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Temporal-Causal Unity as an Operational Framework for Collective Dynamics: Causal-Progress Clocks, Synchronization, and Polarization

Jian Liu, Dong Sun

Year: 2026Venue: arXiv preprintArea: physics.soc-phType: PreprintEmbeddings: 41

Intelligence

Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 90%

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Summary

The paper introduces Temporal-Causal Unity (TCU), a framework linking process philosophy to operational models of collective dynamics. It defines a causal-progress coordinate (tau) based on event intensity, distinct from chronological time. The authors propose a stochastic phase-activation model for agents, deriving a conditional synchronization threshold for Kuramoto-like systems and distinguishing between consensus and bipolar polarization using harmonic order parameters. The framework emphasizes falsifiability and separates ontological claims from measurable operational constructs.

Entities (8)

Temporal-Causal Unity · framework · 98%Dong Sun · person · 95%Jian Liu · person · 95%Causal Progress · concept · 92%Kuramoto Model · model · 90%Synchronization Threshold · parameter · 88%Polarization · phenomenon · 85%Process Philosophy · philosophical-tradition · 80%

Relation Signals (6)

Jian Liu → authored → Temporal-Causal Unity

confidence 95% · Jian Liu1 and Dong Sun1... This paper develops temporal-causal unity (TCU)

Dong Sun → authored → Temporal-Causal Unity

confidence 95% · Jian Liu1 and Dong Sun1... This paper develops temporal-causal unity (TCU)

Temporal-Causal Unity → defines → Causal Progress

confidence 95% · Causal progress is defined by τ(t)=∫0tλ(s∣Hs)ds

Temporal-Causal Unity → utilizes → Kuramoto Model

confidence 90% · For the all-to-all noisy Kuramoto special case... synchronization begins at the conditional threshold

Temporal-Causal Unity → distinguishes → Polarization

confidence 85% · First- and second-harmonic order parameters separate consensus from bipolar polarization.

Temporal-Causal Unity → isbasedon → Process Philosophy

confidence 80% · connecting a process-philosophical thesis -- time is the ordered unfolding of causal change

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Abstract

Abstract:This paper develops temporal-causal unity (TCU), a framework connecting a process-philosophical thesis -- time is the ordered unfolding of causal change -- to an operational model of cognitive and social dynamics. The framework deliberately separates three claims: an interpretive thesis about becoming, a measurable causal-progress coordinate, and a stochastic network model. Causal progress is defined by $\tau(t)=\int_0^t\lambda(s\mid\mathcal H_s)\,{\rm d}s$, where the nonnegative event intensity $\lambda$ must be specified independently of the outcome. Agents carry an orientation phase and an activation amplitude; weighted interaction, heterogeneous drift, external input, anchoring, and diffusion govern their evolution in $\tau$. First- and second-harmonic order parameters separate consensus from bipolar polarization. For the all-to-all noisy Kuramoto special case with Lorentzian drift width $\Delta$, synchronization begins at the conditional threshold $K_c = 2(\Delta + D)$, not at a universal constant. Reproducible numerical illustrations illustrate (not empirically demonstrate) this threshold, causal-clock curve collapse, and the consensus-polarization distinction. Six historical episodes are treated as scope probes rather than validation data. The paper derives falsifiable hypotheses and an out-of-sample protocol for comparing causal-progress and chronological-time models. TCU is therefore offered as a disciplined bridge between process ontology and complex-systems modeling, not as a replacement for spacetime physics or as an empirically established identity between time and causation.

