Assessment160: learned Boolean parity cells

8 October2026. Ordinary research, scheduled66. One written native extension, substantive108/window4 after reassessment104. Next67 is research; the dedicated multi-paper review remains70. No root admission or literature-priority claim.

Result and proof ownership

Reader40 in the maintained relation-composition reader owns the complete proof. For each fixed r, arbitrary Boolean unions of cells of an input-learned common rank-at-most-r parity partition are now recognized and composed natively. Right intermediate and target names have one checked width n; records may interleave and repeat; left fanout is unbounded; endpoints arrive as ordinary late names. The construction checks the row and shared-rank conditions itself.

The initial missing affine quarter target was carried through its directly useful generalization. Let U=2^n and B=U/2^r. A final row of size mB has all Fourier coefficients on the B grid. At q_m=mB-B/4+1 distinct points, its prefix spectrum differs by less than B/4 at every frequency. Heavy-coefficient discovery at B/2 and nearest-grid rounding recover the exact final spectrum. There are only constantly many heavy prefix energies at each level. Exact binary arithmetic and all cube-word lookups cost O_r(nU), fitting the remaining ordinary input at a fixed service rate. The remaining records also fund old-row reinterpretation.

This gives an explanatory sufficient condition for composition and reuse: a discrete determining invariant must be recoverable before the last records, and its residual input budget must cover both classifier construction and repair of old answers. Finite semantic size by itself supplies neither operation.

Primary comparison and reuse classification

Eyal Kushilevitz and Yishay Mansour, Learning Decision Trees Using the Fourier Spectrum, SIAM Journal on Computing22(6),1331-1348 (1993), DOI10.1137/0222080. Primary paper copy. Read the complete selected section3 mechanism: Lemma3.2’s prefix projection, Lemma3.4’s Parseval bound and Figure1’s exact energy recursion; also compared the sampled replacement and Lemmas3.5-3.9/Theorem3.10. Their recursion keeps frequency prefixes with sufficient squared mass. Parseval bounds the number per depth, and a heavy coefficient keeps every ancestor. Their efficient query learner estimates these masses by sampling. Reader40 instead evaluates them exactly from a paid cube table, exploiting the geometric decrease of table lengths to charge bit arithmetic. Its positive stream supplies no target membership oracle.

Mahdi Cheraghchi and Piotr Indyk, Nearly Optimal Deterministic Algorithm for Sparse Walsh-Hadamard Transform, ECCC TR15-076 (2015), later SODA2016. Primary report. Compared section1.1 and Theorem1’s exact quantifiers and encoding: deterministic nonadaptive query access, polynomial-length coefficient bits, and an l1/l1 approximation to the largest k transform entries. Its approximate sparse-recovery guarantee does not directly deliver exact recovery of our noisy prefix spectrum, and its query model does not pay native name routing or enforce our deadline. Its full condenser/recovery proof was not reviewed here and is not imported.

Primary files retrieved9 October2026 UTC and retained privately under the data custody directory. PDF SHA256 values:

• Kushilevitz-Mansour: 2e931b4a48f7227216920d63d8bfea4e84ed7a724c774b049d4507536f39d635.
• Cheraghchi-Indyk: 0969a24c32f38db9d82d5d5d8fcf6c5433c859f0d1efb919779037471f4c6c23.

The first PDF’s text map was unusable, so selected pages6-12 were rendered and OCR-read. The actual exact-energy proof and sampled replacement were recovered; no abstract-only claim of proof review is made. This targeted comparison is part of ordinary research, not another dedicated-review count.

Scholarly judgment: an extension using known Fourier-learning methods, with high confidence in that classification. The Lab’s added content is exact recovery from near-complete positive input and its full persistent deadline accounting, including rebuilding completed masks. Broader priority for that combined native statement is not asserted. No new Fourier-learning theorem is attributed to this assessment.

Actual-draft audit

The writing agent audited the actual draft; no separate reviewer is claimed.

