Six families per word: where the bank ladder tops out, and what the bit rank-6 rung would need (model, no new κ)

[maxime-fleury]

No new κ — a MODEL over the pinned frontier certificate, not a construction. This package
prices the bank-absorption ladder exactly on the frontier certificate's own child ledgers,
screens it one family deeper than before, and works out item by item what instantiating the
cheapest rung would still need. No bank, chart, normalizer or witness is built here, and no
number below is a record claim.

Package: research/frontier-beat-2026-10-09/ (README.md, INSTANTIATE.md,
beats.py → beats.json, beats-six.json, instantiate.py → instantiate.json,
check_docs.py, SOURCE.json).

1. Six families per word: the ladder does not top out on both sides together

The strict ladder (families whose volume tiles whole width-m banks) is screened
exhaustively one family further on both words: 10,318 admissible family sets on the bit word
and 2,725 on the complex one, against 3,512 and 991 at five families.

families	best bit saving	best complex saving
5	[1,3,13,21,22] → 1.38023409118239e-3 (+101.89%)	[1,2,3,4,19] → 1.63407154980676e-3 (+133.13%)
6	[1,3,13,20,21,22] → 1.52739614292328e-3 (+123.42%)	[1,2,3,8,15,18] → 2.30539885681005e-3 (+228.91%)

The six-family removals take 59.7% of the bit ledger (33,599 banks) and 65.6% of the complex
ledger (7,895 banks), but the headline is the smaller budget, so the model's best pairing is
bit six families with complex four:

kappa = 1.5250667618335e-3  (+123.23% over the certificate's 6.831904550366e-4)

11 new residual types, binding bit, removing 59.7% of the bit ledger and 33.8% of the complex
one. The complex side's extra ~100 points are unspendable while the bit side binds. The strict
tier's own best rung (bit ranks 6 and 8 with complex rank 11, +3.674%) is far below the wider
screen's, and the difference is exactly the block shapes the word does not already use.

2. What instantiating the bit rank-6 rung would take

Availability has a one-line form: a rank-r family fills width-m banks only when its own
volume is a whole number of banks (m % r == 0). On the certificate's two words exactly four
families qualify — bit ranks 6, 8, 18 and complex rank 11 — while rank 4 (the certified block
size) and rank 1 (the largest mass) do not.

The bit rank-6 rung is the closest of the pinned targets to a construction:

quantity	value
children	12,096 over three copies = 4,032 per copy
blocks per bank	12 (stride 6 on width 72) — exactly one block per absorbed child
banks	1,008 over three copies = 336 per copy, integral (no replicas needed)
stock W	56,402 → 55,394
rank mass	4,055,136 → 3,982,560
deficit	5,808, unchanged
retained ledger	21 families, max child 22
certified saving	6.83657523331095e-4 → 6.98484415006068e-4
kappa	6.831904550366e-4 → 6.979968750671e-4 (+2.1672%)

The vendored bank proof already covers every item it states for any disjoint projector
partition: the algebra S_P S_Q = S_(P+Q), S_P² = I, chart-conjugation telescoping, the 144
two-bank address-map columns over F2 and over Z for the endpoint and its inverse (rejecting
a missing or repeated last block), the unchanged moment envelope (rare-class fraction 10^-16,
32·72² children per edge, the positive atom/row-adapter toll on its own 10^-24 grid) and the
47-constraint/7-margin assembly. A stride-6 partition is a new instance of that construction,
not new algebra — and here the per-copy allocation is integral, so the rung does not need the
nine physical replicas the certified rank-60 bank used.

What is missing is the normalizer's child-to-block map (rank-6 dirt is interior and has no
registered gauge address), stride-6 charts and per-block Gram/prime witnesses, and — the real
gap — an inventory: no pinned package enumerates the word's rank-6 children, not their frame
or chain occurrences and not a bijection onto the 4,032 blocks per copy, exactly as the complex
word's rank-11 family has never been enumerated either. The conditional interfaces (all-input
weighted compiler, completed-core routing, stage cover, local-ring precision/recovery, prime
supply, uniform setup, all-size analytic/tape contracts) stay conditional; this package
discharges none of them.

