Assessment 220: compact differences still need a paid extension test

12 October 2026. Dedicated primary-proof review. Written semantic arguments and a conditional reader contract; native construction 74 and root claims are unchanged.

What this brings closer

Two missing targets do not require an ever-growing disjunction of row readers. Three suitably chosen original rows always represent a selected union. The obstacle is choosing those rows on time. Exact row-class equality, which was enough for construction 74, cannot perform that selection by itself.

The primary comparison identifies a narrower next target: an input-paid strict-extension test. It decides whether the next row adds any target to the current union. Decoding and materializing the missing names is one possible implementation, not a necessary interface requirement. The conditional schedule below uses three committed readers and one pending reader for two omissions.

Two primary arguments and their resource boundaries

Minsky, Trachtenberg and Zippel: polynomial reconciliation

Source: author-hosted manuscript, 25 April 2000, 17 PDF pages, including both appendices. This is the inspected edition, not an assertion about the later journal text.

Section 3 reconstructs a bounded subset difference from leading characteristic-polynomial coefficients. Its convolution proof cancels the common set. Incremental coefficient maintenance uses a fixed number of field operations when the difference bound is fixed. Section 4 instead reconstructs a rational quotient from evaluations; tagged evaluations handle zeros, and Appendix B proves uniqueness of a reduced rational function by cross-multiplication and a degree bound. Existence comes from the actual set quotient. Unknown difference bounds use probabilistic verification. Appendix A checks square-freeness and splitting, then uses randomized factorization with expected field-operation cost.

The useful mechanism is cancellation of shared content. The limitation is the unit of work: field elements encode whole target names, and reconstruction, arithmetic, field setup and root finding remain computations. Small communication does not supply deterministic per-symbol decoding. The native final degree gate supplies a known omission bound, so unknown-bound sampling is unnecessary here; randomized root finding is still not a native deadline theorem.

Driscoll, Sarnak, Sleator and Tarjan: persistence

Source: Making Data Structures Persistent, JCSS 38 (1989), 86–124. All 39 pages were independently reviewed.

Sections 2–3 convert linked structures using fat nodes, node copying and node splitting. Bounded ephemeral in-degree controls recursive predecessor repairs; a potential argument gives amortized constant update overhead, while access overhead is worst-case constant for copying/splitting. Information fields, version comparisons and indexed access pointers are stronger than finite native labels and ordinary-name routing. Full persistence branches from an old version; it does not itself merge two versions.

Section 5 is the useful structural control: displaced changes obey path invariants that bound the extra record a reader carries. Its search-tree operations still require logarithmic search/backtracking. The write-once observation in section 6 is a total-time simulation, with an access-pointer exception, not a per-symbol immutable-old-top compiler.

Thus persistence can retain an implemented predicate, but cannot supply its semantic operation or deadline. The special bounded-reader invariant is relevant; importing the general amortized transformation as a strict deadline guarantee is not. The paper’s historical open directions are not asserted to remain open today.

A bounded basis exists, but equality cannot find it

Fix a common finite support $U$ and an integer $k\ge0$. Each selected row is $X_i=U\setminus H_i$ with $|H_i|\le k$. An absent alias is EMPTY and can be ignored; the statement concerns nonempty families of eligible observed rows.

Written semantic proposition. Every nonempty selected union equals the union of at most $k+1$ selected rows. More strongly, retain the first row, and retain a subsequent row exactly when it strictly enlarges the retained union. This process never deletes a representative and retains at most $k+1$ rows.

Proof. After the first row the complement of the retained union is its omission set, of size at most $k$. A strict enlargement removes at least one element from that complement. Removed elements never return. There are at most $k$ later strict enlargements. Skipping a row contained in the current union preserves the union invariant. On a support of $k+1$ points, the $k+1$ singleton rows meet the omission bound and all are necessary; hence the bound is tight. This also covers $k=0$.

This is an existence and oracle algorithm, not a charged producer. Its predicate is

$$E(B,X)=\left[X\not\subseteq\bigcup_{Y\in B}Y\right] =\left[\left(\bigcap_{Y\in B}(U\setminus Y)\right)\setminus(U\setminus X)\ne\varnothing\right].$$

Written interface obstruction. No deterministic equality-only selector can always choose a sufficient fixed-size basis of original rows for all two-omission families.

Precisely, fix a capacity $r\ge1$. Tokens are opaque and the selector is invariant under their renaming. It may compare tokens for equality and keep arbitrary equality bookkeeping, but receives no literal probes, code bits, identities, allocation metadata, containment table or information from store construction. It must finish selecting at most $r$ original rows before final readout, which is their Boolean OR.

Feed $r+1$ distinct tokens. Choose an index $j$ omitted by its output. Take distinct names $h,a_1,\ldots,a_{r+1},b,c$ in $U$, and define

$$H_i=\{h,a_i\}\ (i\ne j),\qquad H_j=\{b,c\},\qquad X_i=U\setminus H_i.$$

Every equality comparison has the same answer as before. All retained rows omit $h$, whereas the omitted row contains it. Their OR therefore fails at $h$. All classes are distinct and have exactly two omissions. Their full union is $U$, including when $r=1$.

