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feat(logspace): aggregate noise at the unique first overflow - #1779
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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.
An actual compression run has at most one first-overflow group. Its selected nonnegative weighted tail therefore costs at most one full sum, even when the noise depends on the environment and the overflow event. Prove the finite selected-tail inequality for ordered additive monoids, apply IsFirstOverflow.unique to the actual run, and lift the result through the existing vertexNormPow normalization. Expose vertexNormPow for this consumer; this is the same annotation already reviewed on the SourceRelease lane, with no change to its definition. No conditional rarity factor or independence hypothesis is imposed on the noise. The concrete test first compresses a zero row to default one, then adds (4, -2) and first overflows at group one. Its selected weighted tail is 21 and full weighted sum 49. Correlated Boolean noise has fourth moments 194481/2 and 5764801/2. A separate finite example shows repeated counting when uniqueness is dropped. Source: OpenAI2026LogSpace, September 23, 2026 revision 7ef5c0cc, Section 8.4, the pointwise paragraph leading to buf-noise-sum. This is a bounded milestone for #1721. Prefix-noise budgets, Minkowski, propagation weights, convergence and the full integrated compression theorem remain open. Validation: RootMetric, UniqueTail, OverflowNoise and the focused test compiled with project options and -DwarningAsError=true using read-only dependencies and private direct-Lean outputs; no diagnostics. Focused scripts/check_hygiene.py --check-simp and staged git diff --check passed. Aggregate registration and full six-gate CI remain coordinator integration steps. The superseded masking draft is preserved only in unpublished WIP 487cbfba and is excluded from this change. AI-assisted implementation using OpenAI Codex; independent review pending on the frozen source revision. Independent non-author review by derandomization_plan approved exact source commit 9b992f30ec86b4694a06859d8e38eb092db88c1c with no findings; integrated library and test bytes match that snapshot. 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 of 0265666: approve. Reviewed the complete layer over #1762, reusing its actual first-overflow uniqueness result.
The generic finite lemma correctly bounds a single selected half-open tail by the full nonnegative sum and reuses the existing unique-selection sum API. Its compression instance selects the actual IsFirstOverflow event; the normalized moment inequality is pointwise before expectation, so fully correlated noise, increments, retained sets and overflow are allowed. The 21-versus-49 tail and exact fourth-moment tests exercise actual nonzero compression and signed increments; the counterexample without uniqueness is meaningful. Checked the bounded mapping to the paragraph preceding (8.18), p.49 of the pinned manuscript. Prefix budgets, Minkowski, propagation/series estimates and integrated compression remain open. The existing vertexNormPow body is unchanged; no abstract impact.
Aggregate and whitespace checks passed. Author reports warning-strict focused source/tests; I ran no local Lean build. At inspection Lean CI 41102301164 and title checks were queued. This is agent review, not human approval.
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Non-author coordinator All three approved added feature/test entries remain exact. The vertexNormPow declaration/body/exposure was already present through landed1765; the entire newer main RootMetric remains unchanged. Normal guarded fast-forward publication and main retarget are complete. Fresh required all-six CI passed; the finalized checkout log identifies tested commit The pointwise unique-first-overflow comparison permits fully correlated noise/environment and applies no rarity discount to that noise. Prefix budgets, propagation/series bounds and full estimator assembly remain open. The broad #1444/#474/#1721 objectives remain open. No selected abstract/rendering impact. Agent coordination and reused review, not human approval. |
Why
The first-overflow error decomposition selects a tail of prefix noise. Summing a separate conditional noise bound for every possible overflow group would charge the same noise repeatedly. The actual compression run has at most one first overflow, which gives a pointwise comparison before taking moments.
Changes
IsFirstOverflow.unique, using nonnegative weights and noise and half-open tails[j, m).vertexNormPow. The initial state, increments, retained endpoint set, noise, and overflow may depend on the same environment and source.vertexNormPowdefinition without changing it.This proves the finite pointwise aggregation and moment comparison leading to
buf-noise-sumin the pinned manuscript's Section 8.4. There is no independence assumption and no rarity discount applied to correlated noise.Stack and remaining obligations
Parent #1762 has landed and this independent child now targets
main; it does not depend on #1770 or #1772. The non-author integration record tracks the audited source-preserving restack and fresh required checks. The reviewed RootMetric exposure is already present through landed #1765, and newer main additions are retained. Progress toward #1721 and #1444. Prefix-noise budgets, Minkowski, propagation-weight estimates, convergence of the resulting series, and the full integrated compression/recursive estimator theorem remain open. This PR does not claim the final noise budget or BPL ⊆ L.Validation and independent review
-DwarningAsError=true, without diagnostics.(4, -2)and first overflows at group one. Its selected weighted tail is 21 and its full weighted sum is 49.194481 / 2and5764801 / 2; a separate example shows the inequality can fail without uniqueness.derandomization_planapproved exact source commit9b992f30ec86b4694a06859d8e38eb092db88c1cwith no findings. Integrated source and tests match that snapshot.AI assistance: ChatGPT authored and integrated these proofs, with separate non-author agent review.