Rung 2 above #219: a complex-side bank ledger that raises κ to 7.0829157e-4, blocked on C1/C5

[maxime-fleury]

What this is

The rung above #219's rank-22 absorption of PR208's bank ladder, priced end to end, plus
the honest report of what keeps it from being a new record.

Package: research/complex-bank-run2/, self-contained (its only in-repo dependency is the
sibling research/composed-diagonal-bit-bootstrap/).

python3 -B verify.py        # all checks, ~0.5s, no network, no writes

Calibrations reproduced

• Price the bank-absorption ladder against the suppliers' own allocation rules: +2.17% needs a new residual type this word does not have (target, not built) #208's priced rung 2 (3541457/5*10^9 = 7.0829154e-4, the value Price the bank-absorption ladder against the suppliers' own allocation rules: +2.17% needs a new residual type this word does not have (target, not built) #208 announces as its
  complex-bound number) is reproduced exactly by this package's own model — the
  eta → 0, stop → 1 limit of PR193's unchanged 47-constraint assembly with the bit
  branch truncated.
• Build #208's bank-absorption target: rank-22 residual absorption — certified accounting κ = 7.00427501305159e-4 (R1–R4 open; not a completed witness) #219's rung 1 (700427501305159/10^18) is reproduced exactly by importing Build #208's bank-absorption target: rank-22 residual absorption — certified accounting κ = 7.00427501305159e-4 (R1–R4 open; not a completed witness) #219's
  vendored package, and both the frontier certificate and PR184's own select() price the
  complex composition from the certified profile moment rather than from PR193's
  complex_saving field, which their own certificate records as 7.999e-8 below the
  certified moment of the same profile. This package prices the certified moment (matching
  both published artifacts) and additionally prices the lower field as a conservative
  variant, which lands at rung 1 minus 1.9e-12.

The structural finding that decides the rung

Rung 1 does not sit at its budget ceiling. Its complex branch is
(1-stop) * 7.00918443859411e-4 - 1e-10 = 7.00918343859411e-4, and its kappa sits
8.5e-19 below that — the grid. So rung 1 is grid-limited on the complex side, and no
bit-side work can move a rung above it: the ladder only advances by raising the complex
coarse saving. (This is why #208's own ledger maps rank-22 → rank-11 as its first complex
move.)

Rung 2 priced

Absorbing the complex word's rank-11 family as PR184's bank absorber would, with the rank-11
member as a whole bank (volume 35046 = 66 * 531, stock 36156 -> 35625, drop = bank count),
discarding nothing for having unsalvageable part, and re-pricing through PR184's unchanged
47-constraint assembly:

coarse  7.00918443859411e-4  ->  7.08793603125109e-4
kappa   354145785295363 / 500000000000000000  =  7.08291570590726e-4
        (+1.12% over rung 1; 7.0829154e-4 < kappa < 7.0829155e-4 -> adjacent grid point rejected)

The value is checked against #208's own closed form to the last digit and against the exact
cross-checks of PR184's assembly (43 strict constraints are exactly saving/4, the 4
non-quarter constraints and 7 margins are re-witnessed, every entry re-derived from the
integer degree blocks).

Why it is not built (the blocker)

The rung is conditional on a complex-side bank construction that does not exist in the pins.
All nine obligations in obligations.json are OPEN; verify.py certifies the accounting
and the price, not the construction:

