Independent check of the f7 functional classification, plus the (5,14) domain (no new kappa)

research/f7-functional-independent-check/README.md

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+# Independent check of the `f7` functional classification, and the `(5, 14)` domain
+
+**Independent verification (no κ) of a finite certificate, plus new coverage the
+checked package itself names as its next question. No new `f(7)` bound, no
+supplier, no selected-frontier change.**
+
+## Subject of the check
+
+[`research/f7-bit0-functional-classification`](../f7-bit0-functional-classification/README.md)
+(branch `research/w5-b21-todo-20261018`, head `13db47d8d31603c1391572009e1eb7a22675a219`,
+PR #3699) reports, on the PR943 singleton prefix walk, that the retained engines'
+fixed `L/2` budget substitution is not a coordinate-bit-0 artefact: it falsely
+prunes universal words for every functional of low Hamming weight and for none of
+high weight, on nine `(n, L)` domains.
+
+This package recomputes that certificate from scratch and extends it to three
+domains the published table does not contain, including `(5, 14)`, which the
+published package lists as its open next question 2.
+
+The claim under check is fixed at that package's own publication head
+`13db47d8d31603c1391572009e1eb7a22675a219`, where its `README.md`,
+`RESULT.json` and `verify.py` have SHA-256
+`8176c8a889671f67ebcd6f1b9e9efddae55cfbbff4b486442a9b63977e9c4844`,
+`5938908a76c497bad4c301eed0f12822fc16b0fbbbff2c0bc5150d82f2052105` and
+`a0c93985734fa6855c5561a1b4eb730ef0af6ddd25bfdac1ef8b61d840efbe83` and Git
+blob ids `4618c68ffc94be30055537bca4e49696946437dc`,
+`84017d1265858fe976bd317c95d287c92fc4cb7e` and
+`f554ec10c8ac5e30e067b01adf9b68b2eb80906a`. Those three objects are not
+ancestors of this package's publication commit (the checked PR is still open),
+so they are recorded here rather than in `SOURCES.json`, whose rows must be
+ancestors.
+
+## What is reproduced, and how
+
+`verify.py` is a separate implementation. It imports, executes and copies no
+predecessor code. The published numbers are embedded as asserted constants; the
+prefix walk, the simplicity/restoration/universality predicates, both collision
+excess routes, the enumeration and the classification are written here.
+
+Every number in the published table is reproduced **exactly**, on the first run,
+by a different enumerator:
+
+| `n` | `L` | simple closed | universal | substituted budget | false prunes | firing weights |
+|---:|---:|---:|---:|---:|---:|---|
+| 3 | 6 | 4 | 4 | 3 | 0 | — |
+| 3 | 8 | 2 | 2 | 9 | 0 | — |
+| 4 | 8 | 29 | 22 | 5 | 44 | 1, 2 |
+| 4 | 10 | 110 | 110 | 13 | 190 | 1, 2 |
+| 5 | 10 | 358 | 0 | 5 | 0 | — |
+| 5 | 12 | 3695 | 828 | 15 | 2856 | 1, 2, 3 |
+| 6 | 12 | 6525 | 0 | −1 | 0 | — |
+| 6 | 14 | 123704 | 0 | 11 | 0 | — |
+| 7 | 14 | 161936 | 0 | −21 | 0 | — |
+
+Beyond the column totals, the check also asserts equality of the **per-functional
+false-prune counts** (24 masks at `(5, 12)`, 10 at `(4, 10)`, 9 at `(4, 8)`) and of
+the published bit-0 column (`6`, `20`, `248` false prunes for the coordinate-bit-0
+functional). The reported `01021323` fixture is reconstructed from the word: the
+nine prefix states `0,1,3,2,6,4,12,8,0`, the Hamming-weight split `(4,4)` with
+excess `0`, the bit-0 split `(6,2)` with excess `8`, the substituted budget `5`
+and the corrected budget `9`.
+
+## New coverage: three further domains
+
+The published table jumps from `(4, 10)` to `(5, 10)` and stops at `L = 14` for
+`n = 6, 7`. This package classifies three further domains:
+
+| `n` | `L` | simple closed | universal | substituted budget | false prunes | firing functionals | sound functionals |
+|---:|---:|---:|---:|---:|---:|---:|---:|
+| 3 | 10 | 0 | 0 | 17 | 0 | 0 of 7 | 7 of 7 |
+| **4** | **12** | **314** | **314** | **23** | **150** | **13 of 15** | **2 of 15** |
+| **5** | **14** | **33922** | **24787** | **27** | **56686** | **25 of 31** | **6 of 31** |
+
+Two measured statements follow.
