Research audit: exclude five bootstrap levels at PR387’s grid point

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"
+  }
+}

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": "7809632235752217223397984069134872080344661788142045715703921686365500598489972267307645283328850847355449234299019626319276212790325537255683197037589565669221047/7820309632235752217223397984069134872080344661788142045715703921686365500598489972267307645283328850847355449234299019626319276212790325537255683197037589565669221047",
+  "six_round_published_margin": "499816463088141900297205128072064209357872762473754060265120273141700647703658743382805420527724846975592877478909642980124495569641960257795177119462673369997408872199010767526949918234737985071151720661720420744910146430561047807811523991448063834848293602940951288905774813618317819206478221072341490762115432396362439256337/500499816463088141900797021591160206110154894353633960170420014626775660818123673370158466238651398217134059116130307860114183611699949820371978788819412493741976197691611504509503147609846242494574299330507962915319209476938523963127021000929972026961869294532912978250775069346531296069981547567603637560743663000000000000000000",
+  "six_round_published_gap_in_1e18_grid_units": "203257822613118398423649615681901870759564534919346455435991204641474679249319106317255015496856267698446990959463149304201942079706973256457266051997904055515334919253659334946851857064957214315753006973902315165650031655939434900585134186471885293069757653463627109101445243283972340126413782953852860034627732/500499816463088141900797021591160206110154894353633960170420014626775660818123673370158466238651398217134059116130307860114183611699949820371978788819412493741976197691611504509503147609846242494574299330507962915319209476938523963127021000929972026961869294532912978250775069346531296069981547567603637560743663",
+  "five_round_initial_saving_threshold_at_upper_endpoint": "1829162061115734711204612179343928598464103977181042174100307447672956554957404185362705705270413198411228029494077049541958522935180842925337/3116153670871394950162303005422192000421791303523182298584729275131891133832382909145893145266471267647512375990279342640500000000000000000000000",
+  "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."
+}

research/bootstrap-depth-audit/PROOF.md

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+# 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<eta<1/2`. Thus
+
+```
+M < q/(1+q) < a/(1+a) <= A_j/(1+A_j) = C_j.
+```
+
+The function x/(1+x) is increasing for x>=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<u` already makes the optimistic five-level ceiling exceed κ;
+the verifier checks this as a negative control on any unqualified depth claim.
+
+Different starting suppliers, stopped-leaf recurrences, outer assemblies or
+sharper analysis are outside this result. All-size compilation, dirty-register
+restoration, completed spectators, fixed tapes, finite cutoffs, precision and
+analytic transfer remain inherited obligations. The source's full moment
+bracket is assumed rather than established by this small arithmetic audit.
+
+Isaac Grannis (@BagMan9), with substantial OpenAI Codex assistance. Apache-2.0.
+Credit Rohan Arun's PR185/PR387, the supplier lineage and the outer-assembly
+authors listed in README.md and the preserved notices.