Unified Delay Theory of Entanglement--Consciousness--Time: Fourfold Bridging of Spectral--Scattering--Information--Discounting and Cross-Modal Verifiable Scales

Unied Delay Theory of EntanglementConsciousnessTime: Fourfold Bridging of SpectralScatteringInformationDiscounting and Cross-Modal Veriable Scales Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore Abstract We propose a unied delay theory spanning from quantum scattering to conscious time perception to social delay discounting. First, we establish a scale identity based on spectral shift phasegroup delay, providing regularization via KontsevichVishik (KV) and relative determinants for countable channels and innite-dimensional cases, with explicit almost everywhere dierentiability and de-singularization schemes in neighborhoods of thresholds and embedded eigenstates. Second, we characterize the contraction of local distinguishability rate via monotonicity of quantum Fisher information, giving an operational denition of subjective duration and introducing Petz recovery as the necessary and sucient condition for equality. Third, we provide unied expressions for eective horizon width via exponential, hyperbolic, and quasihyperbolic discounting with monotonicity criteria, establishing discount parameter mappings through the future-self/other-overlap factor. Fourth, we introduce a latent coupling strength parameter to cross-align microwave scattering group delays, behavioral thresholds, and discount curves across layers, proposing joint experiments and error budgets. These three domains close in engineering as a cross-scale veriable framework of coupling enhancementdwell increase horizon extension. Appendices provide detailed proofs of thresholdpole regularization, QFI equality conditions, discount generalization, and complete error budgets. Keywords : WignerSmith group delay; BirmanKren spectral shift; KV/relative determinant; quantum Fisher information; Petz recovery; subjective time; delay discounting; hyperbolic and quasi-hyperbolic; selfother overlap; multi-layer structural equations 1 Introduction & Historical Context The formation of time manifests in three complementary languages: group delay and density of states change in quantum scattering, distinguishability rate and subjective duration dilation in conscious time perception, and discounting with eective horizons in social decision-making. On the scattering side, work by Wigner and Smith established the connection between phase derivatives and dwell times, precisely dening group delay through the lifetime matrix; the BirmanKren formula links the scattering determinant to the spectral shift function, forming a closed loop of phasespectral shiftDOSgroup delay (with Yafaev's framework as rigorous background). In scattering on networks and graphs, Friedel summation can fail due to dark states, requiring corrected counting for accessible channels. On the consciousness side, quantum Fisher information serves as the intrinsic metric for parameter estimation, satisfying monotonicity under CPTP maps; its equality and equivalence with 1 recoverability/sucient statistics are systematically characterized by Petz theory and subsequent work. Human time perception exhibits reversible fastslow dilation phenomena under modulation by emotional and reward pathways, with neural mechanism evidence from classical reviews and multiple neural surveys. On the social side, empirical facts of delay discounting are widely tted as hyperbolic or quasihyperbolic forms ( β  δ model), signicantly correlated with future-self continuity/other overlap. This paper unies the three domains under the common scale of couplingdwellhorizon, providing cross-modal joint tests and error closure. 