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feat(logspace): instantiate stationary polynomial atom rows - #1766
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The Phase 4 estimator needs to retain only a bounded list of exceptions without losing stability when that list changes. Formalize the native two-coordinate compression update of OpenAI2026LogSpace Section 8.2, equations (8.5)-(8.6), and all conclusions of Lemma 8.1. Prove small-entry replacement, signed large-entry preservation, coordinatewise interpolation, monotone defaults, strict candidate capacity and containment, and uniform 28-Lipschitz stability. Also prove the local 6u change bound used by the subsequent preloss argument. Add concrete tests of partial clipping, signed values, positive-default padding, empty rows, candidate selection and perturbations. This advances #1721 and #1444. Iterated propagation and its slack conversion (8.7), rarity estimates, recursive estimation and machine-space realization remain open. No selected formal abstract or roadmap endpoint changes. Validation on Lean v4.36.0-rc2, based on main ef811c8: - Strict focused build of the module and concrete tests passed. - scripts/check.sh passed all six gates: strict full build, mk_all check, strict full tests (including the global axiom audit), lint, lint-style, and checkInitImports. - check_changed_files.py, check_tidy_surface.py, and check_hygiene.py passed. - Focused hygiene with --check-simp and git diff --check passed. Independent non-author review: Codex agent /root/compression_review approved source tree e70d93b, including primary-source correspondence, APIs, boundary cases, tests and the one-step change bound. No requested fixes remain. This is agent review, not human approval. Worker: codex-row-compression-20261030, branch feat/logspace-compression-update. AI assistance: OpenAI Codex implementation and independent Codex review.
Add concrete increment-and-compression states and schedules. Prove width-independent 28^j propagation, exact restarts, a fifth-power slack bound under an explicit group-count condition, and accumulated row/default loss before overflow. This establishes the update-only part of OpenAI2026LogSpace (8.7) and the unweighted part of Lemma 8.2; weighted preloss and the final root functions remain separate. Validation on base c3f7a58: direct Lean 4.36.0-rc2 checks of both new modules with project options passed without diagnostics, using isolated output files and the parent checkout dependencies read-only. Concrete tests cover a two-group nonzero-loss execution, satisfied no-overflow bounds, an actual overflow after lowering capacity, and distinct initial-state perturbations. Focused proof hygiene including --check-simp and git diff --check passed. Aggregate regeneration and full project gates are delegated to integration. AI assistance: ChatGPT implemented and checked these proofs as worker codex-compression-iteration-20261030. Independent agent source/API review: codex-source-release-20261030 approved the library and tests at 7d08e44bbdb626b74adaa9c4d63e07faa28ce525. Publication adds only the generated aggregate import. Integrated targeted build, tests, aggregate check, hygiene and whitespace checks pass. Full gates run in PR CI.
Use decreasing power-of-two slacks and the explicit floors eta * (1/1024)^slack. Derive consecutive floor separation, both finite-sum bounds from source equation (8.8), and the local 8*rho <= eta margin. Bound propagation for the actual remaining group count, distinguishing updates after the current group from those including it. The source uses unscaled inverse powers of two; this mathematical witness explicitly scales by eta and does not choose an exponent for the full integrated estimator. Validation: direct Lean 4.36.0-rc2 checks of Floors.lean and RowCompressionFloors.lean with project options passed without diagnostics, using private outputs and parent dependencies read-only. Tests compute nonzero floors, check halving and empty schedules, and apply the sum theorem to a concrete no-overflow compression execution. Focused hygiene including --check-simp and git diff --check passed. Base is iteration milestone 7d08e44bbdb626b74adaa9c4d63e07faa28ce525; aggregate and full project validation remain integration tasks. AI assistance: ChatGPT implemented and checked the proofs as worker codex-compression-iteration-20261030. Independent agent source/API review: codex-source-release-20261030 approved module/tests at bd14fc247724f63e650ced31e7b0e6c4d30fcee1. Publication adds only generated import registration. Integrated strict module/test build, focused hygiene and whitespace checks pass. Full gates run in PR CI.
