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    "content": "# D-020 \u2014 E-010 analysis: zeros of Suzuki's W(a,\u03b8;z) \u2014 lattice law, no \u03b3-locking yet\nDate: 2026-08-17 | Session: 003 | Task: 3.4c cont. | Status: complete\nInputs: E-010 (numerics), D-018/D-019 (scoping/analysis), Suzuki\narXiv:2606.09096v1 (eq. 1.11, Thm 1.5, Cor 1.6).\n\n---\n\n## 1. What E-010 computed\nDeficiency vectors v_\u00b1 from the E-009b-validated Galerkin machinery:\n(T_a v_\u00b1)(x) = e^{\u00b1x} solved as bordered mean-zero systems\n(Q \u2212 \u03bbM)c = b_\u00b1; W(a,\u03b8;z) per eq. 1.11 using v\u208b(x) = v\u208a(\u2212x)\n(symmetry verified to 7e-14; T-norms agree to 1e-13; \u03b8=0 real gauge\nexact). Zeros on (0, 60] by sign-change + brentq with direct W\nevaluation. Configs: a \u2208 {2,4,6,7}, \u03b8 \u2208 {0, \u03c0/2, \u03c0}, \u03bb \u2208 {\u22120.5,\u22121,\u22122},\nh-convergence at a=4.\n\n## 2. Findings\n\n**(F1) Lattice law.** Zeros of W(a,\u03b8;z) form a near-perfect lattice:\n  z_j(a,\u03b8) \u2248 s(a)\u00b7j + \u03ba(a,\u03b8),   s(a) = (\u03c0/a)(1 + \u03c1(a)),\nwith \u03c1(a) \u2248 0.0063 (a=2), 0.0014 (a=4), 0.00066 (a=6), 0.00079 (a=7)\n\u2014 a g-induced renormalization of the free i d/dx lattice (spacing \u03c0/a),\ndecaying roughly like C/a\u00b2 with C \u2248 0.02\u20130.03. \u03b8 shifts the offset\n\u03ba(a,\u03b8) as expected for a boundary-phase condition (\u03b8=0: \u03ba\u22480;\n\u03b8=\u03c0/2: +0.20; \u03b8=\u03c0: +0.34 at a=4 \u2014 consistent with \u03ba sweeping one full\nspacing over \u03b8 \u2208 [0, 2\u03c0)).\n\n**(F2) \u03bb-independence.** Zeros at a=4 move by \u2264 2e-4 between\n\u03bb \u2208 {\u22120.5, \u22121, \u22122} \u2014 consistent with the abstract expectation that the\nextension spectra do not depend on the arbitrary shift \u03bb (small drift\nattributed to discretization; h-convergence: z\u2081 rel change 4e-4 between\nh = 0.01 and 0.005).\n\n**(F3) No \u03b3-locking at a \u2264 7.** Lattice residuals (z_j \u2212 fitted lattice)\nare concentrated at LOW z (|res| max at z \u2272 9) and are smooth; at the\npositions j \u2248 a\u03b3\u2081/\u03c0 there is no enhanced structure for a = 4, 6, 7\n(a=2 shows a single 6e-2 blip at j\u22489 \u2248 a\u03b3\u2081/\u03c0 \u2014 likely a small-a\ntransient; does not persist). For \u03b8 \u2208 {0, \u03c0/2, \u03c0}, no zero is attracted\ntoward \u03b3\u2081 = 14.13, \u03b3\u2082 = 21.02 within the measured range.\n\n**(F4) Density bookkeeping (important for interpreting Cor 1.6).**\nW(a,\u03b8;\u00b7) has ~2aT/\u03c0 zeros in [0,T] \u2014 far MORE than z\u00b2\u03be(\u00bd\u2212iz)/\u03be\u2032(\u00bd\u2212iz)\n(~T log T/2\u03c0). Since e^{\u03c6} cannot remove zeros unless \u03c6 has logarithmic\nsingularities, the \u03c6(a,z) of Cor 1.6 must effectively act as a canonical\nfactor cancelling all but the zeta-aligned zeros (or moving them off\ncompacts). Consequence for numerics: raw zero positions of W are NOT\nthe right observable for the Cor 1.6 limit; the right one is a\nnormalized ratio such as\n  R_a(z) := W(a,\u03b8;z) / W(a,\u03b8;z\u2080)  (or log-derivative W\u2032/W)\nevaluated at z = \u03b3_j, tracking whether R_a(\u03b3_j) \u2192 0 with a for the\nRIGHT \u03b8(a). Requires: a candidate \u03b8(a) rule (Q-S3 remains the crux).\n\n## 3. Interpretation / honesty notes\n- E-010 validates the FULL pipeline (deficiency vectors, W, real zeros)\n  against every internal check \u2014 the machinery is sound and reusable.\n- The negative content (no \u03b3-locking at a \u2264 7, fixed \u03b8) is EXPECTED\n  in hindsight given F4: the zeta content of W is in the fine structure\n  of the lattice perturbation and in the \u03b8-dependence, not in raw\n  positions at small a.\n- The a=2 blip near \u03b3\u2081 is noted but discounted (no persistence at\n  larger a; would need \u03b8(a)-tracking to confirm).\n\n## 4. Next steps on the Suzuki line (queued, prioritized)\n1. **Q-S3 attack (the crux):** find a principled \u03b8(a). Candidates:\n   (i) match the normalization of Cor 1.6 directly \u2014 demand\n   e^{\u03c6}W \u2192 z\u00b2\u03be/\u03be\u2032 in L\u00b2 on a compact set and READ OFF \u03b8(a), \u03c6(a,z)\n   (least-squares in \u03b8, \u03c6 = polynomial in z of growing degree \u2014 the\n   density bookkeeping F4 says \u03c6 must supply the cancellation);\n   (ii) consult Suzuki's \u00a77\u20138 for any canonical \u03b8 prescription;\n   (iii) Kim et al. (2607.24830) may have a specific extension choice \u2014\n   check their numerical setup.\n2. Zero-tracking diagnostic: follow the zero nearest \u03b3\u2081 across\n   a = 2..12 for several \u03b8 rules; measure |z(a) \u2212 \u03b3\u2081| decay.\n3. If (1) yields \u03b8(a): test Cor 1.6 convergence directly on compacts.\n\n## 5. Status\nE-010 complete and archived. Suzuki line state: machinery validated;\nlattice law measured; the path to Cor 1.6 runs through Q-S3 (\u03b8(a)).\nNo obstruction \u2014 this is an OPEN, well-posed next problem, not a wall.",
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