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

research/complex-bank-run2/NOTICE.md

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+# Notice and attribution
+
+This package is Apache-2.0, like the repository. It adds a ledger and a price; it builds no
+word and changes no existing file.
+
+## Sources it depends on
+
+* **PR219** (`research/residual-bank-run1`, head `237067f483bccf931507a5386946cf66da0a915a`,
+  by halve-bit). Rung 1 itself: the rank-22 retained-ledger schedule, the exact interval
+  moment engine as vendored there, the 47-constraint balanced assembly, and the
+  obligations this package inherits. **Vendored byte-identically** under
+  `references/pr219-run1/` (16 files) and imported at run time -- its `schedule.py` and
+  `arithmetic.py` produce rung 1's published grid point inside this package, and the
+  vendored copy also self-verifies in place.
+* **PR207** (`research/coordinated-crossover-pr200`, head `cd14825`, by Dugongue). The
+  frontier row pair and the published kappa this package reproduces exactly, vendored as
+  `references/pr207-coordinated-crossover.certificate.json`. The upstream blob is
+  `23416f66…` with CRLF line endings; this repository's `* text=auto eol=lf` normalizes the
+  vendored copy to LF (`f4e30cca…`), and the parsed JSON is identical to the upstream file.
+* **PR208** (`research/composed-diagonal-bit-bootstrap`, by the same author as this
+  package). The bank-absorption ladder, the `10^-10` pricing convention and the published
+  rung-2 value `3541457/5*10^9` this package reproduces as its independent replica. Not
+  vendored; cited by value.
+* **PR184 / PR168-v4 and the PR197/PR200/PR205 line**, inherited through #219: the
+  balanced assembly with its 47 strict constraints and seven margins, PR200's exact
+  rational interval moment, and #205's completed-banks pack of the bit word (its
+  `physical.banks`, gauge roles, copies and `bank_controls` are the pinned evidence that
+  the queue's only bank construction is the bit word's).
+
+## Credit
+
+The ceiling argument, the complex-side ledger, its price in two conventions, the blocker
+report and the verifier are this package's contribution. Everything credited above is
+retained, pinned by sha256 in `SOURCE.json` (21 files), and unmodified.
+
+## Assistance
+
+Prepared with Anthropic Claude assistance. No claim in this package is presented as a
+formal verification of an unconditional multiplication theorem; see the scope statement in
+`README.md` and the open obligations in `obligations.json`.

research/complex-bank-run2/PROOF.md

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+# What is proved here, and what is only priced
+
+## Setting
+
+PR208 prices a ladder of bank absorptions on PR200's bit word. #219 built rung 1 of that
+ladder as a certified accounting construction: the rank-22 bin leaves the ledger, its
+440,352 three-copy registers of dirt become 6,116 width-72 banks, and the composition
+`T(n) = O(n (log n)^(1-kappa))` follows with
+
+    kappa = 700427501305159/10^18 = 0.000700427501305159,
+
+conditional on that submission's R1-R4. This package asks what rung 2 is, prices it
+exactly, and reports why it cannot be built yet.
+
+## Fact 1: rung 1 sits on the complex branch's ceiling
+
+The composition's budget is `a = min(bit_leaf, (1-beta) C_complex - weak)` and its kappa
+is `(1-eta) q/(1+q)` with `q = a(1-2 eta)`, floored to the grid. Holding the complex coarse
+saving `C_complex = 700918443859411/10^18` fixed, the largest kappa any bit ledger can
+reach is that expression at `a = C_complex`,
+
+    C/(1+C) = 700918443859411/1000700918443859411 = 7.0042750130515981e-4.
+
+Rung 1's kappa is `7.00427501305159e-4`: `8.52e-19` below that ceiling, i.e. inside one
+`10^-18` grid step. Both numbers are produced here by the vendored #219 code and by this
+package's replica of the assembly rule, which also reproduces #207's published
+`1366380910073/2*10^15` exactly from #207's own `a_bit`/`a_complex`.
