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: A Parameter-Free Framework for Emergent Time via Quantum Optimal Transport and Witten Deformation Overview Configuration Space Temporality (CST) is a theoretical physics framework proposing that time is not a fundamental parameter but an emergent quantity: the cost of optimal transport between quantum configurations. Rather than asking "given t, what is ρ(t)?" (the Lindbladian question), CST asks "given ρ₁ → ρ₂, what cost accumulates?" — time as consequence, not coordinate. This deposit contains the complete theory in its parameter-free form (v16), all previous versions, the first empirical validation on IBM Quantum hardware, and numerical implementations of the new transport-geometric framework. What's New in v16 Version 16 represents a fundamental advance: CST now has zero free parameters and makes quantitative predictions distinguishable from standard quantum mechanics. The Complete Framework in Three Lines Component Formula Origin Axiom V(ρ) = S(ρ || I/N) Unique by Umegaki's theorem (1962) Parameter s = 1/(N−1) Tomita-Takesaki modular flow Time t_CST = ∫ e^{V/(N−1)} ds_Bures Arc length in Witten-deformed metric The configurational potential V is the quantum relative entropy with respect to the maximally mixed state — the only functional satisfying unitary invariance, normalization, monotonicity under CPTP maps, and additivity. The deformation parameter s is fixed by the modular inverse temperature of the democratic reference state. The base metric is the Bures (SLD Fisher) metric, the unique monotone metric on the space of density matrices. Key New Results Quantum Wasserstein ≠ Bures — We computed the Carlen-Maas transport metric for qubit systems and proved numerically that transport distance ≠ distinguishability distance. For gain media, the CM metric is 3–6× smaller than the Bures metric, confirming that "time of transport" and "time of distinguishability" are genuinely different quantities. The Time Ratio R — We derive a closed-form expression for R = t_CST / t_QM along any radial transit: R(ρ₁→ρ₂) = ∫ e^{V(ρ)} / √(1−ρ²) dρ / ∫ 1/√(1−ρ²) dρ This is a pure number depending only on the purity trajectory, not on the speed of the Lindbladian. R diverges between gain (1.18), passive (1.42), and depolarizing (1.23) systems. Morse-CST Flow Divergence — The CST gradient flow (descent of V in the Bures metric) diverges from the Lindblad flow by up to 180° for passive decay. The Lindbladian pushes toward pure states (ascending V); CST pulls toward the maximally mixed state (descending V). This is not perturbation — it is fundamental bifurcation. Application to the Toronto Negative-Time Experiment — We apply CST to the 2023 experiment by Daniela Angulo et al. (Steinberg group, U. Toronto), where photons traversing a Rb-87 gain medium exhibited τ_eff < 0. CST interprets this as a geometric subsidy: the pump pre-pays the configurational asymmetry cost, reducing the transport cost of the probe transit by 46% relative to a passive medium. Negative time is not an illusion — it is the signature of pre-paid configurational cost. Previously Established Empirical Results (v13–v14) We constructed the CST generator operator (Witten Laplacian on the configuration torus T²) for coupled transmon qubits and computed its spectrum using calibration data from 772 qubit pairs across 6 IBM Quantum processors (Sherbrooke, Brisbane, Osaka, Kawasaki, Kyiv, Quebec). Result Value Significance Spectral collapse CV = 1.4% Universal structure across 6 processors Orthogonal information r = 0.205 (p < 0.001) Signal invisible to standard calibration Witten R² (5-fold CV) 0.134 vs 0.097 (raw) 38% improvement over raw features Signal amplification 28× over R_ij baseline Heat kernel vs raw coupling Coherence sensing ρ = −0.76 vs ΔT₁ Intrinsic physics, not firmware Phase boundary ℏ_eff ≈ 0.42 Semiclassical transition on real hardware Three Independent Confirmations Spectral Universality (CV = 1.4%) — Rescaled eigenvalue ratios collapse across all six processors. Despite different fabrication and noise environments, the Witten spectrum has universal structure, consistent with CST's prediction that configuration-space geometry imposes universal constraints. Orthogonal Information Content (r = 0.205, p < 10⁻³) — After removing ALL variance explained by 10 standard calibration parameters, the Witten spectral features still predict hardware behavior. The configuration-space operator sees what component-level characterization cannot. Inter-Processor Coherence Sensing (ρ = −0.76) — The relative Witten index correlates with T₁ relaxation-time differences but NOT with gate-length differences. The spectral geometry senses intrinsic hardware physics (coherence/decoherence), not calibrated firmware. Semiclassical Phase Boundary — Z(β) diverges below ℏ_eff ≈ 0.42 (E_J/E_C ≈ 87) on IBM Kawasaki hardware, revealing a phase transition observable in the Witten Laplacian spectrum. Previously Established Mathematical Results Christoffel-Bloch theorem: Γ^c_ab = r_c g_ab, verified to 10⁻⁸, confirming the S³ isometry of the qubit state space. Entropy production decomposition: dS_vN/dt = −arctanh(ρ)(r̂ · v_D), exact. Entropy production-velocity relation: σ ∝ v²_Bures with r ≈ 0.75–0.84. Extended Koide relation: CST mass formula applied to heavy quarks (c, b, t) yields Q = 0.6695, 0.42% deviation from 2/3. Standard Model correspondence: CST's cascade structure maps onto SM gauge group hierarchy. Files File Description CST_v16_Configuration_Space_Temporality.pdf Current version. Parameter-free framework: Umegaki axiom, Witten-Bures metric, closed-form time ratio, Toronto application, three testable signatures. Configuration_Space_Temporality_v14.pdf Revised axioms, corrected metric signature, refined predictions. Configuration_Space_Temporality_v13.pdf Full theoretical framework: axioms, spectral operator construction, eight falsifiable predictions (P1–P8). witten_transmon_paper.pdf "Witten Laplacian on the Configuration Space of Coupled Transmons" — Empirical test on 772 qubit pairs from 6 IBM Quantum processors. Code Repositories GitHub: github.com/davidangularme/cst — CST theory and initial tests GitHub: github.com/davidangularme/sdsq — Full spectral diagnostics (Phase 2 results) All code reproduces in ~60 seconds with pip install qiskit-ibm-runtime. No IBM Quantum account needed. Three Testable Signatures (v16) Signature 1: Time Ratio R ≠ constant The ratio R = t_CST / t_QM depends on the purity trajectory. For a qubit (s = 1): Transit ρ₁ ρ₂ R Interpretation Gain → mixed 0.90 0.20 1.26 Low cost (geometric subsidy) Passive → pure 0.10 0.98 2.16 High cost (configurational surcharge) Depol. → center 0.90 0.01 1.22 Reference Moving toward purity — what QM treats as natural relaxation — becomes expensive in CST units. Signature 2: Angular Divergence The Lindblad flow and the Morse-CST flow diverge by up to 180° for passive decay. The Lindbladian pushes toward ρ_ss (dynamics-dependent); CST pulls toward I/N (geometry-dependent). Maximal when ρ_ss is far from I/N. Signature 3: Transport Efficiency CST gradient flow achieves η = 0.98 (Bures distance per unit Witten cost). The Lindbladian achieves only 0.54–0.85 depending on the system. The Toronto Interpretation The 2023 experiment by Daniela Angulo et al. measured τ_eff < 0 for photon-atom interaction in a Rb-87 gain medium. CST interprets this as follows: The gain medium maintains atoms at ρ_ss ≈ 0.20 (low configurational potential V ≈ 0.02) A passive medium has ρ_ss ≈ 0.98 (high potential V ≈ 0.64) The Witten cost factor e^V is 1.02 (gain) vs 1.89 (passive): 46% reduction The pump pre-pays the configurational asymmetry cost Negative time = configurational cost below vacuum baseline This is not time running backward. It is a geodesic through curved configuration space being shorter than the flat-space path — the same logic as gravitational time dilation, but in information geometry. Significance This work represents: The first parameter-free theory of emergent time — V fixed by Umegaki's theorem, s fixed by modular flow, base metric fixed by monotonicity. No tuning. The first empirical test of emergent-time predictions — Witten Laplacian spectrum on IBM Quantum hardware confirms three independent observables. A concrete reinterpretation of negative time — Toronto's τ_eff < 0 as geometric subsidy in configuration space, with quantitative predictions. A bridge between quantum information geometry and foundational physics — Connecting Bures metric, quantum optimal transport (Carlen-Maas), Witten deformation, and Tomita-Takesaki modular theory into a unified framework. The theory does not modify quantum mechanics. It changes the grammar: from time as scaffold to time as conserved quantity emerging from transport inefficiency. Authors Frédéric David Blum — Theory, physics, direction (Catalyst AI, [email protected]) Claude (Anthropic) — Computation, analysis, validation, mathematical implementation Catalyst AIS — Conceptual analysis, strategic framework, interpretive synthesis Keywords configuration space, emergent time, Witten Laplacian, quantum optimal transport, Bures metric, Carlen-Maas metric, Witten deformation, negative time, Toronto experiment, Umegaki theorem, Tomita-Takesaki modular flow, transmon qubits, quantum computing, spectral theory, IBM Quantum, information geometry, quantum speed limit, CST Related Zenodo DOI: 10.5281/zenodo.18779189 SDSQ paper: github.com/davidangularme/sdsq