer g(0^j1z)=z gives the three target names with fixed j. This keeps normalized degree one and fixed dyadic span. There is no claim that a fixed normalizer expands a constant-length target to an unbounded key. Thus deferring this isolated-summary validation to one late record fails for unbounded m. This does not exclude background preparation during the earlier input, richer shared receipts, another representation, or direct recognition without this certificate. In particular, it does not transfer entry 129’s matching-port theorem to branching persistent stores.

6. Consequence for the next attack

The successful structural operation is transfer of already certified complete blocks through shared refinement. The missing operation is exact restricted intersection for an incomplete historical block, selected by a late target without revisiting all its earlier names. A useful next construction must prepare a target-correlated receipt while those names are present, and expose its current interpretation within the later target spelling. Merely retaining a raw predecessor list, copying its count, or opening its root is insufficient.

Next test: a target-indexed persistent incidence representation with a shared row-refinement action. Start with two final target types and the growing C_0/C_1 family above. Track whether predecessor-first input can build the required target index without reversing a name for free, and whether a query can validate or bypass the intersection certificate in its actual name budget. Any obstruction must explicitly allow shared receipts before claiming more than this isolated-summary failure.

Evidence and integration disposition

Python checker compares incremental debt, coverage and generation tags with independent literal-set oracles. It tests nested adoptions, forbidden adoption with debt, duplicate and fresh edges, retained versions, exact composed answers, the early split, inherited complete cells, the false-zero reset, and balanced-split indistinguishable transcripts. Partitions are supplied to this finite diagnostic only. It is not a native implementation or timing experiment.

No primary-source intake. Root CLAIMS.yaml unchanged. This note owns the written local argument and failure contract; the standalone diagnostic owns finite evidence. The live program, agenda and checkpoint name the same intersection-receipt target. Site reader 26, homepage and public program entry are unchanged because no new native endpoint was established. Publication of this supporting investigation is through GitLab only; a later coherent reader advance may incorporate its useful boundary. No paper, edition, audience, embargo or installed reverse-writer status changes.