Skip to content

research: exclude 15-call five-b UT4 splits across dirty workspaces #2828

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Merged
Mister-Meeseeks merged 1 commit into from
Oct 16, 2026
Merged

research: exclude 15-call five-b UT4 splits across dirty workspaces #2828

Show file tree
Hide file tree
Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
331 changes: 331 additions & 0 deletions research/ut4-fifteen-call-five-b/PROOF.md
View file
Open in desktop

Large diffs are not rendered by default.

151 changes: 151 additions & 0 deletions research/ut4-fifteen-call-five-b/README.md
View file
Open in desktop
Original file line number Diff line number Diff line change
@@ -0,0 +1,151 @@
# Complete 15-call five-b exclusion and the beta <= 3 min(alpha, gamma) price cone

No fifteen-call word with five `b` calls realizes `diag([z], [uz], I_rest)` in
`F2[UT4(F2)]` on any finite number `r >= 2` of arbitrary restored dirty registers.
All three 15-call five-`b` count splits `(6, 5, 4)`, `(5, 5, 5)`, and `(4, 5, 6)`
explicitly handed off by Gareth Davidson (`@bitplane`) in
[PR #2314](https://github.com/CrocSwap/integer-mult-bounds/pull/2314) and
[PR #2343](https://github.com/CrocSwap/integer-mult-bounds/pull/2343) are excluded.

Consequently, every 15-call word has `n_b >= 6`, and every word with `n_b = 5`
requires at least eleven outer (`a`/`c`) calls (`n_a + n_c >= 11`, total length
`L >= 16`). For nonnegative per-letter prices `(alpha, beta, gamma)`, the direct
fourteen-call `(4, 6, 4)` word is globally optimal throughout **`beta <= 3 min(alpha, gamma)`**,
with exact price `4 alpha + 6 beta + 4 gamma`. At `(alpha, beta, gamma) = (1, 3, 1)`,
cost `25` is impossible and `26` is the exact minimum. This is a formal register-model
theorem and does not provide a native bit supplier or improve `kappa`.

## Model and evidence

Work in the full group algebra `R = F2[UT4(F2)]` with adjacent generators
`a = I + E12`, `b = I + E23`, `c = I + E34` and targets `u = I + E13`, `z = I + E14`.
Each paid call is `I + ([g] + 1) x y^T` with `g in {a, b, c}` and `x y^T` a nonzero
binary rank-one coupling over `F2` (idempotent or nilpotent). Every occurrence,
including restoration, is charged; invertible `GL_r(F2)` mixing is normalized into
the couplings. Extra roles start arbitrary dirty and must be restored to identity
without contaminating the active outputs.

[PROOF.md](PROOF.md) proves the uniform arbitrary-workspace reductions:

1. **Two-role compression for `(6, 5, 4)` and `(4, 5, 6)` (Lemma 1):** Because
`n_b = 5` forces `e_b^x = e_b^y = 0` and `n_c = 4` (or `n_a = 4`) forces
`e_c^x = e_c^y = 0` (or `e_a^x = e_a^y = 0`), only one outer generator label
can have dirty row or column support. Any interior dirty contraction
`v_{i_m}(u_{i_{m+1}}) != 0` therefore occurs between two calls of the same
generator label, which vanishes identically in `R` by `([a] + 1)^2 = ([c] + 1)^2 = 0`.
Active-corner projection preserves the `2 x 2` active block over `R`, and
predecessor exclusions at length `<= 14` prevent any projected factor from
vanishing.
2. **Three-role active-plus-dirty decomposition for `(5, 5, 5)` (Lemma 2):**
All common-orientation excess profiles compress to two roles. The only remaining
15-call five-`b` excess profiles are the two opposite-orientation three-role
cases (`Case I`: `(e_a^x, e_a^y, e_c^x, e_c^y) = (1, 0, 0, 1)`, and `Case II`:
`(0, 1, 1, 0)`). In `Mat_3(Q)`, the 14 basis monomials without adjacent `{A, C}`
decouple completely from the dirty role on `W_HH` (restricting the active projections
to the exact `5,664` compatible `(5, 5, 5)` profile pairs), the off-diagonal
blocks `W_KH = 0` and `W_HK = 0` have zero interior dirty contractions (yielding
exact homogeneous linear equations on the dirty prefix profiles), and `W_HH` on
`AC, ACB, BAC` receives only the rank-one cross-role contraction.
3. **Complete 5-gap rank-sum profile characterization (Lemma 3):** A 5-gap prefix
profile `P in Mat_2(F2)^5` with gap increments `S_g = P_{g+1} + P_g` is realizable
by `n in {4, 5, 6}` nonzero rank-one couplings if and only if
`sum_{g=0..5} rank(S_g) <= n`, yielding `9,721`, `47,521`, and `174,670` profiles
(`55,566`, `907,200`, and `15,471,918` realizations).

### Complete two-role quotient censuses

| Split `(a, b, c)` | `a` / `c` profile passes | Key pairs | Profile pairs | Expanded words | First fail `AC` | First fail `ACB` | `Q` survivors |
| --- | ---: | ---: | ---: | ---: | ---: | ---: | ---: |
| `(4, 5, 4)`, retained PR #2175 control | 89,052 / 1,201,320 | 68 | 152 | 5,536 | 1,480 | 4,056 | 0 |
| `(5, 5, 4)`, retained PR #2314 control | 236,224 / 1,201,320 | 144 | 944 | 211,968 | 125,636 | 86,332 | 0 |
| `(4, 5, 5)`, retained PR #2314 control | 89,052 / 2,899,368 | 136 | 912 | 209,920 | 117,308 | 92,612 | 0 |
| `(6, 5, 4)`, **new** | 542,428 / 1,201,320 | 312 | 3,376 | 6,049,824 | 3,729,888 | 2,319,936 | **0** |
| `(5, 5, 5)`, **new** (2-role) | 236,224 / 2,899,368 | 288 | 5,664 | 8,655,168 | 5,508,044 | 3,147,124 | **0** |
| `(4, 5, 6)`, **new** | 89,052 / 5,883,552 | 312 | 3,280 | 5,874,848 | 3,496,252 | 2,378,596 | **0** |

The three retained predecessor controls reproduce their exact counts and transcript
digests from PR #2175 and PR #2314. The new two-role total is **20,579,840 expanded
words**, with zero survivors.

