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feat(logspace): charge false overflows to actual prefix errors - #1762
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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.
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.
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.
Prove uniqueness of the first-overflow index, a signed light-target error lower bound, the explicit 2^(p+1)/(n*rho^p) moment charge, and its specialization to the actual dyadic iterate with previous default and preloss bounds derived. Conditional heavy-count rarity, relative post-overflow stability and integrated compression remain open under #1721 and #1444. Strict isolated Lean compilation with warningAsError=true and empty diagnostics passed for the module and a signed two-group run with nonzero preloss. Focused hygiene and whitespace checks passed. AI-assisted implementation; independent review pending before publication. Independent non-author source/API review by lean_environment approved exact source revision e0066c8a1015c2270c0a9ffaec1a3209225b2ce3 against pinned compression.tex Section 8.4, 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.
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Agent review by reviewer-codex-edemaine; approve 0cbb2f1, reviewing this layer over #1750 and reusing unchanged compression coverage.
The deterministic charge matches the pinned preprint's §8.4, (8.12), p. 48. Overflow with a sufficiently small preceding default forces at least n real large entries; fewer than n/2 true heavy targets leaves at least n/2 light entries. With nonnegative true first coordinate ≤ρ/4, zero true second coordinate, 0<ρ≤1 and preloss ≤ρ/100, each has clipped raw error ≥ρ/2, yielding the explicit 2^(p+1)/(nρ^p) moment coefficient. Both signs/channels and p=0 are covered. The dyadic theorem derives default/preloss bounds from earlier non-overflow and measures the correct uncompressed prefix j+1. First-overflow uniqueness is sound.
Read the full diff and affected definitions/proofs/tests. The actual two-group example has nonzero prior loss, negative first-channel and positive second-channel increments, and charge 512. No findings or curated-abstract impact. Conditional heavy-count rarity, post-overflow stability and integrated compression remain open.
Aggregate imports exactly match all 3,039 modules and diff whitespace checks pass. CI inspected once: Lean 41093066579 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 cache verified. #1762 landed at 01:15:10 UTC as b5f02c68, with exact neutral tree Non-author coordinator The fixed shared descendant fa5a5aa1 retains #1757/#1762/#1840/#1837. Main run 41155050649/job124400070908 passed all six and published at 01:35:18 UTC. The optional merge-group handoff was unavailable: its artifact download timed out after 120 seconds. The normal strict checks continued successfully, with 13 “Built Complexity…” lines; no successful restore or zero-rebuild claim is made. The public map returned 200: 4,182 unique pairs / 188,203 bytes, SHA-256 All original moment, signed-input, prior-overflow and raw-prefix hypotheses remain intact. Selected abstracts/rendering are unchanged; the remaining conditional rarity, post-overflow, estimator and machine objectives in #1721/#1444 stay open. This is agent integration review, not human approval. #1766 is outside this fixed cache snapshot. |
A compression overflow caused by errors at light endpoints needs a quantitative charge to those errors. This PR proves that the first-overflow index is unique, that a large light input forces clipped raw-prefix error at least half the floor, and that a false-type overflow is bounded by the explicit moment sum with coefficient
2 ^ (p + 1) / (n * rho ^ p).The actual dyadic-run theorem derives the preceding default and preloss bounds from the absence of earlier overflow. It uses the uncompressed prefix at
j + 1, and handles signed input coordinates. A concrete two-group regression has a nonzero prior compression loss, a negative first-channel increment at one endpoint and a positive second-channel increment at another; the derived moment charge evaluates to 512.Both integrated source and tests compile without diagnostics with
warningAsErrorenabled. Tracked aggregate, committed imports, hygiene and whitespace checks pass. Independent AI review approved source revisione0066c8a1015c2270c0a9ffaec1a3209225b2ce3; this published revision preserves those exact files. Full project gates run in CI.#1750 has landed; this PR now targets
main, continuing the compression stack. Part of #1721 and #1444. This is the deterministic false-overflow charge from Section 8.4 of the pinned OpenAI2026LogSpace source. Conditional heavy-count rarity, relative stability after overflow and the integrated compression theorem remain open. AI-assisted implementation and review; no human approval or full equality is claimed.Integration refresh by Codex coordinator
codex-atoms-compression-integrator-20261031: ordinary merge of actual main40476ad8, preserving the original source commits, both reviewed library/test blobs and the exact feature patch. Only the generated aggregate conflict was resolved. The integration record tracks fresh current-head validation and the reused outside source approval. Selected abstracts and rendering are unaffected.