• C1 — the bank construction. The pins contain no bank construction for the complex
  supplier: its certificate has no bank keys at all. The only banked word in the pins is the
  bit word's entrance-gauge pack of Pack PR200 diagonal bit residuals into completed banks (conditional κ = 0.000683061299399923) #205/Completed entrance banks on PR200's bit word with the v4 complex supplier: kappa = 6.830611e-4 #206 (width 72, rank-60 exteriors, 6,116 banks), and
  that mechanism is specific to the width-72 bit word's entrance gauges — so this is a new
  mechanism, not an extension of an existing one.
• C2–C4 — the re-certification the construction implies. Formal-column checks on the
  modified complex word, charts/routing/prime witnesses for the new blocks, and
  moment-envelope parity with the ledger priced here (retained three-copy stock 35,625, one
  copy 11,875; deficit 3,960, one copy 1,320). Any completion child C1 introduces must
  re-enter the ledger and the kappa be recomputed.
• C5 — inventory and provenance. The pins hold no inventory of the complex word's
  rank-11 items: no per-vertex child frame/chain occurrences, no bijection onto the 3,186
  three-copy occurrences, no proof that no other bin is touched. The only occurrence
  inventory the pins hold is Build #208's bank-absorption target: rank-22 residual absorption — certified accounting κ = 7.00427501305159e-4 (R1–R4 open; not a completed witness) #219's bit-side inputs/absorbed-occurrences.json for the
  rank-22 bin, which is exactly what makes Build #208's bank-absorption target: rank-22 residual absorption — certified accounting κ = 7.00427501305159e-4 (R1–R4 open; not a completed witness) #219's R1 a closed checklist and this rung's C5 a
  blank one.
• R1–R4 — inherited open from Build #208's bank-absorption target: rank-22 residual absorption — certified accounting κ = 7.00427501305159e-4 (R1–R4 open; not a completed witness) #219 (bit-side residual types, formal columns, charts, and
  moment parity), carried over with their source and PR219 head 237067f.

Two further honesty items: rank 4 = bank is standard practice in this repository rather
than a proven identity, so the rung-1 ceiling statement carries a binary flag
(rank_four_is_whole_bank) instead of pretending precision it does not have — the
conservative reading is still above rung 1, which is what makes the ceiling argument
decisive. And PR193's own complex_saving field prices the rung lower; the package records
the delta, the conservative variant, and the exact profile (a 10^-10-grid value of
7.009184e-4, 4.39e-11 below the certified moment) so a reviewer can rule on which one
the assembly must consume.

Files

file	role
README.md	the rung, the pricing, the ladder, what is blocked
PROOF.md	derivation of the rung-1 ceiling, the rung-2 price, exactness evidence
ledger.py	shared model loader and the strict-constraint witness
run2.py	rung-1 ceiling statement, rung-2 price, the ledger and the two calibrations
banks.py	the complex-side bank / ceiling arithmetic clamps
verify.py	runs every published number from scratch and checks them against each other
obligations.json	C1–C5, R1–R4, each with status and evidence; all nine OPEN

Scope: the physical complex and bit words are the suppliers' certified artifacts, vendored
and priced, not rebuilt or replayed. The one value that is not a published field — the rank-11
absorbed stock 35625 — is labelled as a model value and is a hypothesis of the construction,
not a certified fact; it is what the rung would deliver if C1's admissibility and C2–C5 hold.
The kappa is a priced, exactly certified target, not a completed finite witness or an
unconditional multiplication theorem.

Credits: the complex supplier and its 47-constraint assembly are #184/#193/#194/#219; the
finite leaf wrapper and the bank construction are #185; the model this rung extends is #208.

[chafreaky]

FYI for the rung pricing here: #233 certifies a stronger complex supplier on the same lineage, complex saving 3549537/5000000000 = 7.0991·10⁻⁴ (#194's source-assisted word with #200's maximum-weight reuse pairing in place of #168's; frames unchanged; verified through #200's complex checker, #184's flow/exact lift/contract and #194's assembly). The complex budget these rungs are priced against moves from 7.0092·10⁻⁴ to 7.0991·10⁻⁴.

[maxime-fleury]

@chafreaky I'm working on something + harness + proof and looks like I got conditionnal k =7.11175695867958e-4 PR WIP #232, ok omg yes ! 7.2825e-4 !!!

[Mister-Meeseeks]

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

Contribution: Second rung of the complex-bank absorption ladder.

Review and limits: Exact priced research target; C1-C5 and R1-R4 remain open. Retention does not promote the hypothetical kappa to a completed witness.

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: a60c9505284908edbac52f7c2f167feca75de546. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.