[jwr404]

Result

This answers PR #4938 open question 1: does one ordinary H-bit interchange fit O(V H^{1-a} + V) at the actual #4174 width H = r_max G?

BIT.md §4 runs the elementary digit-swap algorithm when 2k >= e or the body width is below 64. Here q^k >= K(w)^{D(e)} and w = ceil(e^c). Evaluated exactly:

1. The elementary branch runs exactly for q-ary widths e <= 311308. Every larger width is recursive. In binary, recursion starts at h = 27706412 bits. At D = 19 the stock is k = 155654, so 2k = 311308.
2. Since H <= L/100, a recursive reservoir interchange needs L >= 2770641200. The production plan has L = 1540096, which is 1,799 times too short. Its H = 15360 bits is 173 q-digits. The smallest L <= p layout from Bind the constant-1 CRT evaluator to accepted reservoir layouts (no new κ) #4938 has H = 10752, which is 121 q-digits. Both are elementary.
3. So the bill holds, but only through the constant. BIT.md's statement is uniform, and on every elementary width E^{a2} < 2^{0.033}. At the actual Audit CRT reservoir placement and split-axis coordinate descriptors #4174 widths the H^{-a} saving is not realized.

No new κ. The selected exponent stays 60403/20000000. This does not accept #4100.

Method

• D <= 20 widths: the stock digits are computed exactly against β = 269241600 |GL_64(F_q)|, where q^4096 < β < q^4097.
• Below the D = 20 block: every width is checked against the literal test.
• Up to 2^576: a depth-block induction, using k <= 8193 D while w = 2.
• Beyond (w >= 3): a short written bound.

Both completed stages share the threshold, because the test does not depend on a.

Task and overlap

Card uniform-transfer-audit, scope refined to the callee's branch at the reservoir width. Base: canonical main cef4b52f2a59009865bf8eabde1be3f159063703, priority revision 7. find_overlap.py, the PR search for "interchange" and the handoff thread found no other claim. CLAIM.

Not covered:

• the constants C and c_el, which BIT.md does not state;
• the three-field placement (Audit CRT reservoir placement and split-axis coordinate descriptors #4174);
• the guard scan and the sparse repair;
• the complete CRT or multiplication theorem.

Test plan

python3.12 -B scripts/research_package.py --package research/crt-interchange-width-threshold --check-only
python3.12 -B scripts/research_package.py --package research/crt-interchange-width-threshold
python3.12 -I -O -W error -B research/crt-interchange-width-threshold/verify.py
python3.12 -B scripts/check_research_commit.py --package research/crt-interchange-width-threshold

• All four pass locally on Linux with Python 3.12. The verifier takes about 20 s.
• --check-source-history was not run locally because the checkout is shallow; it is left to hosted CI.
• Mutating the q-ary boundary (311309), the binary boundary (27706411) or the production q-width (172) makes the verifier fail.

Open next questions

At which reservoir length does the composed CRT bill first beat the elementary-callee bill, for a stated constant ratio C/c_el? Below L = 2770641200 any CRT-layer saving must come from elsewhere. See TASK.json.

Credit

• Cinna Davis (Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100), for CRT.md, BIT.md and the wrapper test.
• Hiroki KUMAZAKI (Audit CRT reservoir placement and split-axis coordinate descriptors #4174), for the constructor and the production plan.
• JoyboyBrian (Bind the constant-1 CRT evaluator to accepted reservoir layouts (no new κ) #4938), for the open question.
• The interchange lineage credited in BIT.md.

Prepared by James Reed with substantial Anthropic Claude (Claude Code) assistance. Apache-2.0. No independent human review.

🤖 Generated with Claude Code