diff --git a/research/bootstrap-depth-audit/CLAIM.json b/research/bootstrap-depth-audit/CLAIM.json new file mode 100644 index 0000000000..fbe362e6f5 --- /dev/null +++ b/research/bootstrap-depth-audit/CLAIM.json @@ -0,0 +1,70 @@ +{ + "track": "adversarial mathematical audit", + "assignment_seed": 73, + "seed_command": "python3 -c 'import secrets; print(secrets.randbelow(100))'", + "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60", + "started_utc": "2026-10-13T05:05:03Z", + "claimed_utc": "2026-10-13T05:09:38Z", + "expected_finish_utc": "2026-10-13T05:35:00Z", + "verification_reserved_from_utc": "2026-10-13T05:27:30Z", + "coordination": "GitHub pinnedIssues is empty; only open issue is #650 (Discussions request). Local claim per sprint instructions.", + "candidates": [ + "#621 uniform cyclic-inverse norm bound: already has multiple arithmetic audits and rounded follow-ups", + "#387 finite bootstrap: universal five-round ceiling with independently variable per-level slack", + "#479 streamed chirp residues: exact carry and periodicity obligations" + ], + "selection": "73 modulo 3 = 1 selects the second candidate; scope differs from supplier/resultant audits and the broad #657 assembly wall.", + "object": "PR387 at d9fb9410b1ae6f6ada376a1e7fd12bc176e476ea: PROOF.md fixed-depth recurrence, five-round limitation, and retained outer margin.", + "hypothesis": "The published final grid point needs at least six recurrence levels even with independent choices of positive slack at every level and any coarse saving below the certified upper endpoint.", + "method": "Monotone recurrence envelope plus exact rational cross-products at depths 0 through 6; compare the six-round published witness; adversarial altered-input controls.", + "deliverable": "Scoped mathematical audit, source-pinned independent standard-library verifier and deterministic exact receipt; no new kappa.", + "stopping_rule": "One recurrence family and the pinned root interval only; no supplier search or global theorem claim.", + "overlap_snapshot": { + "open_prs": 567, + "latest_pr": 748, + "relevant": [ + 387, + 657, + 621, + 628, + 666, + 668, + 736, + 479, + 684 + ] + }, + "result": { + "status": "completed scoped mathematical audit / negative result; no new kappa", + "pr_url": "https://github.com/CrocSwap/integer-mult-bounds/pull/768", + "contribution_commit": "a7c07b84290462385695b2b1390882dda9d66ead", + "submitted_utc": "2026-10-13T05:22:22Z", + "verification_started_utc": "2026-10-13T05:15:11Z", + "finding": "At fixed a0=384599/10^10, all zero-through-five-level certificates under PR387 recurrence accounting fall below its target, even allowing independent per-level savings below the inherited upper coarse endpoint. The six-level compact margin clears the target.", + "scope": "Root bracket and completed supplier interfaces inherited; no algorithmic runtime lower bound or full construction replay.", + "completed_coverage": { + "exact_caps_depths": [ + 0, + 1, + 2, + 3, + 4, + 5, + 6 + ], + "negative_controls": 9, + "tests": 7, + "finite_varying_schedules": 1536, + "strict_grid_cases": 2400, + "source_pins": 10 + }, + "final_overlap_snapshot": { + "open_prs": 586, + "latest_pr": 767, + "audited_head_unchanged": true, + "matching_depth_audit_found": false, + "prior_review_credit": "ahrazzle, PR387 reviews 5482259865 and 5482278356" + }, + "handoff": "README.md, PROOF.md, EXPECTED.json, SOURCE.json and VALIDATION.md in this package" + } +} diff --git a/research/bootstrap-depth-audit/EXPECTED.json b/research/bootstrap-depth-audit/EXPECTED.json new file mode 100644 index 0000000000..8de318e016 --- /dev/null +++ b/research/bootstrap-depth-audit/EXPECTED.json @@ -0,0 +1,61 @@ +{ + "status": "PASS_SCOPED_BOOTSTRAP_DEPTH_AUDIT", + "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60", + "audited_head": "d9fb9410b1ae6f6ada376a1e7fd12bc176e476ea", + "assignment_seed": 73, + "initial_ordinary_saving": "384599/10000000000", + "inherited_coarse_interval": [ + "999632926176283805553/1000000000000000000000000", + "499816463088141902777/500000000000000000000000" + ], + "target_kappa": "199726931618167/200000000000000000", + "optimistic_depth_grid_caps": [ + { + "depth": 0, + "largest_strict_1e18_grid_numerator": 38458420892978 + }, + { + "depth": 1, + "largest_strict_1e18_grid_numerator": 997675755024218 + }, + { + "depth": 2, + "largest_strict_1e18_grid_numerator": 998633699540676 + }, + { + "depth": 3, + "largest_strict_1e18_grid_numerator": 998634657132637 + }, + { + "depth": 4, + "largest_strict_1e18_grid_numerator": 998634658089877 + }, + { + "depth": 5, + "largest_strict_1e18_grid_numerator": 998634658090834 + }, + { + "depth": 6, + "largest_strict_1e18_grid_numerator": 998634658090835 + } + ], + "five_round_outer_cap": "15619264471504419470996663651127289709891544680842060693484107900149717665547749813564212594438992404905305609368496240116408599295191126511/15640619264471504419470996663651127289709891544680842060693484107900149717665547749813564212594438992404905305609368496240116408599295191126511", + "five_round_exclusion_cross_product": "17093151025493879485608650219856916363470819258266709352149686182243115351393241219152455694927792401046314682647285050519391569851808429253370000000000", + "five_round_gap_in_1e18_grid_units": "8546575512746939742804325109928458181735409629133354676074843091121557675696620609576227847463896200523157341323642525259695784925904214626685/15640619264471504419470996663651127289709891544680842060693484107900149717665547749813564212594438992404905305609368496240116408599295191126511", + "six_round_optimistic_outer_cap": 