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PLR as Involutions on 24 Triads
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EP01
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D₁₂ Acts Simply Transitively
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EP01
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Tonnetz Torus Identification
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EP01
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Tymoczko Orbifold Quotient
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EP01
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Wave Equation Derivation
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EP02
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D'Alembert General Solution
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EP02
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Normal Modes and Harmonic Series
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EP02
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Fourier Series Completeness
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EP02
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Energy Conservation by Orthogonality
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EP02
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Cut Vertex via DFS Back-Edges (Tarjan)
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EP03
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Whitney's Theorem (k-connectivity)
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EP03
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Menger's Theorem
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EP03
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Simple Transitivity of Polyphony Graph
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EP03
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Klein V₄ as Row Symmetry Group
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EP04
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Orbit-Stabilizer Theorem
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EP04
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Burnside's Lemma (Chord Counting)
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EP04
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All-Interval Row Constraint (sum mod 12)
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EP04
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Turán's Theorem
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EP05
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Mantel's Theorem (r=2 case)
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EP05
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Orbit-Stabilizer Theorem
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EP05
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Perfect Graph Theorem (Lovász 1972)
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EP06
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Interval Graphs are Perfect
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EP06
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Greedy Coloring on Interval Graphs
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EP06
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Brooks' Theorem
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EP06
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Shannon Entropy Maximum (Jensen's Inequality)
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EP07
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Source Coding Theorem
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EP07
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Entropy Rate of Markov Chains
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EP07
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Pitch Helix Projection (Shepard 1964/1982)
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EP08
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Tritone Paradox (Deutsch 1986)
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EP08
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Quotient Topology of Chroma Circle
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EP08
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Nonlinear Frequency Generation
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EP09
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Combination Tone Formula mf₁ ± nf₂
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EP09
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Cubic Difference Tone Dominance
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EP09
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Distance Preservation Under Reflection
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EP10
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D₁₂ as Symmetry Group of the Chromatic Scale
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EP10
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Negative Harmony Reflection Formula
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EP10
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Irrationality of log₂3 and log₂5
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EP11
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Syntonic Comma as an Obstruction
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EP11
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Continued Fraction Approximation of log₂(3/2)
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EP11
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7 Generates ℤ₁₂ (gcd Condition)
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EP12
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Subgroup Lattice of ℤ₁₂
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EP12
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Circle of Fifths as Quotient of the Spiral
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EP12
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LCM Periodicity of Polyrhythm
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EP13
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Rhythm-Pitch Duality (~20 Hz threshold)
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EP13
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Helmholtz Consonance = 1/LCM
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EP13
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Discrete Hodge Theorem (ker L_k ≅ H^k)
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EP14