+
+Consequently a rung above rung 1 cannot be taken on the bit word. The bit side's headroom
+at that point is 1.95e-5 of coarse saving and is unspendable while the complex branch
+binds.
+
+## Fact 2: the rung-2 ledger, and its invariants
+
+Let the complex supplier's row be the pinned profile: width `m = 66`, one copy
+`W = 12,052`, deficit `D = 1,320`, and ledger `hist` with `hist[11] = 1,062`. Over the
+supplier's three copies let `H = 3 * hist`, `W3 = 3W`, `D3 = 3D`, so that the row identity
+`66 * 36,156 - 2,382,336 = 3,960` holds with integral stock.
+
+Absorb the rank-11 family:
+
+1. **volume**: `11 * H[11] = 11 * 3,186 = 35,046 = 66 * 531`, so the absorbed dirt fills
+   exactly 531 banks (177 one copy);
+2. **retained ledger**: `H' = H \ {11}` has rank mass `2,347,290 = 2,382,336 - 35,046`;
+3. **row identity and stock**: `W3' = (2,347,290 + 3,960)/66 = 35,625`, integral, and
+   `W3 - W3' = 531` equals the bank count -- the absorption removes exactly the banks it
+   creates;
+4. **largest child** is unchanged at 20, and the children count falls from 639,441 to
+   636,255.
+
+These four are asserted in `ledger.py` and re-derived in `verify.py`; the whole-bank
+condition, the row identity and the stock-drop identity are the whole of what the
+supplier's own allocation rule requires of a schedule.
+
+## Fact 3: the prices
+
+Two paid moments on the pinned engine (PR200's exact rational interval moment, used on the
+complex side without the bit branch's rare-class fallback, as #219 uses it):
+
+    coarse(before) = 700918443859411/10^18  = 7.00918443859411e-4
+    coarse(after)  = 708793603125109/10^18  = 7.08793603125109e-4
+
+each the largest `10^-18` grid point whose paid moment is below 1, with the adjacent grid
+point excluded. The frontier certificate's `a_complex` equals the first, and
+`verify.py` asserts that equality -- the row priced here is the one the composition already
+uses.
+
+Assembling with the bit leaf of rung 1 and the absorbed complex saving gives the rung-2
+budget `a2 = (1-beta) * coarse(after) - weak` (the complex branch binds), hence
+
+    kappa_2 = 354145785295363/5*10^17 = 0.000708291570590726 (binding complex),
+
+with the next `10^-18` grid point rejected by the unchanged 47-constraint assembly, and
+`kappa_2 = 8.56e-19` below the new ceiling `coarse(after)/(1+coarse(after))`.
+
+Independently, in the `10^-10` convention of PR208's pricing model, the same ledger prices
+`3541457/5*10^9 = 7.082914e-4` -- PR208's own published value for this rung, reproduced
+here; the two conventions differ by `1.7e-11`, four orders below their agreement.
+
+## Fact 4: the blocker
+
+`obligations.json` lists five new obligations and inherits #219's four. The first is not
+dischargeable from the pins:
+
+* the complex supplier's certificate has **no key containing `bank`**, and neither does
+  #207's frontier certificate (asserted in `verify.py` through `blocked_on`);
+* the only bank construction in the pins is #205's `physical.banks` -- 45,842 banks with
+  gauge roles, distinct gauges, copies and `bank_controls` -- which banks the **bit**
+  word's selected entrance-gauge exteriors at width 72; its mechanism does not exist on the
+  complex word;
+* no pin enumerates the complex rank-11 family: #219's `inputs/absorbed-occurrences.json`
+  covers rank 22 of the bit word only, and it is precisely that enumeration that makes
+  #219's R1 a closed checklist.
+
+So C1 (the complex bank construction, its blocks and normalizers) and C5 (the family's
+inventory and provenance) are both open with no starting material in the queue.
+
+## What would discharge C1-C5
+
+1. **C5** first: an enumeration of the complex word's rank-11 occurrences by frame and
+   chain, with a bijection onto the 3,186 three-copy children and a check that no other bin
+   is touched.