+
+1. **At `(5, 14)` the substitution falsely prunes for exactly the 25 functionals
+   of Hamming weight 1, 2 and 3, and for none of weight 4 or 5.** This is the
+   domain the published package names for exactly this test. Its own `(5, 12)`
+   reading carried one exception (mask `11010`, weight 3, did not fire); at
+   `(5, 14)` there is no exception at all, so the weight threshold `n - 1` with
+   "every weight `<= n - 2` fires" is clean there. That is evidence about the
+   heuristic, not a proof of it: nothing here bounds the threshold for general
+   `(n, L)`.
+2. **The `(4, L)` firing set grows with `L` and is monotone on the measured
+   range.** `(4, 8)` fires for 9 of 15 functionals, `(4, 10)` for 10, `(4, 12)`
+   for 13: at `(4, 12)` three of the four weight-3 functionals fire (`1011`,
+   `1101`, `1110`), and only `0111` (weight 3) and `1111` (weight 4) stay sound.
+   Compared with `(4, 10)`, where every weight-3 and weight-4 functional is
+   sound, the unsound set is strictly larger at the longer length. So the old
+   question "is bit 0 special?" is answered with an even wider unsound class at
+   this length, and the fix cannot be a per-weight threshold that is independent
+   of `L`.
+
+`(3, 10)` has no simple closed word at all: `q_0..q_9` would be ten distinct
+states of `F2^3`, which has eight. It is recorded as an empty domain rather than
+dropped, so the enumeration's domain check is exercised on a degenerate case.
+
+## Controls, all run on every invocation
+
+- **Fixture.** The published word's prefix states, splits, excesses and both
+  budgets are recomputed and asserted, including the identity
+  `C(s,2) + C(t,2) - (2^(n-1) - 1) = base + d^2`.
+- **Two excess routes.** The collision excess is computed both as a difference
+  multiplicity sum and as a direct support-size difference, and the two must
+  agree for every universal word and every nonzero functional of every domain.
+- **Budget sensitivity.** Shifting the substituted budget by `-2, -1, +1, +2`
+  must leave every non-firing domain at zero and must move the count on each of
+  the three firing domains. The check therefore depends on the budget's exact
+  value, not only on class sizes.
+- **Second enumeration route.** On six domains the census is also produced by an
+  unconstrained walk (no first-occurrence ordering) whose leaves are canonicalised
+  afterwards; the resulting word sets must be equal. This is the check that would
+  catch a pruning error in the primary enumerator.
+- **Predicates on every enumerated word.** Every emitted word is asserted
+  restored and simple; universality is recomputed as `|S + S| = 2^n`.
+
+## What this does not establish
+
+- It is a **finite certificate on ten bounded `(n, L)` domains**, not a theorem
+  about the weight threshold, and not a census of the seven-label regime. The
+  `(7, 18)` and `(7, 20)` domains that would change a maintained decision are
+  not entered.
+- It does not re-derive the *necessity* of the exact budget `base + d^2`; that
+  implication is inherited from the checked package and from PR2973's `MATH.md`.
+  What is checked here is the classification and the counts built on it.
+- No `f(7)` value, no pruning repair, no source change, no κ and no supplier is
+  claimed. A false prune in this model does not by itself invalidate any reported
+  final bound.
+- Independent human review and formal proof-kernel acceptance are unclaimed.
+
+## Reproduce
+
+```sh
+~/git/integer-mult-bounds/.venv/bin/python3.12 -B research/f7-functional-independent-check/verify.py
+~/git/integer-mult-bounds/.venv/bin/python3.12 -B research/f7-functional-independent-check/test_verify.py
+```
+
+`verify.py --write` regenerates `RESULT.json`, which is the exact finite receipt.
+Standard library only, no Git objects, no network. Runtime is about 30 s on the
+recording host; `(5, 14)` alone is a few seconds.
+
+## Credit
+
+The classification question, the model, the nine published domains and the nine
+pinned numbers are Nick Petalas's, in
+[`research/f7-bit0-functional-classification`](../f7-bit0-functional-classification/README.md)
+— including its named open question about `(5, 14)`, which this package answers
+for the universal-word domain. Nicholas Stone wrote the word-census correction
+(#2973) and its `MATH.md`; James Reed wrote the bit-0 small-domain audit (#3096);
+Ross Richardson supplied the PR943 singleton prefix-walk model. This package's
+re-implementation, the three added domains and the controls are by Nick Petalas,
+produced with AI assistance. Apache-2.0.
+
+## Open next questions
+
+- **Is the firing set monotone in `L` for fixed `n`?** Measured on `(4, 8) <
+  (4, 10) < (4, 12)` and `(5, 12) < (5, 14)`, but not proved, and `(5, 10)` has no
+  universal word to compare against.
+- **Can the weight `>= n - 1` soundness be proved rather than measured?** The
+  published alternation lemma explains why only the all-ones functional is
+  *justified* by the substitution; it does not explain why high-weight functionals
+  do not fire.
+- **The `(6, 16)` domain.** `(6, 14)` has 123,704 simple closed words and no
+  universal word; the next even length is the first that could change that, and
+  it is outside this package's budget.