Spectral Constraints on Schrödinger Encodings of the Riemann Zeros: Eight No-Go Results and the Surviving Framework

Title: Diagnostic Constraints on Schrödinger Encodings of the Riemann Zeros: Eight Tests, Eliminations, and the Emergent Fractal Half-Line Framework Files for PDF merge: Riemann_Zeros_Spectral_Constraints.pdf (main paper) Appendix_D_Emergent_Framework.txt (new Appendix D) Authors: Frederic David Blum¹ and Claude 4.6² (AI Numerical Assistant)¹ Independent Researcher, TLV, Israel. dvid75113@gmail.com² Claude 4.6 AI (numerics and redaction assistance under human oversight) Abstract The Pólya–Hilbert conjecture proposes that the non-trivial zeros of the Riemann zeta function are eigenvalues or spectral data of a self-adjoint operator. We investigate this programme by constructing an explicit one-dimensional Schrödinger operator $H_\varepsilon = -d^2/dt^2 + v_\varepsilon(t)$ whose potential is built from the logarithmic derivative of the Riemann xi function via a supersymmetric (Miura) transformation: $v_\varepsilon(t) = g_\varepsilon(t)^2 + g_\varepsilon'(t)$, where $g_\varepsilon$ is a Gaussian-mollified version of $-\mathrm{Re}[\xi'/\xi(1/2+it)]$. We subject this operator to a comprehensive battery of eight independent diagnostic tests — spectral trace comparison (HTI), Krein–Birman scattering trace, Numerov integration (two-point and least-squares Jost extraction), Born/Volterra expansion, Calogero variable-phase method, Dirac comb optimisation, and full transfer matrix with unitarity verification — yielding uniformly negative results for encoding the zeros in discrete spectrum or scattering data on $\mathbb{R}$. This systematic elimination yields seven principal clarifications and applications of known results: (1) First-order no-go (Thm. 1): Gauge equivalence to the free operator eliminates $H = -i d/dt + V(t)$ and naive $xp$ quantisations (§3).(2) Unitarity bound: $ |A(k)| \geq 1 $ rules out real-axis Jost zeros $F(\gamma_n)=0$ for bilateral real potentials, refuting Conjecture 6.2 (§5.7, App. C).(3) Coupling pincer (Obs. 2): $|v_\varepsilon|_1=355 \gg 1$ (non-perturbative) yet $|R(k)| \sim 0.001$ (quasi-transparent), trapping mollified potentials (§6).(4) Mollification dilemma: $\varepsilon>0$ smooth/empty; $\varepsilon=0$ singular (§8.2).(5) Artefact diagnosis: Two-point "5/5 match" debunks to $P \approx 0.34$ via $\det=2i\sin(k\Delta t)$ (§5.4, App. B).(6) Diagnostic protocol (Table 3): Reusable six-test filter for Hilbert-Pólya candidates (§8.1).(7) Surviving architecture: Half-line fractal $d=1.5$ (Wu-Sprung [6,7]); PT/adèlic open (§7, Table 2). These negatives form a "negative photograph" (§App. D): emergent mold of fractal diffraction resonances ($d=1.5$), hidden/broken SUSY (L/R asymmetry), Connes absorption (pincer-resolved), and topological phase ($\varepsilon \to 0$). RH-neutral yet constraining: proofs must fit fractal half-line $\otimes_p H_p$. Our survey advances Hilbert-Pólya by clarifying obstructions, quantifying failures, and mapping viable paths (Marchenko/adèlic). Code: GitHub [link upon publication]. Keywords: Riemann Hypothesis; Hilbert–Pólya conjecture; Schrödinger operator; Jost function; inverse scattering; supersymmetric QM; unitarity; coupling pincer; mollification dilemma; fractal potential (d=1.5); Wu-Sprung; Connes adèlic; diagnostic protocol; spectral gaps; SUSY breaking; topological phase; Numerov; Calogero; transfer matrix AMS Subject Classifications: 11M26 (Nonreal zeros of $\zeta(s)$; RH); 81U05 (2-body scattering); 34L25 (Inverse scattering); 81Q60 (SUSY QM); 35P25 (Scattering PDEs); 28A80 (Fractals) PACS: 02.10.De (Number theory); 03.65.Nk (Scattering); 02.30.Zz (Inverse problems); 05.45.Df (Fractals) Corresponding author: Frederic David Blum, dvid75113@gmail.com