[pinterbanget]

Summary

This adds research/pr4100-crt-repair-tariff/. It audits the cost of the sparse exception repair in Cinna Davis's #4100 CRT.md §4: "sort those holes by this full key on fixed tapes, and merge them back … It does not sort the full input", and "bad-key evaluation, O(p)-bit keys, radix passes, counters and clearing all fit O(V)".

Gap. Every open CRT audit excludes this tariff:

• Audit PR4100 arbitrary-source CRT repair with an explicit inverse #4169 (kumagi): "sequential head movement/sorting costs";
• Audit CRT reservoir placement and split-axis coordinate descriptors #4174: the "sorting tariff";
• Audit conditional CRT last-K two-slice streaming tariff #4187: the repair integration;
• Lower admitted-stencil CRT guard scan to a literal three-tape program #4201 and Produce a one-target CRT guard stencil from paid unary geometry #4208 (ToikY): "exception extraction, inverse keys, sorting, payload routing, reinsertion";
• Audit the compact-switch coefficient and guard contract. #4321 (unpaidmoon): "bad-key repair".

The mechanism is jmbarrancoml's merged PR2250, but its cost O(V + M A³ + δ M A(R+A)) is absorbed there only because its records are superpolynomial. In CRT.md R and L are both Θ(p), so M A³ = Θ(p²)V. CRT.md says not to import that tariff unchanged, but nobody had checked its replacement.

Claim type: a scoped proof audit (per-step accounting with explicit constants and exact absorption at #4100's parameters), checked against a literal tape run around #4100's own compact fixture. No new κ.

Result

The tariff holds. Per color:

• Dense part, ≤ 9MR + 5M. One split scan driven by the guard scan's hole mask, and one reinsertion scan. No key, inverse or arithmetic work touches a good record.
• Sparse part, ≤ N(L+R)(9L+6) plus N key evaluations. Keys T(S^-1(x)) are computed only at holes, followed by L two-bucket stable LSD passes at nine sweeps each, and keyed reinsertion.
• Counter, ≤ 8M + 4LN + 4L.
• Absorption. With δ ≤ r(2/B + 8B/2^K) at Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100's G = 64(ell+⌈log₂(p+2)⌉), the sparse moves are ≤ p³(p+2)^-64·V, which is o(V) at every p. The exact values run from 2^-640 at p=16 to 2^-37,326 at p=2^64. CRT.md's 2^-G E ≤ 2^(-60ell)(p+2)^(-48) line also holds exactly.

Caution, not a defect. Reinsertion's check "key equals current address" only proves the keys permute the holes. Keys S^-1(x) without T also pass it while misplacing every hole record. Correct keys rest on the inverse evaluator: #4169's reverse word composed with T.

Evidence

verify.py uses only the standard library and takes about 6 seconds.

• Fixtures. S and T come from Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100's pinned check_batched_crt.py (compact, good). There are five fixtures: B ∈ {4,8,16}, guard 6 or 8, three source functions, M up to 65,536.
• Correctness. The repair realizes T on every record, and keys are evaluated only at holes. The measured density never exceeds 2/B + 8B/2^K.
• Costs. The dense part is at most 7.23·MR; the sorted part is 9.3–9.5·L(L+R) per hole.
• Controls, each of which must fail:
  • A full-input sort evaluates 16,384 keys instead of 11,776 and moves 262·MR, which is Θ(L)·V.
  • Keys S^-1(x) misplace 5,888 records without the reinsertion check noticing.
  • Without repair, 2,048 records are wrong.

Commands run locally (macOS, CPython 3.12):

python3.12 -B scripts/research_package.py --package research/pr4100-crt-repair-tariff --check-only
  PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
python3.12 -B scripts/research_package.py --package research/pr4100-crt-repair-tariff
  PASS_REGISTERED_RESEARCH_PACKAGE research/pr4100-crt-repair-tariff outcome=completed
python3.12 -B scripts/check_research_commit.py --package research/pr4100-crt-repair-tariff --check-source-history
  PASS_RESEARCH_COMMIT_BYTES {... "files": 8, ...}

Update: exact two-target check (head b3ea5ee525b131273d190747b892e5f3667d286f)

This closes the package's earlier open question 2.

• Exact r-target density. Active targets use distinct digits and disjoint guard windows, so on a complete enumeration the good events are independent and δ_r = 1 − [(1−1/B)²(1−8B/2^K)]^r holds exactly. CRT.md's r(2/B+8B/2^K) follows. The identity is checked exactly on all five one-target fixtures and on every one of the 2^20 addresses of Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100's two-target layout (positions=(0,6), B=4, guard 6). That case gives 965,632 holes, density exactly 943/1024. The union bound evaluates to 2 at toy B, so the exact identity is what is checked.
• Two-target repair. With Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100's own source_pair and a second pair reading both guards, the repair realizes the two-target switch on all 1,048,576 records. Costs are 4.65·MR dense and 9.25·L(L+R) per hole, both within the written bounds. Keys without T misplace 724,224 and 603,520 records respectively, undetected by the key check.
• Sweep-level repair. It runs the 2^20 case and matches the per-cell repair exactly (meters and layout) on all five one-target fixtures.
• Premise control. A source that reads the other active target breaks S = T on 55,296 good addresses, and S stops preserving the good set. A sparse repair cannot fix that. This is the two-target counterpart of Audit PR4100 arbitrary-source CRT repair with an explicit inverse #4169's counterexample.
• Runtime. About 45 seconds; the registered replay and the committed-byte preflight pass at the new head.

Not covered

• The E(p,ell) evaluator's own cost (Audit uniform Beneš compiler with charged sequential arena scans #4192 covers its Beneš part).
• Producing the hole mask (Lower admitted-stencil CRT guard scan to a literal three-tape program #4201/Produce a one-target CRT guard stencil from paid unary geometry #4208).
• The macro's ordinary interchanges, the last-K branch (Audit conditional CRT last-K two-slice streaming tariff #4187), and reservoir placement (Audit CRT reservoir placement and split-axis coordinate descriptors #4174).
• r ≥ 3 targets are not enumerated (2^(10r) addresses). The exact formula rests on field disjointness at the actual coloring, which is now open question 2.
• There is no proof-kernel check.

Task and overlap

• Card: uniform-transfer-audit, refined to this repair-cost implication (recorded in TASK.json), priority revision 7.
• Base: canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244. Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100 is pinned at 337d04939571a637aeab54a0f58f54fde811663a.
• Advance claim.
• Overlap: I checked every CRT PR by ToikY, unpaidmoon and kumagi, the DONE: PR4100 CRT reservoir placement and coordinate descriptors #4173 thread and Handoffs and requests for independent checks #1166 at 09:10 and again at 09:17 UTC. No claim or PR covers this tariff.

Credit

• The construction and audited claims are Cinna Davis's Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100 (OpenAI Codex assistance).
• The repair mechanism is jmbarrancoml's PR2250, building on Douglas Colkitt's compact-control construction.
• kumagi's Audit PR4100 arbitrary-source CRT repair with an explicit inverse #4169 supplies the reverse-word keys, and ToikY's Lower admitted-stencil CRT guard scan to a literal three-tape program #4201/Produce a one-target CRT guard stencil from paid unary geometry #4208 the hole mask.
• This tariff audit was prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance. Apache-2.0.

🤖 Generated with Claude Code