1. Spectral support lies in the normal space by kernel translation. Fibre grouping gives the integer B grid even when actual rank is smaller than r. Conversely, Fourier inversion proves that support span is the least possible normal space. Therefore the final shared-rank test is necessary and sufficient for this class.
2. Missing positive points perturb every unnormalized coefficient by at most their count. Threshold B/2 separates exact nonzero support from noise; rounding is unambiguous. The zero-frequency coefficient fixes the proposed cardinality.
3. The energy denominator is 2^k U, not U alone. Prefix tables split into two halves, not adjacent lexicographic entries. Frequency words are reversed into native readout orientation by charged scans.
4. Each level keeps at most 44^r prefixes even on malformed input. Candidate lists have twice that bound. Signed additions, squaring, long accumulators, power-of-two comparisons and table traversals are included. Summing 2^(n-k)(k^2+n+k) gives O_r(nU), including abandoned tables. No unit-cost growing integers or random-access array operations enter the native proof.
5. Inverse values must be Boolean and have the tier’s exact cardinality. Frozen prefix containment is checked explicitly, not inferred from a spectral promise. Captures include duplicates and the crossing word. Reversing captured word lists, where required, is included in their O_r(n) membership cost.
6. A failed tier is provisional. Each later snapshot contains all old distinct input. On a valid size-mB row, q_m is the last tier, so its determining sample and every later containment check succeed. Exact certification never triggers a basis change and may occur on the last right symbol.
7. The distinct counter uses constant many phases of input-built B/4 capacities. Small widths use one finite table. Constants depend only on fixed r, not n or an advice family. Initialization, all retained versions and input orientation are explicit in reader40 and its reused reader39 compiler.
8. Coefficient-index batches complete during actual intermediate spellings; candidate and index snapshots publish together. The final notice leaves more than nB/8 eligible bits. A full O_r(nU) rebuild, including old rows, finishes before composition. Earlier jobs are charged when performed.
9. Candidate truth tables are reinterpreted in the current shared basis. Merely retaining masks would be wrong after coordinate changes. Index Overflow is provisional; final support span, complete certification and compiler readiness gate acceptance. Pointer identities, addresses and timing encode no answer.
10. Every final certified row equals its candidate by containment and cardinality, including on unpromised input. Common-width checking and absent-row empty masks are retained. Late source and target spellings pay their own routing and syndrome computation. The final output has no EOF pass.

Finite evidence

tools/research/check_parity_cell_learning.py passed:

• All65,536 subsets of the width-four cube: pruned heavy coefficients exactly match direct character sums, including arbitrary unpromised prefixes.
• 4,368 literal cell rows at widths5-8 and ranks1-3, with13,104 determining-prefix orders and77,136 recovered coefficients. These include64 missing-quarter rows.
• 46,080 product queries and384 retained compiled versions after basis changes; 12 examples rejecting individually valid rows with excessive shared rank.
• 504 symbolic horizon inequalities,40 interleaved histories with real duplicate records in half the cases, and160 index-ready notices.

Literal point sets and direct character sums are independent semantic oracles. The native cost proof is written separately; Python big integers are not evidence of constant-time arithmetic. Deadline tests use explicit worst-case release bounds and actual remaining record counts, not measured interpreter speed.

Failed routes, controls and seven-surface disposition

The positive affine hull of a missing-quarter row is the full cube. That learner cannot recover its hole. Spectral support plus a Boolean truth table repairs this specific failure. The standard full transform’s arithmetic-operation estimate also leaves bit costs unpaid; the exact pruned table recurrence supplies the required geometric bound. Neither failed route establishes native impossibility.

Reader39 supplies the charged index and global compiler. Assessment141’s adoption debt is paid by actual old-row observations;154’s ready-answer requirement is met before composition;156’s final-rank warning is handled by provisional overflow and charged restarts. Completed replay slices, affine-fragment joins, persistent causal algebra, unary coherent algebra and native SAT controls remain unchanged. Entry129’s imported matching and positive-path theorems retain their own exact contracts and growing alphabet; no port-to-native lower-bound transfer is claimed.

Source custody: primary PDFs/OCR remain private. Claim mapping: written extension108, CLAIMS.yaml unchanged. Proof/evidence: one maintained reader40 plus the independent checker. Failure ledger: affine-hull information loss and unpaid full-transform bit costs, each with its specific repair. Frontier: decoupling intermediate-key width from target width, then genuinely heterogeneous targets and independent joins. Reader synthesis: existing program and reader. Site representation: existing homepage entry, same Site audience and routes; parity receipt in integration state.

Validation and next decisive test

Full make verify:1490 passed,1 skipped,5 failed in246.31s. The unchanged failures are four custody-provenance tests and the unavailable-lake frontier test; later make targets were not reached. The semantic checker, Python syntax and initial Site build/scoped checks passed. Final source checks and publication parity are recorded in integration state. No browser handoff is needed for this background publication.

First remove equality of intermediate-name width and common target width. The current whole-dictionary rebuild costs O_r(n2^n) because there are at most2^n intermediate names. Arbitrary intermediate names remove that bound while one late row still supplies only Omega(n2^(n-r)) target bits. Test an occupied-key compiler with a checkable service condition, or isolate the exact stale-mask/readout obstruction. A final small rank alone does not pay rescanning arbitrarily many old keys. Unequal target widths and independent-store joins remain subsequent frontiers. Preserve all SAT/scaffold results and cadence70.