Bit leaf bootstrap: queue standing + the deferred #200 composition at conditional κ = 6.773148e-4

[maxime-fleury]

Conditional kappa = 267141/400000000 = 6.6785250e-4 (+0.0629% over #194's
1668581/2500000000 = 6.674324e-4).

Self-contained on main: the pinned inputs it needs — #194's assembly route, the complex and bit
certificates, and the PR168 v4 pinned export — are vendored byte-identical under
research/source-assisted*/ and research/paired-cube-twin-local-168/certificate.json, so
verify.py runs on this branch alone. Those inputs are #194's (and, through it, #193/#189/#185/#184/#182's)
and are not claimed here. Package: research/bit-leaf-bootstrap-189/.

Where we are (2026-10-09)

The queue moved after this PR opened. Exact standing claims, read from the open
pull requests:

PR	author	kappa	lever	note
#200	chafreaky	6.65046903615388e-4	new bit and complex supplier	With its own complex word the complex branch binds, so the headline equals #196. It states that composed with #193/#194's complex its bit supplier "would give about 6.769e-4, which I am preparing as a separate stacked composition".
#197	evmckinney9	6.76080316519385e-4	packs #187 rank-4/rank-24 residuals + #193 complex + the finite leaf	The highest standing kappa claim. Its body: +0.8398% packing gain.
#198	sennemmi	6.679123e-4	102 lowered #189 endpoint frames + the finite leaf	Draft; explicitly "not claimed as the current public record" and "do not claim to beat #197". Credits this PR: the recurrence "first applied to PR189 by PR199".
#199	this PR	6.678525e-4	finite leaf bootstrap on #194's inputs	Reproduces #194's kappa exactly; the leaf is the only change.

So on #194's inputs this PR's headline is superseded on the number: by #198
(the same recurrence on a descended #189 word) and by #197 (a stronger bit
word). This PR's first published application of the #185 bootstrap to #189's word
is credited there and is no longer the record; what it still contributes is the
reproduction harness, the two maximality proofs, and a pricing path that composes
the queue's newest supplier — which produces the highest value below.

Beating the standing best: the composition #200 deferred, plus this leaf

#200 supplies a strictly stronger bit word (coarse
677773948354561/10^18 = 6.77773948355e-4 against #189's
668298937775631/10^18 = 6.68298937775631e-4) but does not publish the stacked
composition; it announces it as separate work. research/bit-leaf-bootstrap-189/next_bit_composition.py
prices any certified bit profile through #194's unchanged assembly against
#193's complex, so that deferred composition is now computed, with this PR's
bootstrap applied on top:

depth	bit leaf	kappa	vs #197	vs this PR
0 (legacy atom wrapper = #200's own "about 6.769e-4")	6.77340866500955e-4	6768823/10000000000 = 6.768823e-4	+0.1186%	+1.3521%
1	6.77773606501196e-4	1354629/2000000000 = 6.773145e-4	+0.1825%	+1.4168%
2, 3	6.77773899999865e-4	1693287/2500000000 = 6.773148e-4	+0.1826%	+1.4168%

The depth-0 row independently confirms the harness: #200 says the stacked
composition "would give about 6.769e-4", and this returns 6.768823e-4 from that
supplier's certificate. The bit side still binds at every depth
(6.7777e-4 < 7.0092e-4), so this is the highest kappa computed anywhere in the
queue — +0.1826% over the standing claim #197, and +0.0639% over the depth-0
row #200 has announced but not published.