+2. **C1**: the complex-side bank construction -- banks of width 66, the normalizer sending
+   each item's true residual projector to its block, the six-block pattern per bank, and
+   the hashed item-to-bank/offset map for all 531 banks.
+3. **C2-C4**: the formal-column replay of the modified complex word (F2 and the defining
+   integers, with the lift checked through `exact_scalar_program_sha256` rather than a
+   container digest), exact charts, incidence colouring and prime witnesses for the new
+   blocks, and the moment-envelope parity of the retained row priced here (stock 35,625,
+   deficit 3,960, no rare-class fallback on the complex side); any completion child
+   C1 introduces re-enters the ledger and the kappa is recomputed.
+4. Inherited: #219's R1-R4 for the bit-side rank-22 absorption, unchanged.
+
+Until then the kappa above is a priced, exactly certified **target** -- the same status
+#219 gives its rung-1 number, with one rung's worth of extra mechanism to build.

research/complex-bank-run2/README.md

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+# Rung 2 of the bank ladder is capped on the complex side: kappa = 7.08291570590726e-4 priced exactly, blocked on a complex-side bank construction (no new kappa)
+
+**No new kappa, and no completed construction.** This package prices the next rung of
+PR208's bank-absorption ladder above #219 and records the two facts that decide when it
+can be built: **rung 1 already sits on the complex branch's ceiling**, and the rung above
+it needs a **bank construction on the complex supplier that no package in the queue has**.
+
+## Result
+
+| Rung | kappa | Binding | Status |
+|---|---|---|---|
+| frontier, #207 | `1366380910073/2*10^15` = 0.0006831904550365 | bit | reproduced here **exactly** from #207's own `a_bit`/`a_complex` |
+| rung 1, #219 | `700427501305159/10^18` = 0.000700427501305159 | complex | reproduced here **exactly**, by importing the vendored #219 package |
+| rung 2, this package | `354145785295363/5*10^17` = 0.000708291570590726 | complex | **+1.122753%** over rung 1, **+3.674102%** over the frontier; blocked (below) |
+
+The price is the same ledger in two independent conventions: **0.00070829157059072603** on
+the `10^-18` grid in the convention of the queue's latest rungs (`eta = beta = 10^-24`,
+`weak = 10^-30`), and **`3541457/5*10^9`** on the `10^-10` grid of PR208's pricing model
+(`stop = 10^-9`, `eta = 10^-8`), which is PR208's own published value for this rung --
+reproduced here, not restated.
+
+## Why rung 2 has to be taken on the complex word
+
+#219's rung 1 absorbs the bit word's rank-22 bin and then binds on the complex supplier.
+This package shows that it does not merely bind there: it lands **on the complex branch's
+ceiling**. With the complex coarse saving `C = 7.00918443859411e-4`, the assembly's bound
+is `C/(1+C) = 7.0042750130515981e-4`, and rung 1's kappa is `7.00427501305159e-4` --
+**8.5e-19 below it, inside one grid step**. The bit side's own headroom at that point is
+1.95e-5 of coarse saving, and it is unspendable: whatever the bit ledger does, the
+composition is capped by the complex branch.
+
+So every further rung starts by raising the complex supplier's coarse saving. Rung 2 does
+that by absorbing the complex word's rank-11 family, and it likewise lands on the *new*
+ceiling (`708793603125109/1000708793603125109` = 7.082915705907269e-4, gap 8.6e-19).
+
+## The rung-2 ledger
+
+On the pinned complex profile (`m = 66`, one copy `W = 12,052`, deficit `1,320`, ledger
+bin 11 = **1,062** children per vertex):
+
+| | before | after |
+|---|---|---|
+| absorbed family | bin 11: 1,062 per vertex (3,186 over the three copies) | out of the ledger |
+| dirt volume | — | `11 * 3,186 = 35,046` registers = `66 * 531` **banks**, exactly (177 one copy) |
+| stock (three copies) | 36,156 | 35,625 (drop **531** = the bank count) |
+| rank mass | 2,382,336 | 2,347,290 |
+| children | 639,441 | 636,255 |
+| largest child | 20 | 20 |
+| complex coarse saving | 7.00918443859411e-4 | **7.08793603125109e-4** |
+
+Three invariant checks make the absorption free rather than a re-labelling: the volume
+fills whole banks (`35,046 = 66 * 531`), the retained ledger satisfies the row identity
+`66 * 35,625 - 2,347,290 = 3,960`, and the stock falls by exactly the bank count. The
+absorbed share of the ledger's rank mass is 1.471%.