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Hodge Decomposition (exact ⊕ coexact ⊕ harmonic)
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EP14
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Poincaré Duality (cell count bijection)
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EP14
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Boundary-squared = 0
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EP14
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Orchestra Spectral Attenuation (−9 dB/octave)
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EP15
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Singer's Formant Clustering (Sundberg 1974)
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EP15
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Fletcher-Munson Perceptual Weighting
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EP15
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Sabine Reverberation Formula
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EP15
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Total Penetration Index (49 dB)
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EP15
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CFG Representation of Solita Forma
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EP16
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Ensemble Interaction Growth (C(n,2))
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EP16
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Kolmogorov Compression of Code Rossini
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EP16
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Klein V₄ Structure of Motivic Transformations
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EP17
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Ensemble Interaction Graph Growth
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EP17
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Götterdämmerung Trajectory Compression
|
EP17
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Nash Equilibrium (Carmen backward induction)
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EP18
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Bayesian Updating (Don Giovanni persuasion)
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EP18
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Von Neumann Minimax Theorem (Xenakis Duel)
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EP18
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Folk Theorem (Zorn Cobra)
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EP18
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Jacobi-Anger Expansion (FM → Bessel sidebands)
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EP19
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Instantaneous Frequency of FM Signal
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EP19
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Carson Bandwidth Rule
|
EP19
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Formant Sweep Mechanism
|
EP19
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Cut Vertex Disconnection
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EP20
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Betweenness-Centrality Bridge Detection
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EP20
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Modularity Maximization
|
EP20
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Markov Chain Stationary Distribution (Perron-Frobenius)
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EP21
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Softmax Attention as Dynamic Transition Matrix
|
EP21
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DDPM Forward Process Reparameterization
|
EP21
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Score Matching Equivalence
|
EP21
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RVQ Successive Approximation
|
EP22
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Cross-Attention as Conditional Generation
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EP22
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Delay Pattern Causal Ordering
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EP22
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Classifier-Free Guidance Interpolation
|
EP22
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t-SNE KL Minimization
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EP23
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Information Bottleneck Optimality
|
EP23
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Linear Probe Sufficiency
|
EP23
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Emergent Specialization under Prediction Loss
|
EP23
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Surprisal as Self-Information
|
EP24
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LZ Compression as Kolmogorov Upper Bound
|
EP24
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Mutual Information Decomposition (Melody–Harmony)
|
EP24
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Inverted-U Complexity–Preference Curve
|
EP24
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Temporal Difference Prediction Error
|
EP25
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Value Function Bellman Equation
|
EP25
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Schultz Dopamine Analogy (Three Cases)
|
EP25
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Nash Equilibrium of Pop Music Formula
|
EP25
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Wiener–Khinchin Theorem
|
EP26
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Fractional Brownian Motion PSD
|
EP26
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Long-Memory Autocorrelation Decay
|
EP26
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Fractal Dimension via Hurst Exponent
|
EP26
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Surprisal Variance Decomposition
|
EP27
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Same-Entropy Different-Variance Lemma
|
EP27
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Melody–Harmony Mutual Information
|
EP27
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Nash Equilibrium of the Pop Formula
|
EP27
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Transfer Function of the Damped Harmonic Oscillator
|
EP28
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Resonance Peak Condition (ζ < 1/√2)
|
EP28