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"negative_controls_rejected": [ + "one_grid_lower_is_not_excluded_at_depth_five", + "depth_six_is_not_excluded", + "stronger_initial_supplier_escapes_five_round_ceiling", + "unpaid_square_term", + "off_by_one_closed_form", + "zero_assembly_slack", + "next_grid_not_reached_by_witness", + "nonstrict_grid_rounding", + "changed_source_byte" + ], + "scope": "Ceiling for the retained recurrence accounting with independently variable per-level savings, fixed initial supplier and inherited coarse-root upper bound. Six-round compact margin replay only; no supplier replay or full assembly/multiplication proof." +} diff --git a/research/bootstrap-depth-audit/PROOF.md b/research/bootstrap-depth-audit/PROOF.md new file mode 100644 index 0000000000..ca0467e997 --- /dev/null +++ b/research/bootstrap-depth-audit/PROOF.md @@ -0,0 +1,144 @@ +# A ceiling with independent savings at each bootstrap level + +This argument audits the scalar accounting in Rohan Arun's PR387 at +`d9fb9410b1ae6f6ada376a1e7fd12bc176e476ea`. It assumes the inherited completed +ordinary-supplier contract and certified coarse-root bracket. It proves a +limit of that accounting, not a lower bound on an algorithm's actual runtime. + +## 1. The retained recurrence + +The stopped-leaf estimate in the pinned proof, with saving s and preceding +ordinary saving a, is + +``` +T(e)/V = O(e^(1-s) + e^((1-s)^2+s(1-a)) + e^s log(e)). +``` + +The middle term gives the accounted saving + +``` +F_s(a) = 1 - ((1-s)^2+s(1-a)) = s(1-s+a). +``` + +The other terms require positive gaps; ignoring those restrictions can only +enlarge the set of certificates considered here. In particular, we may allow +`a_j <= F_(s_j)(a_(j-1))` at each level, with independent `0 < s_j < u`. +This is an optimistic relaxation, not a construction of every such schedule. + +The initial saving a0 is fixed and satisfies `0 <= a0 < u < 1/32`. +In PR387 every legal s is below its true complex coarse root, which the +inherited moment certificate places below u. Its bit-supplier restriction, +positive backoff and beta restrictions only reduce the permitted set. + +## 2. A uniform envelope + +Set `A_0=a0` and `A_j=F_u(A_(j-1))`. For `0 <= a <= A` and +`0 <= s <= u < 1/2`, the exact identity + +``` +F_u(A) - F_s(a) + = u(A-a) + (u-s)(1-u-s+a) +``` + +has a nonnegative right-hand side. In our domain its second factor is +strictly positive, since `1-u-s+a >= 1-2u > 0`. Induction therefore gives +`a_j <= A_j`; with the legal strict bound `s_j < u`, it is strict for j>=1. +Skipping a level or reverting to the preceding certificate cannot evade the +bound, because `A_j > A_(j-1)`. + +Indeed, + +``` +u-A_j = u(u-A_(j-1)), +A_j = u-u^j(u-a0), +A_j-A_(j-1) = u^(j-1)(1-u)(u-a0) > 0. +``` + +These identities prove the envelope for all finite depths and all real +per-level saving choices in the stated interval. No grid search is being +substituted for this argument. + +## 3. Transfer to the compact assembly margin + +PR387 uses `q=a(1-2 eta)` and `M=(1-eta)q/(1+q)`. Its legal parameters have +`a>0` and `0=0: its difference between y and x +is `(y-x)/((1+x)(1+y))`. Any certificate that exceeds κ in every performance +margin must in particular have `M>κ`. Consequently `C_j<=κ` excludes that +certificate. This reasoning does not assume that the compact margin remains +the minimum after independently changing parameters. + +For the pinned values + +``` +a0 = 384599/10^10 +u = 999632926176283805554/10^24 +κ = 199726931618167/(2*10^17), +``` + +exact integer cross-multiplication gives + +``` +0.546434 * 10^-18 < κ-C_5 < 0.546435 * 10^-18. +``` + +Since C_j is increasing with j, depths 0 through 5 are excluded. The use of +the **upper** coarse endpoint means more precision within the inherited root +bracket cannot rescue five levels. The argument permits zero supplier slack +as a limiting relaxation and independent choices at each level; it is stronger +than the predecessor's five-level test at its one published slack tuple. + +The exact integer implementation is separate from the rational recurrence. +If `u=p/q` and `a0=h/k`, put + +``` +N_j = k*p*q^j - p^j(k*p-h*q), +D_j = k*q^(j+1). +``` + +Then `A_j=N_j/D_j`, and `C_j=N_j/(D_j+N_j)`. For `κ=K/V`, the verifier +records the strictly positive integer `K(D_5+N_5)-V*N_5`, as well as the +fully reduced rational gap. No floating-point result is used for admission. + +## 4. The sixth level and the starting-supplier boundary + +For the published lower coarse endpoint l, the source chooses +`eta=beta=10^-18`, `backoff=10^-24` and `s=(1-beta)l-backoff`. Recomputing its +six steps from a0 gives the exact pinned chain and positive recurrence gaps. +Its compact margin satisfies + +``` +0.406 * 10^-18 < M_6-κ < 0.407 * 10^-18. +``` + +This confirms the compact-margin part of the published six-level certificate. +It does not establish the inherited supplier construction or independently +replay the other assembly constraints. There is no claimed new κ. + +Solving `C_5>κ` for the initial saving gives the necessary condition + +``` +a0 > u - (u - κ/(1-κ))/u^5 = a_required. +0.000586993 < a_required < 0.000586994. +``` + +This threshold is exact for the optimistic scalar envelope at u. At equality, +strict assembly slack still prevents admission. The real root and supplier +conditions can require a larger starting saving. A changed starting saving +`a0=3/5000=0.0006:`, in addition to the reproducible + SHA256 and Git-blob checks in the verifier. No predecessor code was imported. +- The repository's selected-record checker passed with + `51110230636731/31250000000000000`. +- `upstream/`, the selected record and