2 Model & Assumptions Variables and measures : Fix units ℏ= 1 , uniformly measure in frequency variable ω ; all derivatives, density of states, and spectral shift are taken with respect to ω . Scattering-side assumption (H sca ) : Self-adjoint operator pair (H, H0) satises H−H0∈ S1 or relative trace class; wave operators exist and are complete; energy-layer scattering matrix S(ω) is dierentiable except on zero-measure sets of thresholdspolesembedded eigenstates. In the innite-dimensional case, dene detKV S(ω) and phase Φ(ω) = arg detKV S(ω) via KV/relative determinants; the Koplienko case corresponds to second-order spectral shift under HilbertSchmidt perturbations. Thresholdpole de-singularization is performed via Jost functions and resolvent expansions, explicitly specifying almost everywhere dierentiability and principal value integral interpretation. Network and channels (H net ) : Channels decompose into accessible subspace Hacc and dark state subspace Hdark . All counting laws are stated on Hacc , with local-form corrections to DOS when necessary. Consciousness side (H cog ) : Global evolution ρAB(θ) = e−iθH ρABeiθH , measurable only on A , local channel Λ = TrB . Quantum Fisher information FA Q(θ) is dened by monotone metrics (Petz class), satisfying data processing inequality and necessary-sucient equality conditions. Social side (H soc ) : Discount weights adopt unied weight function V(t) , including exponential V(t) = γt , hyperbolic V(t) = (1 + kt)−α , and quasi-hyperbolic V(0) = 1, V (t≥1) = βδt . Dene eective horizon width T∗=Pt≥0wt ( wt normalized weight). Let future-self/other overlap index be C∈[0,1] , mapping to model parameters (e.g., γ, k, α, β, δ ) monotonically. 3 Main Results (Theorems and Alignments) Boxed reconciliation (unied factor throughout) : For any unitary S(ω) and its phase Φ(ω) = arg det S(ω) , we have tr Q(ω) = ∂ωΦ(ω) , and φ(ω) = 1 2Φ(ω) . Thus the unied scale is 1 2πtr Q(ω) = φ′(ω) π=ρrel(ω) (where ρrel =−ξ′ ). This equality holds in innite dimensions and under relative determinants, interpreted via almost everywhere derivatives and de-singularization regularization. Theorem 1 (Theorem 1: Applicability Domain and Regularization of Spectral ShiftPhaseGroup Delay) . Under (H sca ), except for zero-measure sets of thresholds/resonances/embedded eigenstates, for almost all ω we have φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω) . If H−H0∈S2 rather than S1 , replace with Koplienko spectral shift and second-order determinant to obtain the corresponding second-order version. Threshold neighborhood corrections and dark state corrections are in Appendix A. Theorem 2 (Theorem 2: QFI Monotonicity and Equality Condition for Local Time Scale) . Let ρAB(θ) and local channel Λ = TrB . For any FQ corresponding to a Petz monotone metric, we have FA Q(θ)≤FAB Q(θ) . Equality holds if and only if Λ is sucient for the state family, i.e., there exists a 2 Petz recovery Rσ such that Rσ◦Λ[ρAB(θ)] = ρAB(θ) (also holds for equivalent characterizations such as Rényi classes). This condition is the necessary and sucient criterion for local non-decrease. Proposition 3 (Proposition 3: Operationalization of Subjective Duration and Entanglement Monotonicity) . Dene subjective duration tsubj(τ) = Rτ 0(FA Q(t))−1/2dt . If coupling/entanglement enhancement leads to ∂EFA Q(t)≤0 almost everywhere, then ∂Etsubj(τ)≥0 . Behavioral agents are given by the quantum CramérRao lower bound ∆tmin ≥[mFA Q]−1/2 , so (FA Q)−1/2 can be estimated from psychophysical thresholds. Theorem 4 (Theorem 3: Dwell Law and Area for Single Poles and Few Channels) . The Breit Wigner approximation gives τg(ω) = Γ[(ω−ω0)2+ Γ2]−1 , with integral area RRτg(ω)dω =π ; if feedback reduces eective decay Γ = Γ(g) monotonically decreasing with coupling, then τg(ω0) = 1/Γ(g) increases monotonically. In