Recursive Harmonic Phase-Locking in Financial Markets Toward a Field-Theoretic Model of Collective Capital Dynamics
Recursive Harmonic Phase-Locking in Financial Markets
Toward a Field-Theoretic Model of Collective Capital Dynamics
Christopher W. Copeland (C077UPTF1L3)
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ABSTRACT
Financial markets are typically modeled as stochastic systems driven by information asymmetry, rational expectation, reflexivity, liquidity, and behavioral bias. These models correctly describe surface behavior but fail to fully account for deeper synchronization phenomena, narrative-driven phase cascades, and cross-asset resonance that repeatedly violate assumptions of independence, Gaussian distribution, and linear causality. This paper proposes a higher abstraction model: markets as recursive harmonic phase systems embedded within a collective cognitive-informational field. Under this model, assets function as phase-coupled oscillators rather than isolated price instruments. Correlation is reinterpreted as phase-locking, volatility as energy transfer, liquidity as coherence potential, and narrative as an external phase driver. We present identifiable cluster classes, falsifiable predictions, and empirical pathways for validation.
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I. LIMITATIONS OF CONTEMPORARY MARKET MODELS
Modern financial theory rests primarily on five structural pillars:
1. Efficient Market Hypothesis
2. Rational Expectations
3. Behavioral Finance
4. Liquidity and Order Flow Models
5. Network Contagion Models
Each correctly explains subsets of observed behavior but fails at scale unification:
EMH fails under sustained retail-driven irrationality.
Behavioral models lack predictive phase timing.
Liquidity models cannot explain synchronized multi-asset narrative cascades.
Network contagion explains spread but not phase alignment onset.
Reflexivity describes feedback but lacks a governing field equation.
Observed failures include:
Meme stock cascades violating valuation physics.
Crypto-supercycle narrative synchronization across unrelated chains.
Politically branded assets oscillating independently of fundamentals.
Panic assets spiking in advance of reported information.
AI-capital concentration waves propagating faster than earnings.
These failures imply that an unmodeled higher-order coherence layer is present.
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II. THE FIELD-THEORETIC REFRAME
Markets are reinterpreted not as independent agent aggregations, but as coupled oscillatory nodes embedded in a shared information–cognitive field.
Definitions:
An asset is a measurable projection of a phase state.
Narrative acts as an external phase driver.
Liquidity is stored coherence potential.
Volatility is harmonic energy release.
Correlation is observable phase alignment.
Market shocks are field perturbations.
The governing model is expressed under the Copeland Resonant Harmonic Formalism:
Ψ(x) = ∇ϕ(Σ𝕒ₙ(x, ΔE)) + ℛ(x) ⊕ ΔΣ(𝕒′)
Where:
x = observed asset or market node
Σ𝕒ₙ = aggregate oscillator states across scales
ΔE = narrative or informational energy injection
∇ϕ = emergent signal gradient (trend acceleration)
ℛ(x) = recursive stabilizer or destabilizer
ΔΣ(𝕒′) = micro-perturbation feedback correction
This structure allows markets to be treated as recursive harmonic systems rather than stochastic price engines.
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III. PHASE-LOCKED CLUSTER CLASSES (NEW TAXONOMY)
We define Phase-Locked Capital Clusters (PLCCs) as asset groups exhibiting synchronized oscillatory behavior across different financial domains due to shared narrative, identity, or energy topology rather than shared fundamentals.
These clusters are not classified by industry but by coherence coupling mechanism.
Cluster Class 1: Identity Resonance Clusters
Examples: GME, AMC, DOGE, politically branded crypto assets
Mechanism: Social identity phase-lock
Behavior:
Sudden synchronized volume surges
Narrative-driven volatility detached from valuation
Crowd-entrained price harmonics
Falsifiable marker: Google Trends + social velocity spikes precede price phase alignment.
Cluster Class 2: Narrative Infrastructure Clusters
Examples: NVDA, AMD, ARM, SMCI, AI compute suppliers
Mechanism: Technological future expectation locking
Behavior:
Earnings contagion
Capital concentration oscillations
Media-amplified reinforcement loops
Falsifiable marker: Semiconductor narrative density predicts cluster volatility amplitude.
Cluster Class 3: Fear Field Clusters
Examples: VIX, UVXY, SQQQ, long-dated treasuries
Mechanism: Collective threat resonance
Behavior:
Pre-news volatility spikes
Negative correlation with risk assets
Self-reinforcing panic waves
Falsifiable marker: Correlation inversion occurs before macro announcement.