all pre-existing tracked files remain + unchanged. No global artifacts were regenerated. + +The first development run rejected an overly narrow display enclosure for +the sixth-level slack. Exact evaluation gives approximately +`0.4061096846138577 * 10^-18`; the final asserted enclosure is +`(0.406, 0.407) * 10^-18`. This was a new audit-display error, not a failure +of the predecessor witness. The five-level exclusion was unaffected. + +## Checks not claimed + +`make selected-record-check` could not start because `make` is absent in this +environment; its exact Python recipe was run successfully as shown above. +The repository-wide `make verify`, the expensive selected-supplier cold replay, +PR387's full supplier suite, its two moment engines and the other assembly +constraints were not rerun. This addition changes none of their inputs. +No remote CI pass, alternate-interpreter execution, formal proof-assistant +check, separate agent review or human review is claimed. + +The source-pinned moment bracket and completed ordinary-supplier interfaces +remain assumptions of this audit. The new evidence is the written uniform +recurrence envelope and its exact evaluation, not a replay of those assumptions. + +## Reviewer entry point + +Read the identity in PROOF.md section 2, the necessary compact-margin inequality +in section 3 and the source-bracket limitation. The exact exclusion is one +positive integer cross-product in EXPECTED.json. The stronger-initial-supplier +control demonstrates why the starting saving is part of the claim. Reproduce +with the first command above; no network or third-party dependencies are needed. + +The result is a scoped negative result within PR387's certificate accounting. +It supplies no new κ and makes no change to the selected frontier. diff --git a/research/bootstrap-depth-audit/sources/pr387/EXPECTED-COARSE-PRECISION.json b/research/bootstrap-depth-audit/sources/pr387/EXPECTED-COARSE-PRECISION.json new file mode 100644 index 0000000000..587037bf60 --- /dev/null +++ b/research/bootstrap-depth-audit/sources/pr387/EXPECTED-COARSE-PRECISION.json @@ -0,0 +1,16 @@ +{ + "status": "PASS_TWO_EXACT_MOMENT_ENGINES", + "coarse_lower": "999632926176283805553/1000000000000000000000000", + "coarse_upper": "499816463088141902777/500000000000000000000000", + "original_coarse": "999632926176283/1000000000000000000", + "coarse_gain": "805553/1000000000000000000000000", + "grid": "1/1000000000000000000000000", + "density": "1/10000000000000000", + "fallback_rank_one_calls": 52928992224, + 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"negative_controls_rejected": [ + "next_grid", + "zero_eta", + "zero_beta", + "unit_beta", + "zero_bootstrap_gap", + "invalid_constant_bound" + ], + "finite_envelope": { + "proof": "PROOF.md", + "representation": "Unchanged PR384 inherited finite envelope; fixed-depth PR185 ordinary wrappers restore internal rows", + "semantic": { + "G": { + "base": 2, + "exponent": 30000 + }, + "E": { + "base": 2, + "exponent": 100000 + }, + "B_upper": { + "base": 2, + "exponent": 100001 + }, + "C0": { + "base": 2, + "exponent": 210000 + }, + "C1": 1, + "strict_literal_gap": 1, + "induction_gap_lower": 1 + }, + "rows": { + "coefficient": 20161, + "degree": 1000000, + "suffix_slope": 4000000, + "degree_gap": "23971789/25" + } + } +} diff --git a/research/bootstrap-depth-audit/sources/pr387/LICENSE b/research/bootstrap-depth-audit/sources/pr387/LICENSE new file mode 100644 index 0000000000..261eeb9e9f --- /dev/null +++ b/research/bootstrap-depth-audit/sources/pr387/LICENSE @@ -0,0 +1,201 @@ + Apache 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If You + institute patent litigation against any entity (including a + cross-claim or counterclaim in a lawsuit) alleging that the Work + or a Contribution incorporated within the Work constitutes direct + or contributory patent infringement, then any patent licenses + granted to You under this License for that Work shall terminate + as of the date such litigation is filed. + + 4. Redistribution. You may reproduce and distribute copies of the + Work or Derivative Works thereof in any medium, with or without + modifications, and in Source or Object form, provided that You + meet the following conditions: + + (a) You must give any other recipients of the Work or + Derivative Works a copy of this License; and + + (b) You must cause any modified files to carry prominent notices + stating that You changed the files; and + + (c) You must retain, in the Source form of any Derivative Works + that You distribute, all copyright, patent, trademark, and + attribution notices from the Source form of the Work, + excluding those notices that do not pertain to any part of + the Derivative Works; and + + (d) If the Work includes a "NOTICE" text file as part of its + distribution, then any Derivative Works that You distribute must + include a readable copy of the attribution notices contained + within such NOTICE file, excluding those notices that do not + pertain to any part of the Derivative Works, in at least one + of the following places: within a NOTICE text file distributed + as part of the Derivative Works; within the Source form or + documentation, if provided along with the Derivative Works; or, + within a display generated by the Derivative Works, if and + wherever such third-party notices normally appear. The contents + of the NOTICE file are for informational purposes only and + do not modify the License. You may add Your own attribution + notices within Derivative Works that You distribute, alongside + or as an addendum to the NOTICE text from the Work, provided + that such additional attribution notices cannot be construed + as modifying the License. + + You may add Your own copyright statement to Your modifications and + may provide additional or different license terms and conditions + for use, reproduction, or distribution of Your modifications, or + for any such Derivative Works as a whole, provided Your use, + reproduction, and distribution of the Work otherwise complies with + the conditions stated in this License. + + 5. Submission of Contributions. 