RESULTS: controls for the F1 discovery engine

Exploratory. Nothing in hunts/ is a result or evidence (hunts/README.md). What follows is measured, at the sizes stated, with the instruments in this directory. The claims it corrects are in docs/15-the-f1-discovery-engine.md, whose Corrections section carries the same three findings for a reader who arrives there instead.

Grade: measured. Two of the three findings are arithmetic statements about a finite matrix and could be hardened; none of them is about zeta.

The setting

docs/15 writes up seven prototypes and closes with a Reality Check of five critiques, sourced to outside checkers, that dial its own rhetoric back. The Reality Check is the honest half of that page. It had also never been tested: five assertions in prose, one of them a closed-form arithmetic law, one of them a claim about a prototype's behaviour, and no test anywhere in the tree touching either. The prototypes are frozen (AGENTS.md), so everything here reads them by path and modifies nothing.

Finding 1: the ln 2 multiplicity is dim ker C

Reality Check item 5 explains the degenerate eigenvalue at ln 2 in ontology/04_transcendental_matrix.py as a truncation artifact "with multiplicity pi(N/2) - pi(N/3) - 1". The artifact reading holds. The formula is the special case in which every dead-end node is prime.

Derivation and mechanism: ln2_law.py, module docstring. In short, the nodes u with 2u <= N < 3u carry a private ln 2 rung at the top of the graph; the eigenvalue equation is automatic on their support and constrains only the hubs u/q below them; so the multiplicity is the kernel dimension of the hub incidence matrix C. Prime dead ends share the single hub 1, giving |D| - 1 = pi(N/2) - pi(N/3) - 1. Composite dead ends normally pin their own coefficients to zero, until they close into a pattern.

• The law agreed with the eigenvalue count at all 251 sizes N = 10 … 260 and at a coarser scan through N = 800. No disagreement.
• Both rank routes (40-digit elimination, double-precision LAPACK) agree at every size checked.
• The measured cluster sits within ~1e-15 of ln 2: exact degeneracy, not a near-degeneracy that a tolerance manufactured.
• The quoted formula first undercounts at N = 338, where the semiprimes 121 = 11^2, 143 = 11 x 13, 169 = 13^2 give three columns over the two hubs 11 and 13.

Raw scan: results_ln2.json.

Finding 2: the density gap widens under refinement

Reality Check item 2 says the prototype has "dozens of frequencies below 17.7, where zeta only has one zero". Counted at the 400-node truncation of docs/15 section 6: 171 modes below 17.7 against 1, and 159 modes below the first ordinate 14.1347 against 0.

The refinement response is the part worth keeping. Spurious low modes accumulate at roughly 0.35 N, while the top of the spectrum creeps like log N, about 3.7 units of frequency per doubling of the node count.

Extending the trend puts the 20th ordinate (77.145) somewhere near 10^7 nodes, with millions of modes below zeta's first zero. That extrapolation is a trend line and is labelled as one; the finding does not rest on it.

Raw scan: results_density.json.

Finding 3: the gauntlet's criticism was aimed at the wrong half

Reality Check item 4 calls 07_the_imposter_gauntlet.py vacuous because "the construction never actually consults zeta". Measured, that splits three ways.

1. True of the operator. build_polya_hilbert_operator takes N and nothing else, so no candidate function can enter it. "The geometry structurally rejects the imposter" is not about the geometry.
2. False of the predicate. What 07 runs is a coefficient multiplicativity test, which the tree owns as zeta.epstein.claim_multiplicativity. It consults its input: through the three linear-combination rivals of zeta.epstein.battery (shift=None), it passes zeta and fails Davenport-Heilbronn and both discriminant -23 Epstein rivals, distinguishes: True, shared_with: ().
3. The real defect in 07's conclusion. The standing battery (default arguments) also carries the symmetric shifted product W_a(s) = zeta(s+a) zeta(s-a), a rival with an Euler product of its own, and the predicate admits it: distinguishes: False, shared_with: ('shifted_product',). That pass is the repository's own refutation of "immune to false positives". Both mod-5 Dirichlet L-functions whose combination is Davenport-Heilbronn pass the predicate too, as does every primitive L-function with an Euler product. The test separates Euler products from linear combinations of them. That is what battery's docstring says the linear-combination rivals detect, and it is a real answer to docs/11 gate #3. It is not immunity to false positives, and it does not single out zeta.

The branch that first measured this (2026-08-21) predates the shifted-product rival, which reached main afterwards; its single recorded verdict was the shift=None reading. Both readings are now measured and recorded.

Raw output: results_gauntlet.json.

The third kill condition, read against today's battery. MISSION.md names "the coefficient predicate turns out not to discriminate on the standing battery" as a kill condition. It was written when the standing battery was the three linear-combination rivals, and on that set the predicate discriminates. On the default battery as it now stands, with W_a, it does not. The kill condition is left as written. What it guarded, that the "ignores its input" charge is false of the predicate, survives on the shift=None reading, since a predicate that rejects three rivals and admits a fourth is consulting its input. What the W_a pass adds is the stronger refutation of 07's "immune to false positives".

What was not corrected

docs/15's two headline numbers still come out of the prototypes unchanged: Mode 5 at 8.169 on the transcendental prime grid, Mode 20 at 1.937 for the 400-node prototype. Reality Check items 1 and 3 are judgments, not measurements, and are left as written.

Disposition

The three findings are pinned by tests/test_f1_engine_controls.py (32 tests, about 40 s serial on a 32-vCPU box, fast tier). docs/15 carries a correction notice at its head and the detail at its foot. The stale discovery/ paths that survived the rename to ontology/ are repaired in docs/15 and docs/09, and tests/test_docs_paths.py now fails if any document in docs/ points a reader at a Python file that is not there.

No claim here goes to the funnel or the battery: none of it is about zeta.