Research note (no κ; bit headroom): helper-deletion ladder under optimal window-respecting re-hosting — ≈ +0.082% c per deletion (d=0..6, certified)

[thakrarsagar]

Research note (bit headroom only); no κ claim. This answers an open question from #2423: what is each further width-8 helper deletion worth on the bit supplier, once the freed slack is re-hosted optimally while respecting stage windows? #1043's R438 is d = 1. #1250 shows that helper 470 is the only removable helper on the fixed word, so d ≥ 2 prices changed words.

d	R	slack S	certified optimum (unique)	c TEMPLATE (#1043's measured delta)	vs #589
0	439	45	9×(42,5)	0.001733427039589844234749	0
1	438	54	10×(42,5)+1×(42,4)	0.001734802047828389852783	+0.079%
2	437	63	12×(42,5)+1×(28,3)	0.001736238589146810207048	+0.162%
3	436	72	14×(42,5)+1×(28,2)	0.001737652736279473456500	+0.244%
4	435	81	16×(42,5)+1×(14,1)	0.001739118212671177105905	+0.328%
5	434	90	18×(42,5)	0.001740604775296424266918	+0.414%
6	433	99	19×(42,5)+1×(42,4)	0.001741991196114600260000	+0.494%

Shape of the result.

• About +0.082% c per deletion, nearly linear. The steps alternate with S mod 5, which sets the partial slot.
• The optimum is always fully idle 42-slots first, with one partial slot.
• Two more illustrative per-deletion models are reported, RANK1 {1:−8} and RANK7 {7:−1, 1:−1}. Neither is an envelope, as the README explains.

How it was checked (deletion_ladder.py, reusing the pinned verify_rehost machinery of #2423):

• Layouts: an exact top-3 DP, cross-checked by exhaustive enumeration for every row (up to 728,466,365 layouts at d = 6).
• Windows: all window splits are feasible by max-flow and by Hall, and the placements are audited.
• Endpoints: 21 certified endpoints, each accepted at c and rejected at c + 1e-24.
• Controls: the run reproduces the endpoints of Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589 (d = 0), Finite BIT construction: R438 source boundary with full hosted invoice #1043's certified profile and Bit construction (no κ; conditional on #1043): R438 + one transport + optimal window-respecting re-hosting → c = 0.0017348463 (+0.082% vs #589); fixes two #1345 atlas crashes #2423's re-hosting (d = 1), and Bit construction (no κ; conditional on #1043): R438 + one transport + optimal window-respecting re-hosting → c = 0.0017348463 (+0.082% vs #589); fixes two #1345 atlas crashes #2423's transported word (d = 1, RANK7). It has 88 negative controls.
• Independent check: an adversarial AI verifier recomputed all 21 rows with its own code (exhaustive layouts, Decimal endpoint bisection, a 512-subset Hall test) and matched every digit. Its three presentation fixes are applied.
• Packaging: --check-only, the 3.10 grammar check and the research tests pass. Run on CPython 3.14.4; the registered 3.12 replay was not run.

python3 -I -B research/bit-deletion-ladder/deletion_ladder.py --check research/bit-deletion-ladder/RECEIPT.json   # ~80 s
# PASS_BIT_DELETION_LADDER d=0..6 exhaustive=21/21 windows_feasible=7/7 c_template(d=6)=0.001741991196114600260000 ... no_new_kappa

Scope.

• No word exists for d ≥ 2, so the local models are declared rather than derived.
• W, the Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589 frames and the windows are held fixed.
• The slot model is width-8 only, with no banks.
• The native lane was not run.
• The bit supplier does not bind κ.
• Base: main 0fdd7d82. Card selected-packing, scope changed; the assignment is the bit-supplier track, seed 14.

Credits.

• Finite BIT construction: R438 source boundary with full hosted invoice #1043 (Ali Romman).
• Bit construction (no κ; conditional on #1043): R438 + one transport + optimal window-respecting re-hosting → c = 0.0017348463 (+0.082% vs #589); fixes two #1345 atlas crashes #2423 (this author).
• Research note: PR589 bit hosting is width-8 optimal; width-7 residuals worth up to +17-23% bit headroom (no new kappa) #1023.
• The seed72 and seed14 packages.
• Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589 / Bit c = 0.001733052823: #545's punctured unit on the R420 skeleton (111 live ports, +4.79% over #543) #555.
• Scoped negative result: arbitrary source masks add no R439 helper deletion #1250 / bit-source-substitution-seed11.

Prepared with substantial AI assistance (Claude Code, Anthropic, Claude Opus 5.5): an AI builder and an adversarial AI verifier. No independent human review is claimed. Apache-2.0.

🤖 Generated with Claude Code