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We also recommend that a + file or class name and description of purpose be included on the + same "printed page" as the copyright notice for easier + identification within third-party archives. + + Copyright [yyyy] [name of copyright owner] + + Licensed under the Apache License, Version 2.0 (the "License"); + you may not use this file except in compliance with the License. + You may obtain a copy of the License at + + http://www.apache.org/licenses/LICENSE-2.0 + + Unless required by applicable law or agreed to in writing, software + distributed under the License is distributed on an "AS IS" BASIS, + WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. + See the License for the specific language governing permissions and + limitations under the License. diff --git a/research/bootstrap-depth-audit/sources/pr387/NOTICE.md b/research/bootstrap-depth-audit/sources/pr387/NOTICE.md new file mode 100644 index 0000000000..af96013742 --- /dev/null +++ b/research/bootstrap-depth-audit/sources/pr387/NOTICE.md @@ -0,0 +1,9 @@ +# Attribution + +Fixed six-level extension of the reviewed PR185 bootstrap, exact witness and coarse-precision refinement and retained-family grid ceiling: Rohan Arun with OpenAI Codex assistance. Apache-2.0. + +All local constructions, paid profiles and pinned suppliers are from Chafik Boukhalfa / chafreaky's PR #384, with substantial Anthropic Claude assistance. Its original notices credit Jacob Sussman, Zhihao Chen / jacklightChen (#377), Ali Romman / aliromman (#365), DaysSky, Dylan Crow / utcorvusvolat-dotcom, and earlier contributors. + +The unchanged outer assembly is the PR #315 engine by eumemic with Anthropic Claude assistance, retaining icekylinx's PR #144 constraints adapted from Zhihao Chen's PR #23 and the reviewed balanced prefix from the RaD project (hipotures), as credited in its source header. This refinement changes the caller's witness parameters and retains additional certified coarse-root precision using the two unchanged moment engines. + +The pinned main-branch community review records Rohan Arun's PR185 stopped-leaf bootstrap and its completed-interface conditions. The finite-cutoff accounting source is retained with its original surrounding attribution and scope. The six-level extension is new here; the prior review covered three levels. diff --git a/research/bootstrap-depth-audit/sources/pr387/PARAMETERS.json b/research/bootstrap-depth-audit/sources/pr387/PARAMETERS.json new file mode 100644 index 0000000000..bbc4ad22f6 --- /dev/null +++ b/research/bootstrap-depth-audit/sources/pr387/PARAMETERS.json @@ -0,0 +1,6 @@ +{ + "bootstrap_rounds": 6, + "eta": "1/1000000000000000000", + "beta": "1/1000000000000000000", + "saving_backoff": "1/1000000000000000000000000" +} diff --git a/research/bootstrap-depth-audit/sources/pr387/PROOF.md b/research/bootstrap-depth-audit/sources/pr387/PROOF.md new file mode 100644 index 0000000000..915d08efd8 --- /dev/null +++ b/research/bootstrap-depth-audit/sources/pr387/PROOF.md @@ -0,0 +1,60 @@ +# Fixed six-level bootstrap and the retained outer ceiling + +## Pinned interfaces and acyclic composition + +The base completed ordinary saving is a0=384599/10^10. The bit and complex constructions and paid profiles are unchanged from #384. The complex coarse saving is selected with finer rational precision as established below. Use s=(1-beta)b-10^-24 with beta=eta=10^-18. Exact comparison gives a0=1 dominating its already fixed primitive, recurrence, wrapper, row, setup, padding and previous ordinary constants. There is no claim that the executable has measured C. Let ell>=1 be an integer with C<2^ell, and delta the minimum stage gap. Choose + +`L=ceil(max(1,36/delta^2,2*(4+ell)/delta))`, `e0=2^L`. + +For t=log2(e)>=L, `log2(1+t)<=3*sqrt(t)<=delta*t/2` and `log2(C)+4<=delta*t/2`. Hence `C(1+log2(e))*e^(-delta)<=1/16`. Each lower-order term is absorbed; their fixed multiplicity may be included in C. Below e0, use the preceding completed algorithm. A finite next constant can be chosen at least `C_large+C_old*2^(L*(a_j-a_(j-1)))`. This choice is sequential, not circular, and independent of input width. + +The checker implements this cutoff formula, asserts both sufficient inequalities, and exercises representative bit bounds 1, 80 and 210001 at every stage. Those instances check the implementation, while the algebra above proves the statement for any fixed dominating C. Zero gap and invalid bit bound are rejected. + +## Restored rows and fixed tapes + +Each inner completed call borrows from its two supplied chunks, keeps all outer fields as complete spectators, and restores its row coordinate before returning. Binary-radix padding and tape/program counts multiply only through six fixed levels. The parent therefore receives the same completed endpoint interface and does not need an external row multiplier. No raw reversal or free weighted operation is assumed. The retained external 252 allowance is unused slack, not an estimate of the new algorithm's internal reserve. The complex finite envelope and external row coefficient 20161 remain unchanged. + +This is