[pinterbanget]

Summary

This adds research/pr4100-guard-coloring/. It decides the disjoint-guard premise in Cinna Davis's #4100 CRT.md §4: "a margin 2B from both ends of each K-bit guard … Color targets by current position modulo K, so guard intervals are disjoint". It also checks that premise on #4174's actual placement descriptors.

Three audits take the premise as given. #4321 (unpaidmoon) assumes "the K-bit guards are pairwise disjoint" and notes that no audit decides it. #4328's exact r-target density needs it. #4174 (kumagi) supplies the descriptors.

Claim type: a scoped proof audit with a counterexample to the literal text, plus a census of #4174's production descriptors. No new κ.

Result

1. Lemma. Under position-mod-K coloring, (K−1)-bit guards are pairwise disjoint and free of active targets for every descriptor. K-bit guards are pairwise disjoint too, but one contains another active target exactly when two same-color targets are K apart.

2. The literal K-bit reading fails on Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100's own gadget. I ran all 2^20 addresses of its two-target fixture (targets 0 and 6, so exactly K = 6 apart).

   • With 5-bit windows, as the code's good(…, guard=6) uses, S = T on all 82,944 good addresses for both sources.
   • With 6-bit windows, S ≠ T on 3,888 and 2,388 good addresses, and T leaves the good set on 31,104 and 15,552. For example, (0,1,1086) goes to (0,5,1087) instead of (0,1,1087): a carry in one target's window flips the other target.
   • So the premise must read "(K−1)-bit guards", or K-bit guards with coloring mod K+1.
   • This matches Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100's code and its 8B/2^K density term, which counts 4B bad values in K−1 bits. It is a presentational correction to CRT.md's "K-bit guard" and to Audit the compact-switch coefficient and guard contract. #4321's wording. No bound changes.

3. Audit CRT reservoir placement and split-axis coordinate descriptors #4174's production descriptors. I censused all 104,864 plans of Audit CRT reservoir placement and split-axis coordinate descriptors #4174's own production layout: d = 65,536, ell = 24, widths alternating 24/23, K = 196,608, r_max = 5. That is 524,288 active-axis visits.

   • Every active axis lies in y after placement.
   • No two active axes of any plan can share a color under any Beneš bit schedule. No fragment-pair difference range contains a nonzero multiple of K. The aligned schedule confirms this over 12.3M target visits: maximum multiplicity 1, and no exact-K pair.
   • So at production parameters the premise holds vacuously, and every compact macro handles one target. The multi-target cancellation, and Audit #4100 CRT.md's sparse exception repair tariff at Θ(p)-bit records: O(V) dense, o(V) sparse at every p (no new κ) #4328's r ≥ 2 density, are not exercised there.
   • The elementary last-K branch is material. 16,225 plans (15.5%) have active axes in the top K−1 bits of y, and up to all 5 targets of a layer can be exceptions (Audit conditional CRT last-K two-slice streaming tariff #4187 audits that branch).

Update: eight more width patterns, two adversarial ones (head a3b9095c536c404e11d2ed868699fc4fcd4d4f57)

The census now runs all 11 patterns at d = 65,536, each accepted by #4174's Layout:

• Nine regular patterns have zero possible same-color pairs under any Beneš schedule, and aligned multiplicity 1. They are: alternating (production), blocked both ways, uniform 24, uniform 23 (r_max = 4, 131,072 plans), three seeded random patterns, and alternating 32/31 at K = 229,376. Last-K exception axes appear in 13–16% of plans throughout.
• The vacuity is pattern-specific. Uniform 24 misses a multiple of K by one bit. At level 14, batch 1638 pairs axes 16383 and 24576, which straddle a skipped left half and lie 8,193 axes = K + 24 bits apart.
• An admitted adversarial pattern hits it. Uniform 24 with ell−1 axes 16384..16407 puts two active targets of one color exactly K apart under the aligned schedule, at 24 layers. Literal K-bit guards fail there, which is the §2 gadget failure; the (K−1)-bit windows are checked directly to be target-free. A single short axis at 20000 reaches exact K only for some other bit schedule.
• So the K−1 correction is needed on admitted layouts, not just in principle.

The runtime is now about 2 minutes. The registered replay and committed-byte preflight pass at the new head.

Evidence

verify.py uses only the standard library and takes about 18 seconds. It covers 3,266 exhaustive lemma cases, the gadget on 2^20 addresses for two sources and two window lengths, and the full production census.

Source bindings: SHA256 pins of CRT.md and check_batched_crt.py, CRT.md phrases, good()'s guard−1 window expression, and the byte hash of the vendored allocator.

#4174's allocator.py is vendored unmodified, with a byte check and a vendor/NOTICE.md (head 5fec853f, blob 4b3ad896, Apache-2.0, Hiroki KUMAZAKI). It is vendored because that PR head is not reachable from main, and package pins resolve only against reachable objects.

python3.12 -B scripts/research_package.py --package research/pr4100-guard-coloring --check-only
  PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
python3.12 -B scripts/research_package.py --package research/pr4100-guard-coloring
  PASS_REGISTERED_RESEARCH_PACKAGE research/pr4100-guard-coloring outcome=completed
python3.12 -B scripts/check_research_commit.py --package research/pr4100-guard-coloring --check-source-history
  PASS_RESEARCH_COMMIT_BYTES {... "files": 9, ...}

Not covered

• Only d = 65,536 and ell ∈ {24, 32} are censused (11 patterns). The lemma covers every descriptor with (K−1)-bit guards.
• The correctness of the elementary last-K branch (Audit conditional CRT last-K two-slice streaming tariff #4187).
• Control-function supports (Audit PR4100 arbitrary-source CRT repair with an explicit inverse #4169, Audit CRT reservoir placement and split-axis coordinate descriptors #4174).
• There is no proof-kernel check.

Task and overlap

• Card: uniform-transfer-audit, refined to this guard premise (recorded in TASK.json), priority revision 7.
• Base: canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244. Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100 is pinned at 337d04939571a637aeab54a0f58f54fde811663a. Audit CRT reservoir placement and split-axis coordinate descriptors #4174 is read at 5fec853f0c048871de3b8098d1e30d4ee19cda23.
• Advance claim.
• Overlap was refreshed at 09:35 UTC. Audit the compact-switch coefficient and guard contract. #4321 is unchanged since 07:49 UTC, and no other PR or claim decides this premise.

Credit

The construction and audited text are Cinna Davis's #4100 (OpenAI Codex assistance). The placement descriptors are kumagi's #4174, vendored unmodified. #4321 (unpaidmoon) states the hypothesis decided here. Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance. Apache-2.0.

🤖 Generated with Claude Code