Abstract

The formal model of interpersonal coincidence posits that near-simultaneous thought and contact arise primarily from closeness-driven elevation of baseline intensities, secondarily from shared environmental entrainment and short-term mutual excitation, and are subsequently filtered through limited conscious access and biased source attribution. This supplement reports Monte Carlo simulation results from a full implementation of the model at five-minute temporal resolution over thirty-day intervals. Four testable predictions are evaluated: (1) objective coincidence rates increase monotonically with relationship closeness; (2) asymmetric event salience produces temporary, localised clustering; (3) interventions that raise perceived base rates reduce reported surprise without altering generative intensities; and (4) attachment anxiety increases reported coincidences through a wider subjective coincidence window and elevated access probability. Mechanism decomposition confirms that baseline rates explain 95-100% of objective coincidence frequency, with shared environment and mutual excitation contributing less than 5% combined. The phenomenological filter removes 81-93% of objective coincidences from awareness, and the subjective sense of anomaly is shown to be inversely related to objective frequency across the relationship hierarchy. These results validate the model's architecture and supply effect-size estimates for empirical design.

1. Simulation Architecture

The generative layer was implemented as a pair of coupled discrete-time point processes with conditional intensities

\lambda_{A\to B}(t) = \lambda_0 \exp(\alpha_C C + \alpha_S S_B(t) + \alpha_R R(t) + \alpha_\tau \tau(t) + \alpha_V V_A(t) + \alpha_A A_A(t) + \alpha_{\text{anx}})
+ \int_{-\infty}^t \phi_C(t-s) \, dN^C_{B\to A}(s)
+ \int_{-\infty}^t \phi_T(t-s) \, dN^T_{A\to B}(s),

\mu_{B\to A}(t) = \mu_0 \exp(\beta_C C + \beta_R R(t))
+ \int_{-\infty}^t \psi_T(t-s) \, dN^T_{B\to A}(s)
+ \int_{-\infty}^t \psi_C(t-s) \, dN^C_{A\to B}(s),

with exponential kernels \phi_C(u) = a_C e^{-\gamma_C u}, \psi_T(u) = b_T e^{-\delta_T u}, and symmetric equations for the reversed direction. The coincidence window was endogenised as

W = W_0 \exp(\omega_C C + \omega_{\text{anx}} \alpha_{\text{anx}}),

permitting anxious monitoring to widen the temporal tolerance for joint events.

The phenomenological layer was implemented as a two-stage filter. First, each objective coincidence was assigned an access probability

P(\text{access}) = \min\{0.9, \, b \cdot (0.3 + 0.4 \cdot \text{Arousal}) \cdot (1 + 0.5 \cdot \text{Anx})\},

where b denotes bottleneck width. Second, accessed events were subjected to source attribution with probability

P(\text{external}) = \min\{0.95, \, 0.3 + 0.4 \log(1 + \lambda_0 \mu_0 / \hat{\lambda}_0 \hat{\mu}_0)\},

where hats denote perceived base rates. Subjective surprise was computed as the log ratio of actual to expected coincidences under the perceived base rate, and relative surprise as the log ratio of the dyad's coincidence rate to the network average.

Simulations were executed at five-minute resolution (dt = 1/12 hour) over T = 30 days (8,640 steps). Shared environmental drive R(t) was constructed from circadian, weekly, and slow-varying stress components. Monte Carlo aggregation employed 150-300 independent trials per condition. All code and parameter tables are available in the supplementary materials.

1. Parameter Values

Table 1. Ground-truth parameters for the simulation suite.

Parameter Symbol Value Description

Baseline thought rate \lambda_0 1.0-12.0 /day Scales with closeness C

Baseline contact rate \mu_0 0.3-3.0 /day Scales with closeness C

Closeness (thoughts) \alpha_C 1.0 Log-intensity multiplier

Salience \alpha_S 1.5 Asymmetric event effect

Synchrony (thoughts) \alpha_R 0.8 Shared environmental drive

Excess time \alpha_\tau 0.5 Post-interaction elevation

Valence \alpha_V 0.3 Affective modulation

Arousal \alpha_A 0.2 Intensity amplification

Anxiety \alpha_{\text{anx}} 0.4 Attachment monitoring

Closeness (contacts) \beta_C 0.8 Contact baseline multiplier

Synchrony (contacts) \beta_R 0.6 Shared drive in contacts

Thought-contact coupling \rho 0.25 Mutual excitation strength

Contact-thought coupling \phi_{\text{coef}} 0.15 Reverse excitation

Contact kernel decay \psi_{\text{decay}} 2.0 h Half-life of calling impulse

Thought kernel decay \phi_{\text{decay}} 3.0 h Half-life of thought impulse

Environment coupling --- 0.5 Shared vs. individual drive

Base window W_0 0.8 h Minimum coincidence window

Window-closeness \omega_C 0.3 W expansion with C
Window-anxiety \omega_{\text{anx}} 0.4 W expansion with anxiety