Supply a computable rational degree-seven transition and its rescaling, with real endpoint, monotonicity, and value/complement fourth-root Lipschitz laws. Exact rational cast lemmas connect the arithmetic definitions to the real analysis. Concrete tests cover interior values, affine rescaling, clamping, equal endpoints, and root perturbations across both transition boundaries. This advances issue #1721's fourth-moment route. The pinned primary source OpenAI2026LogSpace, atoms.tex Section 5.1 at revision 7ef5c0cc, uses the same normalized-integral construction with s = 8. This implementation specializes it to s = 4 and does not claim eighth-root regularity, complete hierarchy admissibility, or machine resource bounds. No OpenAI Lean was consulted. Validation on base 2d1638c with the two added files: both modules compile under the repository's Lean options with warnings as errors using isolated temporary outputs and read-only existing dependencies; focused check_hygiene.py --check-simp and git diff --cached --check pass. The final documentation-only wording change was inspected after those builds. Full repository integration gates and independent review remain for coordinator integration; no aggregate file has been edited. Authored by worker codex-logspace-polynomial-cutoffs-20261030 with ChatGPT. Independent agent source/API review: codex-compression-iteration-20261030 approved source a6967d13aafee47ae16b61ac2b12f533426f9971 and freshly compiled both files. Publication adds only the generated aggregate import. Integrated strict module/test builds, hygiene and whitespace checks pass. Full gates run in PR CI.
Remove three aggregate imports that came from untracked source copies during integration. The polynomial switch module and tests are unchanged. Validate import targets against the committed Git tree before publication.
Prove the weighted preloss assertion of OpenAI2026LogSpace Lemma 8.2 for the actual compression recursion, with explicit constant12(1+eta)/eta. Apply it to the concrete dyadic floors, discharging numerical small-floor conditions. Stationary-reference preservation, relative stability after overflow, and the complete estimator remain separate obligations. Independent agent source/API review: codex-logspace-polynomial-cutoffs-20261030 approved599c5e8a and the dyadic application delta4cc6581b. Published library/tests are identical to the reviewed source, with generated aggregate registration. Strict private Lean source/test checks, hygiene and whitespace checks pass. Full integration gates run in PR CI. Worker codex-compression-preloss-20261030; AI-assisted implementation and review.
Define the padded first-coordinate rank and prove concrete threshold, default, row-bound and rank preservation results for stationary references. Above the flat region, the rank stays fixed; inside it, the rank remains in the flat region. A gated row output with separated entry support therefore survives each update and any finite zero-increment schedule with floors at most eta and capacities at least the fixed positive rank K. This formalizes the stationary-output assertion of OpenAI2026LogSpace, Section 8.3, Lemma 8.2, revision 7ef5c0cc. It does not assert equality of whole rows, a root-discrepancy estimate, or post-overflow stability. The explicit support and gate hypotheses do not assume the preservation conclusion. Validation on base 7d08e44bbdb626b74adaa9c4d63e07faa28ce525: direct Lean 4.36.0-rc2 checks of both new files passed without diagnostics, using private outputs and existing parent-checkout dependencies read-only. Tests exhibit an above-flat rank fixed at 1/2 while a zero entry moves to 3/8, an arbitrary stationary iteration retaining nonzero output 21/2, and a low rank moving from zero to 1/8 while output 41/4 is preserved. Focused proof hygiene including --check-simp and staged git diff --check passed. Aggregate regeneration and the full project gates remain integration work. AI assistance: ChatGPT implemented and checked these proofs as worker codex-compression-iteration-20261030. Independent agent source/API review: codex-logspace-polynomial-cutoffs-20261030 approved 10a516829a8cf8ea4fd0da6a65abe00147525b46. Publication changes only dependency ancestry and generated aggregate registration. Integrated strict module/test builds and committed import/hygiene checks pass. Full gates run in PR CI.
