A frame-layout ceiling for #200's bit word: coarse < 6.964e-4 under the banked ledger, so κ < 6.959e-4 on #205–#207's word (no new κ)

[chafreaky]

Adds research/bit-frame-ceiling-200/ and one workflow; no existing file changes and no new κ. This carries DaysSky's #192 frame-floor method from #168's complex word to the physical bit word that #200, #205, #206 and #207 share. #200 is not on main, so the package carries the eight files of #200 it reads (frozen physical bit word, frames, partner chronology, graph, profile, sinks, certificate and the retained PR168-v4 exact integer checker) as a pinned archive baseline-pr200.tar.gz (1.4 MB), every sha256 pinned in SOURCE.json to commit a1175449; reviewers can confirm with git show a1175449:<path> | sha256sum or rebuild the archive byte for byte with git archive --format=tar a1175449 <paths> | gzip -n -9.

Result

For every admissible choice of operation frames on this word (DAG, operations, sources, gauges, 1,760 reuse pairs, partner chronology and 34 sinks fixed):

Ledger	Floor on C	Coarse saving ceiling	κ ceiling	Standing
#200's own (entrance tails charged)	2,804,210	6.904e-4	6.900e-4	#200's 6.7777e-4 is 98.2% of it
completed entrance banks (#205/#206/#207)	2,780,144	6.964e-4	6.959e-4	#207's bit coarse 6.8366e-4 is 98.2% of it
without coupling the two registers of an operation		7.056e-4 / 7.118e-4

Consequences:

• Frame refinement on this word is worth at most +1.8% more on the bit side, under either ledger. Conditional kappa 6.831904550365e-4: coordinated frame optimization and completed entrance banks #207's 302 coordinated changes are already inside this margin.
• The ceiling 6.964e-4 lies below the complex supplier those PRs use (Compose the v4 source-assisted complex word with the PR200 bit supplier: conditional κ = 6.768823e-4 #202's source-assisted word, 7.009e-4, itself at A κ ceiling of 7.010e-4 for every frame layout of #168's word (no new κ) #192's ceiling for its frames). So on this bit word the complex branch can never bind, and every κ assembled from it stays below 6.959e-4 whatever the bit frames. Going further needs a different bit word (circuit, reuse, gauges or sinks), a bank construction that removes other children than the entrance tails, or a different design (A κ ceiling of 2.5592e-3 for every word on #144's paired-cube design (no new κ) #201 bounds the design at 2.5592e-3).

Argument

D = 2v − 3ℓ = 1,936, C = Σ n_r·r·ln(m/r), m = 72.

1. Every certified coarse saving satisfies a < D/C (A κ ceiling of 7.010e-4 for every frame layout of #168's word (no new κ) #192 Lemma 1; on A stronger bit supplier: fixed-coordinate and face-diagonal bit local circuit on PR168 v4 (effective bit saving 6.7734e-4) #200 the gap is 0.088%).
2. Every retained assembly accepts κ only below a/(1+a) of the binding supplier (Add an exact κ-ceiling check for the exponent assembly (no new κ) #183), the atom wrapper and the finite stopped-leaf bootstrap of Finite leaf bootstrap above PR182: conditional kappa = 0.000660627334074291 #185/Pack #187's bit residuals: conditional κ = 0.000676080316519385 #197/Conditional kappa 6.831904550365e-4: coordinated frame optimization and completed entrance banks #207 stay below the coarse saving they start from.
3. Splitting a frame climb into more steps only costs more (r·ln(m/r) is subadditive).
4. No frame layout of the word pushes C below a floor: joint bounds propagated through both registers of every operation, an exact per-register dynamic programme over frame dimensions, and Lagrangian coupling of the two registers (1,200 integer subgradient rounds; each round is itself a valid floor).

The terminal record of #200's bit/terminal.py substitutes exactly one child of width 3 (the sink's root-to-full step, present in every layout) and one target child of width 21 per sink and stage and keeps the sink's other steps charged on the pivot; the script checks this against the certificate's child_delta and the pre/post target histograms, so the floor of the terminal ledger is the pre-sink floor minus that fixed cost. The banked ledger drops only the entrance tails 3·dim σ, which is the only child change the completed-bank constructions make; their bank proofs are not re-verified here and the ceiling does not depend on them.

PROOF.md has the statements and proofs.

Verify

python3 research/bit-frame-ceiling-200/ceiling.py --check     # stdlib only, about 2.5 minutes; -O is refused

The script rebuilds the pinned #200 files after checking the archive and every file hash, prepares the retained exact integer checker's geometry, and checks itself: #200's layout lies inside every derived frame bound; its slot chains reproduce #200's certified pre-sink ledger child for child (W 20,668; rank 1,486,160) and the −102/−102 terminal substitution; every per-slot floor and every Lagrangian round lies below #200's actual cost. All bounds use exact integers with rounded-down rational logarithms, so the printed ceilings are upper bounds. --tree PATH runs on a #200 checkout; --overrides FILE prices another layout of the same word: on #207's frames/opframe-bases.json it confirms the 302-frame layout lies inside every bound and above the floor, with banked D/C = 6.8427e-4 (98.3% of the ceiling).

Run before opening: --check passes from the archive alone and on a #200 checkout (Python 3.14, macOS), with and without #207's overrides. The new workflow runs --check on Python 3.11 and 3.14.

Scope and limits

• One word only: constructions that change the DAG, pairs, gauges, sinks or the bank construction are not bounded.
• Elementary proofs, not machine-checked beyond the script's exact arithmetic and self-tests; no one else has reviewed them.

Credits

DaysSky (#192 method, #201 design ceiling), romainhedouin (#183 assembly ceiling), icekylinx and eumemic (paired-cube bit word and exact checker, #144/#168), James Chang (#166 terminal accumulation), Dugongue, Evan McKinney, Rohan Arun and EcmaXp (#186/#197/#205/#206/#207 bank ledger), ikeboy (#202 complex supplier). Prepared by Chafik Boukhalfa with substantial Anthropic Claude assistance; Apache-2.0, attribution in NOTICE.

[Mister-Meeseeks]

Thank you @chafreaky. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Frame-layout ceiling for PR200's fixed bit word.

Review and limits: Prior exact-head scoped frame-floor replay. A change of scalar word or endpoint structure can escape the stated restriction.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: 9b6ad038f8690206c14ed8f28fb90ec6b3a79ae1. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.