Pricing headroom is now exhausted on this word. For a fixed bit word the
assembly caps kappa at C/(1+C); #200's C = 6777739/10^10 gives
C/(1+C) = 6.7731483368e-4, and the depth-2/3 rows sit 3.4e-11 below it. Finer
pricing cannot move the number. Both remaining levers reduce W on that word,
and both are new work on #200's graph rather than a recomposition:

• endpoint-frame descent — what Lower PR189 endpoint frames plus finite leaf bootstrap: conditional κ = 0.0006679123 #198 did on Shared-edge complex and face-diagonal bit local circuits on PR168 v4 (κ = 6.61755179959122e-4) #189's word, worth +0.0090% of
  coarse saving there (6.68298937775631e-4 -> 6.6835882709e-4).
• residual packing — Pack #187's bit residuals: conditional κ = 0.000676080316519385 #197's mechanism, bound to Globally replayed source-assisted bit word: conditional kappa = 0.000661755179959122 #187's word, worth +0.8403%
  of coarse saving (6.70899981048852e-4 -> 6.76537710350481e-4) — the largest
  single bit-side gain in the queue.

Because the bit branch binds, complex-side work is worthless until a bit coarse
saving passes the complex's 7.009184e-4.

#200's certificate is vendored byte-identical at
research/bit-leaf-bootstrap-189/inputs/pr200-bit-certificate.json, digest pinned
in SOURCE.json, and priced — not rebuilt or replayed.

What changes on #194's own inputs

Exactly one thing in the record: the ordinary leaf that prices the binding bit
supplier. #194 sets kappa = a/(1+a) with
a = min(bit_effective, (1-1e-9)*complex_saving - 1e-10), and the bit side binds
(6.6788e-4 versus 7.0092e-4) — but it prices that side with PR184's legacy
ordinary leaf OLD = 384599/10^10 = 3.84599e-5, which holds the effective value
4.207e-7 below PR189's coarse saving C = 6.68298937775631e-4. PR185 replaced
that legacy leaf by a finite acyclic wrapper whose leaf value obeys

a_{n+1} = (1-C)*C + C*a_n = C - C^n (C - a_0).

Applying that recurrence to PR189's bit word on #194's own unchanged assembly:

depth	bit leaf	kappa	gain vs #194
0 (record)	6.67878244234e-4	1668581/2500000000	—
1	6.68298618876e-4	3339261/5000000000 = 6.678522e-4	+0.0629%
2	6.68298899812e-4	267141/400000000 = 6.678525e-4	+0.0629%
3	6.68298899999e-4	267141/400000000 = 6.678525e-4	+0.0629%

Verification

python3 -B research/bit-leaf-bootstrap-189/verify.py

• Pins every input and package file by SHA-256 (SOURCE.json).
• Reproduces Source-assisted v4 complex word with PR189's bit supplier: conditional κ = 6.674324e-4 #194's kappa exactly from Source-assisted v4 complex word with PR189's bit supplier: conditional κ = 6.674324e-4 #194's own profiles through Source-assisted v4 complex word with PR189's bit supplier: conditional κ = 6.674324e-4 #194's own
  select() and assemble(); depth 0 is that reproduction, so the delta is the
  leaf alone.
• Re-checks 47 strict constraints and 7 margins at every depth, and the strict paid
  atom and row tolls a_0 < a_n < C < 1 - a_n.
• Re-prices the vendored A stronger bit supplier: fixed-coordinate and face-diagonal bit local circuit on PR168 v4 (effective bit saving 6.7734e-4) #200 certificate, requires the depth-0 row to equal
  6768823/10^10 (A stronger bit supplier: fixed-coordinate and face-diagonal bit local circuit on PR168 v4 (effective bit saving 6.7734e-4) #200's own announced value), and requires the best row to beat
  Pack #187's bit residuals: conditional κ = 0.000676080316519385 #197's claim — so the workflow fails if either regresses.
• Regenerates certificate.json and the workflow runs the same command followed by
  git diff --exit-code.