Use Q(t) = a*u^4/(1+u)^3 with u = max(0,(t-theta)/a). For a>0, prove nonnegativity, the upper ramp bound, lower defect theta+3*a, monotonicity, and global Lipschitz constant one. The fourth root is globally Lipschitz with explicit constant |a^(1/4)|*|a^-1|; the proof also bounds its derivative at arbitrarily large positive inputs. Definitions use exact rational arithmetic and have real cast lemmas. This is an alternative rational witness for the correction-ramp conditions of OpenAI2026LogSpace Section 4.1 at revision 7ef5c0cc, adapted to issue #1721's fourth-moment route. The primary source uses eighth roots. Full hierarchy threshold/cutoff instantiation is a separate next component; no machine resource bound is asserted and no OpenAI Lean was consulted. Validation atop a6967d13: the module and focused tests compile with the repository Lean options and warnings as errors, using private temporary outputs and read-only existing dependencies. Tests cover exact rational values, negative and large inputs, real cast compatibility, uniform ramp bounds, ordinary perturbation, and the explicit fourth-root Lipschitz constant 8 when a=1/16. Focused hygiene --check-simp and diffcheck pass. Independent review and full integration gates remain for the coordinator. Authored by worker codex-logspace-polynomial-cutoffs-20261030 with ChatGPT. Independent agent source/API review: codex-source-release-20261030 approved d3d62d9ff7fe1e54fabbf8b3ce779f929736f6ba. Publication preserves source/tests and adds generated import registration. Integrated strict module/test builds and hygiene/committed import checks pass. Full gates run in PR CI.
Extend HierarchyParams with the strict core threshold ladder, a positive iteration count, strict copy-count inequality, monotone bounded cutoffs, row-gate closure, and root/complement regularity. Prove the strengthened bundle nonempty using an actual rational ramp and polynomial cutoffs. The construction chooses theta = 1/(128q), width 1/(384q), theta2 = 1/(64q), copy thresholds 1/(8q) and 1/(4q), then the factorial-tail iteration count, copies, and detector ranks in the primary source's dependency order. This advances #1721's fourth-moment hierarchy gap. It is an explicit s=4 adaptation of OpenAI2026LogSpace Sections 4.1-4.4 and Remark 4.6 at pinned revision 7ef5c0cc; the source uses s=8. It does not prove eighth-root regularity, every later atom/access-score margin, a recursive estimator, or a machine-space bound. Existing HierarchyParams APIs are unchanged. Validation atop d3d62d9f: new module and focused tests compile with pinned Lean and repository options, warnings as errors, using private outputs and read-only dependencies. Tests extract actual witnesses at q=2, prove nonzero ramp and exact plateau values, use row-root perturbation at its closed endpoint, and feed the underlying parameters into the existing stage API for a nonzero two-vertex matrix. Focused hygiene --check-simp and diffcheck pass. No aggregate or existing source file was modified. Independent review and full integration gates remain for the coordinator. Authored by worker codex-logspace-polynomial-cutoffs-20261030 with ChatGPT. Independent agent source/API review: codex-compression-iteration-20261030 approved c7ac6dea11574e721ded46aeb0f76c2ccaf79df9 and independently compiled both files. Publication preserves the reviewed source/tests and adds only generated import registration. Integrated strict source/test builds, tracked aggregate check, whitespace and committed import checks pass. Full integration gates run in PR CI.
Construct the reference state from the actual current default and true row plus accumulated compression preloss. Derive its small-negative and second-coordinate bounds from no-overflow dyadic floors, then preserve gated outputs through the remaining capacities and floors at their original indices with zero increments. Add a separate weighted root comparison with the true zero-padded output. Sensitive targets use the existing weighted preloss bound at eta/2; uniform small loss and lower clipping remove the low targets from the rank comparison. The theorem assumes scalar support, flatness and root-Lipschitz laws and a raw-prefix error envelope only at sensitive targets, not the final output estimate. It is exponent-parametric. Instantiating all source row functions and comparison to a subsequent incremented or post-overflow calculation remain separate. Source mapping: OpenAI2026LogSpace, Section 8.3, Lemma 8.2, revision 7ef5c0cc. This connects the reviewed Preloss and Reference layers using the documented margin-scaled dyadic witness. Existing files and aggregates are unchanged. Validation: pinned Lean 4.36.0-rc2 checked the new module and tests without diagnostics, using private outputs and read-only shared dependencies. Tests exhibit a nonzero increment, nonzero actual preloss, preserved nonzero output through the shifted remaining schedule, and a strictly positive fourth-root discrepancy. The review-requested low-target regression raises the padded rank from zero to 1/1048576 while a rank-dependent fourth-root output remains exactly one under a zero sensitive-target error envelope. Focused hygiene with --check-simp and staged diff checking pass. Full project gates and aggregate regeneration remain integration work. AI assistance: ChatGPT implemented and checked this milestone as worker codex-compression-iteration-20261030. Independent reviewer codex-logspace-polynomial-cutoffs-20261030 requested the low-target regression and is reviewing the frozen revision. Independent non-author source/API review by compression_review approved exact source revision c7873d034846fbab13b83d3d9a4d596343c9d191, including a rank-dependent low-target regression. Reviewed source files are unchanged. Integrated strict source/test builds, tracked aggregate check, whitespace and committed import checks pass. Full integration gates run in PR CI.