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Temporal–Causal Unity as an Operational Framework for Collective Dynamics Causal-Progress Clocks, Synchronization, and Polarization Jian Liu1 and Dong Sun1 1Xintong Zhima (Beijing) Technology Co., Ltd., Beijing, China liujian@bonideas.cn aihubxpro@gmail.com (July 20, 2026) Abstract This paper develops temporal–causal unity (TCU), a framework connecting a process-philosophical thesis—time is the ordered unfolding of causal change—to an operational model of cognitive and social dynamics. The framework deliberately separates three claims: an interpretive thesis about becoming, a measurable causal-progress coordinate, and a stochastic network model. Causal progress is defined by τ(t)=∫0tλ(s∣ℋs)dsτ(t)= _0^tλ(s _s)\,ds, where the nonnegative event intensity λ must be specified independently of the outcome. Agents carry an orientation phase and an activation amplitude; weighted interaction, heterogeneous drift, external input, anchoring, and diffusion govern their evolution in τ. First- and second-harmonic order parameters separate consensus from bipolar polarization. For the all-to-all noisy Kuramoto special case with Lorentzian drift width Δ , synchronization begins at the conditional threshold Kc=2(Δ+D)K_c=2( +D), not at a universal constant. Reproducible numerical illustrations illustrate (not empirically demonstrate) this threshold, causal-clock curve collapse, and the consensus–polarization distinction. Six historical episodes are treated as scope probes rather than validation data. The paper derives falsifiable hypotheses and an out-of-sample protocol for comparing causal-progress and chronological-time models. TCU is therefore offered as a disciplined bridge between process ontology and complex-systems modeling, not as a replacement for spacetime physics or as an empirically established identity between time and causation. Keywords: time and causation; process ontology; causal progress; synchronization; collective dynamics; polarization; social networks 1 Introduction Time and causation are usually represented by different objects. A physical or social process is indexed by a time coordinate, while causal relations are encoded by laws, interventions, mechanisms, or directed dependencies. This division is productive, but it leaves a conceptual question: if change is what gives temporal order empirical content, can temporal passage and causal unfolding be treated as two descriptions of a single process? Process philosophy and phenomenology motivate such a view (Heidegger, 1962; Whitehead, 1978); work on the thermodynamic arrow emphasizes the relation between temporal asymmetry and irreversible processes (Reichenbach, 1956; Prigogine, 1980; Price, 1996). None of these traditions, by itself, supplies an operational model for collective cognition. At the same time, coupled-oscillator, threshold, and opinion-dynamics models provide precise accounts of coordination, cascades, and collective order (Kuramoto, 1984; Granovetter, 1978; DeGroot, 1974; Watts, 2002; Castellano et al., 2009). Their ordinary time coordinate, however, often hides an important empirical fact: equal calendar intervals can contain very different amounts of interaction, exposure, decision, and institutional change. A week with one salient event and a week with thousands of mutually reinforcing events need not represent equal progress through a collective transition. This paper develops temporal–causal unity (TCU) as a bridge between these levels. Its central interpretive thesis is: Time is not an empty container in which causes operate; experienced temporal order is the ordered unfolding of causal change. The thesis is not presented as a theorem of physics. It becomes scientifically useful only after an independently measurable progress coordinate and a model with falsifiable consequences are supplied. The paper makes four contributions. 1. It separates an ontological interpretation, an operational causal-progress clock, and a stochastic dynamical model. This separation prevents a metaphor from being mistaken for a measurement claim. 2. It replaces an unconstrained time–cause plane with a monotone event coordinate τ whose intensity must be estimated without using the outcome it is meant to explain. 3. It formulates a phase–activation network model and corrects two common category errors: interaction weights are not automatically causal effects, and high synchrony is not automatically polarization. 