Configuration Space Temporality: A Unified Theory of Time as Energy-Driven State Transition in Configuration Space — with the Transition Potential, Energy Unification via Partial Traces, Cascade Dynamics, Derivative Structure, Energy Propagation Equa

Configuration Space Temporality — Complete Research Archive v33Corrections, a Resolution Law, and the foundations of CST among timeless formulations Frédéric David Blum · Catalyst AI Research · Haifa, Israel · DOI: 10.5281/zenodo.18859602 · October 2026Computations and drafting assisted by Claude (Anthropic). What is CST?Configuration Space Temporality (CST) is a framework in which time is not a background parameter: it emerges as the accumulated cost of energy-driven transitions between configurations of a physical system. Three axioms (configuration uniqueness, energy-driven transition, bounded transition rate) define a tensor of transition G on configuration space (the Bures metric on quantum configurations, with the Umegaki relative entropy as potential), a transport distance d_CST satisfying the Carlen–Maas axioms of a quantum Wasserstein distance, the Witten Laplacian H_W whose thermodynamic limit is a Lindblad generator, and the cascade number κ = ΔE·Δτ/(ħ ln 2). STATUS — October 2026Every claim in this archive that the Riemann Hypothesis has been proved, reduced to a single assumption, or approached through the Bost–Connes Hamiltonian, the Fisher–Rao action functional, the rigidity matrix or the confinement constant α is WITHDRAWN. The physical framework is kept, placed on its exact foundations, and given one law. What's new in v33 Corrections. A re-audit with every number recomputed found four errors. (i) The Fisher pole identity |I(s)|·|s−ρ|² = 1 holds at every simple zero of every analytic function, on or off the critical line; it is not a quantization condition and carries no information about RH. (ii) The rigidity matrix R has an exact zero eigenvalue (its rows sum to zero), so α = μ₁/μ₀ is undefined; the collinear configuration is a transverse maximum of the log-gas energy, not a minimum; and R is built from ordinates already on the line, so no property of R can be equivalent to RH. (iii) Exponential dominance fails: μ₁/μ₀ grows only from 10^6.4 (N = 45) to 10^7.8 (N = 200), α(N) → 0, and GUE, Poisson and evenly spaced control points give the same values as the zeros (the tiny eigenvalues come from the cosine basis, not from arithmetic). (iv) The Bost–Connes Hamiltonian has spectrum {log n}; Assumption A, Assumption B and the "zeros as resolvent poles" claim are withdrawn, and the CMP v25–v27 "complete proof chain" is withdrawn in full. Of 17 RH-related results: 3 hold as stated, 3 hold but are trivial or known, 1 is falsified numerically, 1 stays open (Li positivity, equivalent to RH), 9 are withdrawn. The full audit, with the status of every claim and the superseded files listed, is the document "CST/FDBC Archive v33 — Corrections and a Non-Circular Weil-Form Test". Foundations. CST is placed explicitly on the three established timeless formulations of physics: Jacobi geometry and Barbour's relational dynamics (G is a Jacobi metric, CST time is the Jacobi arc length, i.e. the Maupertuis action, and κ_cl = S₀/(ħ ln 2)); Wheeler–DeWitt and the Page–Wootters mechanism (time as clock correlation); and the thermal time of Connes–Rovelli (time as the modular flow of the state — the correct use of the KMS machinery in CST). These formulations are empirically equivalent to their timed counterparts; a new law must therefore live where state-based and timed descriptions differ. The Resolution Law (new). For a quantum trajectory ψ(t) = e^{−iHt/ħ}ψ of duration Δτ whose energy distribution is flat on a band of width Ω_E, the number of modes of the trajectory distinguishable at level ε isd_ε = Ω_E Δτ/h + (1/π²) ln(Ω_E Δτ/4ħ) · ln((1−ε)/ε) + o(ln c),and no mode beyond twice the Slepian parameter c = Ω_EΔτ/4ħ is distinguishable above e^{−1.5c}. The leading