Define actual rational polynomial preparation, row-gate, auxiliary-score and broad-witness functions. Prove their range and closed threshold laws, flat rank regions, incoming-score compatibility, joint Lipschitz fourth roots, and exact rational-cast identities. Preserve the existing exponential AtomCutoffs API and Christian Reitwiessner's source credit. This is the fourth-moment adaptation for #1721 of the threshold template in OpenAI2026LogSpace (5.5), and the witness in the support-margin proof and compression-threshold table of Section 5.5, revision 7ef5c0cc. The source uses eighth roots; these degree-seven cutoffs only establish fourth-root regularity. The actual functions are proved regular, rather than an assumed regularity bundle. Normalized main outputs, access enumeration and machine-resource bounds remain separate. Validation: the new module and full focused test file compile without diagnostics with Lean 4.36.0-rc2 and repository strict options, using private outputs and read-only dependencies. Executable rational tests evaluate all three row functions to 1/4 with both switches in their transition regions. Further tests check signed inputs, closed endpoints and the witness's strict 82 > 3*27 margin, real/rational agreement, nonzero existing atomPrep and auxScore expressions on a concrete matrix, and nonconstant fourth-root variation. Focused --check-simp hygiene and staged diff checking pass. Independent review and full integration gates remain pending; aggregate regeneration belongs to the coordinator. AI assistance: implemented by worker /root/compression_review on feat/logspace-polynomial-atom-cutoffs, based on the reviewed hierarchy lane. The Stationary specialization will be a separate cross-lane change. Independent non-author source/API review by lean_environment approved exact source revision a4eae07bdd9993e038a83aea84cc3c4b31c794d9 against pinned atoms.tex, with no findings. Reviewed source files are unchanged. Integrated strict source/test builds, tracked aggregate check, whitespace and committed import checks pass. Full integration gates run in PR CI.
Dependency join for the concrete polynomial atom-row consumer. Inputs are PR #1757 and PR #1750. This is a feature-branch merge, not a merge into main. All compression source/test blobs are identical to the reviewed second parent; the first parent is retained unchanged. The tracked aggregate checker passes. No full estimator endpoint is claimed.
Add the concrete preparation, auxiliary-score, and broad-witness row consumers of the stationary-reference compression results for issue #1721. The three-constructor PolynomialAtomRow selects actual rational polynomial functions; no scalar support or regularity contract is assumed. Prove strict retained support, zero default output, rank flatness, stationary preservation, and a weighted fourth-root comparison with a constant quantified before the endpoint type, row, rank, schedule, and error envelopes. Bridge the existing Matrix.atomPrep and Matrix.auxScore expressions at their zero-padded ranks. Source mapping: OpenAI2026LogSpace revision 7ef5c0cc, atoms.tex Section 5.5 support margins and compression-threshold table, and compression.tex Section 8.3 Lemma 8.2 stationary preservation/root comparison. This is the explicit fourth-moment adaptation, not the source's eighth-root endpoint. The sufficient scale margin is 0 < eta <= theta/10000. Normalized main atoms, incoming-gate assembly, access enumeration, overflow stability, and machine resource bounds remain open. No curated abstract changes. Tests construct nonzero actual preloss, move the true witness rank into its polynomial transition, prove positive output and positive root discrepancy, and apply the uniform weighted theorem. A remaining compression group changes the default and erases a nonzero second coordinate while preserving the witness. A separate K > cardinality example distinguishes positive dummy padding from zero padding, with a signed raw coordinate. Validation: library and tests independently compiled without diagnostics using Lean 4.36.0-rc2 and -DwarningAsError=true under project linter options, private outputs and read-only shared dependencies. Focused hygiene --check-simp and staged git diff --check pass. Full integration gates are coordinator-owned. No aggregate changes. Authored with AI assistance by worker codex-logspace-polynomial-cutoffs-20261030; independent review pending. Dependencies: local integration merge 35e8a8ec combines independently reviewed PolynomialAtomCutoffs a4eae07b and Stationary c7873d03 unchanged. Only the two new PolynomialAtomRows files belong to this implementation. Independent non-author source/API review by lean_environment approved exact source revision 5925d6e06da530885fbf5144a844439fcf8084a6 and its unchanged reviewed cutoff/compression dependencies, with no findings. The two implementation files are unchanged after joining the published dependency lanes. Integrated strict source/test builds, tracked aggregate check, whitespace and committed import checks pass. Full integration gates run in PR CI.
