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feat(logspace): prove exact sentinel compression invariance - #1770
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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.
Prove Finset.kthLargest_eq_of_eq_or_gt in the existing order-statistic module, preserving Christian Reitwiessner's credit. Values may change arbitrarily when both versions remain strictly above a nonnegative ceiling; if one positive-order rank is at most that ceiling, the actual ranks agree. The boundary rank = ceiling and zero padding are included. Define actual first-coordinate sentinel replacement, leaving every second coordinate unchanged. Derive equality of padded first ranks, size thresholds, and new defaults. The bounded modified rank implies marked cardinality < K. Under the explicit 3R separation, both channels of marked entries are in the compression identity region, and replacement commutes with the concrete update at every retained endpoint. No output equality is assumed. Source mapping for issue #1721: OpenAI2026LogSpace revision 7ef5c0cc, compression.tex Section 8.3, proof of Lemma 8.3, sentinel replacement and removal in the fewer-than-K large-entry case. This is the exact local rank/update argument, independent of the root exponent. Relative-error control through later increments, the iterated coupling, scalar upper saturation, heavy-count closure, and full post-overflow stability remain open. Sentinels are proof devices, not an algorithmic enumeration claim. No curated abstract or catalogue impact; no aggregate changes. Tests use [1000000,4,2,-1] with sentinel 100, K=n=3, positive old default 1/2, and floor 1. Rank and threshold equal the ceiling 2; smaller signed entries actually compress, while a marked second coordinate of -17 is preserved. Further cases exercise K above the retained cardinality and demonstrate why changing a value exactly at the ceiling can change the rank. Validation: KthLargest, Sentinel, and RowCompressionSentinel tests compile without diagnostics under Lean 4.36.0-rc2, warningAsError=true and project linter options, using isolated private outputs and shared dependencies read-only. Focused hygiene --check-simp and staged git diff --check pass. Full integration gates and independent exact-revision review are left to the coordinator. Base is reviewed Reference 10a5168; only these three files belong to this milestone. AI assistance: ChatGPT worker codex-logspace-polynomial-cutoffs-20261030 authored and validated the change. Independent non-author source/API review by agent derandomization_plan approved source revision 68a162675ea79cd7e6e29b0831c725d0dca9bbb2 with no findings. Integrated source and tests are byte-identical to that reviewed revision. 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 bb98219, reviewing this layer over #1762 and reusing unchanged compression coverage.
The new general order-statistic lemma preserves the actual zero-padded rank, including equality with the nonnegative ceiling and K above the retained cardinality. Its strict-above-ceiling condition is necessary. Applied to first coordinates and entry sizes, it gives exact rank, threshold and new-default invariance; the modified rank bound derives marked cardinality <K. The explicit 3R separation puts both versions of each marked entry in the existing compression identity region, proving one-step commutation on retained endpoints without assuming an output identity. Checked the bounded mapping to the sentinel-removal step of the pinned preprint's Lemma 8.3, p. 47.
Read the complete four-file diff and relevant definitions/proofs/tests. The signed example reaches the ceiling exactly, changes smaller entries under compression, and preserves the marked negative second coordinate. Padding and the failed boundary-replacement control are meaningful. General reuse is placed in KthLargest with original credit preserved. No findings or curated-abstract impact. Iterated coupling, relative-error control, saturation and the full post-overflow estimate remain open.
Aggregate imports exactly match all 3,040 modules and diff whitespace checks pass. CI inspected once: Lean 41096555784 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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Non-author coordinator All three approved feature/test entries remain exact. Normal guarded fast-forward publication and main retarget are complete. Fresh required all-six CI passed; the finalized checkout log identifies tested commit This is the conditional local sentinel/rank/compression step; iterated coupling, relative-error ceilings, saturation and the full post-overflow estimate remain open. The broad #1444/#474/#1721 objectives remain open. No selected abstract/rendering impact. Agent coordination and reused review, not human approval. |
The post-overflow compression argument needs to replace very large first coordinates by finite sentinels while preserving the actual ranks and compression update. This proves that local step, including a general padded-order-statistic equality lemma, boundary equality at the ceiling, and ranks above the row cardinality. The modified rank bound also derives that fewer than K entries were marked; the conclusion is not assumed.
Parent #1762 has landed and this PR now targets
main. The non-author integration record tracks the audited source-preserving restack and fresh required checks. Advances #1721 and #1444 using the sentinel step in the pinned OpenAI2026LogSpace compression argument. Tests use signed rows with a million-valued entry, sentinel 100, rank exactly at the ceiling, smaller entries that actually compress, and an unchanged negative second coordinate. The iterated coupling, relative-error ceilings, scalar saturation, and full post-overflow estimate remain open.Validation and independent agent review are recorded in this revision. Strict direct Lean checks passed for the affected order-statistic module, sentinel module, and behavioral tests, with warningAsError=true and project linters; the tracked aggregate and whitespace checks passed. Lean CI passed for the original published source revision. AI assistance: ChatGPT authored the proofs and tests, performed independent agent review, and integrated the change.