multi-channel cases, area is redistributed by coupling weights. Theorem 5 (Theorem 4: Horizon Monotonicity for Exponential Discounting) . Let V(t) = γt and T= (1 −γ)−1 . If γ= Γ(C) is strictly increasing, then dT/dC = Γ′(C)/(1 −Γ(C))2>0 . This monotonicity is consistent with evidence that future-self continuity/other overlap enhances discount factors. Theorem 6 (Theorem 5 ′ : Eective Width Monotonicity for Hyperbolic Discounting) . Let V(t) = (1 + kt)−α , take wt=V(t)/Ps≥0V(s) , dene T∗=Pt≥0wt . Then ∂kT∗<0 and ∂αT∗>0 , consistent with the exponential model in the k→0 limit (corresponding to γ→1− ). Proof in Appendix D. Empirically, hyperbolic ts outperform pure exponential. Theorem 7 (Theorem 5 ′′ : Eective Width for Quasi-Hyperbolic β  δ ) . Let V(0) = 1, V (t≥1) = βδt . After normalization, T∗= 1 + βδ/(1 −δ) , so ∂βT∗>0 and ∂δT∗>0 . Meta-analyses show β  δ is the mainstream model for characterizing present bias. Theorem 8 (Theorem 6: Cross-Modal Identiable Mapping via Latent Coupling Strength) . Introduce latent variable κ to uniformly characterize coupling strength, assuming existence of monotone dierentiable mappings Γphys :κ7→ Γ(g) , Γcog :κ7→ FA Q , Γsoc :κ7→ (γ;k, α;β, δ) . If joint experiments synchronously collect τg(ω0),∆tmin, γ satisfying a confound-free separable structural equation model, the co-directionality (sign consistency) of the three monotonic relationships can be tested and the latent scale of κ estimated. Proof in Appendix E (identication conditions and estimation strategy). 4 Proofs Proof of Theorem 1 (key points) : From the BirmanKren formula det S(ω) = exp{−2πi ξ(ω)} , we get Φ′(ω) = −2πξ′(ω) ; and Q=−iS†∂ωS gives tr Q=−i ∂ωlog det S=∂ωΦ . Thus (2π)−1tr Q= ρrel =−ξ′=φ′/π ( φ=1 2Φ ). For KV/relative determinant cases, replace at the denition of det with KV trace and ζ -regularization; threshold and embedded eigenstates are de-singularized via Jost functions and resolvent expansions, ensuring almost everywhere validity. Proof of Theorem 2 (key points) : Petz-class monotone metrics satisfy data processing inequality for any CPTP map; letting Λ = TrB gives FA Q≤FAB Q . Equality is necessary and sucient for recoverability: there exists a Petz recovery Rσ such that Rσ◦Λ[ρAB(θ)] = ρAB(θ) ; this condition can also be stated in Rényi families and α  z generalizations. Proof of Proposition 3 (key points) : By the quantum CramérRao lower bound ∆tmin ≥ [mFA Q]−1/2 , we know (FA Q)−1/2 is the unit threshold scale thickness. If ∂EFA Q≤0 , then ∂Etsubj(τ) = Rτ 0 1 2(FA Q)−3/2(−∂EFA Q)dt ≥0 . 3 Proof of Theorem 3 (key points) : In BreitWigner form, τg is Cauchy density with area π . If Γ′(g)<0 , then ∂gτg(ω0) = −Γ′/Γ2>0 . Multi-channel decomposition by coupling, area distribution given by partial widths. Proofs of Theorems 45 ′ 5 ′′ (key points) : Exponential case yields monotonicity by direct dierentiation. Hyperbolic case uses integral trial Pt≥0(1+kt)−α≈R∞ 0(1+kt)−αdt =k−1(α−1)−1 ( α > 1 ); after normalization, T∗ is monotone with k↓ and α↑ . Quasi-hyperbolic yields closed form by direct summation with monotonicity. Model t superiority in relevant reviews and meta-analyses. Proof of Theorem 6 (key points) : Assume observation triplet (τg(ω0),∆tmin, γ) is generated by three monotone mappings of latent variable κ with additive noise, structural equations Yj=hj(κ) + εj . If hj are monotone and noise independent, use rank correlation consistency and multi-level SEM to estimate co-directionality of sign(∂hj/∂κ) ; cross-modal co-directionality is the veriable criterion for unied coupling hypothesis. Identication details in Appendix E. 5 Model Applications Physical sidechannel/network dwell measurement : On a two-port vector network platform, measure S(ω) and use robust unwrapping and Cauchy smoothing dierence to compute