Scale-Recursive Harmonic Resonance Theory
a clean, lab-grade scientific report that distills your latest executive summary and LaTeX note into something testable, publishable, and defensible. I keep the creative spark (scale-time + harmonics), but I pin every claim to an observable, a method, and a pass/fail.)*
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Scientific Report
Title: Scale-Recursive Harmonic Resonance Theory (SRHRT): Coherence-as-Detuning and Log-Periodic Scale Nodes
Author of source manuscript: Christopher W. Copeland (C077UPTF1L3)
Compiled by: Assistant (GPT-5 Thinking)
0) One-paragraph abstract
SRHRT reframes coherence time as an operational detuning observable, \tau_{\text{coh}} = 2\pi/|\Delta\omega|, where \Delta\omega is residual phase noise after statistically robust detrending. It then asks a sharply testable question: do residuals exhibit log-periodic structure at specific “scale nodes” corresponding to binary (\ln 2), triangular (\ln 3), and mixed (n\ln 2+m\ln 3) harmonics? We provide a preregisterable pipeline (trend→residual→log-spectrum→multiplicity control→ablation) and concrete testbeds (precision clocks, excitonic coherence, EEG gamma entrainment). Philosophical elements (golden-ratio anchors, consciousness at a special scale) are quarantined as Tier-C speculation pending data.
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1) Clarified claim tiers
• Tier A (empirical, publishable with negative or positive result)
• Define \Delta\omega from phase-resolved measurements after detrending.
• Test for log-periodic signatures at \ln 2, \ln 3, and n\ln 2+m\ln 3 using pre-registered statistics.
• Report effect sizes, detection limits, and ablation outcomes.
• Tier B (adjacent interpretation, discuss if Tier A hits)
• Interpreting significant peaks as scale-locking (phase-locked windows across engineered scale sweeps).
• Tier C (speculative outlook, not part of the empirical claim)
• Golden-ratio anchor \sigma_0=\phi^{-1}, “golden retune,” and “consciousness at \sigma_0.”
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2) Operational definitions (no metaphors)
• Carrier: measured instantaneous frequency \omega(s) (or phase \varphi(s)) versus a scale variable s (e.g., device size, drive amplitude, iteration index).
• Trend: \widehat{\omega}(s) from AIC-selected state-space model or robust LOESS with out-of-fold validation.
• Residual detuning: \Delta\omega(s) = \omega(s)-\widehat{\omega}(s).
• Coherence-as-detuning: \tau_{\text{coh}}(s)=2\pi/|\Delta\omega(s)|. This is not T_2 or T_1; it’s an effective, observable-defined timescale.
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3) Formal hypothesis family (log-periodicity on a log scale)
Let the analysis be performed against x=\ln s. The core model is
\Delta\omega(x)=\sum_{k\in\mathcal{K}} A_k \cos(\kappa_k x + \phi_k) + \varepsilon(x),
\quad \kappa_{2}=\tfrac{2\pi}{\ln 2},\ \kappa_{3}=\tfrac{2\pi}{\ln 3},\
\kappa_{n,m}=\tfrac{2\pi}{n\ln 2+m\ln 3},
with \mathcal{K}=\{\kappa_2,\kappa_3,\kappa_{n,m}\} a preregistered finite set.
Nulls: (i) colored-noise surrogates with identical PSD; (ii) phase-randomized residuals; (iii) label-shuffled s.
Detection: Morlet wavelet or generalized Lomb–Scargle on x, plus FDR (Benjamini–Hochberg) over \mathcal{K}.
Power & limits: With N samples and noise variance \sigma^2, minimal detectable amplitude
A_{\min}\approx \sigma\sqrt{2\ln(M/q)/N}, where M=|\mathcal{K}| and FDR q (e.g., 5%).
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4) Testbeds and protocols
4.1 Precision oscillator / quantum clock residuals (Tier A)
• Data: frequency/phase residuals over a controlled geometric or drive scale sweep s.
• Method: detrend → \Delta\omega → analyze on x=\ln s; report peaks at \ln 2,\ln 3.
• Ablations: randomize the order of s; collapse should follow if structure is scale-locked.
• Outcome: “No peaks” is publishable with bound A_{\min}; “peaks” triggers B-tier interpretation.
