D-001 — Master Equation & Tool Index (living document)

Date: 2026-08-17 | Session: 001 | Task: 0.1 | Status: in-progress (grows with project)

Purpose (per NGUYEN.md §4A.4): the single living index of ALL equations and tools available to the project. Every newly discovered tool — from any source — is added here so no session ever rediscovers it. Each entry is annotated with its role in the RH attack. Relevance claims are tagged [RIGOROUS] / [HEURISTIC] / [SPECULATIVE].

External references: Physics Hypertextbook ToC https://physics.info/ (equation-level reference, consult on demand per §4A.3).

PART I — QUANTUM FORMALISM (from Notes.txt; Track B core)

Q1. Born rule — ρ = |ψ|², ∫|ψ|² dV = 1

• Role: normalization of states. In Hilbert–Pólya modeling, candidate eigenfunctions must be square-integrable on the chosen configuration space — the choice of L²-space IS the choice of spectral problem. [RIGOROUS]

Q2. Time-dependent Schrödinger equation — iħ ∂ψ/∂t = Ĥψ

• Role: defines unitary evolution. Contrast with the de Bruijn–Newman flow, which is a HEAT (dissipative) flow ∂_t f = ∂_x² f driving zeros toward/away from the real line. Unitary vs. dissipative evolution is the structural difference between Track B (spectrum) and Track C (Λ = 0) attacks. [RIGOROUS analogy]

Q3. Time-independent Schrödinger equation — Ĥφ = Eφ

• Role: THE Hilbert–Pólya template. RH ⟺ ∃ self-adjoint Ĥ with spectrum {γ_ρ} (zero ordinates). Every Track B candidate is an instance: specify Hilbert space + self-adjoint Ĥ, compute spectrum, compare to {γ_ρ}. [RIGOROUS]

Q4. Position/momentum operators — p̂ = −iħ∇, x̂ = x·

• Role: building blocks of the Berry–Keating Hamiltonian Ĥ_BK = ½(x̂p̂ + p̂x̂) = −iħ(x d/dx + ½), the dilation generator. Its formal "eigenfunctions" x^(−1/2−iE/ħ) on the half-line connect directly to the Mellin transform and to ζ(½+iE) — the critical line appears as the L²-unitarity axis of the dilation group. [RIGOROUS connection; spectrum problem itself is HEURISTIC — needs boundary conditions, see Phase 2.2]

Q5. Canonical commutation relations — [x̂ᵢ, p̂ⱼ] = iħδᵢⱼ

• Role: algebraic foundation. Any operator model importing CCR structure inherits Stone–von Neumann rigidity (unique irreducible representation), which constrains candidate Hamiltonians. The Weyl (exponentiated) form underlies Gabor frames — see Q6, Q12. [RIGOROUS]

Q6. Heisenberg uncertainty — ΔA ΔB ≥ ½|⟨[Â,B̂]⟩|; Δx Δp ≥ ħ/2

• Role: direct analogue of the paper's critical-density Gabor frame (Lemma 2.1, h = 2π/L): time-frequency localization limits. ALSO the structural reason for the bandwidth-one ceiling ≈ 0.682: Fourier-support restrictions are uncertainty constraints (cf. Benedicks-type uncertainty principles in harmonic analysis). Track A ceiling analysis should mine uncertainty-principle literature. [HEURISTIC bridge, worth formalizing]

Q7. Expectation values — ⟨A⟩ = ⟨ψ|Â|ψ⟩

• Role: the Weil form W(f,g) = Σ_ρ m_ρ f̂(γ_ρ) ḡ̂(γ_ρ) is formally a sum of spectral weights — if W were ⟨f|ρ(Ĥ)|g⟩ for a self-adjoint Ĥ, RH follows. Track B central task: realize W as a genuine spectral quadratic form. [RIGOROUS as equivalence — Weil positivity ⟺ RH; realization is the open problem]

Q8. Probability current / continuity equation

• Role: conservation laws for zero dynamics under flows (de Bruijn–Newman, heat flows). Zeros are conserved under continuous deformations except at collisions — a "current" picture for tracking off-line motion. [HEURISTIC]

Q9. Angular momentum — so(3) algebra, [Lᵢ,Lⱼ] = iħεᵢⱼₖLₖ

• Role: representation-theoretic toolkit. No direct known RH role; kept for completeness. Possible relevance via representation theory of SL(2,ℝ) (Mellin/dilation group) in de Branges theory. [SPECULATIVE]

Q10. Pauli matrices / spin-½ — 2×2 Hermitian units

• Role: linear-algebra toolkit for the paper's 2×2 signature blocks: on-line zeros → rank-one positive blocks P; off-line pairs {ρ, 1−ρ̄} → signature (1,1) blocks Q. Pauli basis diagonalizes any 2×2 Hermitian — useful for inertia computations in the rank–trace inequality. [RIGOROUS utility]

