PREREG: ghost zeros of the DH ground-state transform at the off-line ordinate

Written 2026-10-21 on branch hunt/weil-ghost-zeros (from origin/main fb647bfc7), lane "ghost-zeros" of the existing hunt weil_propagation. This is the smallest check that ../theory/RESULTS.md s4 C3 names. Committed before any ground state, transform value or zero count of this lane exists. Everything below is fixed from this commit on; anything decided after the numbers goes in RESULTS.md under a heading that says so, never here. Nothing here is a claim about RH.

Read before writing this file: ../MISSION.md, ../theory/RESULTS.md (s0, s1, s4 C3, s5), ../numerics/RESULTS.md (s0 to s3, s9), ../numerics/wp_common.py, ../numerics/grid_dh_N128.json (its stored zero-side split), ../numerics/crossing.json, ../numerics/epstein_N128.json, ../numerics/epstein_offline.json. This file is not blind to the stored values of the off-line quadruple term Q1 at the grid windows (../numerics/RESULTS.md s3.3); expectation S2 below was derived from them.

0. Inputs taken as given (sources, not recomputed here)

• CvS (arXiv:2511.23257, CMP 406, 2026), as the theory worker records it in ../theory/RESULTS.md s1: if the lowest eigenvalue of the truncated Weil form on a window is simple and isolated with an even eigenvector v, the transform F_v has only real zeros.
• DH's continuum form is negative on every window c >= 30.617188 (../numerics/RESULTS.md s3.2: hardened cell at N = 256 plus the nesting argument, ordinary and unreviewed).
• The band N = 128 even-sector crossing c*_128 lies in (15691/512, 31383/1024] = (30.646484375, 30.6474609375] (../numerics/crossing.json, both ends hardened).
• Xi_DH(z) vanishes at z_1 = gamma_1 - i delta_1 with gamma_1 = 85.69934848537759..., delta_1 = 0.3085171824566374... (GAMMA_OFF, DELTA_OFF in ../numerics/wp_common.py, 50 digits; the zero is s = 1/2 + i z_1). The second pair z_2 = 114.1633427308 - 0.1508300806 i (GAMMA_OFF2, DELTA_OFF2).
• DH's on-line gap around gamma_1 is (83.10884108474983, 87.64747633255973), consecutive entries of data/dh_zeros_online_T120.json (float64 scan, completeness not checked here).
• zeta: gamma_23 = 84.73549298051705, gamma_24 = 87.42527461312523 (mpmath.zetazero).
• Epstein (1,1,6): off-line zero 0.953260474794661 + 16.2902157203904 i (../numerics/epstein_offline.json, measured), so z_E = gamma_E - i delta_E with gamma_E = 16.2902157203904, delta_E = 0.453260474794661. Its N = 128 even crossing lies in (29.303711, 29.304199] (../numerics/epstein_crossing.json).

1. Objects

• Windows. C = {29, 30, 61/2, 245/8, 31, 63/2, 47}, exact rationals, L = log c. The windows at or before c*_128 are {29, 30, 61/2, 245/8}; the windows after it are {31, 63/2, 47}. (245/8 = 30.625 is above the continuum bound 30.617188 and below c*_128.)
• Kinds. dh (the subject); zeta (negative control at the same height); epstein (rival with its own off-line zero, Q = x^2 + xy + 6y^2); dedekind (negative control sharing Epstein's completion: the Dedekind zeta of Q(sqrt(-23))).
• Matrices. Band N = 128, even sector, the CCM basis of ../numerics/wp_common.py: wp_common.build for dh and zeta, wp_common.epstein_matrices for epstein and dedekind, all ball matrices (Arb). Odd-sector matrices from the same assemblies.
• Ground state. v = the unit eigenvector of the lowest eigenvalue lambda_1 of the even matrix; x = an inverse-iteration vector for it.
• Transform. F_v(z) = wp_common.F_even_acb(v, L, z) = 2 sin(zL/2) [v_0/z + sum_k sqrt2 v_k z/(z^2 - omega_k^2)], omega_k = 2 pi k/L. F_v is entire, even, and real on the real line.

