[petalas]

Result

Answers Open next question 1 of James Reed's PR4942 (research/crt-interchange-width-threshold). Claim type: scoped proof with exact computation, conditional on a stated ratio rho = C/c_el. No κ change; the selected result is untouched.

With BIT.md's exact a2 = c(1-c^2), c = 8667/5000000, and PR4942's branch threshold H_b = 27706412:

• Crossover width. The recursive bill C V H^(1-a2) first beats the elementary bill c_el V H at H*(rho) = max(H_b, h(rho)), where h(rho) is the least integer with h^a2 > rho. This is roughly rho^576.9 once the ratio binds.
• Critical ratio. rho_c = H_b^a2 = 1.030151098331576173576873168658... (30 digits certified). For rho < rho_c the branch threshold binds. Any recursive constant more than 3.02% above the elementary one pushes the crossover past it.
• Exact least reservoir length at the branch threshold. Over every layout the Audit CRT reservoir placement and split-axis coordinate descriptors #4174 constructor accepts, the least L with H = floor(L/(100G))G >= H_b is 2770643200. Witness: d = 2^28, ell = 11, m = 2, G = 832, r_max = 33301. This sharpens PR4942's bound 100 H_b = 2770641200 and answers the question for every rho < rho_c.
• Ratio-bound examples. rho = 1.031: H* = 44561394, L = 4456140800. rho = 1.032: H* = 77956662, L = 7795667200. Exact H* is certified for rho = 1.05, 1.1, 1.5, 2, 4, 10, with 41, 80, 338, 577, 1154 and 1917 bits. At rho = 2 a profitable recursive call needs more than 100 * 2^576 reservoir bits.

Evidence

research/crt-recursive-interchange-crossover/verify.py uses only the standard library:

• Two logarithm engines. A rigorous integer fixed-point interval ln (atanh series, directed rounding, explicit tail bound) certifies h^a2 > rho > (h-1)^a2 for each candidate proposed by correctly rounded decimal ln/exp. An unresolved interval aborts.
• The 30-digit rho_c bracket is certified at both ends, and the width just above rho_c is H_b + 1.
• The pruned layout search agrees with a complete brute (s, ell, m) scan on five fixtures. The verifier also checks that the brute ranges are large enough. Every witness is checked against the accepted domain.
• Five rejection controls. BIT.md constants and placement-domain phrases are read from pinned inputs. RESULT.json is rebuilt and compared byte for byte.

Reproduce from the repository root (interpreter used: ~/git/integer-mult-bounds/.venv/bin/python3.12, CPython 3.12):

python3.12 -B research/crt-recursive-interchange-crossover/verify.py
python3.12 -B scripts/research_package.py --package research/crt-recursive-interchange-crossover
python3.12 -B scripts/check_research_commit.py --package research/crt-recursive-interchange-crossover --check-source-history

All three pass locally. The registered check takes about 1.6 s.

Limitations

• Conditional on BIT.md's bill as stated and on PR4942's threshold. Both are cited, not re-proved. BIT.md gives neither C nor c_el.
• Per-layer bills only. A common multiplicity (PR4944's 6 calls) cancels. Padding that differs between branches would change the effective rho.
• Exact least lengths are computed only for H* <= 10^9. Larger ratios report L >= 100 H*.
• No complete CRT bill, supplier admission or multiplication theorem. All five framework assumptions remain. No Lean, Docker or selected replay was run.

Open next questions

1. Bound the actual constants C and c_el, including padding, at w = 2, D = 19. Above rho_c the crossover grows like rho^577, so a definite ratio settles whether any physical reservoir profits.
2. Exact least lengths for larger H* via a divisor search near 64 ceil(H*/64) instead of the linear k scan.

Related open work

No open PR studies this crossover, including this account's PRs opened since the claim. #4187 (conditional CRT last-K streaming tariff) concerns a different CRT quantity.

Credit

Builds on PR4942 (James Reed), PR4944 (JoyboyBrian), PR4174 (Hiroki KUMAZAKI) and PR4100 (Cinna Davis). The new analysis and verifier were produced by Nick Petalas. Claim: #1166 (comment)

CI notes

Both hosted jobs passed at head ec5f3b0119aaae28e494f987dafcdf07a23b873d: plan and replay (research/crt-recursive-interchange-crossover) in run 41671993815. This account's PRs opened since the claim (#6663 to #6674) study other objects, so none share a number with this result.