[jwr404]

Result

This answers PR #4954 open question 1. The old-leaf reserve S_old of the selected #4470 ordinary-BIT stage is 1.

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

1. The stage retains a_old = 0.0000384599 (query-thinned-controllers PROOF).
2. In certificates/copied-centers-network.json, that saving belongs to the copied-centers BIT supplier, with W_b = 188181929, m_b = 575 and largest child 529. copied-centers-bit.tex sets a_b = 384599/10^{10}, and its all-width interface comes from batched-bit-rows.tex.
3. In batched-bit-rows.tex, "Supplying the bit rows from existing digits", the interface borrows ρ = O(log e) high digits from its own two chunks and restores them. In the note's words: "The row digits were already part of the two input chunks".
4. All old-supplier call sites in the stage notes take their rows from their own chunks: weighted leaves, cover routing, peeled and stopped leaves, and the high-part exchange and residual digits. These are the four uses PR185 lists.

So the bridge's external old-leaf factor is S_old = 1, and k = k_par + 1.

Conservative fallback. If the multiplication-assembly row supply of copied-centers-assembly.tex is used instead, then log2 S_old ≤ 252·⌈log2 e'⌉, from (t_b, w_b) = (9, 28). This is the "252 ordinary-leaf allocation" in PR185 and the #1094 row bill. At #4954's fit threshold it gives k_old ≤ 103 digits.

Consequences.

• Bound the selected #4470 BIT-stage stock on both sides: every single-cover-level width overdraws, fit from log2 e = 52182660963735 (no new κ) #4954's fit side, log2 e ≥ 52182660963735, now needs only c_B ≤ N².
• lockstep-live-role-audit's assumed log2 A_old ≤ 64 becomes A_old = 0, or ≤ 2^9 under the fallback.
• log2 C is unchanged, because the complex share dominates.
• The six-bit window now depends only on B.

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. The overlap search for S_old found only #4954's posts. CLAIM.

Not covered:

• B and the window;
• the old supplier's own finite word and fixed tapes, which are taken as the pinned notes state them;
• the complex row stock, C, precision and supplier replays.

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

Test plan

python3.12 -B scripts/research_package.py --package research/selected-old-leaf-reserve --check-only
python3.12 -B scripts/research_package.py --package research/selected-old-leaf-reserve
python3.12 -I -O -W error -B research/selected-old-leaf-reserve/verify.py
python3.12 -B scripts/check_research_commit.py --package research/selected-old-leaf-reserve

• All four pass locally on Linux with Python 3.12, in under a second.
• Fifteen inputs are pinned, and 20 verbatim phrases are checked.
• Mutating W_b, the fallback digit count, or the batched-bit-rows phrase makes the verifier fail.
• --check-source-history was not run locally because the checkout is shallow.

Open next questions

Close #4954's six-bit window from the compiled role count B. That count is not materialized in the pinned sources.

Credit

• JoyboyBrian (Bound the selected #4470 BIT-stage stock on both sides: every single-cover-level width overdraws, fit from log2 e = 52182660963735 (no new κ) #4954).
• Mark/Matarener (lockstep-live-role-audit).
• zxv (Conditional κ = 0.0023 (+5.690e-4, +32.87% over #4181): productive helper jobs compiled into a uniform supplier #4359/Conditional kappa = 0.00302015 (+0.6304% over #4447): joint helper boundaries #4470 bridge).
• 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).
• The copied-centers, batched-bit-rows and three-stage-cover authors.
• The PR182/PR185 lineage.

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

🤖 Generated with Claude Code