# Noncommutative formula: exact witnesses and existing baselines

The new [structured reference](../../tools/noncommutative_hitting_reference.py) implements the characteristic-zero paper's explicit rational matrix action on the first column. It accepts a visible tree, not an opaque oracle. A nonzero entry is a finite free-algebra witness; zero is source-conditional or unknown. [Full record and ten source hashes](comparison.json), [two tiny matrix presets and scalar baseline](existing-matrix-baseline.json).

## Generated comparisons

| Case | Gates | Source dimension | Full reference outcome | Full call, ms | 32-coefficient call, ms | SymPy expansion, ms | Expanded terms |
| --- | ---: | ---: | --- | ---: | ---: | ---: | ---: |
| ordered_commutator | 9 | 324 | nonzero_polynomial_witness | 5.183 | 0.420 | 0.122 | 2 |
| distributive_identity | 15 | 1,350 | zero_at_source_tuple_conditional | 111.394 | 0.795 | 0.365 | 0 |
| incorrect_square_rewrite | 23 | 2,116 | nonzero_polynomial_witness | 313.176 | 1.120 | 0.474 | 2 |
| rational_scalar | 5 | 100 | nonzero_polynomial_witness | 0.542 | 0.171 | 0.239 | 2 |
| exceptional_variable | 1 | 1 | nonzero_polynomial_witness | 0.010 | 0.003 | 0.004 | 1 |
| product_of_sums_2 | 7 | 196 | nonzero_polynomial_witness | 3.067 | 0.429 | 0.086 | 4 |
| product_of_sums_4 | 15 | 900 | nonzero_polynomial_witness | 83.534 | 0.857 | 1.119 | 16 |
| product_of_sums_8 | 31 | 3,844 | unknown | 136.787 | 1.771 | 27.912 | 256 |
| product_of_sums_16 | 63 | 15,876 | unknown | 71.942 | 3.524 | 14772.280 | 65,536 |

Timings are one local sequential sample per method, without an isolation protocol. SymPy expansion computes all coefficients, whereas a nonzero matrix witness only refutes identity. Two ordinary exact two-by-two presets find all nonidentity examples in the nine-case table, including the sixteen-factor expression in about 0.16 ms per preset. A scalar evaluation already handles that product. No speedup over a competitive witness method or general white-box algorithm follows. Full source-dimension attempts at eight and sixteen factors hit the 8,192-bit rational cap; a separate completed 32-coefficient attempt supplies a witness.

The extra `nilpotent_probe_miss` input is x^2*y-y*x^2. Its first matrix preset gives zero; the second finds nonzero, as does the source prefix with row-four entry -1/48. A zero at a chosen tuple is not identity proof. In a separate model control, x^2-1 is a nonzero free polynomial but vanishes at Pauli X. Target algebra matters as much as arithmetic.

## Source schedule sizes

| n | Tree bound s | Characteristic-zero dimension | Positive-characteristic dense dimension | Positive-characteristic total matrix entries | Inverse-formula dimension | Inverse-formula grid triples |
| ---: | ---: | ---: | ---: | ---: | ---: | ---: |
| 1 | 1 | 1 | 6 | 36 | 32,768 | 92,950,340,097,909,002,166,337,537 |
| 2 | 3 | 36 | 16,306 | 531,771,272 | 8,388,608 | 375,670,790,678,992,431,863,771,923,481,666,519,041 |
| 2 | 9 | 324 | 14,608,288 | 426,804,156,581,888 | 536,870,912 | 10,736,235,695,876,997,810,245,769,403,257,872,033,307,766,554,625 |
| 2 | 31 | 3,844 | 27,090,887,694 | 1,467,832,392,097,841,275,272 | 34,359,738,368 | 1,418,360,178,793,051,471,768,867,131,274,247,759,546,579,101,346,749,457,891,329 |
| 4 | 63 | 31,752 | 15,632,898,923,251 | 977,550,114,978,329,100,753,636,004 | 8,796,093,022,208 | 6,101,746,186,916,524,883,221,652,220,201,965,008,844,081,068,201,712,555,498,398,010,761,543,681 |
| 8 | 127 | 258,064 | 8,492,647,581,757,989 | 577,000,503,583,518,547,648,990,122,592,968 | 9,007,199,254,740,992 | 13,584,460,105,755,699,884,646,190,896,080,844,286,105,505,319,778,108,456,562,136,067,544,260,048,198,964,934,082,561 |

The integer schedules were evaluated without allocating companion matrices or lists. Positive characteristic uses N=1+n*binomial(2s,2), E=1+binomial(N,2)+n*(N-1), d=N*E. The rational-formula generator uses B=2s+1, delta=2*(n+1)^3*B, the least power of two M>B*delta, D=M^2/2 and H=1+3*(s+2)*B*M^3, then visits all H^3 grid triples and retains the nonsingular ones. Grid visits are distinct from actual output count. The selected formal scope is narrower for the characteristic-zero paper and does not link the newer positive-characteristic companion.

## Controls, replay and limits

The reference/paper-coefficient/library checks total 686; the tiny existing-matrix extension adds 32; model/resource boundaries add eight. All 726 passed. Exact JSON inputs, outcome records and source hashes are retained. Existing Python 3.10.19 and SymPy 1.13.1 were reused; no package install was performed.

```sh
python3 research/tools/noncommutative_hitting_reference.py research/prototypes/noncommutative-hitting-comparison-2026-10-09-v1/ordered_commutator-input.json
python3 research/tools/noncommutative_hitting_reference.py research/prototypes/noncommutative-hitting-comparison-2026-10-09-v1/product_of_sums_16-input.json --coefficients 32
```

Input caps: eight variables, 127 tree gates, depth 32, 256-bit canonical scalar components. Execution caps: dimension 16,384, one million selected arithmetic charges, 8,192-bit intermediate rational components. The arithmetic counter excludes bit operations, allocation, parsing, negation and wall time. Exhaustion retains no unfinished witness. Source universal correctness, Python formal correctness and buyer benefit remain open. [Revised dossier](../../opportunities/012-noncommutative-identity-checks/2026-10-09-v2.md).