R-065F29: white-box attack on `urms2-0.51`

Outcome: the attack ran, the claim is not withdrawn, and it found three things about the record that a reader leaning on this claim should know.

The mathematics of the load-bearing new step survives direct numerical attack in the regime the old proof forbade. What does not survive is the supporting apparatus: the "independent" audit route for one gate is a second arithmetic assembly of the same numbers, the recorded rational witness does not select 51/100, four of the six obligations the theorem lists have no gate, and the record's own falsification control runs on a coefficient family that violates the hypothesis of the step it is offered as evidence for.

Nothing here is evidence for or against RH (docs/08).

Reproduce: python3 hunts/r_065f29/probe.py (numpy and sympy only; no zeta import, no mpmath, no network). Numbers below are from results.json, written by that run.

The table

A3: the load-bearing step holds where it was attacked

URMS2-051.md §3 replaces the worst-spacing majorant with the spacing-sensitive Montgomery–Vaughan estimate, and §4 concludes the off-diagonal error is O(x log x) independent of the upper length W. The whole half-band crossing rests on this: it is what converts the old γ < δ < 1 condition, which is infeasible at α = 0.51, into α < δ, which is not.

The probe evaluates the exact block second moment

∫U^{2U} |Σ{n≤W} c_n n^{-it}|² dt = U Σ|c_n|² + Σ_{m≠n} c_m c̄_n K_U(log(m/n)), K_U(θ) = ∫_U^{2U} e^{-itθ}dt

as a closed-form double sum on the author's own frozen level-two coefficient array, with the two-range weights √n/x and x/n^{3/2} of §1, at U = 300, x = e^{0.51·8} = 59.1, and W swept to 2980. This is the regime W ≫ U that the old proof's W/U = o(1) forbade.

The integral saturates to four significant figures. That is the W-independence the claim asserts, measured rather than argued, and it is the strongest positive result of this attack. The deviation from U Σ|c_n|² also stays inside the claimed O(Σ n|c_n|²) throughout, with ratio in [0.400, 0.883].

One caveat, stated because the number invites a wrong reading: that ratio falls across the sweep, which looks like a widening safety margin but is partly its denominator inflating on this particular coefficient family (see A5). The saturation of the integral does not depend on the denominator and is the observation to lean on.

A second caveat: at U = 300 the off-diagonal is the same order as U Σ|c_n|², so this test confirms the bound and the W-independence, not the asymptotic main-term dominance, which lives at U → ∞.

A5: the record's own control does not satisfy its own hypothesis

This is the finding with the most consequence for how the claim should be read.

§4 derives the W-independence from A(y) = Σ_{n≤y}|a_{2,U}(n)|² ≪ y log y, supplied by RAMS2, via partial summation on x² Σ_{x<n≤W}|a(n)|² n^{-2}. The probe checks that mathematics twice.

The mathematics is correct under its hypothesis. On a surrogate family built to satisfy the law exactly (|b(n)|² = log n), the upper-range sum saturates as W grows 67×: 441.6 → 691.3 → 830.4 → 907.1 → 984.7. §4's step does what it says.

The record's control family does not satisfy the hypothesis. On the frozen level-two array from dirichlet_recurrence.py, the family half_band_crossing.finite_height_spacing_experiment runs on, and the source of §9's ℓ = 6, 8, 10 table, A(y)/(y log y) is not bounded. It falls to a trough of 0.0231 at y = 400 and then climbs to 2.036 at y = e^10, a factor of 88. Swept at fixed x, the upper-range sum on that family grows like W^{0.825} (32.1× over the same sweep) instead of saturating.

Why §9's table cannot see this. The ℓ ladder moves x = e^{0.51ℓ} and W = e^ℓ together, holding the effective γ at 1. It therefore reports the ratio spacing cost / (x log x) along a single ray and never varies W at fixed x, which is precisely the quantity §4 claims is W-independent. Its reassuring decreasing sequence (0.409, 0.282, 0.185) is consistent with W-independence and equally consistent with W^{0.825} growth. The probe reproduces that table's ℓ = 8 entry to five digits (0.28176 against the recorded 0.28174), so this is the same object, read along a second axis.

What this does not show: dirichlet_recurrence.py says in its own docstring that it is "a frozen finite model, not a tail theorem", and RAMS2, not the frozen model, is the actual analytic input to §4. So this is not a counterexample to RAMS2 and not a kill. It is narrower and still worth recording: §9's falsification control has no power over the step it is attached to, because the family it runs on violates that step's hypothesis and the ladder is structurally blind to the failure mode. A reader who took that table as numerical support for the W-independence took support that is not there.

A2: gate 6's "independent route" is arithmetic, not independence

URMS2-051-AUDIT.md's verdict table lists six gates, each with an "independent route". For the 0.51 window functional the route is given as "JSON coefficient fixture and rebuilt tail sum", and urms2_051_audit.py's docstring says it "does not call constant_window_bound". Both statements are true. Neither delivers independence.

• The fixture C2_EXTENDED.json reproduces corrected_coefficients(40) exactly, index for index. It is a snapshot of the function the primary route calls, not a second derivation.
• Both routes import fock_upper_coefficient, coarse_upper_coefficient, MAJORANT_CUTOFF and TWO_STEP_RATIO from corrected_form_factor. The tail terms come out bit-identical as exact rationals.
• Perturbing fock_upper_coefficient(41) by one part in 10⁶ moves both denominator_upper values by the identical amount, 4.426081703885579e-27.

