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

research/frontier-beat-2026-10-09/INSTANTIATE.md

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+# What instantiating the bit rank-6 bank rung would take
+
+The cheapest way to beat the frontier is [beats.py](beats.py)'s first strict rung:
+absorb the bit word's rank-6 child family into width-72 banks, worth **+2.1672%**
+(`kappa` 6.831904550366e-4 → 6.979968750671e-4). This note asks what would have to
+exist for it to be a construction.
+
+```
+python3 -B instantiate.py           # ~2 s, writes instantiate.json
+python3 -B instantiate.py --check   # byte-for-byte comparison with it
+```
+
+Everything below is derived from the frontier certificate's own three-copy row; the
+supplier's published bank proof is vendored beside it
+([coordinated-bank-proof.md](references/queue/coordinated-bank-proof.md), hashed in
+[SOURCE.json](SOURCE.json)) and read item by item.
+
+## The schedule, and the condition that decides it
+
+A bank is a width-`m` coordinate-block partition, so a family absorbs into banks only
+when its own volume fills whole banks. For rank `r` on width `m` that has a one-line
+form: `m/r` blocks fill a bank, so the family's child count must be **divisible by
+`m/r`**. On the frontier's two words that leaves exactly four families:
+
+| word | rank | blocks per bank | banks (three copies) | children divisible by `m/r` |
+| --- | ---: | ---: | ---: | --- |
+| bit (m=72) | **6** | 12 | 1008 | yes |
+| bit (m=72) | 8 | 9 | 816 | yes |
+| bit (m=72) | 18 | 4 | 999 | yes |
+| complex (m=66) | 11 | 6 | 177 | yes |
+| bit | 1, 2, 3, 4, 9, 12 | 72, 36, 24, 18, 8, 6 | — | no |
+| complex | 1, 2, 3, 6 | 66, 33, 22, 11 | — | no |
+
+Rank 4 is the certified block size and rank 1 is the biggest mass in the word, but
+neither has a volume that fills whole banks, so neither is available without a mixed
+tiling. The stride-6, 9 and stride-4 partitions are the ones the arithmetic admits.
+
+## The rung's arithmetic, re-derived
+
+| quantity | value |
+| --- | --- |
+| family | bit rank 6 |
+| children | 12,096 over three copies = **4,032 per physical copy** |
+| blocks per bank | 12 (stride 6 on width 72, `12 × 6 = 72`) |
+| blocks needed | 4,032 per copy — **exactly one block per absorbed child**, count for count |
+| banks | 1,008 over three copies = **336 per copy**, integral |
+| removed rank volume | 72,576 |
+| stock `W` | 56,402 → 55,394 (the fall is the bank count, by the row identity) |
+| 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 one-block-per-child identity is the useful structural fact: the normalizer's
+targets and the absorbed children are equinumerous, so no child needs two blocks and
+no block is left half-used at this rank. The certified proof's own absorption is not
+like that — it removes 2,200 rank-60 gauge exteriors whose volume (396,000 over three
+copies) fills 5,500 banks, and it needed nine physical replicas to make its
+stage-private allocation integral. Here the per-copy count is 336, so if the rank-6
+dirt is stage-private in the same three-family way the allocation is 112 banks per
+stage and needs no replicas. **That is an inference**: no pin says the rank-6 dirt is
+stage-private.
+
+## What the published bank proof already covers
+
+The proof is a finite verification of a *different* absorption — the rank-60 entrance
+exteriors, one per selected gauge — but several of its items are stated for any
+partition, so the rung inherits them:
+
+* **The projector algebra.** `S_P S_Q = S_(P+Q)` and `S_P² = I` are proved by direct
+  expansion for *any* finite disjoint projector partition summing to `I`, and chart
+  conjugation telescopes for any common invertible weighted chart. A stride-6
+  partition needs no new algebra; it is a new instance of the same construction.
+* **The column check.** 144 columns of the two-bank address map over `F2` and over
+  `Z`, for the endpoint and its inverse, rejecting a missing or repeated last block in
+  both rings. `144 = 2 × 72` independently of which blocks fill the bank.
+* **The moment-envelope parity.** The rare-class fraction `10^-16`, the fallback
+  `32 × 72²` children per edge and the positive atom/row-adapter toll on its own
+  `10^-24` grid are all unchanged by this rung, and two rational log/exp enclosures
+  certify the accepted moment and reject the adjacent `10^-18` point — which is
+  exactly what `beats.json` and `instantiate.json` already do with the supplier's own
+  enclosure implementation.
+* **The assembly.** 47 strict constraints and 7 margins at PR207's parameters. The
+  rung is bit-binding, so the complex budget is untouched.
+
+## What it does not cover, and what is missing
+
+* **The normalizer.** The certified proof's banked items are the 2,200 rank-60 gauge
+  exteriors, and each one's *gauge itinerary* supplies its address. The rank-6 family
+  is interior dirt with no registered gauge address, so the child-to-block map has to
+  be built rather than inherited.
+* **The inventory — the real gap.** No pinned package enumerates the word's rank-6
+  children: not their frame or chain occurrences, not a bijection onto the 4,032
+  blocks per copy, and not a proof that nothing else in their ledger row moves. This
+  is the same blank shape as the complex rung's inventory obligation, and it is
+  smaller: one family of one word, whose counts, bank schedule and divisibility
+  condition are pinned by this file.
+* **Charts and witnesses.** Integer kernel bases with fraction-free inverses and
+  replayed elementary factors for the stride-6 frames, and distinct integer Gram/prime
+  witnesses for each of the 1,008 three-copy blocks (the certified supply is per block
+  *frame*, not per bank).
+* **The conditional interfaces.** The inherited all-input weighted compiler, the
+  completed-core routing, the stage cover, local-ring precision/recovery, prime
+  supply, uniform setup and the all-size analytic/tape contracts stay conditional; the
+  bank proof discharges none of them, and neither does this package.
+
+## Verdict
+
+The rung is the closest of the pinned targets to being buildable: its algebra is
+inherited from a published proof rather than invented, its allocation is integral
+per copy (so it does not need the replicas the certified bank used), the block
+arithmetic is exact and pinned here, and the saving and kappa are certified with
+tooling that already exists. What stops it is not mathematics but an inventory: the
+dirt it removes has never been enumerated anywhere in the queue, exactly as the
+complex word's rank-11 family has never been. Whoever writes that enumeration — one
+family, one word, 4,032 items per copy — hands this rung to the certified harness with
+the stride changed from 4 to 6.