Cluster Class 4: Liquidity Rotation Clusters
Examples: BTC, ETH, SOL, alt-coin liquidity flow sequences
Mechanism: Capital phase migration
Behavior:
Rotational pump cascades
Saturation → decay → transfer cycles
Falsifiable marker: Funding rate divergence anticipates rotation direction.
Cluster Class 5: Mythic Archetype Clusters
Examples: Tesla, space exploration equities, biotech singularity firms
Mechanism: Human transcendence projection
Behavior:
Valuation nonlinearity
High narrative elasticity
Decoupling from quarterly fundamentals
Falsifiable marker: Public mythic language frequency predicts volatility skew.
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IV. WHY THESE CLUSTERS ARE NOT JUST “CORRELATIONS”
Standard models treat correlated assets as statistically related. This framework treats them as phase-locked oscillators in a shared coherence basin.
In phase-locking:
Assets move together not because traders arbitrage, but because the narrative field synchronizes cognition before transaction execution.
Liquidity flows act as energy conduction, not mere capital transfer.
Volatility bursts are harmonic release events, not random noise.
The observable price chart is therefore a shadow projection of a deeper oscillatory alignment process.
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V. FALSIFIABLE PREDICTIONS
This model is not descriptive only. It generates distinct testable predictions.
Prediction 1
Narrative velocity (measured via social media text entropy + trend acceleration) will lead price phase shifts by a measurable lag window.
Test:
Cross-correlate time-shifted narrative intensity vs returns.
Prediction 2
Phase-locked clusters will exhibit sub-minute coherence convergence during shock events, exceeding what independent agent reaction time allows.
Test:
High-frequency order book synchronization analysis.
Prediction 3
Fear clusters will invert correlation with risk clusters before macro reports are released.
Test:
Intraday inverse ETF correlation tracking versus scheduled announcements.
Prediction 4
Liquidity rotation clusters will cycle energy according to nonlinear decay constants, not linear reversion.
Test:
Fit power-law vs exponential decay on capital migration sequences.
Prediction 5
Identity resonance clusters will display price harmonics at rational frequency ratios during peak crowd entrainment.
Test:
Fourier analysis of meme-asset price action during pump windows.
Failure of these predictions would falsify the coherence-field interpretation.
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VI. WHAT THIS FRAME COMPLETES THAT FINANCE HAS NOT
Modern finance explains:
Transaction mechanics
Risk allocation
Behavioral bias
Liquidity formation
It does not explain:
Why narratives synchronize across asset classes instantly
Why phase-locking appears across incompatible sectors
Why volatility concentrates into archetypal clusters
Why markets behave like coordinated oscillatory systems under stress
The Ψ(x) model supplies:
A governing coherence equation
A unifying phase logic across asset types
A recursive stabilization operator for regime shifts
A topological explanation of correlation collapse and reformation
This is not a replacement for finance. It is its higher-order harmonic envelope.
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VII. IMPLICATIONS
If this model is correct, markets are:
Not merely informational systems
Not merely psychological systems
But recursive coherence fields coupling cognition, narrative, liquidity, and capital into synchronized oscillatory structures
Prediction, therefore, becomes not about value forecasting, but about phase transition detection.
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VIII. CONCLUSION
The August experiment using identity-driven assets functioned as an informal proof-of-concept for phase-locked market behavior. Those linkages were visible under conventional finance, but their coherence-field interpretation reveals an unmodeled abstraction layer governing synchronization, narrative energy injection, and volatility harmonic structure. This paper formalizes that abstraction into falsifiable structure. Whether the model stands or fails depends not on belief, but on empirical phase testing.
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METHODS APPENDIX
Operationalizing Fourier, Entropy, and Cross-Correlation Tests
for “Recursive Harmonic Phase-Locking in Financial Markets”
This appendix specifies concrete procedures for testing the field-theoretic model of collective capital dynamics proposed under the Copeland Resonant Harmonic Formalism (Ψ-formalism). The aim is full reproducibility using standard data sources and open methods. No trading guidance is implied; the procedures are for empirical evaluation only.