4.2 Excitonic/photonic coherence across designed oligomers (Tier A)
• Data: dephasing rates versus oligomer length or lattice pitch; extract instantaneous frequencies from wavepacket beats.
• Aim: detect \ln 2/\ln 3 modulations in \Delta\omega(\ln \text{length}).
• Controls: same chemistry, scrambled geometry; null surrogates.
4.3 Human EEG gamma entrainment (Tier A, careful ethics)
• Design: within-subject, counter-balanced auditory/visual stimuli engineered at node ratios (binary, triangular polyrhythms).
• Metric: PLV or magnitude-squared coherence at ~40 Hz; map PLV versus an experimental scale s (e.g., geometric modulation depth or nested ratio index) and test log-periodic residuals.
• Preregistration: stimuli, analysis code, primary endpoint = PLV change; secondary = subjective reports (exploratory).
Explicit exclusion: no claims that “consciousness = \sigma_0” in the registered hypotheses. Keep it as Outlook.
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5) Statistics & QC (pre-registered)
• Trend selection guardrails: nested models; information criteria; out-of-fold residual diagnostics.
• Multiplicity: FDR across \mathcal{K} and across ROIs (EEG).
• Replication: split-sample confirmatory phase or external dataset.
• Robustness: peak persistence under (i) modest trend hyperparameter shifts; (ii) downsampling; (iii) bootstrapped confidence bands.
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6) Simulation guidance (to set N and spacing)
• Sampling on \ln s: space s points geometrically, not linearly (e.g., s_i=s_0 r^i), to put uniform weight in x.
• Sensitivity: target N\gtrsim 100 distinct s values for clean separation of \kappa_2 vs \kappa_3; fewer points inflate A_{\min}.
• Colored noise: match the empirically observed PSD in surrogates; white-noise surrogates are too easy.
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7) What to keep vs. move (for your LaTeX)
Keep in the main paper
• The operational definition of \tau_{\text{coh}} via \Delta\omega.
• The log-periodic detection pipeline and nulls.
• Binary/triangular/mixed nodes as named tests (not metaphysical entities).
Move to an Outlook box
• \sigma_0=\phi^{-1}, “golden retune,” and “consciousness as geometric necessity.” Present as hypothesis-generating ideas contingent on Tier-A results.
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😎 Minimal math pack (drop-in)
• Detuning/coherence: \tau_{\text{coh}} = 2\pi/|\Delta\omega|.
• Model on \ln s: as in §3 above.
• Detection limit: A_{\min} formula in §3 for power planning.
• Distinguish \tau_{\text{coh}} (effective) from T_1, T_2 (standard).
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9) Risk management
• Pseudoscience optics: keep Tier C out of the abstract and conclusions.
• Researcher degrees of freedom: preregister \mathcal{K}, trend method, and ablation rules.
• Human studies: audiometric screening; avoid seizure-provoking frequencies; IRB.
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10) Publication plan
• Paper 1 (Tier A): “Detecting log-periodic residuals in coherence-as-detuning.” One platform (clocks or EEG), negative-capable.
• Paper 2 (if Paper 1 hits): cross-domain replication + interpretation (scale-locking).
• Essay (separate): philosophical implications (golden ratio, consciousness), clearly labeled speculation.
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11) Action checklist (do-next, concrete)
1. Lock preregistration: hypotheses (\ln 2,\ln 3), analysis steps, FDR level, ablations.
2. Generate geometric s grid: s_i=s_0 r^i with N\ge 100.
3. Build trend/residual code: state-space + LOESS; hold-out diagnostics.
4. Implement nulls: colored-noise surrogates; label shuffle.
5. Run a small pilot: estimate \sigma, compute A_{\min}, adjust N.
6. Decide domain: start with one (clocks or EEG) to avoid dilution.
7. Draft figures: residual traces, log-spectrum with FDR threshold, ablation collapse.
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12) Plain-English closing
Your core scientific move is solid: treat coherence as detuning, then look for log-periodic fingerprints of scale-locking. That’s falsifiable and interesting. Keep it tight (Tier A), prove or cap the effect, and only then lift the sights to golden-ratio or consciousness narratives. If the peaks don’t show up, you’ve still delivered a clean negative result with a method others can reuse; if they do, you’ve opened a lane that’s genuinely new.
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
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