φ′(ω) and tr Q(ω) , verifying (2π)−1tr Q=φ′/π and consistency with DOS counting; in feedback cavities, tune g to t monotone law of Γ(g) , reporting area conservation. For non-minimum phase loops, use Bode/KramersKronig relations for phasemagnitude consistency checks. Consciousness sideslowthick subjective duration : Time reproduction and minimum dierence thresholds in parallel, estimating ∆tmin and subjective ratings; in high-connection contexts, expect ∆tmin ↑ , tsubj ↑ , aligned with internal clockdopamine modulation evidence. Social sidediscountinghorizon : Use adaptive titration to obtain individual γ or (k, α) /( β, δ ), synchronously collect IOS and future-self continuity, verifying monotone mappings of Theorems 4 5 ′ 5 ′′ . 6 Engineering Proposals P1 | Canonical measurement of microwave network group delay : Phase unwrapping threshold setting, frequency grid ∆ω , equivalent noise bandwidth and port mismatch error budgeting; use three-point/ve-point dierence and spline derivative cross-validation; for non-minimum phase, correct parasitic phase via Bode gainphase relations and Hilbert transform. P2 | Subjective durationQFI proxy dual-task paradigm : Oddball event induction and neutral sequence control in parallel, collect ∆tmin , pupil/skin conductance/HRV for multi-modal fusion to isolate arousal confounds; map ∆tmin to Fbeh Q∝∆t−2 min via CRB. P3 | Hierarchical Bayesian t of discount curves : Simultaneously t exponential/hyperbolic/quasihyperbolic and compare with WAIC/AIC/BIC; collect IOS and future-self continuity for mediation analysis; prosocial/trust manipulations under ethical compliance as external validation. 7 Discussion (Risks, Boundaries, Past Work) Applicability domain : The identity holds within S1 or relative trace class; Koplienko case gives second-order version; thresholdsembedded eigenstates require Jost/threshold expansion treatment; dark states in graphs and feedback networks need explicit exclusion or local DOS correction. Provabletestable boundaries : On the consciousness side, subjective duration as behavioral proxy for F−1/2 Q relies on near-saturation of CRB; experimental tests for near-saturation 4 and bias correction. Social-side model heterogeneity controlled via hierarchical Bayes and model comparison. Relationship with existing work : This framework anchors in spectralscattering scales, aligning quantication of conscious and social time on the common geometry of dwelldistinguishability horizon; it does not claim metaphysical identity of time equals entanglement, but provides crossdomain operational equivalence and joint criteria. See references for classical reviews and modern advances. 8 Conclusion On the rigorous foundation of spectralscattering regularization and QFI monotonicity, we provide unied scales and monotone laws for subjective duration and social horizons, establishing the veriable triad of coupling enhancementdwell increasehorizon extension. The common latent variable scale across three domains brings microwave scattering, behavioral thresholds, and discount curves into the same statisticalcausal structure, constituting an engineering path for cross-scale time theory. Acknowledgements, Code Availability Thanks to publicly available literature on spectral shiftgroup delay, quantum Fisher information, and delay discounting. Scripts for S -parameter tting, phase unwrapping, CRB estimation, and hierarchical Bayesian discount tting can be directly implemented from appendix pseudocode. References Wigner (1955); Smith (1960) group delay and lifetime matrix; Yafaev (1992/2010) scattering theory monograph; Pushnitski (2010) BirmanKren; Texier (2001/2003) Friedel on graphs; Kontsevich Vishik (1994) KV determinant; GesztesyPushnitskiSimon (2007) Koplienko; Petz (1996/1988) monotone