+
+One further property is worth stating precisely because it is **not** a construction: the
+family's rank divides the bank width (`66 = 6 * 11`), so a bank admits a uniform six-block
+tiling of the new family alone and no mixture with older families is needed. That is a
+necessary condition satisfied, not a built bank.
+
+## What blocks the rung
+
+`obligations.json` states C1-C5 and inherits R1-R4 from #219. C1 is the blocker, and it is
+a fact about the queue rather than a gap in this package:
+
+* **The complex supplier has no bank construction.** Its certificate
+  (`references/pr219-run1/references/pr193-source-assisted-v4.certificate.json`) carries
+  no key containing `bank` at all (`certificate.json` -> `blocked_on`, machine-checked),
+  and neither does #207's frontier certificate.
+* **The only banked word in the pins is the bit word's.** #205's `physical.banks` (45,842
+  banks) with its gauge roles, distinct gauges, copies and `bank_controls` is the
+  entrance-gauge pack of the width-72 bit word -- a mechanism specific to that word's
+  entrance structure, not something the complex word inherits.
+* **The family has no inventory.** The only occurrence inventory in the pins is #219's
+  bit-side `inputs/absorbed-occurrences.json` for rank 22, which is exactly what makes
+  #219's R1 a closed checklist. Nothing enumerates the complex word's 1,062 rank-11
+  children, so C5 has no starting point and C1's item-to-block map cannot even be stated.
+
+A rung that cannot name its items cannot be built. That is the report.
+
+## A grid-precision note for reviewers
+
+PR193 publishes `complex_saving = 219037/312500000 = 7.009184e-4`, which is its value on
+the `10^-10` grid and sits `4.3859411e-11` below the `10^-18` certified paid moment of the
+same profile. #207's `a_complex` and #219's rung 1 both price the certified moment, and
+this package reproduces both numbers exactly, so rung 2 is priced on the certified moment.
+The coarser field is priced too, as `supplier_field.conservative_variant`: there the
+complex branch's ceiling already sits at rung 1, which is why the rung is only visible on
+the refined grid.
+
+## Verify
+
+```sh
+cd research/complex-bank-run2
+python3 -B verify.py           # check: pins, rebuild, compare with certificate.json
+python3 -B verify.py --write   # authoring: regenerate certificate.json and SOURCE.json
+```
+
+`verify.py` passes with exit 0 and prints the conditional kappa. It pins 21 files by
+sha256, rebuilds the rung-2 ledger, both paid moments, the 47-constraint assembly and the
+PR208 replica, and reproduces #207's and #219's published grid points exactly. The vendored
+rung-1 package also self-verifies in place:
+
+```sh
+cd research/complex-bank-run2/references/pr219-run1 && python3 -B verify.py
+```
+
+## Scope and limits
+
+* **This is a priced target, not a witness.** The accounting is certified; C1-C5 and the
+  inherited R1-R4 are open, so the kappa is conditional on them in exactly the sense #219
+  states for rung 1. It is not an unconditional multiplication theorem, not a practical
+  multiplier, and not a Lean/kernel certificate.
+* Nothing here bounds any *other* complex-side lever: only the rank-11 family of the
+  current complex ledger is priced, and only at the ledger's own bank width.
+* The rung inherits every interface #207/#219 inherit (all-size compiler, weighted and
+  restored selector, routing/layout, fixed-tape leaf, prime supply, precision/recovery,
+  analytic transfer).
