We derive the exponential suppression of off-diagonal coherences predicted in Conjecture 1 of P1 v1.1 from the parent action S_eff[C], using the Feynman-Vernon influence functional within the Schwinger-Keldysh formalism. The coherential field C is decomposed into long-wavelength system modes and short-wavelength environment modes, with the kernel scale l_K providing natural separation. For homogeneous backgrounds, the decoherence exponent reduces exactly to the quadratic sector of D_K with explicit coefficient Gamma = 2 mu^8 kappa_env / C_*^2. In the binary discrete limit, this reproduces the Hamming-weighted exponential decay established experimentally in P0-a (GHZ-Bridge). No modification of quantum mechanics is assumed; no new parameters are introduced. The derivation closes the structural bridge S_conf → D_K → d_H → Hamming Decay for the homogeneous case. Extension to inhomogeneous backgrounds remains open. Part of the Radial Coherential Dynamics (RCD) 2.0 program.
We derive the exponential suppression of off-diagonal coherences predicted in Conjecture 1 of P1 v1.1 from the parent action S_eff[C], using the Feynman-Vernon influence functional within the Schwinger-Keldysh formalism. The coherential field C is decomposed into long-wavelength system modes and short-wavelength environment modes, with the kernel scale l_K providing natural separation. For homogeneous backgrounds, the decoherence exponent reduces exactly to the quadratic sector of D_K with explicit coefficient Gamma = 2 mu^8 kappa_env / C_*^2. In the binary discrete limit, this reproduces the Hamming-weighted exponential decay established experimentally in P0-a (GHZ-Bridge). No modification of quantum mechanics is assumed; no new parameters are introduced. The derivation closes the structural bridge S_conf → D_K → d_H → Hamming Decay for the homogeneous case. Extension to inhomogeneous backgrounds remains open. Part of the Radial Coherential Dynamics (RCD) 2.0 program.
We provide a minimal analytic framework, entirely within standard open-quantum-system theory, that explains the protocol-robust GHZ advantage observed experimentally across heterogeneous quantum platforms. The framework rests on three results: (1) the Hamming Decay Theorem, showing that off-diagonal density matrix elements decay as lambda^{d_H} under any local quantum channel; (2) a binomial derivation of the product-state fidelity F_prod = [(1+exp(-gamma*t))/2]^N under the IBM protocol; and (3) a unification of the two experimental regimes (IBM dephasing-dominated and IonQ amplitude-damping-dominated) via T2-blindness of the GHZ observable. A three-parameter noise model calibrated to IonQ 5-qubit data reproduces the selectivity pattern GHZ+, W-, Cluster- with correct signs. The structural invariant S(rho) captures the channel-independent topology underlying the advantage. Includes Appendix A: Revised Empirical Basis with cross-platform convergence data, selectivity tables, hardware threshold, and kill switches.
The late-time dissipative kernel of the Radial Coherential Dynamics (RCD) program is parabolic and does not generate a Lorentzian causal cone: signature is an input, not an output. This relocates the program's claim to a logically prior proposition — that a coherence field can organize the local relations of influence any later geometric description would have to account for — and motivates the operational question addressed here: can a coherence field organize a finite, switchable influence front in a local quantum dynamics? We study a transverse-field Ising chain and measure the infinite-temperature squared commutator (out-of-time-order correlator) c(x,t) = ||[Z0(t), Zx]||^2_F / (2·2^N) as the primary observable, with frozen pre-registered gates. By exact operator evolution at N = 10, 12, 13 the front velocity in the clean channel is stable without appreciable size drift (v = 2.000 → 1.980 → 1.980 lattice units at theta=0.10, consistent with the exact maximal quasiparticle group velocity 2·min(J,h), distinct from and below the rigorous Lieb–Robinson bound), with OFF collapse to ~1e-13 and robustness across thresholds; a persistent failure of the linearity gate at theta=0.20 is retained, with its cause stated as open (quasiparticle-interference vs temporal-sampling artifact, pending a control). Imposed spatial coherence profiles modulate the front: an incoherent barrier blocks it, a coherence ramp slows it per a local WKB law, a uniform profile recovers the bare velocity. The coherence-to-coupling map J_i = J0·sqrt(C_i·C_{i+1}) is declared a constitutive ansatz, not a derived or unique law. We document an instrumental failure (a reproducible backend-specific crash in tall-skinny complex GEMM that made the N=14 typicality target non-viable on 32 GB hardware, traced through four probes, circumvented by row chunking; precise low-level cause not