the same conditional fixed-depth reasoning in the reviewed PR185 mechanism, extended here from three levels to six. It is not an assertion that the previous review already checked six levels or that an all-size tape compiler was machine-verified. + +## Exact witness and grid ceiling + +Use a=a6, q=a(1-2 eta), epsilon=(1-eta)/(1+q). The pinned outer assembly verifies all 47 strict constraints, including all seven performance margins; their minimum is M=epsilon*q. The result is the greatest 10^-18 multiple strictly below M. The next point, zero eta, zero beta and unit beta are rejected. A separate regression reproduces the former three-level witness at its original coarse input; five levels at the same slacks do not reach the new grid point. + +Every valid witness in this unchanged outer formula has a a_j`) and row-borrowing toll (`c < 1-a_j`) are strictly lower order. All gaps are positive fixed rationals. The very small depth-3 gap affects asymptotic constants and thresholds; it is not a practical speedup claim. + +Independently, `a_j = c - c^j (c-a_0)`. The verifier checks this closed form against every recurrence step. It also checks `kappa < a_j/(1+a_j) < c/(1+c)`. + +## Nested row restoration and fixed tapes + +The outer construction's row stock still has `log_q S(e,w)=O(w log e)`: its unchanged cover recursion supplies that stock, and its new ordinary leaves need no *external* extra stock. When an inner `A_(j-1)` borrows high digits, it borrows from the two chunks passed to that ordinary call, retains all outer fields as complete spectators, and restores its row coordinate before returning. Hence the parent sees exactly the original complete ordinary interchange. The parent does not assume a raw reversed operation preserves a borrowed range. + +The parent high-part exchange has width `k=O(w log e)` and costs `O(V k^(1-a_(j-1)))`, conservatively bounded by `O(Vw log e)`. Its own internal borrowing is included in that completed bound. Nested padding factors and program/tape counts multiply only across three fixed levels, and each completed interface restores its rows. They remain constants independent of input size. This is finite program composition, not a new unbounded stack of tapes or an uncharged row multiplier. + +For fixed positive c and fixed wrapper depth, ancestor and prefix-control descriptors remain polynomial in each active atom width. The complete q^w target fibers absorb their polynomial preparation, as in notes/three-stage-cover-bit.tex. Wrapper constants may depend on the selected level but not on input size; no growing-depth program is used. + +The bridge retains the original external row coefficient and its extra 252 ordinary-leaf allocation conservatively. **252 is not asserted to be the new leaf's actual internal reserve.** The new leaf's additional externally supplied stock is zero because it uses its completed internal-borrowing wrapper. The old allocation remains unused slack. All complex finite-group, scalar, router and precision formulas are rechecked unchanged by the parent validator. + +## Exact assembly and scope + +Only the uniform-bit fields of the bridge change. The validator reconstructs the new leaf saving from the pinned original PR182 certificate and the exact finite recurrence. It rejects a fictitious base, detached saving, unpaid atom exponent or external-row assumption. It separately runs the complete original bridge validator with the original leaf fields, preserving every scalar, complex, finite-group and row formula. The unchanged 47-constraint assembly body then consumes the newly checked ordinary saving. + +Depth 3 gives `kappa = 660627334074291/10^18`, approximately `0.000660627334074291`, a `0.062219284%` improvement over PR182. Both physical supplier moments are recomputed with rational intervals; the next coarse grid point is excluded. All 47 constraints, seven margins, and the next final grid exclusion are checked. The finite physical words themselves are inherited as source-pinned evidence and are not claimed to have been newly replayed by the arithmetic verifier. + +This is a compositional improvement to the retained conditional ordinary-supplier interface. It does not improve the coarse physical profile, prove the analytic/fixed-tape hypotheses independently, or establish global optimality. PR183 records the existing assembly ceiling; it does not perform this finite ordinary-leaf replacement. diff --git a/research/bootstrap-depth-audit/sources/pr387/coarse_precision.py b/research/bootstrap-depth-audit/sources/pr387/coarse_precision.py new file mode 100644 index 0000000000..7f5c506358 --- /dev/null +++ b/research/bootstrap-depth-audit/sources/pr387/coarse_precision.py @@ -0,0 +1,26 @@ +"""Retain 24 certified decimal places in the unchanged paid PR384 moment. +Both independently written rational engines and the original positive fallback +are retained. Decimal arithmetic proposes endpoints; it never admits them. +""" +from pathlib import Path +from fractions import Fraction as Q +import importlib.util +ROOT=Path(__file__).resolve().parent + +def module(name): + spec=importlib.util.spec_from_file_location('precision_'+name,ROOT/'source384/code/engines'/(name+'.py')) + obj=importlib.util.module_from_spec(spec);spec.loader.exec_module(obj);return obj + +def certify(profile,m,stock,original_coarse): + M=module('moment');B=module('base_two_moment');H={int(k):v for k,v in profile.items()} + assert (m,stock)==(73,14727) + assert m*stock-sum(r*n for r,n in H.items())==2496 + root=M.certify(H,m,stock,True);lo=Q(root['lower']);hi=Q(root['upper']);step=Q(1,10**24) + assert hi-lo==step and original_coarse0 