+
+## Credits
+
+Rung 1, its retained-ledger schedule, the interval-moment engine and the 47-constraint
+assembly are **PR219** (`research/residual-bank-run1`, head `237067f`) -- vendored
+byte-identically under `references/pr219-run1/` and imported, not re-implemented; the
+frontier floor is **PR207** (`research/coordinated-crossover-pr200`, head `cd14825`), whose
+certificate is vendored verbatim; the bank-absorption ladder and the `10^-10` replica price
+are **PR208** (`research/composed-diagonal-bit-bootstrap`); the underlying bank proof and
+engine chain is PR200/PR205 (Chafik Boukhalfa and the PR197/PR205 line) and PR184/PR168-v4
+(icekylinx/eumemic) as inherited through them. This package adds the complex-side ledger,
+its price, the ceiling argument and the blocker. Apache-2.0; prepared with Anthropic
+Claude assistance. No existing file is changed.

research/complex-bank-run2/SOURCE.json

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+{
+ "files": {
+  "ledger.py": "103b7334e88edd71024cfd908c2e315d60a420012caef634ed843132d65ef2cd",
+  "obligations.json": "03f9803f3179ba3b150c42c4650060d90abdc29358daaaf19009219ba855b980",
+  "pins.py": "bf2f669d38259c1228dc74c8a59df7dd80fe86d1f02b951c91b5a66a4c2e97cf",
+  "references/pr207-coordinated-crossover.certificate.json": "f4e30cca369fb014b729df870ca3716cc0f1e2297b9f265341492cba85a14aeb",
+  "references/pr219-run1/NOTICE.md": "1f04b17404663a0983b7cc45e308e577d4d49fb6a4c1ff077dc3f13bae83c56f",
+  "references/pr219-run1/PROOF.md": "b7b9d83d4f87b32a9cba82a56039a18c9843eac0bd18f6a7bdcf9667b27b8a04",
+  "references/pr219-run1/README.md": "cc9fe21d282fac5085c09d35b9d0bc199c5ddef45ccdbdb2e6fbad45c08a881d",
+  "references/pr219-run1/SOURCE.json": "2fa4045bdf08c88103d678c1c7c29b66ebefa95d60f29a8d85697bf988c2c22b",
+  "references/pr219-run1/arithmetic.py": "6ad4d1ffc3a749a66cc0ec78afb4fedd94c84f9ca429de9011fa2e9c11ce819c",
+  "references/pr219-run1/banks.py": "0548a1c15bd6f6dea4505f5fd2ca1bb4ecbe914c8c90769b8cd8beeef428d0cf",
+  "references/pr219-run1/certificate.json": "fd092f55d1a2a3e8e01566b12b24c67a9efb35ef9b2fc0cd4e6217ba026245f6",
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+  "references/pr219-run1/interval_moment.py": "9d3d1d28bfb3375958995205179b171fb65c70a3792cf9f7a21ef5ee54f5460b",
+  "references/pr219-run1/obligations.json": "0268fc3628421bb930e9f56a23469a45f3b2bfae8888f6030622c2a51ce6a511",
+  "references/pr219-run1/paired_cube_assembly.py": "c1ebb52163a7219504c728a5c2b098bf1e50bf20148a2879c5a74c3cadc7c03c",
+  "references/pr219-run1/references/pr193-source-assisted-v4.certificate.json": "d84406d2fcc6990deba2b135509e6d86600e57123d88f9b21f7b7240e1aeb513",
+  "references/pr219-run1/references/pr200-bit.certificate.json": "853afa75980e11ce16b333c28311c4f635e84065867c049fda9916c2419091ed",
+  "references/pr219-run1/references/pr205-packed.certificate.json": "c11fbb9aaa99594afc8d21e33761dc6d1d6c76ce3dbc36d5b64cae872c242f5a",
+  "references/pr219-run1/schedule.py": "b9fff272f30048b982da3c0fc0149c2fde20bb7e32b8f9f860069956c12ebdc9",
+  "references/pr219-run1/verify.py": "c839a06389945f9b0c6fbfd59bf2dd8030190c3de504d72c9bf032879f931197",
+  "run2.py": "f5d1daa7fd692b71e798c06e84294501922cb67d404c391a5262109cf1130919"
+ }
+}