[jwr404]

Result

This answers the c_row half of research/lockstep-live-role-audit open question 1. The complex row proportionality in the #4470 outer-reserve constant C is exactly 1. It was previously assumed ≤ 2^64.

The chain, with every link a verbatim phrase checked against a pinned source:

1. notes/batched-complex-rows.tex reserves q_0 = ⌈log2 W⌉·D_c(d) row chunks, which is the same unit as OUTER-TRANSFER-AUDIT §5's q0.
2. AUDIT.md §6 says "The row stock is based on B, not B+O", with child ratio 1/N and f = ⌊e/N⌋.
3. FINITE-PRECOMPILATION says "the only remaining paid children have rank one". So exit batches add no outer depth, and D_c(d) = ⌊log_N d⌋.

Hence q_0 ≤ ⌈log2 B_c⌉·log_{N_c} d, and c_row = 1.

Recomputation. The verifier authenticates lockstep's constant_c.py and reruns it unchanged, reproducing its published values. It then reruns it with exactly one substituted assumption line, log2_c_row = 64 → 0, in memory only.

	assumed	derived
log2 C upper bound	64,215,200,000,088	64,215,200,000,024
fit point log2 p	64,473,100,659,114	64,473,100,659,050
upper − lower complex share	132 bits	68 bits

All of lockstep's internal threshold checks still pass. The A_old term never binds, because C is a maximum dominated by the complex share. #4970 separately derives S_old = 1.

No new κ. The selected exponent stays 60403/20000000.

Task and overlap

Card selected-stock-envelope, scope refined (see TASK.json). Base: canonical main b91956876e5b4122fd89179f1b57557ff8378f15, priority revision 7. CLAIM.

The old-lineage "row coefficient 20,161" was a κ≈0.0015-era allowance for a different reservation. It is not a coefficient of this formula.

Not covered:

• B_c and the compiled ranks;
• the router;
• the BIT share;
• the analytic premises.

This is a source-bound reading of the retained reservation, not a new proof of it.

Test plan

python3.12 -B scripts/research_package.py --package research/selected-complex-row-coefficient --check-only
python3.12 -B scripts/research_package.py --package research/selected-complex-row-coefficient
python3.12 -I -O -W error -B research/selected-complex-row-coefficient/verify.py
python3.12 -B scripts/check_research_commit.py --package research/selected-complex-row-coefficient

• All four pass locally on Linux with Python 3.12, in under a second.
• Three mutations make the verifier fail: the derived log2 C, log2_c_row = 1, and a perturbed depth recursion.
• --check-source-history was not run locally because the checkout is shallow.

Open next questions

Close the remaining 68-bit gap between the upper and lower complex shares of C.

Credit

• Mark/Matarener (lockstep-live-role-audit).
• The batched-complex-rows and kappa-rebuild audit authors.
• Cinna Davis and zxv (Conditional κ = 0.00296 via query-thinned productive controllers (+10.18%) #4440/Conditional kappa = 0.00302015 (+0.6304% over #4447): joint helper boundaries #4470).

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

🤖 Generated with Claude Code