and isinstance(coefficient_bits,int) and coefficient_bits>=1 + L=ceil(max(Q(1),36/delta**2,2*(4+coefficient_bits)/delta)) + assert L*delta**2>=36 and L*delta>=2*(4+coefficient_bits) + return L + +def refine(b,c): + spec=importlib.util.spec_from_file_location('pinned_outer',ROOT/'source384/code/engines/outer.py') + outer=importlib.util.module_from_spec(spec);spec.loader.exec_module(outer) + params=json.loads((ROOT/'PARAMETERS.json').read_text());depth=params['bootstrap_rounds'];assert depth==6 + grid=Q(1,10**18);eta=Q(params['eta']);beta=Q(params['beta']);backoff=Q(params['saving_backoff']) + saving=(1-beta)*b-backoff + assert Q(0)0 + examples={str(bits):str(cutoff_log2(delta,bits)) for bits in (1,80,210001)} + stages.append(dict(level=j+1,ordinary_calls_only_to_level=j,gaps={k:str(v) for k,v in gaps.items()},minimum_gap=str(delta),cutoff_examples_by_constant_bits=examples)) + chain.append(a) + previous=json.loads((ROOT/'THREE-ROUND-WITNESS.json').read_text()) + prior_saving=(1-beta)*Q(previous['complex_coarse'])-backoff + prior_chain=[initial] + for _ in range(3):prior_chain.append((1-prior_saving)*prior_saving+prior_saving*prior_chain[-1]) + assert list(map(str,prior_chain))==previous['bootstrap_chain'] + previous_six=json.loads((ROOT/'SIX-ROUND-18-DIGIT-COARSE-WITNESS.json').read_text()) + a=chain[-1];q=a*(1-2*eta);minimum=(1-eta)*q/(1+q) + scaled=minimum/grid;kappa=grid*((scaled.numerator-1)//scaled.denominator) + bridge=dict(proof='PROOF.md',representation='Unchanged PR384 inherited finite envelope; fixed-depth PR185 ordinary wrappers restore internal rows', + semantic=dict(G=dict(base=2,exponent=30000),E=dict(base=2,exponent=100000),B_upper=dict(base=2,exponent=100001),C0=dict(base=2,exponent=210000),C1=1,strict_literal_gap=1,induction_gap_lower=1), + rows=dict(coefficient=20161,degree=10**6,suffix_slope=4*10**6,degree_gap=Q(10**6)-Q(51*20161,25))) + asm=outer.assembly(a,b,bridge,kappa,eta=eta,beta=beta) + assert len(asm['strict_constraints'])==47 and all(x>0 for x in asm['strict_constraints'].values()) + assert asm['minimum_margin']==minimum + rejected=[] + for name,k,e,be in [('next_grid',kappa+grid,eta,beta),('zero_eta',kappa,Q(0),beta),('zero_beta',kappa,eta,Q(0)),('unit_beta',kappa,eta,Q(1))]: + try:outer.assembly(a,b,bridge,k,eta=e,beta=be) + except AssertionError:rejected.append(name) + else:raise AssertionError('accepted invalid witness '+name) + for name,delta,bits in [('zero_bootstrap_gap',Q(0),80),('invalid_constant_bound',Q(1,2),0)]: + try:cutoff_log2(delta,bits) + except AssertionError:rejected.append(name) + else:raise AssertionError('accepted invalid finite cutoff '+name) + # For the pinned outer formula all valid witnesses have a0; + # consequently the compact margin is Q(previous_six['kappa']) + # Five levels at the same slack values do not yet reach the chosen grid. + a5=chain[-2];q5=a5*(1-2*eta);assert (1-eta)*q5/(1+q5)0 for v in slacks.values()),{k:str(v) for k,v in slacks.items() if v<=0} + assert min(margins.values())==minimum + assert margins['balanced_prefix']-minimum==eta and 1-eps-r==eta/2 + assert 1-eps*(1+c)==eta-eps*eta/4 + return dict(parameters=dict(a_bit=a,a_complex=b,tau=tau,sigma=sigma, + eta=eta,beta=beta,q=q,c=c,epsilon=eps,lambda_=lam, + lambda_prime=lp,alpha_squared_power=r,delta=delta, + C0=bridge['semantic']['C0'],C1=1,kappa=kappa), + strict_constraints=slacks,margins=margins,minimum_margin=minimum, + absorption_gap=minimum-kappa, + recurrence=dict(internal=internal,leaf=leaf,reservations=1-c)) diff --git a/research/bootstrap-depth-audit/test_audit.py b/research/bootstrap-depth-audit/test_audit.py new file mode 100644 index 0000000000..9edbc94736 --- /dev/null +++ b/research/bootstrap-depth-audit/test_audit.py @@ -0,0 +1,114 @@ +#!/usr/bin/env python3 +"""Independent cost-exponent and CLI failure checks for the scoped audit. + +Isaac Grannis (@BagMan9), with OpenAI Codex assistance. Apache-2.0. +""" + +import importlib.util +import itertools +import json +import shutil +import subprocess +import sys +import tempfile +import unittest +from fractions import Fraction as F +from pathlib import Path + +ROOT = Path(__file__).resolve().parent +spec = importlib.util.spec_from_file_location("bootstrap_audit", ROOT / "verify.py") +audit = importlib.util.module_from_spec(spec) +spec.loader.exec_module(audit) + + +class BootstrapAudit(unittest.TestCase): + def test_varying_savings_against_original_cost_exponents(self): + # This uses the three cost exponents, not audit.step. All 1,536 + # four-stage schedules are checked at every intermediate depth. + schedules = 0 + for upper in (F(1, 64), F(31, 1024)): + for initial in (F(0), upper / 2, 7 * upper / 8): + choices = [i * upper / 4 for i in range(1, 5)] + for schedule in itertools.product(choices, repeat=4): + old = initial + for depth, saving in enumerate(schedule, 1): + worst_cost_exponent = max( + 1 - saving, + (1 - saving)**2 + saving * (1 - old), + saving, + ) + # Keeping the previous completed supplier is permitted. + # Ignoring the logarithmic strict gap is optimistic. + old = max(old, 1 - worst_cost_exponent) + cap_n, cap_d = audit.cap_integer(initial, upper, depth) + self.assertLessEqual(old / (1 + old), F(cap_n, cap_d)) + schedules += 1 + self.assertEqual(schedules, 1536) + + def test_monotonicity_domain_is_needed(self): + # Outside u < 1/2, blindly maximizing s can lose the maximum. + upper, saving = F(3, 4), F(1, 2) + from_cost_exponent = 1 - ((1 - saving)**2 + saving) + n, d = audit.cap_integer(F(0), upper, 1) + self.assertGreater(from_cost_exponent / (1 + from_cost_exponent), F(n, d)) + + def test_integer_cap_at_large_depth(self): + # Cover j=0 and