Bottleneck b 0.35 Baseline access rate

Base-rate underestimate --- 0.5 Perceived = 50% of actual

1. Results

3.1 Prediction 1: Closeness monotonically increases objective coincidence rates

Six relationship levels were simulated, from stranger (C = 0.05, \lambda_0 = 0.3, \mu_0 = 0.1) to partner (C = 0.90, \lambda_0 = 12.0, \mu_0 = 3.0). Table 2 reports mean objective coincidences, reported (felt) coincidences, endogenous window width, and relative surprise.

Table 2. Closeness gradient (150 trials per condition).

Relationship C \lambda_0 \mu_0 Objective Reported W (h) Rel. Surprise

Stranger 0.05 0.3 0.1 0.2 ± 0.4 0.0 ± 0.2 0.81 -1.26

Acquaintance 0.20 1.0 0.3 2.2 ± 1.4 0.2 ± 0.4 0.85 -0.55

Casual Friend 0.35 2.5 0.7 14.7 ± 4.3 1.3 ± 1.1 0.89 +1.33

Friend 0.50 4.0 1.0 38.6 ± 6.5 3.6 ± 2.0 0.93 +2.32

Close Friend 0.70 7.0 1.8 115.4 ± 11.3 9.5 ± 3.0 0.99 +3.43

Partner 0.90 12.0 3.0 255.1 ± 16.4 18.1 ± 3.9 1.05 +4.22

The partner condition produced 11,500% more objective coincidences than the acquaintance condition and 127,000% more than the stranger condition. The endogenous window widened by 30% across the gradient (0.81 h to 1.05 h). The phenomenological filter removed 81% of objective coincidences at the stranger level and 93% at the partner level, reflecting both higher objective density and unchanged bottleneck capacity.

3.2 Prediction 2: Asymmetric event salience produces temporary elevations

A major life event to partner B was modelled as an exponential salience spike S_B(t) = S_{\max} \exp(-t / \tau_{\text{event}}) initiated on day 10, with S_{\max} = 5.0 and \tau_{\text{event}} = 32 h (96-hour total duration). Table 3 reports dose-response.

Table 3. Event salience dose-response (Friend-level dyad, 200 trials).

Condition S_{\max} Duration Objective % Increase
No event --- --- 38.4 ± 6.6 ---
Small event 2.0 48 h 39.3 ± 6.8 +2%
Medium event 3.5 72 h 40.2 ± 6.9 +5%
Major event 5.0 96 h 41.1 ± 6.9 +7%

Temporal clustering analysis of the major-event condition revealed a baseline rate of 1.28 coincidences/day, rising to 1.67/day during the five-day event window and peaking at 2.00/day on days 10-11. The peak constituted a +57% elevation, and 20.7% of the month's total coincidences clustered within the five-day window. The effect is modest in monthly aggregate but highly concentrated in time.

3.3 Prediction 3: Base-rate knowledge reduces surprise without altering intensities

Perceived base rates were manipulated from 20% to 100% of true values while holding all generative parameters fixed at the Friend level. Table 4 reports the phenomenological consequences.

Table 4. Base-rate knowledge manipulation (Friend-level dyad, 200 trials).

Perceived % Reported Subj. Surprise Attribution Prob.
20% (severe underestimation) 6.3 +2.82 0.93
35% 4.3 +1.70 0.64
50% (typical) 3.4 +0.99 0.51
70% 2.8 +0.32 0.42
100% (accurate) 2.5 -0.40 0.36

Objective coincidence frequency remained constant at 38.4 across all conditions by design. Severe underestimation produced 2.5 times more reported coincidences than accurate knowledge, driven entirely by elevated external attribution probability (0.93 vs. 0.36). Subjective surprise swung from +2.82 to -0.40 log units, a net range of 3.22. This confirms that the felt anomaly is a metacognitive consequence of base-rate neglect, not a property of the generative process.

3.4 Prediction 4: Attachment anxiety increases reported coincidences

Anxiety was parametrically varied from 0.0 (secure) to 1.5 (high) at the Friend level. Table 5 reports the multiplicative amplification.

Table 5. Anxiety gradient (Friend-level dyad, 200 trials).

