coneFFT Verification Instrument v0.2.1 - Public Technical Note

Introduction

I have released a browser-based implementation for parallel comparison of cone system responses based on R₆ / N-tree / Φ_d against standard FFT.
This implementation does not claim to be a "complete replacement for FFT."
It is designed as a verifier to observe and quantify band responses related to the B13 hypothesis and the presence of false positives.
This article is a public technical memo that summarizes the README and TECHNICAL NOTES.
It is not a declaration of a completed theory, but rather intended to present an observation system that others can interact with and verify.

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What is this?

If a standard FFT is a device that measures "what is at which frequency,"
the cone observer here is an observation instrument for seeing "in which direction/structure the cone system response exists."

At the core of the implementation are the following three operators.

R₆ (Fibonacci Difference)

$$R_6^{(d)}x(p)=x(p+d)-x(p-d)+x(p+2d)-x(p+3d)+x(p+5d)-x(p+8d)$$

• The kernel is the symmetric two-point difference x(p+d) - x(p-d).
• The outer chain is a sign-alternating difference along Fibonacci numbers.
• Used as a difference operator to emphasize responses in the structural direction.

N(a, b) (Dual Nodes)

$$N(a,b)=(a+b,\ a-b)$$

• a+b in the synthesis direction.
• a-b in the analysis direction.
• Becomes the core of the cone conservation law through the identity ab = (s^2 - d^2)/4 (where s=a+b, d=a-b).

Φ_d (Layer Invariant)

$$\Phi_d[R_6x]=\frac{1}{n}\sum_p R_6^{(d)}(x)(p)\cdot R_6^{(d)}(x)(p+2d)$$

• 2d shift correlation after applying R₆.
• Whereas Parseval looks at the norm, this looks at oriented correlation.

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Stance of the Implementation

I want to state this clearly.

The current coneSpec() is not a "final coneFFT transform" in the strict sense.
As an implementation, it is the following hybrid detector:

R₆ response
→ N-tree synthesis
→ DFT magnitude
→ H(θ) weighting

Therefore, the output at this stage is
not the completed final theory, but
an observation instrument for the R₆/B13 structure.

Theoretically, it is more accurate to call it a cone spectrum observer or R₆/B13 detector, but for the sake of continuity, I have maintained the implementation name: coneFFT Verification Instrument.

1. What can it do?

v0.2.1 allows for the following observations:

• Parallel spectrum display of FFT / coneFFT
• DIFF display
• Visualization of the difference between cone-dominant bands and FFT-dominant bands

2. B13 Band Energy Evaluation

• Evaluation of 300Hz ±80Hz and 1000Hz ±80Hz based on actual injection frequencies
• Φ_d session statistics: Mean, Variance, CV, Sparklines
• False positive guard: B13 band false reaction rate against White Noise
• VERDICT: Automatic summary of session results

3. Changing FFT Markers to Actual Frequencies

• The markers in the FFT panel have been mapped to actual injection frequencies instead of theoretical mode ratios:
  • 300Hz
  • 1kHz
  • 1.3kHz

With this, the B13 reproducibility, noise false positive rate, and cone/fft band ratio are now at least meaningful as measured values.

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B13 Test Signal

The test signal is as follows:

x(t) = sin(2π·300t)
     + 0.5 sin(2π·1000t)
     + 0.3 sin(2π·1300t)

This corresponds to an integer ratio injection of 3, 10, 13 when fBase = 100Hz.

The B13 hypothesis stands on the view that "the R₆ operator has structural strong responses at m=3, 10." This test signal is intentionally designed to stimulate that hypothesis.

That a response occurs with the B13 test signal is, to a certain extent, expected by design. The essential verification lies in whether the cone-specific response pattern is reproduced even for arbitrary inputs.

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Evaluation Metrics

• DIFF quantification

  • max Δ mode: argmax_i ( dB(cone[i]) - dB(fft[i]) )
  • max Δ dB: The difference value at the above index
  • cone-dominant bins: The number of bins satisfying dB_cone[i] - dB_fft[i] > 3dB

• B13 band energy

  • Evaluation bands: 300Hz ±80Hz, 1000Hz ±80Hz
  • cone B13 ratio: bandEnergy(cone) / totalEnergy(cone)
  • fft B13 ratio: bandEnergy(fft) / totalEnergy(fft)
  • cone / fft: The ratio of the above two ratios

• Φ_d session statistics

  • Mean: μ
  • Variance: σ²
  • Coefficient of Variation: CV = σ / |μ|
  • Sparkline: Time-series display

• False positive guard

  • B13 reproducibility: The rate at which the cone-side peak stably appears in the expected band when a B13 signal is injected
  • Noise false positive rate: The rate at which the B13 band ratio rises excessively when White Noise is injected

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Verification Procedure

The minimal procedure is as follows:

1. Set the signal to B13 Resonance.
2. Click RESET.
3. Observe for 20 frames or more.
4. Record the following:
   • B13 reproducibility
   • cone B13 ratio
   • fft B13 ratio
   • cone/fft
   • max Δ dB
   • Φ_d CV

Next, switch the signal to White Noise.

Observe for 20 frames or more.

Record the following:

• Noise false positive rate
• Noise response
• cone B13 ratio

Check the VERDICT.

The provisional criteria at this time are as follows:

Metric	Guideline
B13 reproducibility	80% or higher
Noise false positive rate	5% or lower
cone/fft ratio	1.5x or higher
Φ_d CV	Less than 0.1

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Limitations

This implementation has clear limitations:

1. Approximation of cone spectrum: coneSpec() is a DFT-based hybrid detector and is not a pure R₆ recursive spectrum.
2. Decimation of dftMag(): Lightweight processing using step=4 is applied, which may introduce accuracy differences with the FFT side into the DIFF.
3. Asymmetry of coordinate systems: The FFT side contains an Hz axis, while the cone side contains a reading similar to a mode index.
4. Single-frame focus: Spectrogram-like temporal evolution is not implemented.
5. 1D implementation: While R₆ has inherent spatiality, the current implementation remains a 1D approximation.

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Future Challenges

Theory side

• Explicitly state the transformation rules for Φ_d[R₆x].
• Extend the cone conservation law from local theorems to the entire layer.
• Implement hierarchical connections and advance to multi-layer recursion.

Implementation side

• Replace dftMag() with an FFT-based approach.
• Strictly define the cone mode index space.
• Add spectrogram display.
• N=3120 benchmark.
• Create a Python verifier.

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Stance on Publication

The value of this implementation is not a declaration of a complete proof. The value lies in presenting an observation system that others can touch and verify.

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Demo and Full Code

• Demo running in the browser

• Full version code (GitHub)
  https://github.com/morcb13-bit/Paper/tree/main/2026-03-09-coneFFT-Verification

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Conclusion

This implementation does not completely finalize the theory. However, it serves as a foothold in that it makes structural observation via R₆ / N / Φ_d accessible, accompanied by FFT comparison and false-positive checks.

I believe the next steps are to accumulate logs, advance to a time-series observer, and move closer to the transformation rules of Φ_d[R₆x], rather than increasing explanations.