Record a 21-shear one-scale zero-swap word for helper-462 (no κ)

research/helper462-one-scale-21-shear/README.md

@@ -0,0 +1,151 @@
+# helper462-one-scale-21-shear
+
+**Constructive witness. No new κ and no selected-frontier change.**
+
+## Result
+
+For the pinned helper-462 matcher `S`
+(`research/bit-completed-backend-api/science/RESULT.json#matcher`, an 8 by 8
+integer matrix of determinant `-16`), the alphabet of axis swaps,
+single-coordinate rational scales and rational shears contains a zero-swap
+word with exactly one scale and **21 shears**. The scale is the last letter,
+on row 0, with coefficient `-16`. The shear denominators are `1, 2, 4, 8, 16`.
+
+The shortest one-scale zero-swap word recorded before this package has 26
+shears (unpaidmoon, open PR #7054), on the same matrix with the same scale.
+The lower bound is 9 shears (unpaidmoon, open PR #7055). The open interval for
+the one-scale shear length therefore moves from 9 through 26 to **9 through 21**.
+
+Twenty-one is the length of this witness. It is not claimed to be minimal.
+
+### The word
+
+Letters apply first to last as row operations on the identity.
+`t += a*s` means row `t` gains `a` times row `s`. This is the content of
+[WORD.json](WORD.json).
+
+| # | letter | # | letter | # | letter |
+|---:|---|---:|---|---:|---|
+| 1 | `7 += -1*2` | 8 | `0 += 3/4*1` | 15 | `5 += 1/2*1` |
+| 2 | `1 += 1*3` | 9 | `1 += 4*0` | 16 | `4 += 1*7` |
+| 3 | `5 += 1*3` | 10 | `0 += -3/16*1` | 17 | `4 += 16*0` |
+| 4 | `3 += -2*1` | 11 | `2 += -8*0` | 18 | `6 += -1*3` |
+| 5 | `5 += 2*0` | 12 | `0 += 1/8*2` | 19 | `3 += 1*6` |
+| 6 | `0 += -1/2*5` | 13 | `0 += -1/16*4` | 20 | `6 += -1*3` |
+| 7 | `4 += 2*2` | 14 | `1 += 2*3` | 21 | `1 += 1*2` |
+
+Letter 22 is the scale: row 0 times `-16`.
+
+### One word per scale row
+
+A zero-swap word with one scale `c` on row `r` has determinant `c`, so `c = -16`
+and the shears multiply to `D_r^{-1} S`. Conjugating by the scale moves it to
+either end without changing the shear count, so only the row `r` matters.
+[ROW_WORDS.json](ROW_WORDS.json) records one exact word for each row:
+
+| scale row | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
+|---|---:|---:|---:|---:|---:|---:|---:|---:|
+| shears | 21 | 21 | 24 | 21 | 21 | 21 | 21 | 23 |
+
+Each count is an upper bound for that row. The larger counts on rows 2 and 7
+are a search outcome. They are not a proof that those rows need more shears.
+
+## Claim type and limits
+
+- **Finite certificate.** Each frozen word multiplies exactly to `S` over the
+  rationals. That is the whole proof of the upper bounds.
+- **Not a minimum.** The words came from a beam search, which is a heuristic
+  and not a census of rational words. Lengths 9 through 20 stay open.
+- **Lower bound cited, not replayed.** The floor of 9 is the statement of open
+  PR #7055. This package does not recheck it.
+- No supplier admission, no P2 ordinary bill, no tariff, and no
+  integer-coefficient claim. With integer shears no one-scale zero-swap word
+  exists (merged PR #6954); this word uses the denominator 16.
+
+## Evidence
+
+`verify.py` uses the Python 3.12 standard library and exact fractions. It checks:
+
+1. the pinned matcher is 8 by 8 with determinant `-16`;
+2. the 21 shears followed by the scale reproduce `S`, by row operations on the
+   identity and by an independent product of elementary matrices;
+3. the shear part has determinant 1, and no two adjacent shears share a
+   position, so no pair merges into one letter;
+4. the same checks for the eight words of `ROW_WORDS.json`, whose row-0 entry
+   equals `WORD.json`;
+5. 406 failure controls: for the primary word and each row word, deleting any
+   single letter misses `S`, and adding 1 to any single coefficient misses `S`.