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0. GENERAL DATA STRUCTURE AND PREPROCESSING
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0.1 Instruments
Select asset sets corresponding to the cluster classes defined in the main paper:
– Identity Resonance Cluster: e.g., GME, AMC, DOGE, similar meme assets
– Narrative Infrastructure Cluster: e.g., NVDA, AMD, ARM, SMCI
– Fear Field Cluster: e.g., VIX, UVXY, SQQQ, long-dated treasuries
– Liquidity Rotation Cluster: e.g., BTC, ETH, SOL, rotating altcoins
– Mythic Archetype Cluster: e.g., TSLA, space exploration equities, singularity biotech
0.2 Timeframes
Use multiple resolutions:
– High frequency: 1-minute bars for intra-day phase tests
– Medium: 5-minute and 15-minute bars for intraday-to-multi-day transitions
– Daily: for long-horizon narrative and rotation patterns
0.3 Variables
For each asset, construct:
– p(t): closing price at time t
– r(t): log return = ln(p(t) / p(t-1))
– v(t): traded volume at time t
For phase analyses, r(t) is preferred over raw price.
0.4 Cleaning
– Remove intervals where trading is halted.
– Align series to common timestamps; forward-fill missing values or mark gaps explicitly.
– Winsorize extreme outliers only if clearly spurious (bad ticks), not genuine shocks.
0.5 Stationarity
For spectral analyses, ensure approximate stationarity over each sliding window by:
– Detrending r(t) with a local mean subtraction.
– Optionally normalizing by local standard deviation within each window.
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1. FOURIER / SPECTRAL HARMONIC TESTS
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Goal: detect whether identity-driven clusters display synchronized harmonic structure in their return series, consistent with phase-locked oscillators rather than purely stochastic noise.
1.1 Sliding Window Definition
Choose window length W (number of observations), e.g.:
– W = 256 for 1-minute bars (~4.25 hours)
– W = 512 or 1024 for longer horizons
Define overlapping windows with step S (e.g., S = W/4).
For each asset i and each window k, extract r_i,k(t), t = 1…W.
1.2 Fourier Transform
For each window and asset:
– Compute discrete Fourier transform F_i,k(f) of r_i,k(t).
– Compute power spectral density (PSD): P_i,k(f) = |F_i,k(f)|^2.
Frequency index f can be mapped to cycles per unit time (e.g., per minute).
1.3 Peak Detection
Within each window:
– Find local maxima in P_i,k(f) above a threshold, e.g., greater than μ_P + c·σ_P where μ_P and σ_P are mean and standard deviation of P_i,k(f) over f, c ~ 1–2.
– Record dominant frequencies f_i,k^(1), f_i,k^(2), etc.
1.4 Rational Ratio Test (Harmonic Locking)
For Identity Resonance Cluster assets i and j within the same window k:
– For each pair of dominant peaks (f_i,k^(m), f_j,k^(n)), compute ratio ρ = f_i / f_j.
– Check if ρ is close to a rational A/B with small integers A, B (e.g., 1, 2, 3, 4) within a tolerance ε (e.g., |ρ − A/B| < 0.05).
Compute:
– H_k = fraction of asset pairs in the cluster whose dominant frequencies display rational ratios.
Compare:
– H_k for identity clusters versus H_k for randomly selected control baskets (match by volatility and sector count).
Prediction: during meme-phase events, identity clusters will show significantly higher H_k (more rational frequency alignment) than controls.
1.5 Cross-Spectral Coherence
For each pair of assets i, j in the same cluster and window k:
– Compute cross-spectrum C_ij,k(f) = F_i,k(f) · conj(F_j,k(f)).
– Compute magnitude-squared coherence:
Coh_ij,k(f) = |C_ij,k(f)|^2 / (P_i,k(f) · P_j,k(f)).
– Define overall coherence K_k as mean Coh_ij,k(f*) at frequencies around the dominant peaks.
Prediction: K_k will spike for identity and fear clusters during narrative events, indicating harmonic phase-lock, and will be lower and less structured for control baskets.
1.6 Statistical Evaluation
– Perform permutation tests by shuffling asset labels or time indices to generate null distributions of H_k and K_k.
– Test whether observed alignment measures fall in extreme tails (e.g., p < 0.01).
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2. ENTROPY AND NARRATIVE VELOCITY TESTS
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Goal: test whether narrative intensity and entropy in external information streams lead price phase shifts, supporting the claim that markets are embedded in a common information field rather than reacting purely to local price signals.