established) and the resulting procedural substitution to N=13 exact, declared as a deviation from pre-registration. Finally we pre-register the decisive bridge test (KS-C1a, frozen, adversarially reviewed): whether a coherence landscape extracted from the RCD Wright–Fisher kernel — not a synthetic profile — organizes the same front, with paired controls against link-shuffling and the permutation-invariance of the WKB crossing time. We state explicitly what is NOT claimed: no derived metric, no derived time, no physical light cone, no emergent general relativity; the coherence-to-coupling map is not derived from any action; the geometric mean is not unique; coherence does not control velocity in the uniform case (an identity). The object validated is an operational influence front and its modulation by coherence — a necessary, not sufficient, step toward the coupled-dynamics bridge (KS-C1b), which is named as future work. STATUS: Preprint v0.3. Instrument validation and the frozen pre-registration protocol are complete; the decisive test (KS-C1a, Section 8) is a registered placeholder pending execution. References are provisional; a methodological amendment (v0.5) to the KS-C1a protocol is flagged in the text and required before execution. Appendix B reproduces the ratified pre-registration v0.4 verbatim.
The late-time dissipative kernel of the Radial Coherential Dynamics (RCD) program is parabolic and does not generate a Lorentzian causal cone: signature is an input, not an output. This relocates the program's claim to a logically prior, operational question: can a coherence field organize a finite, switchable influence front in a local quantum dynamics, and does the specific spatial order produced by the RCD kernel matter beyond the distribution of coupling intensities? We study a transverse-field Ising chain and measure the infinite-temperature out-of-time-order correlator c(x,t) = ||[Z0(t), Zx]||^2_F / (2·2^N) as the primary observable, with frozen pre-registered gates. By exact operator evolution at N = 10, 12, 13 the front velocity in the clean channel is stable without appreciable size drift (v = 2.000 -> 1.980 -> 1.980 at theta=0.10, consistent with the exact maximal quasiparticle group velocity 2·min(J,h), distinct from and below the rigorous Lieb-Robinson bound), with OFF collapse to ~1e-13. Imposed coherence profiles modulate the front: an incoherent barrier blocks it, a coherence ramp slows it per a local WKB law, a uniform profile recovers the bare velocity. The coherence-to-coupling map J_i = J0·sqrt(C_i·C_{i+1}) is a constitutive ansatz, not a derived or unique law. The decisive test (KS-C1a) was executed on a snapshot of the SEALED 2D Wright-Fisher/Allen-Cahn kernel (de Sitter, H=0.02, seed 20260612, t*=1.0 t_H, central row coarse-grained 120->12). The candidate at t*=0.5 t_H failed the pre-quantum G0 gate; t*=1.0 t_H passed all three amended G0 conditions (CV(J)=0.189, percentile[S(C)]=3.5, xi=2). The result is a PRE-REGISTERED NULL (the RED branch): G1 PASS (real front obeys local WKB, R^2=0.913), G3 PASS (OFF 4.8e-14), but G2 FAIL for all three J-shuffles — cross-prediction differences (dR^2 ~ 0.03) and trajectory differences (Delta ~ 0.03-0.06) fell well below the pre-registered thresholds (0.15 and 0.10) on eight common interior sites. G4 was INDETERMINATE (the coarse-grained frozen snapshot contained no link below J_min=1e-6; declared before execution). A pre-specified diagnostic at row y=32 returned INDETERMINATE by dynamic censorship (extreme weak links left zero common interior sites), documenting an observability limit of the protocol in high-dynamic-range landscapes; implied kappa varied -4.5% to +8.1% across cuts, immaterial to the (differential) null. What survives: the operational front instrument (G1, G3), the perilla as a conditional predictive translator, and the local WKB law. What does NOT survive: the stronger statement that "the coherence-to-coupling map comes from the field." The narrow statement earned by the data is that a static RCD coherence landscape can be inserted into the constitutive ansatz to predict a front locally, but its tested spatial ordering was not operationally distinguishable from shuffled orderings with the same coupling values. Three non-exclusive readings (genuine physical null / coarse-graining loss / cut-dependent dynamic-range limitation) are stated without selection; each requires a new pre-registered follow-up. We claim NO derived metric, NO derived time, NO physical light cone, NO emergent general relativity. The dynamically coupled bridge (KS-C1b) is named as a new hypothesis, not a validated continuation. Execution evidence is sealed by SHA-256 (manifest in the supplementary record). The KS-C1a protocol amendment v0.5 was applied as recorded prior to execution; Appendix B reproduces the ratified pre-registration v0.4 verbatim.