### Complete three-role `(5, 5, 5)` quotient censuses

| Three-role `(5, 5, 5)` mode | Active profile pairs | Dirty profile pairs (`W_KH = W_HK = 0`) | Expanded 3-role words | First fail `W_HH[AC]` | First fail `W_HH[ACB]` | `Mat_3(Q)` survivors |
| --- | ---: | ---: | ---: | ---: | ---: | ---: |
| Case I (`e_a^x = e_c^y = 1`) | 5,664 | 17,864 | 42,447,616 | 25,294,720 | 17,152,896 | **0** |
| Case II (`e_a^y = e_c^x = 1`) | 5,664 | 17,864 | 42,447,616 | 25,294,720 | 17,152,896 | **0** |

The new three-role total is **84,895,232 expanded words**, with zero survivors.

The separate [controls.py](controls.py) script checks:

- all 4,096 literal `4 x 4` binary group products in `UT4(F2)` and all 20,440 words
of length `<= 3` against an independent subsequence-sum quotient oracle;
- the complete `16^5 = 1,048,576` profile rank-sum histogram and exact agreement
with the explicit 4-call (`55,566` realizations, `9,721` profiles) and 5-call
(`907,200` realizations, `47,521` profiles) gap catalogues;
- planted 15-call two-role word reconstruction and quotient evaluation across
`(6, 5, 4)`, `(5, 5, 5)`, and `(4, 5, 6)`;
- 50 same-label nilpotent dirty-compression pairs in `Mat_3(F2[UT4(F2)])` (confirming
Lemma 1 when both dirty rows and dirty columns are present on the same outer
generator label) and detection of a cross-label dirty contraction;
- an independent 3-role subsequence-sum quotient oracle `quotient_3role` in `Mat_3(Q)`
on planted 15-call three-role `(5, 5, 5)` words for both Case I and Case II, plus
perturbation negative controls where violating a single dirty bit breaks `W_KH = 0`
or `W_HK = 0`;
- all 28 span-rank profiles at `L <= 15` and 216 weighted price grid points,
including the `(1, 3, 1)` boundary minimum `26` and the outside-cone `(1, 4, 1)`
negative control (`31 < 32`).

## Scope of the price result

For all nonnegative letter prices `(alpha, beta, gamma)`, every word satisfies the
necessary lower bound

```
min(4 alpha + 6 beta + 4 gamma,
4 alpha + 5 beta + 4 gamma + 3 min(alpha, gamma)).
```

The first branch is attained by the direct 14-call `(4, 6, 4)` word and is the
minimum whenever `beta <= 3 min(alpha, gamma)`. The second branch is a relaxed
count bound (`n_b = 5` implies `n_a + n_c >= 11`); whether any word with `n_b = 5`
exists at `L >= 16` remains open.

## Reproduce

From the repository root (standard library only):

```sh
python3.12 -I -B scripts/research_package.py --package research/ut4-fifteen-call-five-b
# Direct commands:
python3 -I -O -B research/ut4-fifteen-call-five-b/verify.py
python3 -I -O -B research/ut4-fifteen-call-five-b/controls.py
```

Both commands complete in under 10 seconds total. `verify.py --write-result`
regenerates [RESULT.json](RESULT.json).

## Open next questions

1. Does any realizing word with `n_b = 5` exist at all in `F2[UT4(F2)]`, or is
`n_b >= 6` an unconditional structural invariant of `diag([z], [uz], I_rest)`
for every word length `L` and workspace size `r`?
2. At `L = 16` with `n_b = 5` (`(7, 5, 4)`, `(6, 5, 5)`, `(5, 5, 6)`, `(4, 5, 7)`),
the pairing bound from PR #2343 still restricts any realizing word to at most
`r <= 2 + floor((16 - 13) / 2) = 3` roles, and Lemma 1 compresses `(7, 5, 4)`
and `(4, 5, 7)` directly to `r = 2` roles. Can an algebraic invariant in `Q`
explain why `AC = 0` and `ACB = I_2` are incompatible with the 14 non-adjacent-`{A,C}`
equations whenever `n_b = 5`, without requiring a separate census at each outer
length `n_a + n_c`?

## Credit

Prepared by `steph4n-gh` with AI assistance (Antigravity). Apache-2.0.

Full predecessor credit to Gareth Davidson (`@bitplane`, PR #2027, PR #2314, PR #2343)
for the mixed-root tariffs, 14-call weighted boundary, register-pairing compression,
and explicit 15-call `(1, 3, 1)` handoff; Nick Petalas (`@petalas`, PR #2175) for
the mixed-middle rank inequality, 13-call `(4, 5, 4)` census, and retained verifier
library; ToikY (PR #1662, PR #1680) and CinnaDavis / `1337isnot1337` (PR #1541)
for the degree-3 quotient and span-orientation compression; and Aaron Wang
(`@navinor`, PR #1690) for the gap-profile screening method.
Loading
Loading