[JoyboyBrian]

Result

Binds PR #4377's constant-1 evaluator E(p, ell) = 2^{4 ell} (p+2)^{16} to every layout the merged #4174 constructor accepts with L <= p. That is #4328's absorption hypothesis, and it holds at the actual #4100 caller. This answers the first step of #4377's open question 2, which was blocked on placement geometry. It also removes the unevaluated C_E that #4646 and #4921/#4923–#4926 carried.

On every such layout:

1. ell >= 21, p >= 1310720 and G >= 2688. These are sharp at p = 1310720, ell = 21, d = 2^16, widths 20.
2. All Prove the uniform CRT evaluator bound with constant 1. #4377 hypotheses hold. Control groups have W <= p. Batches have r_max <= p/268800 axes and the layout has d <= p/20 axes, both below 8(p+2). So C_E = 1 applies.
3. H >= G and 2^{-H}E <= 2^{-G}E <= 2^{-60 ell - 48 λ} <= 2^{-2268}, with equality at p = 2^21 - 2, ell = 21.
4. The per-record bad-key charge r(2/B + window)E is below (p/89600) 2^{-G} E < 2^{-2263}, for both the sharp and the published windows.

L <= p is necessary. The constructor accepts p = 2, ell = 2, d = 2^19, where r_max = 40 > 32 = 8(p+2) and the root group has W = 2^19 > p. Under the actual caller's stronger L <= b + 1 with p = 6b, the threshold becomes ell >= 24, and the #4174 production plan attains it.

No new κ. The selected exponent stays 60403/20000000. This does not accept #4100. The H-bit interchange callee, sparse sort/reinsertion and #4377's bit-step model are not covered.

Task and overlap

The card is uniform-transfer-audit, with scope refined to the evaluator-to-layout binding (see TASK.json). The base is canonical main cef4b52f2a59009865bf8eabde1be3f159063703, priority revision 7. find_overlap.py and PR/discussion searches for the #4377 evaluator, C_E, H >= G and 8(p+2) found no open claim or PR on this binding. That is a search result, not a novelty proof.

Test plan

python3 scripts/research_package.py --package research/crt-evaluator-layout-binding --check-only
python3.12 -B scripts/research_package.py --package research/crt-evaluator-layout-binding
python3.12 -B scripts/check_research_commit.py --package research/crt-evaluator-layout-binding --check-source-history

All three pass locally on macOS with Python 3.12.2. The verifier also passes with -O and with -I -W error, in well under a second. The 26 onboarding tests also pass. Mutating the exact claims (ell >= 21, the 2^{-2268} equality and r_max = 40) makes the verifier fail.

Open next questions

Does one ordinary H-bit interchange of CRT.md §3–§4 fit O(V H^{1-a} + V) at the actual #4174 width H = r_max G? See the README and TASK.json.

Credit

Cinna Davis (#4100), unpaidmoon (#4377, #4646), #4328, Hiroki KUMAZAKI (#4174), #4650/#4845, James Reed (#4921, #4923–#4926). Prepared by JoyboyBrian with substantial Anthropic Claude (Claude Code) assistance. Apache-2.0. No independent human review.

🤖 Generated with Claude Code