4. It gives conditional analytical results, reproducible numerical illustrations, scope probes, and a protocol by which the framework could fail. The epistemic status is therefore modest but nontrivial. TCU does not derive the arrow of time, reduce relativistic spacetime, import quantum measurement into cognition, or validate a social theory by redescribing historical events. It proposes a precise research program: specify a causal clock before observing the target trajectory, fit a constrained dynamical model, and test whether that clock improves prediction and cross-context invariance. 2 From an Ontological Thesis to an Operational Framework 2.1 Three levels of claim The phrase time is causation can express several logically distinct claims. Conflating them would make the framework immune to evidence. TCU therefore uses the hierarchy in table˜1. Table 1: Three levels of TCU. Only the operational and model levels generate direct statistical tests. Level Claim Evidential status Interpretive Temporal passage and directed causal becoming are two aspects of process. A philosophical postulate assessed by coherence, scope, and relation to existing accounts of time. Operational The progress of a process can be indexed by a monotone accumulation of prespecified events or hazards. Testable through measurement validity, curve collapse, and out-of-sample prediction. Dynamical Orientations and activations evolve through heterogeneous drift, network coupling, input, anchoring, and diffusion in causal progress. Testable against alternative stochastic network models and null networks. The interpretive level has three postulates. TCU-1 (duality). Temporal order is the order in which causal change is realized; causal direction is what distinguishes earlier from later within a process. TCU-2 (trace). Memory, records, institutions, and material structures are present traces of prior process, not literal persistence of the past. TCU-3 (slice). A state at a time is a cross-section of an unfolding process. Spatial, informational, cognitive, and institutional descriptions are different observables on that cross-section. These postulates preserve the original intuition while avoiding the stronger claim that physical space is mathematically eliminable. Relativistic causal structure remains untouched. Results showing that causal order constrains substantial spacetime structure under specific assumptions do not establish a literal identity between time and causation (Malament, 1977). 2.2 The causal-progress clock Let t denote chronological time and ℋtH_t the history of observable events before t. Let λ(t∣ℋt)≥0λ(t _t)≥ 0 be a prespecified intensity of process-relevant events. Definition 1 (Causal-progress coordinate). For a process beginning at t=0t=0, its causal-progress coordinate is τ(t)=∫0tλ(s∣ℋs)ds.τ(t)= _0^tλ(s _s)\,ds. (1) For discrete observations one may use τn=∑m≤nwm _n= _m≤ nw_m, where wm≥0w_m≥ 0 is a preregistered weight for event m. This construction resembles compensator or rescaled time in event-process analysis (Brown et al., 2002), but TCU gives it a specific substantive interpretation: τ measures progress through a hypothesized causal process. Candidate events include exposures, replies, protests, policy decisions, or verified institutional actions. Raw message volume is not necessarily a good clock; it may be duplicated, automated, or endogenous to the response. Proposition 1 (Reparameterization limitation). If λ(t)>0λ(t)>0 is allowed to be chosen freely after the outcome trajectory is observed, replacing t by τ(t)τ(t) has no empirical content. Proof. Strict positivity makes equation˜1 monotone and therefore invertible. Any trajectory y(t)y(t) can then be written y~(τ)=y(t(τ)) y(τ)=y(t(τ)). If λ may depend arbitrarily on the realized y, a desired shape for y~ y can be manufactured by time warping. Empirical content requires λ to be constrained by independent measurements, fixed on training data, or preregistered before the target trajectory is evaluated. ∎ Proposition˜1 is a central safeguard. Successful visual alignment after post-hoc time warping does not support TCU. Support requires improvement on held-out data or invariance across contexts under one prespecified measurement rule. 2.3 Operational translations Several evocative terms in the motivating framework can be retained if they are tied to observables. Table˜2 states those translations and their limits. Table 2: From philosophical language to operational constructs. Motivating term Operational construct Required caution Causal unfolding Accumulated, prespecified event intensity τ The event set and weights cannot