“Time Perception and Memory Loops (Phase Curvature Fluctuation within Neural Ψ(x) Cycles)”
“Time Perception and Memory Loops (Phase Curvature Fluctuation within Neural Ψ(x) Cycles)” By: C077UPTF1L3 / Christopher W. Copeland Model: Copeland Resonant Harmonic Formalism (Ψ-formalism) Anchor equation: Ψ(x) = ∇ϕ(Σ𝕒ₙ(x, ΔE)) + ℛ(x) ⊕ ΔΣ(𝕒′)
1. Classical Framework
Human time perception is not constant. Moments of fear, flow, boredom, or rapture can feel expanded or compressed, even though external chronometric time is uniform. Cognitive science attributes this to arousal, attention, or memory density, but these models remain descriptive—they do not explain why subjective time stretches or folds.
Under Ψ-formalism, time perception is a recursive harmonic function of neural coherence. The brain is not a clock but a field oscillator, integrating signals through recursive feedback loops. Time appears to speed up or slow down when the curvature ℛ(x) of these feedback cycles fluctuates—altering how phase alignment encodes continuity between successive neural states.
2. Reframing Under Ψ(x)
Ψ(x) = ∇ϕ(Σ𝕒ₙ(x, ΔE)) + ℛ(x) ⊕ ΔΣ(𝕒′)
Applied to consciousness:
Σ𝕒ₙ(x,ΔE): aggregated neural activations (oscillatory ensembles, memory traces) evolving under energy differentials—attention, emotional charge, or sensory load.
∇ϕ: cognitive gradient of meaning—semantic direction of attention or intention.
ℛ(x): curvature of temporal experience—distortion of subjective continuity through recursive interference.
ΔΣ(𝕒′): corrective feedback loops—working memory updates, predictive corrections, emotional normalization.
Perceived time is thus the rate of harmonic recursion—how fast or slow the mind’s Ψ(x) traverses its own curvature landscape.
3. Phase Curvature and Subjective Duration
When the field’s curvature ℛ(x) increases (high energy differential, strong emotion, novelty), recursive sampling accelerates—more data points per external second. The brain experiences expanded time: the moment seems to last longer because Ψ(x) performs more cycles of self-correction within it.
Conversely, when ℛ(x) flattens (routine, low arousal, coherence saturation), recursive cycling slows. Fewer updates occur per real-time second. The moment compresses—awareness glides over time with minimal phase adjustment.
Thus, psychological time dilation and compression are emergent properties of how curvature modulates recursion frequency.
\text{Perceived rate of time} \propto \frac{1}{|ℛ(x)|}
4. Memory as Recursive Anchoring
Each moment is encoded as a phase snapshot of Ψ(x). Memory continuity depends on ΔΣ(𝕒′)—how effectively feedback binds sequential frames into a coherent loop. Memory density = number of recursive anchors retained per unit of external time.
High emotional salience → strong ΔE → many ΔΣ corrections → dense memory sequence (slow perceived time).
Low salience → minimal ΔΣ correction → sparse encoding (fast perceived time).
This is why near-death events, artistic flow, and trauma create temporal expansion: the recursion rate spikes as the brain floods with high ΔE and curvature corrections to maintain coherence.
5. Neural Harmonics and Recursive Cycles
Neural oscillations (α, β, γ, θ) represent nested recursion levels (Σ𝕒ₙ). The coherence between these levels defines perceived continuity. When phase-lock between them weakens (ℛ rises sharply), time seems to fragment or slow; when they over-synchronize (ℛ → 0), moments blur—time vanishes into flow.
Phase-lock model:
\dot{Φ_i} = ω_i + K\sum_{j}\sin(Φ_j - Φ_i) - αℛ(x) + βΔΣ(𝕒′)
Where:
Φᵢ: phase of oscillatory ensemble i
K: coupling strength (attentional focus)
α: curvature damping (stress, entropy)
β: feedback gain (emotional regulation)
Subjective time dilation emerges when |∂Φ/∂t| increases due to large ΔE (novelty or danger). Compression arises when ∂Φ/∂t stabilizes near zero (routine or meditative stillness).