arbitrary a0 independently of the source witness. + for initial, upper in ((F(0), F(1, 64)), (F(1, 1000), F(1, 50))): + old = initial + for depth in range(21): + n, d = audit.cap_integer(initial, upper, depth) + self.assertEqual(F(n, d), old / (1 + old)) + old = 1 - ((1 - upper)**2 + upper * (1 - old)) + + def test_strict_grid_boundaries(self): + for numerator in range(1, 41): + for denominator in range(1, 21): + for scale in (1, 10, 100): + point = F(audit.strict_grid(numerator, denominator, scale), scale) + value = F(numerator, denominator) + self.assertLess(point, value) + self.assertLessEqual(value, point + F(1, scale)) + + def test_expected_receipt_in_optimized_isolated_process(self): + with tempfile.TemporaryDirectory() as work: + result = subprocess.run( + [sys.executable, "-I", "-B", "-O", str(ROOT / "verify.py"), + "--check", str(ROOT / "EXPECTED.json")], + cwd=work, capture_output=True, text=True, + ) + self.assertEqual(result.returncode, 0, result.stderr) + self.assertIn("9 controls rejected", result.stdout) + + def test_tampered_receipt_rejected_under_optimization(self): + with tempfile.TemporaryDirectory() as work: + path = Path(work) / "receipt.json" + receipt = json.loads((ROOT / "EXPECTED.json").read_text()) + receipt["five_round_exclusion_cross_product"] = "0" + path.write_text(json.dumps(receipt)) + result = subprocess.run( + [sys.executable, "-I", "-B", "-O", str(ROOT / "verify.py"), + "--check", str(path)], + cwd=work, capture_output=True, text=True, + ) + self.assertNotEqual(result.returncode, 0) + self.assertIn("receipt mismatch", result.stderr) + + def test_changed_source_rejected_under_optimization(self): + with tempfile.TemporaryDirectory() as work: + copy = Path(work) / "audit" + shutil.copytree(ROOT, copy) + path = copy / "sources/pr387/PARAMETERS.json" + params = json.loads(path.read_text()) + params["bootstrap_rounds"] = 5 + path.write_text(json.dumps(params)) + result = subprocess.run( + [sys.executable, "-I", "-B", "-O", str(copy / "verify.py")], + cwd=work, capture_output=True, text=True, + ) + self.assertNotEqual(result.returncode, 0) + self.assertIn("source SHA256", result.stderr) + + +if __name__ == "__main__": + unittest.main(verbosity=2) diff --git a/research/bootstrap-depth-audit/verify.py b/research/bootstrap-depth-audit/verify.py new file mode 100644 index 0000000000..df856a0699 --- /dev/null +++ b/research/bootstrap-depth-audit/verify.py @@ -0,0 +1,228 @@ +#!/usr/bin/env python3 +"""Exact, scoped PR387 bootstrap-depth audit. Standard library only. + +Isaac Grannis (@BagMan9), with substantial OpenAI Codex assistance. +Apache-2.0. Predecessor sources are data: no predecessor code is imported. +""" + +import argparse +import hashlib +import json +from fractions import Fraction as F +from pathlib import Path + +ROOT = Path(__file__).resolve().parent +BASE = "66849e1475c930fd7732b1a0067ffee7e1e95f60" +HEAD = "d9fb9410b1ae6f6ada376a1e7fd12bc176e476ea" +INITIAL = F(384599, 10**10) +TARGET = F(199726931618167, 2 * 10**17) +GRID = F(1, 10**18) + + +def require(condition, message): + if not condition: + raise ValueError(message) + + +def load(relative): + return json.loads((ROOT / relative).read_text()) + + +def check_source(data, record): + require(hashlib.sha256(data).hexdigest() == record["sha256"], "source SHA256") + blob = b"blob " + str(len(data)).encode() + b"\0" + data + require(hashlib.sha1(blob).hexdigest() == record["git_blob"], "source Git blob") + + +def pins(): + source = load("SOURCE.json") + require(source["base_commit"] == BASE, "base pin") + require(source["audited_head"] == HEAD, "PR387 head pin") + for name, record in source["files"].items(): + check_source((ROOT / name).read_bytes(), record) + return source + + +def step(s, a): + return s * (1 - s + a) + + +def chain(initial, schedule): + result = [initial] + for s in schedule: + result.append(step(s, result[-1])) + return result + + +def cap_integer(initial, upper, depth): + """Unreduced integer closed form; independent of Fraction recurrence. + + Return numerator/denominator of A_depth/(1+A_depth), where + A_depth = u - u**depth * (u-a0). + """ + a, b = initial.numerator, initial.denominator + p, q = upper.numerator, upper.denominator + n = b * p * q**depth - p**depth * (b * p - a * q) + d = b * q ** (depth + 1) + return n, d + n + + +def compact_margin(a, eta): + require(0 < eta < F(1, 2), "positive assembly slack") + q = a * (1 - 2 * eta) + return (1 - eta) * q / (1 + q) + + +def strict_grid(numerator, denominator, scale=10**18): + """Largest scale-grid point strictly below a rational number.""" + require(denominator > 0, "positive grid denominator") + return (scale * numerator - 1) // denominator + + +def separated(n, d, target): + gap_numerator = target.numerator * d - target.denominator * n + require(d > 0 and gap_numerator > 0, "depth ceiling does not exclude target") + return gap_numerator + + +def controls(source, hi, witness, params): + rejected = [] + + def reject(name, operation, reason): + try: + operation() + except ValueError as exc: + require(str(exc) == reason, "unexpected rejection: " + name) + rejected.append(name) + else: + raise ValueError("negative control accepted: " + name) + + n5, d5 = cap_integer(INITIAL, hi, 5) + n6, d6 = cap_integer(INITIAL, hi, 6) + reject("one_grid_lower_is_not_excluded_at_depth_five", + lambda: separated(n5, d5, TARGET - GRID), + "depth ceiling does not exclude target") + reject("depth_six_is_not_excluded", + lambda: separated(n6, d6, TARGET), + "depth ceiling does not exclude target") + improved_n, improved_d = cap_integer(F(3, 5000), hi, 5) + reject("stronger_initial_supplier_escapes_five_round_ceiling", + lambda: separated(improved_n, improved_d, TARGET), + "depth ceiling does not exclude target") + wrong = INITIAL + for _ in range(5): + wrong = hi * (1 + wrong) # Deletes the paid s**2 term. + reject("unpaid_square_term", + lambda: require(wrong / (1 + wrong) == F(n5, d5), "recurrence identity"), + "recurrence identity") + reject("off_by_one_closed_form", + lambda: require(F(n6, d6) == F(n5, d5), "recurrence identity"), + "recurrence identity") + reject("zero_assembly_slack", lambda: compact_margin(F(witness["bootstrap_chain"][-1]), F(0)), + "positive assembly slack") + margin = compact_margin(F(witness["bootstrap_chain"][-1]), F(params["eta"])) + reject("next_grid_not_reached_by_witness", + lambda: require(margin > TARGET + GRID, "witness below target"), + "witness below target") + # At exact equality, plain floor would wrongly admit the endpoint. + reject("nonstrict_grid_rounding", + lambda: require(strict_grid(3, 10, 10) == 3, "strict grid boundary"), + "strict grid boundary") + name, record = next(iter(source["files"].items())) + reject("changed_source_byte", + lambda: check_source((ROOT / name).read_bytes() + b" ", record), + "source SHA256") + return rejected + + +def verify(): + source = pins() + precision = load("sources/pr387/EXPECTED-COARSE-PRECISION.json") + params = load("sources/pr387/PARAMETERS.json") + witness = load("sources/pr387/EXPECTED-WITNESS.json") + lo, hi = F(precision["coarse_lower"]), F(precision["coarse_upper"]) + require(0 <= INITIAL < lo < hi < F(1, 32), "envelope domain") + require(hi - lo == F(1, 10**24), "root interval width") + # Receipt consistency only: these signs do NOT reprove the moment bracket. + for engine in ("first", "second"): + require(F(precision[engine + "_engine_lower_residual_upper"]) < 0, + "inherited lower residual sign") + require(F(precision[engine + "_engine_upper_residual_lower"]) > 0, + "inherited upper residual sign") + require(F(witness["complex_coarse"]) == lo, "coarse binding") + require(F(witness["kappa"]) == TARGET, "target binding") + require(witness["bootstrap_rounds"] == params["bootstrap_rounds"] == 6, "depth binding") + eta, beta, backoff = (F(params[k]) for k in ("eta", "beta", "saving_backoff")) + require(0 < beta < 1 and backoff > 0, "positive supplier slack") + saving = (1 - beta) * lo - backoff + require(INITIAL < saving < lo < F(witness["bit_coarse"]), "published saving domain") + require(saving == F(witness["degraded_bit_saving"]), "saving binding") + actual = chain(INITIAL, [saving] * 6) + require(actual == list(map(F, witness["bootstrap_chain"])), "published recurrence chain") + for j in range(1, 7): + a, old = actual[j], actual[j - 1] + require(old < a < saving < 1 - a, "completed recurrence gaps") + require(1 - a - saving * (1 - old) > 0, "remainder gap") + margin = compact_margin(actual[-1], eta) + require(margin == F(witness["minimum_margin"]), "compact margin binding") + require(margin > TARGET, "witness below target") + require(F(witness["absorption_gap"]) == margin - TARGET, "witness gap binding") + + optimistic = chain(INITIAL, [hi] * 6) + rows = [] + for j, ordinary in enumerate(optimistic): + n, d = cap_integer(INITIAL, hi, j) + require(F(n, d) == ordinary / (1 + ordinary), "independent cap identity") + if j <= 5: + separated(n, d, TARGET) + rows.append({"depth": j, "largest_strict_1e18_grid_numerator": strict_grid(n, d)}) + n5, d5 = cap_integer(INITIAL, hi, 5) + exclusion = separated(n5, d5, TARGET) + five = F(n5, d5) + six = F(*cap_integer(INITIAL, hi, 6)) + threshold = hi - (hi - TARGET / (1 - TARGET)) / hi**5 + require(F(586993, 10**9) < threshold < F(586994, 10**9), "base threshold enclosure") + require(F(546434, 10**6) < (TARGET - five) / GRID < F(546435, 10**6), + "five-round gap enclosure") + require(F(406, 1000) < (margin - TARGET) / GRID < F(407, 1000), + "six-round witness gap enclosure") + rejected = controls(source, hi, witness, params) + return { + "status": "PASS_SCOPED_BOOTSTRAP_DEPTH_AUDIT", + "base_commit": BASE, + "audited_head": HEAD, + "assignment_seed": 73, + "initial_ordinary_saving": str(INITIAL), + "inherited_coarse_interval": [str(lo), str(hi)], + "target_kappa": str(TARGET), + "optimistic_depth_grid_caps": rows, + "five_round_outer_cap": str(five), + "five_round_exclusion_cross_product": str(exclusion), + "five_round_gap_in_1e18_grid_units": str((TARGET - five) / GRID), + "six_round_optimistic_outer_cap": str(six), + "six_round_published_margin": str(margin), + "six_round_published_gap_in_1e18_grid_units": str((margin - TARGET) / GRID), + "five_round_initial_saving_threshold_at_upper_endpoint": str(threshold), + "negative_controls_rejected": rejected, + "scope": "Ceiling for the retained recurrence accounting with independently variable per-level savings, fixed initial supplier and inherited coarse-root upper bound. Six-round compact margin replay only; no supplier replay or full assembly/multiplication proof.", + } + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + group = parser.add_mutually_exclusive_group() + group.add_argument("--check", type=Path, help="Compare the full deterministic receipt") + group.add_argument("--output", type=Path, help="Write a new receipt; never overwrite") + args = parser.parse_args() + result = verify() + if args.check: + require(result == json.loads(args.check.read_text()), "receipt mismatch") + if args.output: + with args.output.open("x") as stream: + stream.write(json.dumps(result, indent=2) + "\n") + print(result["status"] + "; depths 0..5 excluded; six-round margin admitted; " + + str(len(result["negative_controls_rejected"])) + " controls rejected") + + +if __name__ == "__main__": + main()