be selected to fit the outcome. Causal direction Phase or orientation θi _i plus a signed response to an intervention Temporal precedence alone does not establish causation. Causal condensation A measured transition from dispersed to persistent state concentration This is classical state locking, not quantum wave-function collapse. Causal freezing Long recovery time or strong anchoring after a perturbation The term is an analogy, not the quantum Zeno effect. Causal depletion High alignment together with low response capacity to novel input High consensus alone can be adaptive and does not imply depletion. Social phase transition A parameter-dependent qualitative change in order parameters Finite social systems may show smooth crossovers rather than thermodynamic singularities. 2.4 Causal effects versus influence weights Modern causal inference defines effects through interventions, counterfactuals, or structural assumptions (Pearl, 2009; Woodward, 2003). A network edge estimated from communication or similarity is not, by itself, a causal relation. In what follows, WijW_ij is therefore called an influence weight. It may be interpreted causally only when an identification design supports that interpretation—for example randomized exposure, a credible natural experiment, or a fully stated structural causal model. This distinction prevents the framework’s ontological use of causal from licensing unwarranted empirical conclusions. 3 A Stochastic Phase–Activation Model 3.1 State variables and dynamics Consider N units, such as individuals, communities, or institutions. Unit i has an orientation θi(τ)∈[−π,π) _i(τ)∈[-π,π) and nonnegative activation ai(τ)a_i(τ). The complex notation ψi=aieiθi _i=a_ie^i _i is a compact representation; it is not a quantum state. The proposed dynamics are dθi= _i= [νi+K∑j=1NWijajsin⁡(θj−θi)+i⊤(τ)−qisin⁡(θi−θiref)]dτ+2DidBi(τ), [ _i+K _j=1^NW_ija_j ( _j- _i)+ b_i x(τ)-q_i ( _i- _i^ref) ]dτ+ 2D_i\,dB_i(τ), (2) daidτ= da_idτ= ai[αi+γiIi(τ)−βiai],βi>0. a_i [ _i+ _iI_i(τ)- _ia_i ], _i>0. (3) Here Wij≥0W_ij≥ 0 is a fixed or slowly varying influence matrix, scaled so that its spectral radius has a stated value; K is global coupling; νi _i is intrinsic drift; x is external input; qiq_i anchors a unit to a reference state; DiD_i is a diffusion coefficient; and BiB_i are standard Brownian motions. Equation (3) is logistic activation with input. It avoids interpreting aia_i as a probability amplitude. If the model is simulated in calendar time, the stochastic time-change rule gives dθi=λ(t)fi(,,t)dt+2Diλ(t)dB~i(t),d _i=λ(t)f_i( θ, a,t)\,dt+ 2D_iλ(t)\,d B_i(t), (4) where fif_i is the drift in brackets in equation˜2. Multiplying the drift but not the diffusion by λ would be an inconsistent change of clock. 3.2 Dimensions and identifiability The phase is dimensionless. If τ is measured in causal-event units, νi _i, KWijajKW_ija_j, i⊤ b_i x, and qiq_i all have units of inverse causal-event units, while DiD_i has the same inverse unit. If aia_i is dimensionless, αi _i, γiIi _iI_i, and βiai _ia_i also have units of inverse causal-event units. Only products such as KWijajKW_ija_j are identified without normalization. Consequently, W, a, and K require explicit scale conventions. In addition, simultaneously estimating a highly flexible λ(t)λ(t) and a highly flexible dynamical drift is generally nonidentifiable. The empirical protocol in section˜6 fixes or cross-fits the clock before estimating the state dynamics. 3.3 Order parameters: alignment is not polarization For nonnegative measurement weights ωi _i with ∑iωi=1 _i _i=1, define circular harmonics Zm(τ)=∑i=1Nωieimθi(τ),rm(τ)=|Zm(τ)|,m=1,2.Z_m(τ)= _i=1^N _ie^im _i(τ), r_m(τ)=|Z_m(τ)|, m=1,2. (5) The first harmonic r1r_1 measures one-cluster alignment. The second harmonic r2r_2 is large for either one cluster or two opposing clusters. A simple bipolarity diagnostic is P(τ)=max⁡0,r2(τ)−r1(τ).P(τ)= \0,r_2(τ)-r_1(τ)\. (6) Thus a uniform distribution has r1≈r2≈0r_1≈ r_2≈ 0, consensus has r1≈r2≈1r_1≈ r_2≈ 1 and P≈0P≈ 0, while balanced opposition has r1≈0r_1≈ 0, r2≈1r_2≈ 1, and P≈1P≈ 1. No threshold such as ⟨|θi|⟩>π/4 | _i| >π/4 is invariant to rotation of the angular origin. More elaborate applications should compare equation˜6 with distributional and group-based measures of polarization (Esteban and Ray, 1994; Bramson et al., 2017). Activation is summarized by a¯=N−1∑iai a=N^-1 _ia_i. A depleted system additionally requires a low response capacity. For a standardized novel perturbation of size ε at τ0 _0, define Ch=1Nε∑i=1N|wrap⁡(θi(ε)(τ0+h)−θi(0)(τ0+h))|.C_h= 1N _i=1^N |wrap\! ( _i^( )( _0+h)- _i^(0)( _0+h) ) |. (7) High r1r_1 and low ChC_h, not high r1r_1 alone, operationalize rigidity or depletion. 