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Mister-Meeseeks merged 1 commit into from
Oct 14, 2026
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Research: exact P112 shared-gauge decoder census #1962

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24 changes: 24 additions & 0 deletions research/coord-root-p112-global-gauge-seed655425/PROOF.md
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# Exact identity and bounded cost obstruction

All root lines and normal vectors are the rational E8 model from the pinned `e8lab.py`. The pinned predecessor verifier supplies exact rational frame solving and the all-pairs identity checker.

Let `b_0,...,b_8` be the coordinate-frame-plus-root normals, with exact dependency

```text
sum_k z_k b_k = 0,
z = (-1,-1,-1,-1,0,0,0,0,2).
```

For every one of the 112 live root lines `r_i`, the verifier solves an exact rational row `c_i` with `sum_k c_ik b_k = 2r_i`. Given any integer vector `w`, define `t_i=<r_i,w>` and `D_ik=c_ik+t_i z_k`. Then

```text
sum_k D_ik b_k
= sum_k c_ik b_k + t_i sum_k z_k b_k
= 2 r_i.
```

Thus the row identity is exact throughout the searched box. The cost of term `k` is computed as the binary rank of the E8 label pairing restricted to the exact row support `X_k={i:D_ik != 0}` and normal support `Y_k={j:<r_j,b_k> != 0}`. The total is the sum of all nine term ranks, with no inactive term silently charged as active.

The executable enumerates the Cartesian product `{-1,0,1}^8` without sampling. Exact rational coefficients are scaled by their common denominator before zero tests; rank and census operations are integer/F2 operations. The result is two cost-64 assignments with an empty term, three fully active cost-72 assignments, and 6,556 fully active cost-73 assignments. The full all-active minimum in this finite family is therefore 72.

This conclusion is only about the pinned frame, puncture, shared-gauge rule and finite ternary box. It is not a lower bound for arbitrary per-root gauges or other frames. Exact all-pairs checks of the five cost-at-most-72 representatives confirm the decoder identities; a changed coefficient row gives a negative control.
61 changes: 61 additions & 0 deletions research/coord-root-p112-global-gauge-seed655425/README.md
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# Ternary global-gauge screen for one coordinate-root P112 decoder

**Scoped exact negative result; no new κ.** On one pinned E8 P112 puncture and one merged nine-normal frame, the full ternary box of correlated coefficient shifts contains no active nine-term decoder below copied-information cost 72. The two cost-64 assignments each remove one term. This does not bound other frames, punctures, independent per-root gauges, or shifts outside the box.

The research seed is `655425ccd826280b13917975e2854d16`, selected on the `complex-mechanisms` track at catalogue priority revision 3. The exact base is `0b1cccf86e2f5132ed286a26ef0a681a495740ea`; the selected-record SHA256 is `10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32`. That base retains selected conditional finite construction κ `836330098538459/500000000000000000` (about `0.001672660197076918`). The review, source manifest, assembly certificate, selector and task catalogue are pinned in [SOURCES.json](SOURCES.json).

## Construction family

The eight deleted E8 lines are the exact coordinate frame with indices `[0, 13, 90, 99, 108, 113, 118, 119]`. Add the first extra root with four nonzero inner products against that frame (index `1`) to make a nine-normal frame. Its unique coefficient dependency is

```text
z = (-1, -1, -1, -1, 0, 0, 0, 0, 2), sum_k z_k b_k = 0.
```

For each live root line `i`, let `c_i` be the exact base row satisfying `sum_k c_ik b_k = 2 r_i`. For one shared integer vector `w ∈ {-1,0,1}^8`, set

```text
D_ik(w) = c_ik + <r_i,w> z_k.
```

The identity remains exact for every `w` because the added row term is a scalar multiple of the frame dependency. The verifier enumerates all `3^8 = 6,561` choices. For each of the nine product terms it forms the row support `X_k` and normal support `Y_k`, then charges the exact `F2` pairing rank `rank_F2(X_k,Y_k)`. This is the full copied-information invoice for this decoder family; it does not include a physical frame or recursive restoration circuit, so no κ conversion is justified.

## Exact result

| Total copied-information cost | Assignments | Nine terms active |
|---:|---:|---:|
| 64 | 2 | 0 |
| 72 | 3 | 3 |
| 73 | 6,556 | 6,556 |

The two cost-64 gauges are `w=(0,-1,0,0,0,0,0,0)` and `w=0`; each has one empty row support, so it is an eight-term decoder. The three active cost-72 gauges are

```text
(-1,-1,0,0,0,0,0,0)
(-1, 0,0,0,0,0,0,0)
( 1, 1,0,0,0,0,0,0)
```

Each has nine term ranks `(8,8,8,8,8,8,8,8,8)`. The verifier checks all `112^2 = 12,544` live root pairs for each of these five low-cost exact identities. A one-coefficient mutation is rejected. The predecessor package records `64/112` as its shipped P112 comparator, so this active family supplies no copy-cost saving to carry into a fully charged profile.