2. The CvS hypotheses, decided per (kind, c) for the band-N matrix

• H1 simple. A ball LDL^T of A - sI, at a shift s with RQ(x) < s, is conclusive with exactly one negative pivot. Then lambda_1 < s < lambda_2.
• H3 isolated. In band N the spectrum is finite, so isolated is H1 with gap at least s - RQ(x). Isolation of the continuum operator is not decided here.
• H2 even (in CvS's sense: the lowest eigenvector of the whole form, both sectors, is even). Holds if the ball LDL^T of A_odd - s_e I is conclusive and positive definite, s_e the upper end of the even lambda_1 bracket. Fails if A_odd - s_l I has a negative pivot, s_l the lower end of that bracket. Otherwise undecided.

3. A derivation written before computing (ordinary argument, unreviewed)

The band-N even space V_N is exactly the set of transforms F(z) = 2 sin(zL/2) p(z^2) / (z q(z^2)) with q(w) = prod_{k=1..N} (w - omega_k^2) and p any polynomial of degree at most N (the map v -> p is a linear bijection, fixed by the values of p at 0 and omega_k^2). For a root w_0 of p, the multiplier (z^2 - conj(w_0)) / (z^2 - w_0) maps V_N (complexified) to itself, and |F(t)| is unchanged for real t. The form is Q(f) = W(f * f~) (../numerics/RESULTS.md s0, Fact A), which depends on F only through |F(t)|^2 on the real line; the norm is unchanged by Plancherel. So if the lowest even-sector eigenvalue is simple, the multiplied function is a second minimiser unless w_0 is real, and every zero of F_v is real or purely imaginary. This is the CvS multiplier argument carried to V_N, and it needs only H1, not H2. Its consequence, tested as S1: in every rectangle below where H1 holds, the argument-principle count equals the number of real sign changes. A failure would be a defect in the code or in this argument, and would be reported as such.

4. The counter

• Eigenvector enclosure. Davis-Kahan from the ball residual eps = ||Ax - RQ(x)x|| / ||x|| and the gap g = s - RQ(x) (H1's shift): ||v - x/||x|| || <= sqrt(2) eps / g, with the sign of v chosen so that <v, x> > 0. Propagated into F by Cauchy-Schwarz, |F_v(z) - F_{x/||x||}(z)| <= eta ||a(z)||_2, a_k(z) the basis transforms.
• Argument principle (Arb). On each boundary segment of a rectangle R = [a, b] x [-h, h], a Taylor model of F at the segment centre (exact centre, coefficients in balls) plus a Cauchy remainder from an explicit bound on |F| in a disk, plus the eigenvector term. A segment is accepted when the model excludes 0 and keeps F inside a sector of half-angle below pi/4; otherwise it is bisected. The arg changes come from endpoint values of F_even_acb; the winding number must be a ball containing exactly one integer, n_AP. The vertical sides avoid every omega_k by at least 1/32 and avoid real zeros: a side is moved inward in steps of 1/512 until both hold and the count closes. The sides used are reported.
• Real-axis sign changes. F_v at exact rationals of step 1/512 on the nominal interval, each sign from a ball that excludes 0 after the eigenvector term; n_SC = number of sign changes. Each bracket is bisected to width 2^-40 for the location. If n_AP = n_SC, every zero in R is real, simple and inside one bracket.
• Second precision. The whole chain (assembly, eigenvector, counts) is rerun at a second precision: 640 bits for dh, epstein, dedekind (primary 400), and 1300 bits for zeta (primary 900; LDL steps at 3200).
• Independent float route. mpmath only, sharing no code with the Arb route below the assembly formulas: galerkin.Truncation matrices (the lab's second assembly), mpmath inverse iteration seeded from a float64 spectrum, the transform in the sinc form F_v(z) = L [v_0 sinc(zL/2) + sqrt2 sum_k (-1)^k v_k z sinc((z - omega_k)L/2)/(z + omega_k)], sign changes plus a sampled winding number with an integrality guard. Working precision: dps 60 (dh, epstein, dedekind), dps 320 (zeta).
• Planted control for the counter. An element of V_N built from a chosen p with known roots in the same rectangles (real roots, a planted non-real pair at gamma_1 +- 0.3 i, and a root within 0.02 of some omega_k) must be counted exactly by the same code. Done in test_ghost.py.