3.4 Conditional synchronization threshold The full network model has no universal critical coupling. An analytical benchmark is available under restrictive assumptions: ai=1a_i=1, qi=0q_i=0, all-to-all weights Wij=1/NW_ij=1/N, common diffusion D, no external drive, and intrinsic drifts distributed according to a Lorentzian with half-width Δ . In the continuum limit, the first-harmonic amplitude obeys (Strogatz, 2000; Acebrón et al., 2005; Ott and Antonsen, 2008) dr1dτ=[K2−(Δ+D)]r1−K2r13. dr_1dτ= [ K2-( +D) ]r_1- K2r_1^3. (8) Proposition 2 (Conditional onset). Under the assumptions above, the incoherent state loses stability at Kc=2(Δ+D),K_c=2( +D), (9) and for K>KcK>K_c the stable nonzero branch is r1∗=1−Kc/Kr_1^*= 1-K_c/K. Proof. Linearizing equation˜8 at r1=0r_1=0 gives growth rate K/2−(Δ+D)K/2-( +D). It changes sign at equation˜9. Setting the right-hand side of equation˜8 to zero yields the nonzero branch; the cubic coefficient makes it stable when it exists. ∎ For a general nonnegative network without row normalization, the leading spectral mode changes the onset approximately to Kc≃2(Δ+D)/ρ(W)K_c 2( +D)/ρ(W), subject to the mean-field assumptions used in the network synchronization literature (Restrepo et al., 2005; Arenas et al., 2008). Degree heterogeneity, correlated drifts, delays, adaptive edges, finite size, or non-sinusoidal influence can shift or remove this transition. The earlier formula 2⟨D⟩/⟨A⟩2 D / A is therefore not retained. 3.5 Activation and anchoring For constant input IiI_i, equation˜3 has equilibria ai∗=0,ai∗=αi+γiIiβiwhen αi+γiIi>0.a_i^*=0, a_i^*= _i+ _iI_i _i _i+ _iI_i>0. (10) The positive branch is locally stable. Activation can amplify social influence through the factor aja_j in equation˜2, but it is conceptually distinct from directional alignment. The anchoring term makes freezing testable without quantum terminology. When qiq_i is large relative to bounded countervailing input and coupling, the state remains near θiref _i^ref and the exit time from that neighborhood increases. Whether repeated evaluation raises qiq_i is an empirical psychological or organizational hypothesis, not a consequence of quantum measurement. 4 Numerical Illustrations Figure 1: Reproducible illustrations, not empirical fits. (A) A finite N=600N=600 Euler–Maruyama simulation of the all-to-all noisy Kuramoto special case with Δ=0.40 =0.40, D=0.15D=0.15, step size 0.0150.015, and fixed seed, compared with proposition˜2. Finite size and truncated Lorentzian tails produce a small precritical baseline. (B) The same mean-field growth curve appears at different rates in calendar time when λ∈0.5,1,2λ∈\0.5,1,2\. (C) The curves coincide when plotted against the prespecified τ=λtτ=λ t. (D) First and second harmonics distinguish diffuse, consensus, and bipolar states. Code is included as an ancillary file. Figure˜1 checks internal implications of the model. Panel A recovers the conditional onset predicted by equation˜9; it does not show that any observed society is a Kuramoto system. Panels B and C integrate equation˜8 in a regime with positive linear growth under three constant event rates. The collapse in panel C is guaranteed because the curves were generated in τ. In real data it becomes informative only if λ was specified independently, as required by proposition˜1. Panel D shows why a single synchrony statistic can misclassify opposition: both consensus and balanced bipolarity have a large second harmonic, but only consensus has a large first harmonic. The simulation uses a deterministic grid of Lorentzian quantiles, clips only the numerically extreme tails, and averages post-burn-in order parameters. The fixed seed and complete script permit exact regeneration. No historical case data enter the figure. 