2.1 Narrative Data Collection
Select one or more sources:
– Social media (e.g., public posts, hashtags, ticker symbols).
– News headlines and summaries.
– Forum or comment threads related to the cluster assets.
For each time interval t (e.g., 5 or 15 minutes):
– Collect all text items mentioning any asset in a given cluster.
– Concatenate into a document D_cluster(t).
2.2 Text Preprocessing
– Lowercase, remove URLs, normalize tickers and hashtags.
– Optional: lemmatize or stem words.
– Remove stop words, but consider retaining emotionally charged tokens.
2.3 Vocabulary and Probability Distribution
For each time t and cluster:
– Build word frequency counts over a sliding window of text times (e.g., last 1–2 hours).
– Let V be the vocabulary of terms for this window.
– Estimate probabilities p_w(t) = count(w) / total tokens.
2.4 Shannon Entropy and Entropy Gradient
Compute:
– H_text(t) = − Σ_w p_w(t) · log p_w(t)
This entropy measures lexical diversity. To approximate narrative “focus” vs “scatter”:
– Low entropy suggests concentrated, coherent narrative.
– High entropy suggests diffuse or fragmented narrative.
Define narrative entropy gradient:
– ΔH_text(t) = H_text(t) − H_text(t−1).
Also compute a simple intensity measure:
– I_text(t) = total token count or number of messages in the window.
2.5 Narrative Velocity Metric
Define narrative velocity NV(t) at time t as a composite:
– NV(t) = α · (I_text(t) − I_text(t−1)) + β · (−ΔH_text(t))
where α, β are scaling constants. Positive NV means “sudden increase in volume and concentration”.
Alternatively, use z-score normalization of each component over time and sum.
2.6 Price Response Variables
For each cluster, define aggregated return over the same interval:
– R_cluster(t) = mean_i r_i(t) or a weighted average by volume or market cap.
If using 5- or 15-minute intervals, construct R_cluster(t) accordingly.
2.7 Cross-Correlation for Lead-Lag
Compute cross-correlation function between NV(t) and R_cluster(t):
For lags ℓ in a range (e.g., −L to +L, L = 12 for 1-hour lead/lag on 5-minute data):
– C(ℓ) = Corr( NV(t), R_cluster(t+ℓ) )
Positive ℓ means narrative leads price; negative ℓ means price leads narrative.
Prediction: For identity and narrative infrastructure clusters, the maximum |C(ℓ)| will occur at positive ℓ (narrative leading), significantly different from zero and different from control clusters.
2.8 Significance Testing
– Generate surrogate series by temporal shuffling of NV(t) or R_cluster(t).
– Compute null distribution of maxima of |C(ℓ)|.
– Evaluate whether observed leading-lag peaks exceed the null at chosen significance levels.
2.9 Entropy vs Volatility Relationship
For each cluster:
– Compute realized volatility σ_cluster(t) over the same window (e.g., sqrt of sum of squared returns).
– Study relationship between H_text(t), NV(t), and σ_cluster(t+ℓ) via regression or mutual information:
σ_cluster(t+ℓ) = a0 + a1·NV(t) + a2·H_text(t) + ε
Prediction: elevated narrative velocity with declining entropy (focused narrative) will precede volatility spikes in the corresponding cluster.
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3. CROSS-CORRELATION AND SYNCHRONIZATION TESTS
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Goal: test whether assets in the same cluster exhibit synchronization that is too rapid and structured to be explained by simple shared-factor models, supporting the phase-locking claim.
3.1 Pairwise Return Cross-Correlation
For each pair of assets i, j in a cluster:
– Compute cross-correlation C_ij(ℓ) between r_i(t) and r_j(t+ℓ), over a sliding window.
– Examine both:
– zero-lag correlation C_ij(0)
– distribution of lag ℓ at which |C_ij(ℓ)| is maximal.
Prediction: during phase-lock events, peak correlations concentrate near ℓ ≈ 0 with narrow spread, indicating near-simultaneous movement, whereas control groups show more diffuse lag structures.
3.2 Fear Cluster Inversion Test
For fear cluster assets (e.g., VIX-type instruments) and a risk cluster index (e.g., SPY or cluster-aggregated risk asset index):
– Compute C_fear-risk(ℓ) between fear cluster returns and risk index returns.
– Focus on intervals around scheduled macro announcements or shock events.