edemaine
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Agent review by reviewer-codex-edemaine; approve c1b1f79. Isolated the new three-file implementation layer over merge 3bee097; verified every inherited non-aggregate change matches the reviewed #1750 files exactly, reusing #1750/#1757 coverage.
The three concrete row functions discharge the entry/gate separation, flat-rank and joint fourth-root laws needed by stationary compression. The common η≤θ/10000 margin is sufficient, including the narrow witness's 82>3·27 separation. Support is strictly below K; default output vanishes even with signed/default-padded inputs. The weighted comparison quantifies its constant before the endpoint type, row and schedule and charges actual uncompressed-prefix envelopes only on sensitive true targets. Checked the mapping to the pinned preprint's Table 1, p. 26, and Lemma 8.2, p. 46, with the fourth-moment adaptation stated explicitly.
Read the complete implementation/tests and affected parent statements. The example has nonzero preloss, an interior witness and positive root discrepancy, followed by a changed default and erased second coordinate with preserved output; the K>cardinality case distinguishes positive default padding from zero padding. Existing atomPrep/auxScore bridges are exact. No findings or curated-abstract impact. Normalized main atoms, incoming assembly, post-overflow stability and the estimator/machine endpoint remain open.
Aggregate imports exactly match all 3,044 modules; diff whitespace checks pass. CI inspected once: Lean 41093991533 was running; title check passed. Author reports warning-strict focused compilation; I did not run a local Lean build. This is agent approval, not human review.
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Landed and usable exact-main cache verified. #1766 landed at 01:46:19 UTC as b48b310f, with exact neutral tree Non-author coordinator The fixed main consumer 41156930619/job 124407776227 passed all six at exact The public map returned 200: 4,192 unique pairs / 188,653 bytes, SHA-256 The original fourth-moment adaptation, scalar margins, stationary-law/model hypotheses and constant quantification before endpoint types, rows and schedules are unchanged. Selected abstracts/rendering are unaffected. Normalized main atoms, incoming assembly, post-overflow stability and recursive estimator/class/machine endpoints in #1721/#1444 remain open. This is agent integration review, not human approval or an expanded theorem claim. |
SamuelSchlesinger
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October 31, 2026 01:21
This was referenced Oct 31, 2026
This was referenced Oct 31, 2026
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The stationary-compression results previously required scalar support, flatness and root-Lipschitz laws as hypotheses. This PR discharges those laws for the actual polynomial preparation, auxiliary-score and broad-witness functions. It proves their preservation along the remaining compression schedule and a weighted fourth-root comparison using the constructed preloss reference. The error constant is quantified before the endpoint type, row and schedule. Direct bridges connect the family to the existing atomPrep and auxScore expressions.
Tests exercise nonzero preloss, a witness strictly between zero and one with positive root discrepancy, a changed default and erased second coordinate while preserving that witness, and the padding case where the rank exceeds the endpoint count.
Dependencies and review
Both prerequisites, #1757 and #1750, have actually landed. This PR now targets
main; its ordinary integration merge preserves original implementationc1b1f794and actual main012a2e02. Only the generated aggregate conflict was resolved. The diff is the original three-file implementation layer, with both reviewed library/test files and their complete feature patch unchanged.The outside source/API approval is reused for identical code, including the implementation previously reviewed at
5925d6e06da530885fbf5144a844439fcf8084a6. The integration record tracks fresh canonical current-head validation and the source-preserving metadata delta. No duplicate local Lean build was run. Selected abstracts and rendering are unaffected. Original source credit is retained; Codex coordinatorcodex-atoms-compression-integrator-20261031prepared only this metadata refresh.Part of #1721 and #1444. This closes three actual scalar-family consumers on the fourth-moment route. Normalized main atoms, incoming assembly, stability after overflow and the recursive estimator/machine endpoint remain open. AI-assisted implementation and review; no human approval or full equality is claimed.