5. Windows and rectangles

The Epstein gap and the on-line zeros of zeta_K = zeta(s) L(s, chi_-23) in [12, 21] are located after this commit: Epstein by sign changes of zeta.epstein.Z_epstein (dps 40, step 1/100, then bisection), zeta_K both from Z_(1,1,6) + 2 Z_(2,1,3) and from mpmath's zeta and Dirichlet L zeros. An F-zero x is accounted for by an on-line zero gamma of its own function if |x - gamma| <= 0.15. A ghost is a real zero of F_v within 0.3 of the off-line ordinate (gamma_1 for dh and zeta, gamma_E for epstein and dedekind) that no on-line zero accounts for.

6. Prediction and decision rule

P (the campaign's prediction, recorded here as given). "A real ghost zero within 0.3 of 85.699 appears at or before the even-sector crossing."

• Confirmed if for at least one c in {29, 30, 61/2, 245/8} H1 holds and the enclosure-carrying sign-change count of F_v on [gamma_1 - 0.3, gamma_1 + 0.3] is at least 1 (dh, N = 128).
• Refuted if for all four of these c, H1 holds and n_AP(R_ghost) = 0.
• Undecided otherwise (H1 fails, or a count does not close).
• Reported with it: the first c in C with a ghost, whether it persists at 31, 63/2 and 47, and its location and distance from gamma_1 at each c.

7. Secondary expectations (exploratory, graded as such)

• S1 (reality, from s3). n_AP = n_SC in every rectangle where H1 holds.
• S2 (ghost and Q1, derived from the stored Q1 values). To first order at the real zero x_g nearest gamma_1, Q1 = (4/L) Re F_v(z_1)^2 ~ (4/L) F_v'(x_g)^2 [(gamma_1 - x_g)^2 - delta_1^2], so sign(Q1) = sign(|gamma_1 - x_g| - delta_1) whenever ||gamma_1 - x_g| - delta_1| > 0.05. A negative Q1 means a ghost within delta_1 = 0.3085 of gamma_1.
• S3 (zeta control). At every c in C: n_AP(R_gap) = 0, no ghost, and in R_ctx exactly one real zero of F_v within 0.01 of each zeta zero in [80.5, 90.5] and no other zero.
• S4 (Epstein, the rival with its own off-line zero). A ghost within 0.3 of gamma_E at every c in C where the Epstein even lambda_1 < 0 (all c >= 30 here); at c = 29 reported without expectation.
• S5 (Dedekind control). No ghost at any c in C, and every real zero of F_v in R_ctx accounted for by an on-line zero of zeta_K.
• S6 (lesion). Replace the DH matrix A by A - M_Q1, where M_Q1 = (4/L) Re(a(z_1) a(z_1)^T) is the matrix of the off-line quadruple term (its value at the stored ground states is the zero_side Q1 of ../numerics/grid_dh_N128.json, checked). At every c where the unlesioned F_v has a ghost, the lesioned ground state has no real zero within 0.3 of gamma_1, or its zero nearest gamma_1 moved by more than 0.1. A sham lesion of the same kind at an on-line ordinate (the DH zero 89.43924..., matrix (2/L) a(gamma) a(gamma)^T) is run alongside and reported without expectation.
• S7 (precision response). Identical counts at the second precision, zero locations agreeing to 1e-20 (dh) and 1e-40 (zeta); the float route gives the same counts and locations to 1e-8.
• Exploratory scan. dh at c in {14, 18, 22, 25, 27, 28} (N = 128), ghost present or not, to see where it first appears below 29. No expectation.

8. Not claimed

The ghost is a statement about the band-128 ground state, not about the continuum minimiser. No convergence in N is claimed. H1 to H3 are decided for the band-N matrix only. Nothing here bears on RH for zeta; DH, Epstein and zeta_K enter as rivals and controls.