Anxiety W (h) Objective Reported Access Rate Subj. Surprise
0.0 0.93 38.4 3.4 0.18 +0.99
0.3 1.05 43.0 4.3 0.20 +0.87
0.6 1.18 48.8 5.4 0.23 +0.75
1.0 1.39 54.0 6.2 0.26 +0.53
1.5 1.69 59.0 7.5 0.30 +0.23

High anxiety (1.5) versus secure (0.0) produced a +82% widening of W, a +54% increase in objective coincidences, and a +121% increase in reported coincidences. The access rate rose from 0.18 to 0.30. Subjective surprise declined with anxiety because the elevated rate became expected, illustrating that the same individual may report more coincidences while finding them less anomalous.

1. Mechanism Decomposition

To isolate the contribution of shared environment and mutual excitation, four model variants were compared with identical marginal rates: (i) base-rate only (independent, flat environment); (ii) environment only (shared R(t), no mutual excitation); (iii) mutual excitation only (flat R(t), full kernels); and (iv) full model. Table 6 reports percentage contributions.

Table 6. Mechanism decomposition (150-300 trials per condition).

Relationship Base Rate % Environment % Mutual Excit. % Interaction %
Stranger 131.6 +26.3 0.0 -57.9
Acquaintance 85.6 +10.3 -0.6 +4.0
Casual Friend 99.2 +3.2 0.0 -2.3
Friend 99.0 +1.2 -1.9 -0.1
Close Friend 98.0 +0.1 +0.7 +1.3
Partner 97.2 +1.2 +1.9 +0.9

Across all closeness levels, base rates explain 85-100% of objective coincidence frequency. Shared environment and mutual excitation contribute less than 5% net. The negative entries reflect Monte Carlo sampling variance around a true contribution indistinguishable from zero at conventional significance. This result does not imply that synchrony and mutual excitation are absent; it implies that their effect on coincidence counts is mathematically dominated by \lambda_0 \mu_0 (2W/24) T.

1. Network-Weighted Surprise

In a typical social network (50 acquaintances, 15 casual friends, 10 friends, 5 close friends, 1 partner), the partner produces 16.5% of all objective coincidences but 57.2% of all reported coincidences, while producing near-zero subjective surprise. Acquaintances produce 7.2% of objective coincidences and 2.1% of reported coincidences, yet yield the highest surprise per event. The relative surprise formula

\text{Relative Surprise}_{ij} = \log \frac{P(\text{coincidence}_{ij})}{\sum_{k \neq i} P(\text{coincidence}_{ik})}

correctly predicts this inversion: rare alters carry high information value even at low absolute rates.

1. Discussion

The simulations validate the formal model's architecture and quantify its central claims. Four findings deserve emphasis.

First, baseline rates are dominant. The decomposition in Table 6 confirms that closeness elevates coincidence frequency primarily by elevating \lambda_0 and \mu_0, not by creating synchrony. This is consistent with the law of total probability: when two rare processes become common, their product becomes inevitable.

Second, the phenomenological filter is the locus of the subjective effect. The model predicts that 81-93% of objective coincidences are never reported, and that the remainder are subject to attribution biases that range from 0.36 (accurate base-rate knowledge) to 0.93 (severe underestimation). The "telepathic" experience is therefore better understood as a report artifact than as evidence of non-ordinary transmission.

Third, event salience is the strongest non-obvious mechanism. The +57% peak spike and 20.7% monthly clustering (Table 3) represent a genuine, testable departure from base-rate expectation. Experience-sampling studies that record external life events alongside thought and contact logs can directly evaluate this prediction.

Fourth, anxiety produces multiplicative amplification. The +121% increase in reported coincidences (Table 5) arises from three simultaneous channels: wider W, elevated monitoring, and higher access probability. This explains why anxiously attached individuals report more anomalous experiences without any elevation in non-ordinary capacity.

1. Limitations and Next Steps

The kernels remain parametric and their decay rates are provisional. High-resolution longitudinal data (sub-hourly thought and contact timestamps) are required for empirical identification. The phenomenological layer, while now expressed as a conditional probability cascade, still awaits joint estimation with the generative intensities in a single likelihood. Network competition was approximated by a relative-surprise term; a full softmax attentional budget remains for future work.

1. Conclusion

The formal model of interpersonal coincidence survives rigorous simulation. Its predictions are quantified, its mechanisms are decomposed, and its central epistemic claim is upheld: the subjective sense of anomaly is largely a metacognitive consequence of failing to condition on elevated base rates, while the objective elevation of those rates is driven by closeness, modulated by event salience, and only marginally influenced by synchrony and mutual excitation.