metrics and recovery; Eagleman (2008) time perception review; Ersner-Hersheld (2009/2011) future-self continuity; Mazur (1987) hyperbolic discounting; Laibson (1997) β  δ . A Rigorous Identity in Innite Dimensions and at Thresholds Poles A.1 KV/relative determinant and almost everywhere derivative : Let detKV S(ω) = exp{TRKV log S(ω)} , phase Φ = arg detKV S . In S1 case, BirmanKren gives Φ′=−2πξ′ ; in S2 case, adopt Koplienko spectral shift to dene second-order equality. Φ′ exists except on sets of thresholdspolesembedded eigenstates. A.2 Jost/threshold expansion : Threshold neighborhoods adopt Jost functions and resolvent expansions, clarifying principal value interpretation and additional terms for ξ′ ; embedded eigenstates leak via Fermi golden rule into regular patterns, see JensenKato and subsequent threshold expansion work. A.3 Scattering on graphs and dark state correction : For local states in Hdark , Friedel counting needs to subtract contributions from inaccessible states; local DOS and injection/emission rates give local version. 5 B QFI Monotonicity, Equality Condition, and Subjective Duration B.1 Data processing and equality : Petz results show that for monotone metric FQ and channel Λ , FQ(Λ[ρθ]) ≤FQ(ρθ) ; equality if and only if Λ is sucient for the state family, recovery Rσ exists. Equality criteria for Rényi and α  z extensions are equivalent to Petz recovery. B.2 Subjective duration : Take tsubj(τ) = Rτ 0(FA Q(t))−1/2dt . CRB gives ∆tmin ≥[mFA Q]−1/2 ; substituting behavioral measure of ∆tmin yields empirical estimation formula; estimate is unbiased when near-saturation optimal measurement exists. C Area Conservation for Single Poles and Few Channels BreitWigner τg is standard Cauchy form, area π independent of Γ ; in multi-channel coupling, area is distributed by partial widths; in cavityfeedback networks, test coupling enhancementdwell increase by tting monotone relation of Γ(g) and τg(ω0) . D Discount Generalization and Eective Width D.1 Hyperbolic family : V(t) = (1 + kt)−α , normalization constant Z=Pt≥0(1 + kt)−α≈ k−1(α−1)−1 ; eective width T∗=Pwt monotone with k↓ and α↑ . D.2 Quasi-hyperbolic β  δ : T∗= 1 + βδ/(1 −δ) ; when β→1 , δ→γ returns to exponential. Literature comparison shows hyperbolic and quasi-hyperbolic outperform pure exponential across multiple categories. E Joint Identication and Statistical Power E.1 Structural equations and identication : Let τg(ω0) = h1(κ)+ε1,∆tmin =h2(κ)+ε2, γ = h3(κ) + ε3 , hj monotone, εj independent. Use multi-level SEM and rank correlation to test codirectionality of sign(h′ 1),sign(h′ 2),sign(h′ 3) . E.2 Statistical power and sample size : Target eect size d∈[0.3,0.5] detection requires tens to hundreds of samples; report multiple comparison correction and post-hoc manipulation tests. Power and eect size reporting follows standard guidelines. F Implementation Details and Error Budget F.1 Microwave network (P1) error closure : (i) Phase unwrapping: set allowable jump threshold and residual detection; (ii) dierence and spline derivative cross-validation, Cauchy smoothing dierence suppresses high-frequency noise; (iii) port mismatch and ENBW correction; (iv) nonminimum phase detection and Bode/Hilbert correction. F.2 Time perception (P2) and CRB mapping : Parallel collection of ∆tmin , subjective ratings, and physiological indicators, isolating arousal/attention confounds; map ∆tmin to Fbeh Q via CRB, reporting condence intervals. F.3 Discounting (P3) model comparison : Exponential/hyperbolic/quasi-hyperbolic simultaneously tted with WAIC/AIC/BIC; hierarchical Bayes mitigates individual heterogeneity; collect IOS and future-self continuity as explanatory variables for mediation regression; ethical and blinding controls for potential prosocial/trust manipulations. 6