+
+Failures use an explicit exit, so the controls stay active under `-O`.
+
+## Reproduce
+
+From the repository root, with CPython 3.12:
+
+```sh
+python3.12 -B research/helper462-one-scale-21-shear/verify.py
+python3.12 -O -B research/helper462-one-scale-21-shear/verify.py
+python3.12 -B scripts/research_package.py --package research/helper462-one-scale-21-shear
+python3.12 -B scripts/check_research_commit.py --package research/helper462-one-scale-21-shear --check-source-history
+```
+
+Expected first line: `PASS_HELPER462_ONE_SCALE_21_SHEAR`. The verifier runs in
+well under a second.
+
+### Search provenance (not a registered check)
+
+`search.py` is the beam search that proposed the words. It reduces
+`D_r^{-1} S` to the identity by row and column shears whose coefficient clears
+one entry or sets one diagonal entry to 1, and keeps the matrices with the
+fewest entries differing from the identity. It writes its output to a
+directory outside the checkout:
+
+```sh
+python3.12 -B research/helper462-one-scale-21-shear/search.py "$OUT" 4000 1 0 "" 24
+python3.12 -B research/helper462-one-scale-21-shear/search.py "$OUT" 300 1 0 "" 30 2,7
+```
+
+The first command took about 11 seconds on ten cores and returns the recorded
+21-shear words for rows 0, 1, 3, 4, 5 and 6; it returns nothing for rows 2
+and 7 within depth 24. The second returns the recorded 24-shear word for row 2.
+The row-7 word came from
+`search.py "$OUT" 1500 2 1.5 "1,-1,2,-2,0.5,-0.5" 23`, seed 1, a run of about
+three minutes that was not repeated. The frozen words do not depend on the
+search: the exact products are the certificate.
+
+## Open next questions
+
+Structured copies are in [TASK.json](TASK.json).
+
+1. **What is the minimal one-scale shear length?** The interval is now 9
+   through 21. First step: run `search.py` with a wider beam and depth cap 20
+   on rows 0, 1, 3, 4, 5 and 6, and separately try to raise the rank-drop floor
+   of PR #7055 from 9 to 10. A search that returns nothing is not a lower bound.
+2. **Do scale rows 2 and 7 admit 21 shears?** The recorded words have 24 and
+   23. First step: a wider or noisier beam on those rows, or an invariant of
+   `D_r^{-1} S` that separates them from the other six rows.
+3. **Does the 21-shear word help on the P2 ordinary bill?** Letter counts do
+   not determine the signed moment, and this word has denominators up to 16.
+   First step: evaluate it under the tariff model of merged
+   `research/helper462-p2-ordinary-bill`.
+
+## Credit
+
+This package was produced by Nick Petalas. New work is Apache-2.0.
+
+Predecessors, none of which is replayed here:
+
+- The matcher and its five-swap elimination are from
+  `research/bit-completed-backend-api` (Chris R / Clear1nsight, PR #4957),
+  using the helper-462 witness of ai-nanachi #4760/#4814.
+- unpaidmoon, merged PR #6954: a one-scale zero-swap word (53 shears),
+  the determinant identity that forces the scale `-16`, and the integer-shear
+  obstruction.
+- unpaidmoon, open PR #7030 (head `98238eeed3faca9f0a5f1d4060db5d5da524a192`):
+  a 29-shear one-scale word and the rank lower bound 8.
+- unpaidmoon, open PR #7054 (head `ae1cf2af174cd4f8c89089a27fc58e470b5452bd`):
+  the 26-shear one-scale word this package shortens. Its word format and
+  application order are reused in `WORD.json`.
+- unpaidmoon, open PR #7055 (head `0a6361ec1b63621ca5f985218892e0cdc79502ae`):
+  the lower bound of 9 shears, which poses the open interval answered in part
+  here.