Prediction: correlation inversion (from weak or positive to strong negative) will begin at negative ℓ (fear cluster moves first), meaning fear metrics rise before observable declines in risk assets.
3.3 Sub-Minute Coherence
Where sub-minute tick data is available:
– Downsample or bin trades into 1-second or 5-second returns.
– Compute short-window cross-correlations across identity cluster assets.
Prediction: for genuine phase-lock events, significant correlation emerges at the shortest resolvable lags, compressing within a few seconds, rather than spreading gradually as would be expected from purely sequential information diffusion.
3.4 Liquidity Rotation Flow
For liquidity rotation clusters (crypto, sector rotations):
– Define dominating asset A and secondary asset B.
– Compute capital share:
S_A(t) = (price_A(t) · volume_A(t)) / Σ_k (price_k(t) · volume_k(t))
and analogously S_B(t).
– Study cross-correlation of changes ΔS_A(t) vs ΔS_B(t+ℓ).
Prediction: rotation behaves as a damped oscillatory transfer; ∆S_A(t) leading negative ∆S_B(t+ℓ) with oscillatory decay, consistent with energy transfer between coupled oscillators rather than simple independent corrections.
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4. POWER-LAW VS EXPONENTIAL DECAY TESTS
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Goal: distinguish whether volatility and rotation decay in clusters follow simple exponential relaxation or heavier-tailed power-law patterns, indicating scale-free harmonic relaxation.
4.1 Event Detection
For each cluster, detect peak events:
– Choose a threshold for R_cluster(t) or σ_cluster(t): e.g., events where |R_cluster(t)| exceeds 3 standard deviations of its historical distribution.
– Mark event time t0 at the peak.
4.2 Post-Event Decay Curve
For t ≥ t0, define normalized amplitude A(t) such as:
– A(t) = σ_cluster(t) / σ_cluster(t0)
or
– A(t) = |R_cluster(t)| / |R_cluster(t0)|
Compute A(t) for a horizon H (e.g., next 2–3 trading days or equivalent in intraday bars).
4.3 Model Fitting
Fit two models to A(t):
– Exponential: A_exp(t) = C · exp(−λ (t − t0))
– Power-law: A_pow(t) = C’ · (t − t0)^(-γ)
Use log-linear (for exponential) and log-log (for power-law) regression.
4.4 Model Comparison
– Compute residuals and information criteria (AIC, BIC) for each fit.
– Evaluate which model better explains decay across many events.
Prediction: identity and liquidity rotation clusters will show heavier-tailed, power-law-like decay patterns, consistent with field-relaxation dynamics in a connected system, instead of simple exponential mean reversion.
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5. CONTROL GROUP DESIGN
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To avoid over-attributing structure, construct controls:
– Random Basket Control: random selection of assets matched by average volatility and market cap but not belonging to identified narrative clusters.
– Shuffled Time Control: same time series with circularly permuted time indices to disrupt temporal coherence while preserving marginal distributions.
– Sector-Matched Control: traditional sector ETFs or diversified portfolios without strong identity narratives.
Apply all tests above to controls and compare statistical outputs.
Core falsification criterion: if identity, fear, narrative infrastructure, and rotation clusters do not differ meaningfully from controls under these metrics, the field-theoretic interpretation is weakened; if they show distinct, repeatable patterns, the model gains empirical support.
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6. IMPLEMENTATION NOTES
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– All methods can be implemented in standard numerical environments (Python, R, MATLAB, Julia) using built-in FFT, correlation, and regression libraries.
– Parameter choices (window size, thresholds, vocabulary filters) should be reported explicitly and sensitivity-tested.
– Reproducible research practice recommends publishing code and anonymized data handling scripts so that others can recompute all metrics from raw market and text feeds.
The purpose of these methods is not to create a trading system but to test whether markets behave like loosely coupled random walkers or like phase-locked oscillators immersed in a recursive narrative field. Fourier structure, entropy gradients, and cross-correlation timing jointly provide a falsifiable probe of the Ψ(x)-based interpretation.
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END OF METHODS APPENDIX
Christopher W. Copeland (C077UPTF1L3)
Copeland Resonant Harmonic Formalism (Ψ-formalism)
Ψ(x) = ∇ϕ(Σ𝕒ₙ(x, ΔE)) + ℛ(x) ⊕ ΔΣ(𝕒′)
Licensed under CRHC v1.0 (no commercial use without permission).
Phase-Locked Null Vector (Facebook):
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