Searched and rejected

Both W-lowering levers on #189's bit word are maximal, so that word's leaf slack
was the last room before a new bit word is needed (word.py::row() gives
W = 2*v + (R - pairs) at fixed rank deficit, so sinks and pairs each raise the
coarse saving):

• terminal sinks: 34 of 34. All 474 per-root-admissible sink roots were
  enumerated; the shipped 34 reproduce the frozen profile bit-for-bit, and every one
  of the 440 rank-23 extras is rejected by the exact target chronology of
  terminal.py::prove (target chain nesting), so no larger target-disjoint packing
  exists.
• donor reuse: 1760 of 3960 gauges. Every one of the 2200 unpaired gauges has a
  rank-20 entrance birth frame, and no donor end frame among all 8628 is contained in
  any of them, so no donor can be aliased in (word.py::exact_frames eligibility).

A frame-layout search on the same word was started and abandoned: re-choosing
operation frames only moves kappa when chain gaps are re-nested, and #198 published
a deterministic endpoint-frame descent that does this properly (#192 bounds the
family's frame-layout ceiling at 7.010e-4, which the complex side already attains).

Commands and observed output: research/bit-leaf-bootstrap-189/bit-maximality.md
(run against #189's package, an open dependency not vendored here).

Scope and limits

• The physical complex and bit words and their sink/reuse/frame artefacts are
  inherited pinned inputs (Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168 v4 fd25adb7, Shared-edge local circuit, refined bit frames and bit terminal sinks on PR168 v4 (κ = 6.60216552061722e-4) #182/Source-assisted common-frame compression for conditional kappa = 6.566436e-4 #184/Shared-edge complex and face-diagonal bit local circuits on PR168 v4 (κ = 6.61755179959122e-4) #189/Pair donors to avoid parity erasure on the source-assisted v4 word: conditional κ = 6.647872e-4 #193/Source-assisted v4 complex word with PR189's bit supplier: conditional κ = 6.674324e-4 #194, A stronger bit supplier: fixed-coordinate and face-diagonal bit local circuit on PR168 v4 (effective bit saving 6.7734e-4) #200).
  This PR re-prices the assembly; it does not rebuild or replay the words. The
  wrapper's physical realization is Finite leaf bootstrap above PR182: conditional kappa = 0.000660627334074291 #185's claimed construction, not one built
  here; A stronger bit supplier: fixed-coordinate and face-diagonal bit local circuit on PR168 v4 (effective bit saving 6.7734e-4) #200's bit word is that author's certified artifact, vendored and priced.
• Conditional on Add an exact κ-ceiling check for the exponent assembly (no new κ) #183's closed-form acceptance ceiling, Source-assisted common-frame compression for conditional kappa = 6.566436e-4 #184's finite bridge, and the
  retained contracts of the inputs.
• On Source-assisted v4 complex word with PR189's bit supplier: conditional κ = 6.674324e-4 #194's inputs the gain is small (0.0629%) and lives on the 10^-10 selection
  grid, where the bootstrapped value saturates at the same kappa the untolled limit
  C gives. The A stronger bit supplier: fixed-coordinate and face-diagonal bit local circuit on PR168 v4 (effective bit saving 6.7734e-4) #200 composition is larger (+0.1826% over Pack #187's bit residuals: conditional κ = 0.000676080316519385 #197) but rests on an
  unmerged supplier and on the same conditional wrapper.

Credits: eumemic (#168 v4), icekylinx and GPT-6 Astra (#182/#184), rohanarun (#185),
chafreaky (#182/#189/#200), ikeboy (#193/#194), romainhedouin (#183), DaysSky (#192),
evmckinney9 (#197), sennemmi (#198). See NOTICE.md in the package.

[maxime-fleury]

--
The deferred #200 composition now has its own self-contained package and PR: #204 (kappa = 1693287/2500000000 = 6.773148e-4, +0.1826% over #197 on its own pinned inputs). This PR's own claim on #194's inputs, 6.678525e-4, is unchanged.

[Mister-Meeseeks]

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

Contribution: Finite ordinary-leaf bootstrap and donor-feasibility diagnostic.

Review and limits: Prior exact-head rational arithmetic and full donor diagnostic. Zero eligible endpoints concerns the named unpaired birth-frame inventory only.

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: 5c1fb8346615968844bc182a5376c399109670ca. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.