Q11. Hydrogen atom — E_n = −13.6 eV/n²

• Role: example of a completely solved spectral problem; template for "guess the potential from the spectrum" inverse problems. No direct RH role identified. [SPECULATIVE — keep for inverse-spectral intuition]

Q12. Quantum harmonic oscillator — E_n = ħω(n+½), Hermite–Gaussian eigenfunctions

• Role: HIGH VALUE. Hermite functions are eigenfunctions of the Fourier transform — the natural self-dual test-function basis for Weil forms (self-dual cones in CGdL20-style SDP, Track A.1.2). Gaussian ground state = extremizer of uncertainty (Q6) and of many Fourier inequalities. Also: oscillator representation of SL(2,ℝ) connects to automorphic forms. [RIGOROUS utility]

Q13. Time-evolution operator — Û = T exp(−(i/ħ)∫Ĥ dt); Heisenberg picture

• Role: framework for flow-based attacks (Track C de Bruijn–Newman): study dγ/dt of zero ordinates under deformations of Ĥ or of the window. Heisenberg vs. Schrödinger picture = evolve operators vs. evolve states — dual viewpoints for moment methods (evolve G̃ vs. evolve test functions). [HEURISTIC bridge]

Q14. Ehrenfest theorem — classical limit of expectations

• Role: gateway to SEMICLASSICS — the deepest physics–RH bridge. Gutzwiller trace formula: quantum spectrum ↔ classical periodic orbits. For the Berry–Keating system, periodic orbits are (conjecturally) at lengths log p — the primes. The Weil explicit formula IS formally a Gutzwiller trace formula with primes as orbits and zeros as spectrum. This is the master heuristic of Track B; making it rigorous = Hilbert–Pólya. [HEURISTIC — the central open problem]

PART II — NUMBER-THEORETIC TOOLS (to be populated in Phases 0.2–0.4)

Placeholder. Will index: Weil explicit formula, rank–trace inequality, Montgomery second moment, window theory R(ψ), Gabor compression Lemma 2.1, Sylvester inertia, SDP self-dual cone machinery, pair-correlation results, Rudnick–Sarnak moment bounds, Davenport–Heilbronn/Epstein counterexamples.

NT-1. Weil explicit formula — MACHINE-VALIDATED conventions (Sess. 003)

For h even, sufficiently decaying, ĥ(ξ) = ∫ h(x) e^{iξx} dx: Σ_ρ ĥ(γ_ρ) = ∫ h(x)·2cosh(x/2) dx − 2 Σ_{n≥2} (Λ(n)/√n) h(log n) + (1/2π) ∫ [Re ψ(1/4+iτ/2) − log π] ĥ(τ) dτ (ψ = digamma; zero sum over both ±γ.) Gaussian test h = e^{−x²/2}: RHS − LHS = 8.9e-16 (E-009 diagnostics, D-019 §1b). CAUTION: the arXiv HTML of Suzuki 2606.09096 renders the prime weight as Λ(n)/n — with our ĥ conventions the validated weight is Λ(n)/√n.

NT-2. Suzuki screw function (arXiv:2606.09096 eq. 1.3) — VALIDATED

g(t) implemented/validated in E-009a (D-019 §1a); eq. 2.2 near-origin expansion confirmed to 25 digits. Operator form (eq. 2.9): Weil form Q_W^a(u,v) = ∫∫ g(x−y) u′(x) v̄′(y) dxdy on H₀¹(−a,a); Galerkin realization validated against the zero side to 4.5e-6 (E-009b, D-019 §1c). Bottom eigenvalue λ_a measured (D-019 F1). Zeta-aligned operator is 𝒟̄_{a,θ} (first-order, in T_a-inner-product), spectrum = zeros of W(a,θ;z) (eq. 1.11, quoted in D-019 F2).

PART III — EXTERNAL DISCOVERIES (web research log, per §4A.3)

2026-08-19 — R7 resources atlas registered (research/resources/, 43 notes)

User-provided RH-equivalence atlas; read in full session 007, formally registered as manifest resource R7 with per-note index + project-use map: research/resources/index.md. Key direct inputs: Schoenfeld 8π + Büthe 2018 (→D-037), Hasanalizade–Shen–Wong S(t) bound (→D-036 alt constants), Li/Lagarias notes (→tasks 3.5–3.8), Hilbert–Pólya/Berry–Keating (→task 2.2). Criteria atlas = standing cross-pollination reservoir (§4A.5).

