[JoyboyBrian]

Result

Scoped written proof with exact finite checks. κ is unchanged. This is about the one ordinary-BIT bootstrap stage of the current selected record #4470 (κ = 60403/20000000). That stage is the delayed-cover producer of uniform rank N = 2^H1, H1 = 25986965159937. It is stopped at atom w = ceil(e^(1/250)) with old ordinary leaves and has ordinary saving 0.0034363538396. Its source reserve (DELAYED-COVER-BRIDGE §8, BIT-ALLSIZE-PARAMETERS.json stock) borrows k = D(N^2(w-1) + ceil(log_q(B|GL_N(F_q)|))) + ceil(log_q S_old) + 1 digits, with D = floor(log_N ceil(e/w)). The source requires k < e/2 "for sufficiently large e".

Results (research/selected-bit-stage-stock-band, PROOF.md):

1. Two-sided parent envelope: D·N²·w ≤ k_par ≤ D(N²w + ceil(log_q B)). This holds for every N, every odd prime q and every B, e, w ≥ 1.
2. Depth classification for N ≥ 64:
   • D = 0 has no parent stock.
   • Every D = 1 width overdraws the chunk (k_par > e).
   • Every width with n ≤ 4N² is infeasible.
   • Every width with n ≥ 16N²+1 fits (k_par < e/4, given ceil(log_q B) ≤ N²). This includes all widths with D ≥ 3.
3. Selected bands, with t = log2 e:
   • t ≤ 26091330481863: no cover level runs.
   • 26091330481866 ≤ t ≤ 52182660963729: the source prescription is infeasible. This holds for every B, q, S_old and every larger admissible depth.
   • t ≥ 52182660963735: it fits, under two explicit hypotheses on B and on the old-leaf stock.

So the source's "sufficiently large e" begins inside a proved six-bit window of log2 e. This confirms #4612's ≈5.2183·10^13 estimate with a matching lower side. The rank-N word first applies at t ≈ 2.609·10^13. Over the ≈2.609·10^13-bit band between that point and the fit point, the conversion must call the retained supplier directly. The threshold is fixed, so exponents are unaffected. This is a precise domain statement, not a counterexample.

The lower side does not depend on the old-leaf reserve S_old. On the fit side S_old stays an explicit hypothesis: any logarithmic envelope with log2 σ_old ≤ 52182660963686 suffices. Its numerical value is not derived here.

Task and overlap

• Card: selected-stock-envelope (adversarial-audit, priority revision 7). Scope is refined to the Conditional kappa = 0.00302015 (+0.6304% over #4447): joint helper boundaries #4470 stage (TASK.json). The card's starting points concern the historical Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094 seven-stage chain.
• CLAIM: Handoffs and requests for independent checks #1166 (comment)
• Base: cef4b52f2a59009865bf8eabde1be3f159063703. Seven inputs are pinned in SOURCES.json.
• Overlap: Audit #4359 BIT row stock and its #4470 atom change (no new κ) #4612 (Mark/Matarener) gives the δ-generic upper envelope and an illustrative threshold. research/lockstep-live-role-audit (Matarener) gives the outer C, with log2 A_old ≤ 64 assumed. This PR adds the lower side, the depth classification and the exact bands. It does not repeat their upper-envelope or C arithmetic. find_overlap.py and a board check before submission found no other claim on this object.

Test plan

python3.12 -I -W error -B research/selected-bit-stage-stock-band/verify.py      # PASS_SELECTED_BIT_STAGE_STOCK_BAND ... controls=7
python3.12 -I -O -W error -B research/selected-bit-stage-stock-band/verify.py   # PASS
python3 scripts/research_package.py --package research/selected-bit-stage-stock-band --check-only   # PASS_PACKAGE_METADATA_AND_PINS
python3.12 -B scripts/research_package.py --package research/selected-bit-stage-stock-band          # PASS_REGISTERED_RESEARCH_PACKAGE outcome=completed
python3.12 -B scripts/check_research_commit.py --package research/selected-bit-stage-stock-band --check-source-history   # PASS_RESEARCH_COMMIT_BYTES
python3.12 -W error -B -m unittest discover -s tests -p 'test_research_onboarding.py'   # 26 tests OK

What the verifier covers:

• Source hashes and the cited sentences.
• 432,000 exhaustive small-rank instances of the lemmas, using exact |GL_N(F_q)|; three orders are brute-forced.
• Exact selected-rank witnesses that bracket the window (last infeasible t=52182660963500, first fit t=52182660963750).
• A reduced-rank big-integer oracle over 869 widths.
• Seven rejection controls: tampered bridge, no high-matrix lift, role count above q^(N²), stale atom 3/1000, rank off by one, a fit slack one bit short, and a claimed fit for a single cover level.

Manual mutation tests made the verifier FAIL in every case: a doubled D=1 inequality, fit slack 5, a tighter fit lemma, a shifted documented threshold, and off-by-one witness exponents.

Not run: any supplier, native, Lean or frontier replay.

Open next questions

1. Derive S_old numerically from the retained completed-leaf contract. Is S_old = 1 admissible under PR185's "no external extra stock" wording? This would replace the A_old ≤ 2^64 assumption.
2. Should the Conditional kappa = 0.00302015 (+0.6304% over #4447): joint helper boundaries #4470 source or review state the stage domain log2 e ≥ 52182660963735 instead of "sufficiently large e"? This is a maintainer decision.
3. Close the six-bit window with the exact B from the compiler realization.

Credit

• zxv: Conditional κ = 0.0023 (+5.690e-4, +32.87% over #4181): productive helper jobs compiled into a uniform supplier #4359 and the delayed-cover bridge and stock formula.
• Cinna Davis and zxv: the Conditional κ = 0.00296 via query-thinned productive controllers (+10.18%) #4440/Conditional kappa = 0.00302015 (+0.6304% over #4447): joint helper boundaries #4470 chain, including the atom change to 0.004.
• Mark (Matarener): Audit #4359 BIT row stock and its #4470 atom change (no new κ) #4612 and lockstep-live-role-audit.
• The three-stage-cover-rows and PR185 completed-leaf authors, as credited in their sources.

Prepared by JoyboyBrian with substantial assistance from Anthropic Claude (Claude Code, Opus 5.5). No independent human review is claimed.

🤖 Generated with Claude Code