5 Historical Episodes as Scope Probes The motivating framework discussed six global episodes. Redescription of these episodes cannot validate the model: a flexible vocabulary can be fitted to almost any sequence after the fact. They are more useful as scope probes that expose measurement choices and possible counterevidence. Table˜3 converts each episode into a prospective design. The cited studies establish relevant empirical context, not support for TCU. Table 3: Prospective operationalizations for the six motivating episodes. Each row is a research design sketch, not a result. Episode Candidate clock fixed before outcome analysis State and observable Evidence against the proposed account Black Lives Matter, especially 2020 Weighted cumulative count of independently verified local protests and cross-community exposures; measurement informed by prior social-media protest work (Freelon et al., 2018). Community orientation, activation, r1r_1, r2r_2, geographic reach. Chronological time predicts held-out mobilization as well as or better than τ; inferred alignment is an artifact of platform sampling. Brexit referendum and implementation Prespecified sequence of campaign exposures, official votes, court decisions, and implementation milestones (Hobolt, 2016). Distributions of policy preference and institutional commitment; bipolarity P. The clock merely encodes the known outcome, or preference changes are better explained by stable covariates and shocks outside the event set. Arab uprisings, 2010–2012 Within-country protest events and verified cross-border media exposures, with country-specific measurement models (Tufekci and Wilson, 2012). Participation, network reach, institutional response, cross-country heterogeneity. One shared clock fails measurement invariance; country outcomes require mechanisms absent from the phase model. COVID-19 communication and policy Separate epidemiological, policy, and information-event clocks rather than one post-hoc composite (Cinelli et al., 2020). Policy attitudes, protective behavior, vaccine attitudes, r1r_1 and P. Apparent synchronization disappears after accounting for disease incidence, policy coercion, bots, or changing sample composition. South Korean candlelight protests, 2016–2017 Weekly protest events plus a separately coded sequence of legislative and judicial steps (Kang, 2019). Participation, impeachment support, institutional state, recovery after the decision. The outcome follows institutional procedure without a detectable coupling transition, or alternative event codings yield incompatible clocks. Metaverse attention, 2021–2023 Product releases, investment announcements, active-device adoption, and developer activity measured separately (Dwivedi and others, 2022). Attention activation a¯ a, adoption, expectation orientation, response capacity. Attention and implementation are not dynamically coupled, or a conventional hype/adoption model predicts better with fewer degrees of freedom. The table also shows why causal potential should not be treated as a single latent energy. Protest events, institutional acts, disease incidence, media exposure, investment, and adoption have different measurement processes. A model may contain several clocks or a vector of event counts; combining them requires a theory and validation data. 6 Falsifiable Hypotheses and Empirical Protocol 6.1 Hypotheses The operational framework yields the following hypotheses. Each is stated with a failure condition. H1: cross-context clock invariance. For processes governed by the same mechanism but exposed to different event rates, trajectories expressed in a preregistered τ are more invariant than trajectories in t. Failure occurs if held-out alignment or likelihood does not improve relative to chronological-time and flexible-time baselines. H2: conditional coupling onset. After estimating heterogeneity, diffusion, and network scale, sustained r1r_1 should emerge near the conditional threshold appropriate to the fitted model. Failure occurs if no parameter-stable onset appears or if null networks reproduce it. H3: distinct consensus and bipolarity. Episodes with two opposed clusters should show high r2r_2, low r1r_1, and high P, whereas consensus should show high r1r_1 and low P. Failure occurs if these observables do not distinguish independently labeled states. H4: anchoring and intervention timing. Controlling for exposure and selection, stronger anchoring qiq_i should predict longer exit times after a standardized counter-message or institutional change. Failure occurs if estimated exit times are unrelated to independently measured anchoring. H5: consensus is insufficient for depletion. Low response capacity ChC_h, not high r1r_1 alone, should predict failure to adapt to novel input. Evidence that aligned groups adapt as readily as dispersed groups would reject the stronger consensus-causes-depletion claim while remaining consistent with the narrower model. 6.2 Measurement and estimation workflow A credible test should be prospective or pseudo-prospective: 1. Define the units, orientation scale, activation measure, network boundary, event types, and event weights without consulting the target outcome period. 