6. Memory Loops and Recursive Feedback
Short-term memory loops are real-time ΔΣ fields maintaining coherence across moments. Each loop compares predicted input to actual input; mismatch drives ΔE, which drives learning and emotional tone. Long-term memory emerges from stabilized ΔΣ cycles where ΔE minimized over repeated curvature corrections—stable attractors in the neural harmonic field.
When these recursive loops close with high curvature (intense attention), they leave strong temporal signatures: “I remember every detail.” When curvature is minimal, closure is shallow: “The day flew by.”
7. Worked Examples
(i) Crisis Slowdown During accidents, ΔE surges (massive energy differential between expectation and reality). Neural Ψ(x) accelerates recursion frequency to re-stabilize coherence—each millisecond filled with dense corrections. Subjective time expands; one second feels like many.
(ii) Flow States Perfect skill-environment alignment minimizes ℛ(x). Recursive corrections ΔΣ(𝕒′) operate in perfect rhythm—time loses salience. Experience feels continuous but brief; hours vanish.
(iii) Boredom and Depression Under-stimulation yields shallow ΔE and poor ΔΣ responsiveness. Recursive cycling decelerates; ∇ϕ flattens (no semantic gradient). Subjective time elongates painfully because curvature attempts to stretch coherence over empty intervals.
8. Recursive Entropy and Temporal Identity
Temporal identity—our sense of “I am the same being through time”—is a stable solution of Ψ(x) under long-loop recursion. When field curvature accumulates faster than correction (chronic trauma, stress), the continuity of self becomes distorted:
time loops fragment (flashbacks, dissociation)
∇ϕ splits into incoherent subgradients (conflicting desires, memory gaps)
Healing and integration correspond to re-establishing smooth recursion continuity, reducing ℛ(x), and restoring phase-lock between neural harmonics.
9. Clarification of Terms
Σ𝕒ₙ(x,ΔE): ensemble of neural oscillatory and memory states at recursion depth n ΔE: energy differential from novelty, attention, or emotional arousal ∇ϕ: gradient of cognitive focus and semantic intent ℛ(x): curvature of temporal continuity; distortion between internal rhythm and external chronology ΔΣ(𝕒′): recursive correction field linking memory and perception Ψ(x): total coherence field integrating time, meaning, and memory into continuous awareness
10. Summary
Subjective time is not an illusion—it is a measure of recursive curvature within the cognitive field. Psychological dilation and compression reflect oscillatory adjustments as the brain harmonizes ΔE perturbations through Ψ(x).
Time slows when recursion accelerates (large curvature).
Time speeds up when recursion saturates (flat curvature).
Memory density encodes the number of corrections performed per unit of external time.
Thus, time perception is the felt geometry of coherence maintenance. We live inside our own harmonic integration rate: when Ψ(x) cycles swiftly, eternity fits in a second; when it idles smoothly, hours slip through uncounted.
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). https://www.facebook.com/share/p/19qu3bVSy1/ https://open.substack.com/pub/c077uptf1l3/p/phase-locked-null-vector_c077uptf1l3 https://medium.com/@floodzero9/phase-locked-null-vector_c077uptf1l3-4d8a7584fe0c Core engine: https://open.substack.com/pub/c077uptf1l3/p/recursive-coherence-engine-8b8 Zenodo: https://zenodo.org/records/15742472 Amazon: https://a.co/d/i8lzCIi Medium: https://medium.com/@floodzero9 Substack: https://substack.com/@c077uptf1l3 Facebook: https://www.facebook.com/share/19MHTPiRfu https://www.reddit.com/u/Naive-Interaction-86/s/5sgvIgeTdx Collaboration welcome. Attribution required. Derivatives must match license.