2026-08-19 — Domain sweep (§4A.5): the residual-equilibrium wall

Walls listed: (i) linear-K equilibrium / residual → Z/(2π) (D-042/44); (ii) WH factorization of ĝ with real poles (D-035); (iii) λ_a unresolvable (G-004). Sweep result:

• Christoffel-function universality (Levin–Lubinsky line; Eichinger–Lukić–Woracek arXiv:2409.18045, 2024): sine-kernel asymptotics ⟺ unique tangent measure = Lebesgue. Our measured K-averages (plateau 2a free law + valleys at γ's) match the universality picture for a limit measure = ac background + atoms. CONSOLIDATION ⟺ limit spectral measure PURE ATOMIC ⟺ tangent measure Dirac everywhere — a spectral-classification question for the underlying canonical system. This REFRAMES the equilibrium as a theorem-class phenomenon with an exact criterion.
• RHP-with-poles / dressing method (integrable systems): standard machinery for pole data — candidate for the WH factorization (W-3).
• Prediction theory (Kolmogorov–Wiener–Krein): ac part ⟹ nondeterministic — the same ac-vs-atomic dichotomy.
• Beurling–Malliavin/Logvinenko–Sereda: candidate for λ_a lower bounds (G-004 revival condition (i)). Next opening (queued): determine the canonical system underlying ℋ(T_a) and test spectral purity (the consolidation criterion).

2026-08-17 — Task 1.4 literature sweep (arXiv API)

• BGSTB25 (arXiv:2501.14545): pair correlation I — all beyond paper's inputs conditional on narrow-box hypothesis. No support>1 unconditional.
• GLSS25 (arXiv:2503.15449): PCC ⟹ 100% simple + on-line (horizontal multiplicity). Conditional on PCC.
• GS25 (arXiv:2511.20059): expository survey, conditional.
• BGSTB24 (arXiv:2306.04799): unconditional Montgomery theorem (= paper's input); support ≤ 1 only.
• Liu–Williams–Zaharescu (arXiv:2407.14656): unconditional pair correlation for non-CM newform L-functions in shifted (macroscopic) ranges — different object, not applicable to zeta bandwidth>1.
• Goldbach exceptional sets: Pintz 2018 E(X) ≪ X^0.72; Zhao 2025 E(X) ≪ X^{7/10} (ineffective); Montgomery–Vaughan 1975 classical power saving. Generic lags only; nothing at short lags h ≲ X/T.
• Survey: Bhowmik–Grimmelt 2026 (explicit major-arc formulas; traces power saving to MV75/Pintz18). CONCLUSION logged in D-012: no unconditional short-lag prime-pair control exists; support-(1+δ) walled at lag scale h ≲ X/T.

Candidate fields for future domain sweeps (§4A.5)

• Moment-problem theory / orthogonal polynomials (Christoffel language)
• Determinantal point processes (Soshnikov, Johansson)
• MaxEnt duality (certificate ceilings as dual values)

2026-08-17 — Task 3.1 literature sweep (arXiv; Track C frontier)

• Suzuki (arXiv:2606.09096, 2026): screw-function unification of Weil-form theory (Yoshida/Bombieri/Connes–Consani/CCM), all unconditional; conjectures Hilbert–Pólya operator as a→∞ limit of self-adjoint nonlocal operators on [−a,a]. ← Track C↔B bridge.
• Groskin (arXiv:2607.02828v3, 2026): finite Guinand–Weil dictionary (truncated values = exact zero sums) + archimedean tail budget B_T ~ (2N+1)ρ log T/(π²T), ρ = 2π/log c; certification rule with indeterminate zone [−B_T, 0). Verified over 512 zeros.
• Kim et al. (arXiv:2607.24830v2, 2026): numerical Suzuki operator; lowest eigenvalue positive, superexponentially decaying; off-line zero injection → exponential blow-up criterion.
• Connes (arXiv:2602.04022, 2026): survey; primes <13 extremization approximates first 50 zeros; approximants proved on-line. Analysis + plan: D-014.

2026-08-17 — Session 003: Suzuki full-text re-fetch (arxiv.org/html/2606.09096v1)

Taken: eq. 1.11 (W(a,θ;z) via deficiency vectors v_±, (T_a v_±)(x) = e^{±x}, deficiency indices (1,1), Lemma 6.2); Thm 1.4 (λ_a = log(1/a)+O(1), small a); Cor 1.6 exact statement (θ(a), φ(a,z) choice; no distinguished θ given); 𝒟̄_{a,θ} = extensions of first-order 𝒟_a in ℋ(T_a) (NOT of B_a); Weil functional rendering (Λ(n)/n — superseded by our validated NT-1 conventions). Use: D-019. Note: Kim et al. 2607.24830 (3.1 sweep) reported "lowest eigenvalue positive, superexponentially decaying" — our F1 λ_a-cliff (D-019) independently reproduces that behavior.