2. Estimate or cross-fit λ(t∣ℋt)λ(t _t) on a training window. Record uncertainty in the resulting τ rather than treating it as exact. 3. Establish that orientation measurements are comparable across groups and time. Compute r1r_1, r2r_2, P, a¯ a, and, where possible, the perturbational response ChC_h. 4. Fit at least three models: M0M_0, the same dynamics in chronological time; M1M_1, TCU dynamics under the fixed causal clock; and M2M_2, a flexible time-warp baseline with equal or penalized complexity. 5. Compare preregistered held-out log likelihood, calibration, forecast error, and cross-context parameter stability. Use temporal block cross-validation rather than random shuffling. 6. Repeat the analysis on degree-preserving, timestamp-shuffled, and outcome-shuffled nulls. Report sensitivity to event weights, missing nodes, bots, and platform sampling. The decisive comparison is not whether M1M_1 fits, but whether it predicts better than M0M_0 without gaining the arbitrary flexibility of M2M_2. 6.3 Identification of influence Interaction data commonly confound influence with homophily and shared exposure. A descriptive fit of equation˜2 may estimate predictive coupling, but it does not identify WijW_ij as a causal effect. Causal interpretation requires a design: randomized message exposure, exogenous platform or policy changes, instrumental variables with defensible exclusion restrictions, or longitudinal structural assumptions accompanied by sensitivity analysis. If those conditions are absent, the paper should use association network and reserve causal for the philosophical level and the prespecified event clock. 6.4 Ethical boundary Models of intervention timing and collective alignment could be used for manipulation. Empirical work should favor aggregate reporting, data minimization, informed consent where individual intervention occurs, and independent ethical review. The framework does not justify steering public opinion toward a preferred direction. Its scientifically legitimate use is to describe, forecast with uncertainty, and test mechanisms. 7 Relation to Existing Theory TCU inherits its process orientation from accounts that treat becoming as fundamental (Heidegger, 1962; Whitehead, 1978). It differs from physical accounts of the temporal arrow (Reichenbach, 1956; Prigogine, 1980; Price, 1996) by making no claim to derive macroscopic irreversibility from microphysics. The causal-progress clock is closer to event rescaling than to a new physical dimension (Brown et al., 2002). Mathematically, equation˜2 belongs to the family of coupled-oscillator models (Kuramoto, 1984; Acebrón et al., 2005; Arenas et al., 2008); the order parameter and conditional threshold are established results used here as a disciplined substrate. The proposed contribution is their integration with a separately measured event clock, an activation layer, anchoring, and explicit philosophical semantics. Threshold and cascade models (Granovetter, 1978; Watts, 2002; Centola and Macy, 2007) may outperform oscillators when adoption is discrete or reinforcement is complex. Such comparisons are part of the falsification program, not competing views to be dismissed. Social-dynamics work has long shown that averaging, network structure, and selective exposure can produce consensus, fragmentation, or persistent disagreement (DeGroot, 1974; Friedkin and Johnsen, 1990; Castellano et al., 2009; Del Vicario et al., 2016). TCU does not claim priority for these mechanisms. Its value, if borne out, would lie in showing that a constrained causal-progress coordinate improves transfer across episodes with different calendar speeds. 8 Limitations and Boundary Conditions First, the ontological identity proposed by TCU is underdetermined by the network model. The same equations can be interpreted without accepting the philosophical thesis. Empirical success would support the operational clock, not prove metaphysical identity. Second, τ is scalar. Real systems may contain multiple asynchronous processes—legal, epidemiological, economic, and informational—that cannot be compressed without loss into one ordering. Partial orders or vector clocks may be more appropriate. Third, circular phase is suitable for orientations with periodic or directional structure but not for every belief. Discrete threshold models, bounded-confidence dynamics, or multidimensional latent spaces may be better for particular domains. Fourth, the mean-field critical value in equation˜9 is conditional. It cannot be transferred to empirical networks by relabeling D as noise and W as causality. Every application must state its normalization and estimate uncertainty. Fifth, historical analogies are vulnerable to hindsight, selection, and survivorship bias. The six cases in table˜3 have not been fitted and do not validate the model. A future study should use event-level data and precommitted alternatives. Finally, terms such as condensation, freezing, and phase transition are useful only when operational definitions accompany them. They do not imply quantum superposition, wave-function collapse, or the quantum Zeno effect in human cognition. 