A check that cannot diverge from what it audits has no power against a fault there. This is WHITEBOX_CHECKLIST entries 2 and 5 verbatim, and the same shape as the docs/25 shared-parsing-layer incident the checklist was built from. The honest description of gate 6 is a second arithmetic assembly of the same numbers, which catches assembly errors and nothing upstream of them. The five decimal digits of 0.0147728663285376 are therefore single-sourced through corrected_form_factor.

A1: the witness does not derive 0.51

half_band_crossing.urms2_051_witness() records four margins at α = 51/100, δ = 3/4, γ = 21/20, ε = 1/100. All four are positive, as recorded. But they do not bind at 51/100:

• At the published (δ, γ, ε), the binding margin is far_tail_margin and the system stays feasible to α = 257/500 = 0.514.
• γ is bounded above by nothing in the recorded system, §4's whole point is that the upper length may cross U, so far_tail_margin can be made positive at any α by raising γ. δ is bounded only by initial_interval_margin > 0, i.e. δ < 1. The probe exhibits feasible witnesses at α = 0.6, 0.7, 0.74, 0.8, 0.9.

So the four margins as encoded permit every α < δ < 1. 51/100 is not derived from them.

Reading hunts/higher_xi/MISSION.md confirms this is by intent rather than by error: the target comes from downstream, where "the downstream window problem has exact bounds 0.5 ≤ β_useful ≤ 0.51" and "bandwidth optimization stops at this first corrected theorem". 0.51 is the first rational past the half band that makes the window functional useful. The claim as worded, that the bandwidth "extends past the half band to 0.51", is a lower bound and is not overstated.

The finding is about the record, and it is sharp: if the encoded margins permitted α up to δ < 1, the method would prove far more than 0.51, and it does not. Something binds that is not written down. A4 says where to look.

A4: where the unrecorded constraint most likely lives

§7 rests URMS2-051 on six obligations. The audit gates two of them. The other four,

• infinite tail o(U log U) by the 9/1000 exponent margin,
• initial height interval,
• height commutator,
• contour localization

• are carried by the sentence "retain their earlier bounds". Those earlier

bounds were established under γ < δ < 1. This proof's entire novelty is γ = 21/20 > 1. Inheriting a bound across the regime change that the proof itself introduces is the correlated-assumption shape the checklist names, and it is the one place this attack could not close: it is a claim about prior derivations that are not in hunts/higher_xi/ in a form the probe can evaluate.

If a constraint genuinely caps the band below δ, the probe's reading is that it is in one of these four, most plausibly the height commutator or contour localization at γ > 1.

What this attack did not settle

• Whether RAMS2 in fact supplies A(y) ≪ y log y for the true a_{2,U}(n). The probe measured the frozen model, which is explicitly not the analytic object. Settling this needs the RAMS2 statement itself, not a finite model.
• Whether the four ungated §7 obligations survive at γ > 1. This needs the earlier derivations, which are not in this tree in evaluable form.
• The asymptotic regime. U = 300 confirms a bound and a saturation, not a main-term asymptotic.
• The claim is not withdrawn. No fault was found in the mathematics of the half-band crossing itself.

Loose threads

1. The α < δ < 1 gap. If §4's mean-value gain is as general as the encoded margins suggest, the recorded system proves URMS2 well past 0.51, up to δ < 1. Either the result is substantially understated or a real constraint is missing from the record. Why it matters: the difference between "bandwidth 0.51" and "bandwidth approaching 1" for a level-two pair statistic is not a rounding of the same result. First step: re-derive the height commutator and contour localization bounds at γ = 21/20 rather than inheriting them, and see which one caps α.

1. docs/25-shaped shared-layer risk across hunts/higher_xi/. A2 found one gate whose independence is arithmetic re-derivation over a shared fixture. The same import pattern (from corrected_form_factor import ...) appears in bandwidth_forensics.py, urms2_051_audit.py and dirichlet_recurrence.py. Why it matters: if any other "independent" gate in that hunt shares the same substrate, the audit's six-of-six pass rate overstates the coverage. First step: run the same one-part-in-10⁶ mutation against each of the six gates and record which ones move.

1. A W-at-fixed-x axis for finite_height_spacing_experiment. The existing function sweeps ℓ only. Adding a W parameter would let the record's own control test the quantity §4 actually claims. Why it matters: this is a two-line change that converts a control with no power into one with power. First step: add W: int | None = None to finite_height_spacing_experiment and truncate integers accordingly. (Not done here: hunts/higher_xi/ is outside this hunt's write scope.)

1. The frozen model's second-moment blow-up may itself be interesting. A(y)/(y log y) on the level-two recurrence is not merely unbounded, it has a clean trough near y ≈ e^{0.55ℓ} and then climbs with a stable slope. Why it might matter: if that trough location tracks ℓ predictably it is a property of the level-two coefficient family, not an artifact, and would say something about where the frozen model stops proxying the real object. First step: locate the trough at ℓ = 8, 10, 12 and check whether log(y_trough)/ℓ is stable.