FINDING: Decoherence is the mechanism by which quantum superpositions become classical mixtures, driven by entanglement with the environment; the key mathematical object is the reduced density matrix and its off-diagonal decay. | MATH: Master equation for reduced density matrix ρ_S: dρ_S/dt = −(i/ℏ)[H_S, ρ_S] − Γ(t)(ρ_S − ρ_diag), where Γ(t) is the decoherence rate. For a superposition |ψ⟩ = α|0⟩ + β|1⟩, the off-diagonal element ρ_01(t) = αβ* exp(−Γ(t)t) e^{−iωt}. In the Feynman-Vernon influence functional, decoherence rate Γ ∝ ∫ dω J(ω) coth(ℏω/2k_BT) (spectral density J(ω)). For Ohmic bath J(ω) ∝ ω, Γ ∝ k_BT/ℏ. For super-Ohmic J(ω) ∝ ω^n (n>1), non-Markovian memory effects appear. | CONNECTION: The decay of off-diagonal coherence follows an exponential with rate Γ — no direct golden-ratio or base-60 link. However, the spectral density J(ω) for a bath of harmonic oscillators maps to a lattice of modes; in 1D, the mode spacing is uniform (Δω = 2πv/L), a linear lattice — related to crys Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
The phenomenological decoherence principle introduced in Ryabokon (2026a) expresses coherence loss as an exponential of the integral of the squared effective field F. In this paper we analyze the statistical foundations of stretched-exponential decay, which arises when F possesses scale-invariant (fractal) statistics. Starting from a power-law spectrum Φ(k) ∝ k^{-p}, we derive explicit relations between the spectral index p and the stretched-exponential exponent β. For Gaussian fluctuations we obtain: β = p-1 for the field itself, and β = p-2 for integrated quantities (e.g., wave phase). The domain of validity and the role of non-Gaussian statistics are discussed. This publication is part of a broader conceptual program aimed at providing a unified phenomenological description of decoherence and attenuation processes. The framework is fully consistent with standard quantum mechanics and should be viewed as an organizing perspective rather than a proposal of new dynamics.
This work introduces the geodesometry of coherence, a minimal discrete dynamical extension of the Hamming Decay Theorem in quantum state spaces. The Hamming Decay law establishes that off-diagonal elements of a density matrix decay exponentially with Hamming distance under local noise channels. However, this metric does not account for the physical structure through which coherence is lost. In this work, the Hamming hypercube is endowed with anisotropic physical weights on its edges, representing local noise, coupling asymmetry, or entropic cost. A minimal 3-qubit model with non-uniform weights (3, 2, 1) demonstrates that pairs of states with equal Hamming distance can exhibit different accumulated transition costs, leading to a falsifiable prediction: equal Hamming distance does not necessarily imply equal decoherence rate under anisotropic conditions. Explicit GO/KILL conditions are defined, and a minimal experimental protocol is proposed. The formulation is deliberately restricted to the quantum discrete sector and does not claim extension to gravity, cosmology, or gauge theories. It is presented as a falsifiable intermediate step between: Hamming geometry (metric structure), geodesometry (discrete dynamics), and a potential continuous effective action Seff[C]S_{\mathrm{eff}}[C]Seff[C] within the Radial Coherential Dynamics (RCD) program. This work is intended as a minimal, testable framework rather than a complete theory.