The output restriction matters: this is a basis-selector obstruction, not a proof against a global FULL shortcut or a richer readout. Literal probes, structured code bits and precomputed relations are stronger interfaces whose cost must be charged. The statement does not rule out native recognition. It explains why replacing count/XOR by an exact class identifier alone does not extend the previous selection rule.

Polynomial coefficients expose the stronger operation

Here is a direct algebraic specialization, independent of a root-finding implementation. Inject actual names into a field, and write $P_S(z)=\prod_{a\in S}(z-a)$. For $X\subseteq U$ with at most two omissions,

$$P_U=P_XP_{U\setminus X}.$$

Let $e_1(S)$ be the sum of its elements and $e_2(S)$ the sum of products of unordered distinct pairs. Coefficient comparison gives

$$e_1(H)=e_1(U)-e_1(X),\qquad e_2(H)=e_2(U)-e_2(X)-e_1(X)e_1(H),\qquad H=U\setminus X.$$

The known degree $|H|$ distinguishes zero, one and two omissions. In the two-omission case the exact polynomial is $z^2-e_1(H)z+e_2(H)$. If $G$ is the monic polynomial for the current common omissions, adjoining a row replaces it by $\gcd(G,P_H)$. Strict extension means this gcd has smaller degree. Factoring into names is unnecessary for that algebraic test.

This eliminates one proposed dependency, not the native implementation gap. A whole-name field multiplication, division or gcd step is not constant local work. Choosing a field uniformly as the width grows is also charged. A target insertion changes $e_2(X)$ by $w e_1(X)$; the paper’s incremental formula alone does not show that this finishes during the actual target. At the right boundary, a support birth can change the interpreted omission polynomial of every untouched row, so eager rowwise decoding cannot be assumed free. Finally, an $O(n)$ coefficient scan after an alias of width $\ell\ll n$ does not fit that alias’s budget. No arithmetic lower bound is claimed.

The alternative is to prepare an exact extension interface before selection. An arbitrary table of relations between row classes is not free preprocessing. It must be built from ordinary right records with all routing, storage and old observations counted.

Conditional native consumer and next decisive test

Retain the exact syntax, orientation, common widths, one-sided funding gate and immutable store contract of construction 74, replacing only the final gate by at most $k$ omissions. The minimum-degree argument extends directly to checking $m\ge g-k$ for fixed $k$; this does not create the extension predicate.

Conditional transfer. Suppose an independently charged right-store producer supplies ready row references, direct literal readers and a deterministic extension procedure for a basis of at most $k+1$ references and one further row. Once those references are available, the procedure must finish within the next $\ell$ actual alias symbols at a fixed service rate, with bounded degree/radius, finite labels/control, immutable old-top updates and $O(\ell)$ work, observations and allocation. Initialization and any required descriptors must already have been paid by actual input. No random failure, free table, side channel or EOF work is permitted.

Then a source capsule keeps a committed greedy basis and one pending row. During its next selection’s alias, finish the old pending extension test in parallel with the new ordinary alias lookup. Append the old pending row exactly on strict extension; at the endpoint install the newly looked-up row as pending. Absent aliases contribute EMPTY. No job is charged to symbols before its references are available. A source that is not selected again keeps its pending row without forcing completion.

An immediate final query ORs the at most $k+1$ committed literal readers and the one pending reader. For $k=2$ this means three committed readers plus one pending, not an unjustified three-reader deadline claim. Actual source and target pins pay routing and traversal. This is one fixed consumer for fixed $k$ and fixed service constants. The missing producer remains an explicit premise, so no new native theorem is admitted. Recognition would transfer to a PEG for the reversed checked language only.

The next mathematical test is now precise: construct this short-alias strict-extension procedure for two omissions, or isolate a failure of a concrete proposed producer. Polynomial gcd avoids root extraction semantically; structured class relations may avoid scanning long coefficients. Neither is yet charged. The next editorial pass preserves this target. Removing the width gate and joining independently prepared stores remain separate.

Evidence and continuity

The finite audit checks greedy prefix unions and literal answers, adversarial basis-selector assignments, exact difference-polynomial evaluations, coefficient cancellation, polynomial gcd and the extension identity. Python sets and field arithmetic are oracle operations; this is not a native execution test. Independent review approved the semantic bound and the equality-only obstruction, including the one-reader edge case and the restriction excluding FULL shortcuts.

Primary PDF custody remains outside public source. Source hashes and actual validation/publication receipts are retained in integration state. Prior replay slices, affine joins, causal sharing and coherent unary algebra remain controls: none supplies the missing extension producer. Imported entry 129 retains its growing alphabet and conditional state/port bounds.

Scheduled completion advances from 124 to 125 once. This is a primary review with supporting arguments, not a new native construction or separate mathematical-result count: mathematical 137, window three, last reassessment 134 and native 74 remain. Next 126 is editorial; next dedicated primary review is 130. No root admission or external priority claim.