9 Conclusion Temporal–causal unity begins from a strong intuition: time is experienced not as an empty axis but through the directed realization of change. This paper turns that intuition into a constrained scientific proposal by separating interpretation from measurement and measurement from dynamics. The causal-progress coordinate τ is meaningful only when its event intensity is specified independently; the network model is meaningful only when its states, scales, and influence design are explicit; historical episodes are informative only when they generate risky predictions rather than retrospective labels. Under those constraints, TCU offers a coherent program for studying collective tempo, alignment, polarization, anchoring, and adaptive capacity. Its next step is not a broader metaphysical claim but a narrower empirical test: preregister one clock, compare it with chronological and flexible-time baselines, and evaluate prediction on held-out processes. Failure of causal time to improve invariance would count against the operational framework. Success would not replace established physics, but it would show that the amount of causally relevant activity is a useful clock for collective dynamics. Acknowledgments and tool-use disclosure The conceptual framework originated with the human author. OpenAI Codex was used to assist with English drafting, mathematical consistency checks, LaTeX preparation, and simulation-code scaffolding. The human author must verify the claims, references, code, and final wording and assumes responsibility for the submitted work. Generative software is not an author. Appendix A Derivation and Invariance Notes A.1 Mean-field fixed points Writing equation˜8 as r˙1=r1[A−(K/2)r12] r_1=r_1[A-(K/2)r_1^2] with A=K/2−(Δ+D)A=K/2-( +D) gives the incoherent fixed point r1=0r_1=0. If A>0A>0, the nonzero fixed point satisfies r12=2AK=1−2(Δ+D)K.r_1^2= 2AK=1- 2( +D)K. At K=KcK=K_c, the branch emerges continuously. This is a supercritical onset for the stated special case; discontinuous or hysteretic transitions require other distributions, coupling rules, adaptation, inertia, or correlations. A.2 Rotation invariance Under a global change of angular origin θi↦θi+ϕ _i _i+φ, one has Zm↦eimϕZmZ_m e^imφZ_m, so rmr_m and P are unchanged. In contrast, ⟨|θi|⟩ | _i| changes with ϕφ and is therefore not an intrinsic measure of extremity on the circle. A.3 Clock scaling If all event weights are multiplied by c>0c>0, then τ′=cτ =cτ. Parameters with inverse-τ units transform as νi′=νi/c _i = _i/c, K′=K/cK =K/c, qi′=qi/cq_i =q_i/c, and Di′=Di/cD_i =D_i/c. Predictions are unchanged only if this scale convention is propagated. A paper must therefore report the event-weight normalization before comparing parameters across datasets. Appendix B Traceability to the Motivating Hypotheses Table 4: Status of the twelve motivating hypotheses after operationalization. Original Motivating statement Status in the present framework H1 Time and causation are two aspects of one essence. Retained as TCU-1, explicitly interpretive rather than proven. H2 Space is a synchronic slice of temporal–causal unfolding. Retained only as a descriptive state-slice interpretation; no elimination of physical space. H3 Causation is the tangent vector of temporal unfolding. Replaced by the measurable event intensity and directed state drift; not a differential-geometric claim. H4 Cognition is a complex amplitude on a time–cause plane. Recast as the classical state ψi=aieiθi _i=a_ie^i _i; no quantum meaning. H5 Temporal–causal unfolding obeys a dynamical equation. Retained as the explicitly scoped stochastic model (2)–(3). H6 Synchrony r is the system order parameter. Refined to r1r_1, r2r_2, P, activation, and response capacity. H7 Cognition is subjectivized temporal–causal unfolding. Retained as interpretation; empirical units require validated cognitive measures. 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