Reproduce and verify

python3 -B beats.py                                      # ~91 s, writes beats.json
python3 -B beats.py --check                              # byte-for-byte against it
python3 -B beats.py --max-types 6 --out beats-six.json   # ~157 s, one family deeper
python3 -B beats.py --max-types 6 --out beats-six.json --check
python3 -B instantiate.py                                # ~2 s, writes instantiate.json
python3 -B instantiate.py --check
python3 -B check_docs.py                                 # 41 quoted numbers, 44 required claims

• The frontier is reproduced on the certificate's own rows under each side's own convention:
  a_complex to the last digit, a_bit to sixteen significant digits, with the residual
  recorded rather than asserted away. The kappa grid and the assembly rule are discovered by
  search, not assumed.
• Every number is exact rational arithmetic through the pinned interval/exponential enclosure
  module (.py files hashed in SOURCE.json on every run); no floating point enters a check.
• Both six-family headlines and the headline pairing were re-derived by a separate script
  written against the certificate rows, independently of the model code, and all three
  artifacts reproduce byte for byte when the package is copied to a fresh checkout of
  main.
• check_docs.py keeps the prose honest: it re-reads every number quoted in README.md and
  INSTANTIATE.md and the required claims against the three artifacts.

Scope and limits

• MODEL, NOT CONSTRUCTED. Each rung is a priced target. The bank construction, its charts
  and its witnesses belong to the supplier's harness; nothing here is a new bound on the
  exponent and nothing here should be read as a κ claim.
• A rung's cost is measured in new residual types (families absorbed) because that is what the
  supplier's harness would have to instantiate; chart, normalizer and witness work is ignored
  entirely by that metric.
• The wider screen treats an absorbed family's rank as free, which is certified precedent only
  for an exterior of the rank-60 kind; the strict tier is the only tier that needs no block
  shape the word does not already use.
• The pinned ladder in one earlier package prices both sides with the rare-class fallback, so
  its values sit a few 1e-10 relative below the convention-matched values recorded here; both
  are given for every rung.
• No contributor verifier, CI run or upstream proof package is re-executed: the rows are the
  certificate's own recorded ledgers and the arithmetic is this package's.
• Not run here: upstream CI. The local suite on this Windows host has six failures/errors
  (ModuleNotFoundError: No module named 'resource', no C++ compiler in PATH, and three
  frozen-certificate mismatches in test_copied_reversed / test_copied_fixed_reversed /
  test_batched_network) which reproduce identically on a pristine main worktree and are
  unrelated to this package.
• One defect in this package was found and fixed while verifying it: --check printed a
  hardcoded label and so could not distinguish the five- from the six-family artifact. It now
  names the file and the depth it compared, and the README's runtime claims were corrected to
  the measured 91 s / 157 s.

Credits

The pinned inputs are not mine. The frontier certificate is PR207's coordinated crossover, and
the bank proof is the coordinated-frames-and-entrance-banks text of the #186 → #205/#206/#207
lineage; the interval and exponential enclosures that compute every number here are PR200's own
arithmetic module. The contraction and bank models, their validation against the queue's
published banked rows and the closed-form bank proof are PR184/PR185/PR197/PR200/PR205/PR206/PR207's.
This package re-implements only the arithmetic it prices with, and modifies no pinned artifact.

Prepared by Maxime Fleury with Codebuff assistance.

🤖 Generated with Codebuff

[Mister-Meeseeks]

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

Contribution: Six-family bank ladder and bit-rank limits.

Review and limits: Scoped priced research, with prior exact arithmetic/model checks. No completed bank-absorption witness is inferred from family counts.

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: cf64f1cb29f7af951db375f460ad683ec6d16419. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.