## Scope and overlap

The scope is a correlated, one-vector gauge family on one puncture. The exact support alphabet is the ternary coordinate box; this is not a search over arbitrary real or integer `w`.

The selected nine-normal frame comes from merged [PR #1224](https://github.com/CrocSwap/integer-mult-bounds/pull/1224), whose `research/nine-term-decoder-normal-form-seed42/` package scans per-root ratio-class choices in several frames and punctures. This search instead binds all per-root gauge parameters to the same eight-coordinate vector. Direct nearby work was also checked before claiming: PR #1923 / Discussion #1622 treats a symmetric ten-term Householder family; PR #1614 treats A8 simplex punctures; PRs #1779 and #1817 concern rank-16 diagonal batches and their interleaving. None matches this object and parameter family. The live claim is in [Discussion #1167](https://github.com/CrocSwap/integer-mult-bounds/discussions/1167#discussioncomment-18926689).

The result is a finite census for this family. It is not a universal lower bound for nine-term decoders, a circuit implementation, a supplier-compatible local word, or a new conditional multiplication exponent. The selected record is unchanged.

## Reproduce

```sh
python3 -I -B research/coord-root-p112-global-gauge-seed655425/verify.py
python3 scripts/research_package.py --package research/coord-root-p112-global-gauge-seed655425 --check-only
```

The first command performs the exact census and writes `RESULT.json`. The registered replay uses Python 3.12 and reruns the same check. `verify.py` refuses to run if the two source modules from merged PR #1224 have drifted from their pinned SHA256 values.

## Open next questions

- Does a larger bounded global-shift box contain an active decoder with a lower copy invoice on this puncture? Before expanding the search, derive the optimistic target invoice and claim a new explicit box.
- Can independently chosen per-root gauges be constrained by a meaningful correlation model that is not already covered by the predecessor's ratio-class census?
264 changes: 264 additions & 0 deletions research/coord-root-p112-global-gauge-seed655425/RESULT.json
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{
"schema_version": 1,
"seed": "655425ccd826280b13917975e2854d16",
"base_commit": "0b1cccf86e2f5132ed286a26ef0a681a495740ea",
"selected_record_sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32",
"puncture_indices": [
0,
13,
90,
99,
108,
113,
118,
119
],
"extra_root_index": 1,
"normal_dependency": [
-1,
-1,
-1,
-1,
0,
0,
0,
0,
2
],
"live_roots": 112,
"global_shift_domain": [
-1,
0,
1
],
"assignments": 6561,
"all_cost_histogram": {
"64": 2,
"72": 3,
"73": 6556
},
"all_active_cost_histogram": {
"72": 3,
"73": 6556
},
"best_all_active": {
"cost": 72,
"w": [
-1,
-1,
0,
0,
0,
0,
0,
0
],
"term_ranks": [
8,
8,
8,
8,
8,
8,
8,
8,
8
],
"term_support_sizes": [
56,
76,
76,
76,
56,
56,
56,
56,
56
],
"exact_pairs": 12544
},
"low_cost_exact_controls": [
{
"cost": 72,
"w": [
-1,
-1,
0,
0,
0,
0,
0,
0
],
"term_ranks": [
8,
8,
8,
8,
8,
8,
8,
8,
8
],
"term_support_sizes": [
56,
76,
76,
76,
56,
56,
56,
56,
56
],
"exact_live_pairs": 12544
},
{
"cost": 72,
"w": [
-1,
0,
0,
0,
0,
0,
0,
0
],
"term_ranks": [
8,
8,
8,
8,
8,
8,
8,
8,
8
],
"term_support_sizes": [
76,
56,
76,
76,
56,
56,
56,
56,
56
],
"exact_live_pairs": 12544
},
{
"cost": 64,
"w": [
0,
-1,
0,
0,
0,
0,
0,
0
],
"term_ranks": [
8,
0,
8,
8,
8,
8,
8,
8,
8
],
"term_support_sizes": [
76,
0,
76,
76,
56,
56,
56,
56,
56
],
"exact_live_pairs": 12544
},
{
"cost": 64,
"w": [
0,
0,
0,
0,
0,
0,
0,
0
],
"term_ranks": [
0,
8,
8,
8,
8,
8,
8,
8,
8
],
"term_support_sizes": [
0,
76,
76,
76,
56,
56,
56,
56,
56
],
"exact_live_pairs": 12544
},
{
"cost": 72,
"w": [
1,
1,
0,
0,
0,
0,
0,
0
],
"term_ranks": [
8,
8,
8,
8,
8,
8,
8,
8,
8
],
"term_support_sizes": [
56,
88,
88,
88,
56,
56,
56,
56,
56
],
"exact_live_pairs": 12544
}
],
"negative_control": "adding 1 to coefficient row 0, term 0 fails at a live pair",
"scope_limit": "This is only the fixed frame, puncture, global coefficient shift, and ternary box; it does not bound independent per-root gauges or other shifts."
}
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