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Cardioid Polar Pattern Decomposition
|
EP28
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Dynamic vs Condenser Frequency Response Ratio
|
EP28
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Room Mode Frequency Formula
|
EP29
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Schroeder Frequency Crossover
|
EP29
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Helmholtz Resonator Natural Frequency
|
EP29
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Flat-Magnitude DFT of QRD Sequence (Gauss Sum)
|
EP29
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Nyquist-Shannon Sampling Theorem
|
EP30
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Quantization SNR Formula (6.02n + 1.76 dB)
|
EP30
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TPDF Independence Theorem
|
EP30
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First-Order Noise Transfer Function
|
EP30
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Four-Pole Ladder Transfer Function (−24 dB/oct)
|
EP31
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Barkhausen Oscillation Criterion (k=4)
|
EP31
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Odd Harmonics Theorem (tanh / odd functions)
|
EP31
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Zero-Delay Feedback Newton-Raphson Convergence
|
EP31
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Chebyshev Identity: T_n(cos θ) = cos(nθ)
|
EP32
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Harmonic Generation Theorem
|
EP32
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Drive Amplitude Power Law
|
EP32
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Intermodulation Spectrum Theorem
|
EP32
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|
D'Alembert General Solution (wave equation)
|
EP33
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Karplus-Strong Fundamental Frequency (f = fs/N)
|
EP33
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Digital Waveguide Loop Transfer Function
|
EP33
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Scattering Junction Coefficients
|
EP33
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|
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Inharmonicity Formula (f_n = n f0 sqrt(1+B n^2))
|
EP33
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|
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Jacobi-Anger Expansion (FM → Bessel sideband series)
|
EP34
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|
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Carson's Rule (BW ≈ 2(I+1)fₘ)
|
EP34
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|
|
Bessel Energy Conservation (Parseval for FM)
|
EP34
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|
|
Feedback Fixed-Point and Bifurcation
|
EP34
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|
|
STFT Phase Advance Formula
|
EP35
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|
|
Instantaneous Frequency Estimation
|
EP35
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|
|
COLA (Constant Overlap-Add) Condition
|
EP35
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|
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Phase Vocoder Pitch-Shift Theorem
|
EP35
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Phase Correction for Time Stretching
|
EP35
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|
|
Heisenberg-Gabor Uncertainty Principle
|
EP36
|
|
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Gaussian Optimality Theorem
|
EP36
|
|
|
Balian-Low Theorem
|
EP36
|
|
|
Biquad Transfer Function and Stability
|
EP37
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|
|
Bilinear Transform: Stability Preservation
|
EP37
|
|
|
Frequency Warping Correction
|
EP37
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|
|
Magnitude Response from Pole-Zero Distances
|
EP37
|
|
|
Serial Biquad Product Theorem
|
EP37
|
|
|
FDN Stability via Unitary Feedback Matrix
|
EP38
|
|
|
Hadamard Matrix Orthogonality
|
EP38
|
|
|
Coprime Delays and Uniform Mode Density
|
EP38
|
|
|
Theorem 39.1: K-Weighting Cascade Response
|
EP39
|
|
|
Theorem 39.2: Block Loudness Formula
|
EP39
|
|
|
Theorem 39.3: Dual-Gate Integrated Loudness
|
EP39
|
|
|
Theorem 39.4: True Peak via Sinc Oversampling
|
EP39
|
|
|
TDAC Perfect Reconstruction Theorem
|
EP40
|
|
|
Princen-Bradley Window Condition
|
EP40
|
|
|
MDCT Inversion Formula
|
EP40
|
|
|
SMR Bit Allocation Optimality
|
EP40
|
|
|
NMF Multiplicative Update Monotonicity (Lee–Seung 1999/2000)
|
EP41
|
|
|
Wiener Mask MMSE Optimality (additive Gaussian model)
|
EP41
|
|
|
Non-negativity Preservation of Lee–Seung Updates
|
EP41
|
|
|
Viterbi Optimality (DP on trellis)
|
EP42
|
|
|
Log-space Equivalence
|
EP42
|
|
|
Chroma Octave Folding
|
EP42
|
|
|
HMM Emission Factorization
|
EP42
|
|
|
Bjorklund/Euclidean Algorithm (GCD redistribution)
|
EP43
|
|
|
Burnside's Lemma (rotation fixed-points)
|
EP43
|
|
|
Fixed-point count = 2^gcd(r,n)
|
EP43
|
|
|
Swing Ratio Continuum
|
EP44
|
|
|
Participatory Discrepancy (PD) Decomposition
|
EP44
|
|
|
Kilchenmann–Senn Perceptual Threshold Model
|
EP44
|
|
|
Phase Shift Orbit Theorem (Z12 on beat pattern)
|
EP45
|
|
|
Emergence via Overlap Structure
|
EP45
|
|
|
Discrete-Continuous Interference Duality
|
EP45
|
|
|
Perfect Tiling (direct-sum factorization of ℤₙ)
|
EP46
|
|
|
Hajós Conjecture Failure (Sands 1974)
|
EP46
|
|
|
Vuza Canon Construction (subgroup chain method)
|
EP46
|
|
|
Coven–Meyerowitz T1/T2 Conditions
|
EP46
|
|
|
CSP Formalization of Counterpoint
|
EP47
|
|
|
Arc Consistency Pruning
|
EP47
|
|
|
Hard vs Soft Constraint Dichotomy
|
EP47
|
|
|
Constraint Optimization for Style
|
EP47
|
|
|
Harmonic Series as Fourier Decomposition
|
EP48
|
|
|
Plomp-Levelt Roughness Maximum at 1/4 Critical Bandwidth
|
EP48
|
|
|
Spectral Interpolation Linearity
|
EP48
|
|
|
Fibonacci Growth from Parallel Rewriting
|
EP49
|
|
|
Hausdorff-Besicovitch Dimension of L-System Curves
|
EP49
|
|
|
Golden Angle Optimality (Phyllotaxis)
|
EP49
|
|
|
Gaussian Pitch Distribution
|
EP50
|
|
|
Poisson Process for Note Density
|
EP50
|
|
|
Exponential Inter-event Times
|
EP50
|
|
|
Rule 30 Cellular Automaton
|
EP50
|
|
|
Best Rational Approximation (Convergent Optimality)
|
EP51
|
|
|
Pythagorean Comma (irrationality of log₂(3/2))
|
EP51
|
|
|
Stern–Brocot Tree (mediant enumeration)
|
EP51
|
|
|
Pareto Optimality of 12-EDO
|
EP51
|
|
|
Kantorovich Duality (primal = dual)
|
EP52
|
|
|
Wasserstein Metric (triangle inequality)
|
EP52
|
|
|
LP Optimality (extreme-point transport plan)
|
EP52
|
|