term is the Landau count (the action budget in units of h), of the same kind as the Margolus–Levitin and Mandelstam–Tamm speed limits and identifying the cascade number (κ = (2π/ln 2) × Landau count); the second term is a universal logarithmic "soft horizon" (Landau–Widom plunge); the cutoff is a rigorous non-asymptotic bound. The law is proved as a theorem of time–band limiting — the Gram operator of a quantum trajectory is Slepian's time–band limiting operator with the energy distribution as symbol — with two-sided bounds for bounded energy densities, verified numerically (prolate eigenvalue counts at c = 10, 30, 100), and given a falsification protocol on a many-level system (cavity mode or quantum simulator) through the survival amplitude. It replaces the earlier prediction P2 of a sharp reversibility threshold at κ = 1, and it is the first prediction of CST stated with a theorem behind it. A tested negative. The conjecture that the cascade number is tied to Barbour–Koslowski–Mercati shape complexity by a universal power law was tested on E = 0 three-body solutions: the Janus structure is reproduced, but the exponent is not universal (0.8–1.9); the CST clock is the ephemeris time of the bound pair. Recorded so that it is not tried again. Companion workScale-Resolved Weil Positivity (preprint, DOI 10.5281/zenodo.23229271; code: github.com/davidangularme/scale-resolved-weil-positivity): the positive mathematics that came out of the audit — a prime-side study of the truncated Weil form, with certified positivity on the window [1/2, 2] (margin enclosed between 1.38×10⁻¹³ and 7.6×10⁻¹³; an independent certification, at L = log 2, of a theorem of X. Zhu, arXiv:2608.24827, valid for L ≤ 0.8), certified rigidity of the prime data, the detection and reconstruction laws, and the structure of the minimiser as Riemann's Φ times a window. Nothing there bears on RH beyond the finite scales treated. Validated and untestedValidated: the H_W spectral diagnostic on 768 qubit pairs of six IBM Eagle r3 processors, r = 0.365 after projecting out calibration parameters (a diagnostic correlation; it does not test the Resolution Law). Untested: P6 (no dark-matter particle), P9 (gravity as collective mode), P16, and the Resolution Law itself. Withdrawn: P2 (cascade criticality at κ = 1), IG-P1 to IG-P5. Files added in v33CST_v33_Corrections_and_a_Resolution_Law.pdf (the single document of this version) · cst-fdbc-archive-v33-corrections.pdf (the full audit) · v33_cst_code.zip (prolate.py, lw_check.py, janus.py and results). Earlier files remain for the record; those superseded are listed in the audit document: CMP_Paper_FDBC_v25–v27, FDBC_v2_CMP_Submission, FDBC_v2_Complete_Proof_Chain, FDBC_v2_Proof_Chain_CORRECTED, FDBC_v2_Assumption_B_PROVED (both), FDBC_v2_Spectral_Gap_PROVED, FDBC_v2_No_Gap_Final, Information_Geometric_Confinement_Riemann_Zeros_Blum_2026, blum-claude-weil-minimiser-2026 (its table stands, its conclusions do not). The submission CIMP-D-26-00460 is being withdrawn. Previous versionsv33 (current): corrections and withdrawals; foundations; the Resolution Law; tested negative on complexity. v32: information-geometric confinement (withdrawn in v33). v31: exponential isolation of the Weil minimiser (table stands, conclusions withdrawn). v30: Fourier-side ΨDO identification. v29.1–v25: see the archive. CitationBlum, F. D. (2026). Configuration Space Temporality: Complete Research Archive v33 — Corrections, a Resolution Law, and the foundations of CST among timeless formulations. Zenodo. https://doi.org/10.5281/zenodo.18859602 Keywordsemergent time · configuration space temporality · Jacobi metric · relational dynamics · Page–Wootters · thermal time · quantum optimal transport · Bures metric · Witten Laplacian · cascade number · quantum speed limit · Margolus–Levitin · time–band limiting · prolate spheroidal functions · Landau–Widom · resolution law · transmon qubits · Weil positivity · corrections · retraction