DH_GATE_PREREG_2: a second stage for the Davenport-Heilbronn gate, preregistered

Written and committed 2026-10-31, at landing, before any stage-2 box was decided. Stage 1 (DH_GATE_PREREG.md, commit 2ba8cbba3) is not changed by this file: its six boxes, its parameters, its decision table, its verdict rule, its list of faults that must be caught and its recorded run (dh_gate_results.json, ../RUNS.md E7) all stand as they are. That file says that a change after the run is a new preregistration with a new run; this is that, and it adds boxes rather than changing any.

Why a second stage

Stage 1 recorded no_margin and no_M2 as not caught: both give PASS. The cause, read from dh_gate_results.json and the stage-1 boxes:

1. A principal argument fails only when the zeros on one side of a segment subtend more than a half turn in total. H_t is even and entire of order 1, so by its Hadamard product its argument change along a segment is the sum, over its zeros, of the angle the segment subtends at each, signed by the side the zero is on (the lab's reading, unreviewed). A single zero subtends less than a half turn at any distance, and zeros on opposite sides partly cancel. So a margin-free count can only go wrong where two or more zeros lie close to the same side of one segment.
2. No stage-1 segment met that condition. Z2, Z3 and Z4 each hold one zero at least 0.083 from every edge, with its conjugate more than half a unit below the box. Z1 is the one box beside a close pair (its zero and the conjugate, 0.041 apart), and both lie below its top edge. But Z1 is centred on the zero, whose abscissa is 0.0001 from a vertex of the four-point mesh, so the pair's angle is split over two top-edge segments, about 142.5 degrees each, and the count stays right. Had the pair sat mid-segment, that segment would have subtended about 225 degrees. The fault removes a guarantee; on the stage-1 boxes nothing needed it.
3. no_M2 drops a second-order term. Where H is close to linear or quadratic along a segment (an isolated zero, or a close pair), |H'| along the segment is the distance to a point on a line times a constant, which is largest at an endpoint, so the endpoint maximum already bounds |H'| on the segment and the dropped M2 h/2 cannot change a decision. It can only matter where |H'| peaks strictly inside a segment, which needs a third zero (a second zero of H') close to the segment. This file knows no such configuration at heights where hunt #61's truth step decides a box: that step uses one M2, a supremum over Re z >= x_lo, and |H| falls like e^{-pi x/8}, so its boxes are a few units wide at most, while the real zeros near x = 480 are about 2.4 apart on average (an estimate from the mean spacing 2 pi / log(5 gamma / 2 pi) at gamma = 240, doubled for the wide frame; not computed here).

The configuration stage 2 uses

Hunt #61 located a zero of the narrow-frame flow at t = 36/625 near z_n = 240.3451714460939 + 0.01018313776460141 i (winding_results.json, zstar_re, zstar_im; a measured locating pass inside a decided box). In this evaluator's wide frame it is z = 480.6903 + 0.0204 i at t = 144/625, and its conjugate 480.6903 - 0.0204 i is also a zero (real coefficients). The pair is 0.041 apart: two zeros close to one side of any segment that passes just above the upper one.

The stage-2 boxes share the x-range [480.39, 481.19], so that the four-point mesh of no_margin (vertices 480.39, 480.59, 480.79, 480.99, 481.19) has one segment centred over the pair, and the clean mesh (eight points) has a vertex directly above it. A segment of half-length 0.1 at height 0.0096 above the upper zero and 0.0504 above the lower one subtends about 2 arctan(0.1/0.0096) + 2 arctan(0.1/0.0504), about 169 + 127 = 296 degrees.

The boxes (wide frame, exact rationals)

"Expected" is a prior: the truth step (hunt #61's decided count, as in stage 1) decides each box.

What is unchanged

• The pipeline: dh_gate.dh_barrier exactly as at 2ba8cbba3, with stage 1's parameters (8 initial points per edge, max_depth 6, at most 600 evaluations per box run, precision dh.split_prec(x_hi)).
• The truth step: dh_gate.truth at 420 bits.
• The faults: the five of stage 1, unchanged.
• The decision table and the verdict rule: dh_gate.classify and dh_gate.verdict, unchanged. Caught still means a FAIL verdict.

The stage-2 code (GATE_BOXES_2, run_stage2_boxes, assemble_stage2, combined_verdicts, main_stage2) only names the new boxes, runs them and applies the stage-1 rules to their rows.

Verdicts reported

1. The stage-2 verdict of each run (clean and each fault): the stage-1 rule on the stage-2 rows alone.
2. The combined verdict of each run: the stage-1 rule on the union of the stage-1 rows (read from dh_gate_results.json, not rerun) and the stage-2 rows.

Predictions, fixed here

Stage 2 is read as showing that the gate can see a margin-free count if no_margin gives FAIL on the stage-2 boxes and the clean run has no ACCEPT and no INCONSISTENT row there. Any other outcome is recorded as it comes out. A stage-2 result does not change the stage-1 verdicts, which stay on record.

What was computed on stage-2 points before this file was committed

Point evaluations only, never a box decision and never a run of dh_barrier or truth on a stage-2 box (dh.H_ball, dh.Hprime_ball at 433 bits, and dh_gate.M2_wide):

M2 (wide frame) is 2.729e-79 for Z5 and 2.795e-79 for C3, about 60 times the local |H''| of about 4.5e-81 implied by the pair. The arguments along y = 0.03 at 480.59, 480.69 and 480.79 fall through -0.568, -3.107 and (one turn down) 0.555, a change of about -5.16 radians, -296 degrees, as the geometry predicts. With that M2, the clean rule accepts a segment ending at (480.69, 0.03) only once h is about 0.0031 (five bisections of the initial 0.1), and one ending at (480.69, 0.01) only at about 0.0016 (six, the depth cap). These are estimates from the endpoint values above, not runs of the rule.

Cost estimate (before the run)

From stage 1's Z1 and C1, at the same t and height: truth 5 to 10 s, clean 15 to 40 s, narrow_frame up to its 600-evaluation budget (about 170 s), the other faults under 20 s each. About four to five minutes per box; the two boxes run as two processes at nice 10, under ten minutes of wall time.

Grades

As stage 1: the truth and the pipeline's decisions are enclosure-carrying, given hunt #61's analytic steps (the shifted-contour M2 derivation, its Phi_DH representation and kappa's functional equation), which are not re-reviewed here. The location of the pair is hunt #61's measured locating pass; the truth step, not that location, decides each box. The stage-2 verdicts are statements about these two boxes and this rule, not about any de Bruijn-Newman constant, and Theorem 1.3's error terms are still not exercised (on DH, E = 0).