Independent Mathematical Peer-Review & Comparative Evaluation (v3.5 vs. v2.5): The Singularity Taxonomy38 v3.5 Architecture

1. Executive Summary of Findings

Evaluating The Singularity Taxonomy38 v3.5 Research Monograph alongside our prior audit of Taxonomy38 v2.5, Edition v3.5 represents a major mathematical leap from first-order linear/uncoupled physical estimators to fourth-order dispersive, coupled multi-degree-of-freedom PDEs, robust redescending M-estimation, invariant state-space manifolds, and collective mean-field psychoacoustic control.

Most impressively, the v3.5 monograph does not merely propose higher-order differential equations; it rigorously proves why naive v3.0 implementations of those equations fail under adversarial boundary conditions (polyphonic chord leverage $k^4$, $192\text{ kHz}$ sample-rate scaling, rank-1 Hankel matrices on single-headed drums, right-half-plane all-pass poles under positive phase slope, $\cot^2(\omega_0 t)$ quadrature blowups, soprano formant tuning $f_0 \approx F_1$, and square-wave biphase degeneracy $\mathcal{B}_{3,1} = -1$) and constructs closed-form mathematical regularizations for each.

At the same time, our independent mathematical audit of v3.5 uncovered four specific analytical and editorial items—most notably a subtle squared-energy vs. RMS-amplitude exponent discrepancy in the Theorem 3.2 split-doublet beat-curvature ratio (where our Taylor expansion proves why the code's RMS amplitude ratio succeeds to $\mathcal{O}(\alpha^4)$), a sign-transposed interval bound and two-multiplier vs. four-multiplier Gray–Markel unitarity distinction in Theorem 2.3, and truncated HTML anchor IDs in the Table of Contents.

Summary Scorecard Across All 13 Theorems of Taxonomy38 v3.5

2. Compare-and-Contrast Analysis: Taxonomy38 v2.5 vs. Taxonomy38 v3.5

Comparing the mathematical foundations of Taxonomy38 v2.5 and Taxonomy38 v3.5 reveals seven structural upgrades where v3.5 directly resolves either a first-order physical limitation of v2.5 or an open theoretical frontier identified during our v2.5 audit:

3. Detailed Section-by-Section Mathematical Audit of Taxonomy38 v3.5

3.1 Stream A (#0..#10 Guitars): Theorems 2.1, 2.2, and 2.3

1. Well-Posedness & Sign Audit of the 4th-Order Damped PDE (§2.1.1):

The governing equation in §2.1.1 is the Boussinesq / Chaigne–Askenfelt damped stiff-string PDE:

$$\rho A \frac{\partial^2 u}{\partial t^2} = T_0 \frac{\partial^2 u}{\partial x^2} - E I \frac{\partial^4 u}{\partial x^4} - 2\sigma_0 \rho A \frac{\partial u}{\partial t} + 2\sigma_1 \rho A \frac{\partial^3 u}{\partial t \partial x^2}$$

Substituting a hinged spatial eigenmode $u_k(x,t) = q_k(t)\sin(k\pi x / L_{\text{eff}})$ flips the sign of $\partial_x^2 \mapsto -(k\pi/L_{\text{eff}})^2$ and preserves the sign of $\partial_x^4 \mapsto +(k\pi/L_{\text{eff}})^4$, yielding the modal ODE:

$$\ddot{q}_k + 2\left(\sigma_0 + \sigma_1 \frac{k^2\pi^2}{L_{\text{eff}}^2}\right)\dot{q}_k + \frac{T_0}{\rho A}\frac{k^2\pi^2}{L_{\text{eff}}^2}\left(1 + \frac{\pi^2 E I}{T_0 L_{\text{eff}}^2} k^2\right) q_k = 0$$

Note that the $+2\sigma_1\rho A u_{txx}$ term on the right-hand side of the PDE is essential for forward-time parabolic well-posedness: a minus sign would produce negative high-wavenumber damping $-\sigma_1(k\pi/L_{\text{eff}})^2$ (backward-heat instability). Furthermore, under equal-temperament fretting $L_{\text{eff}}(n_{\text{fret}}) = L_0 2^{-n_{\text{fret}}/12}$, the inharmonicity coefficient $B(n_{\text{fret}}) \propto L_{\text{eff}}^{-2}$ scales as $B_{\text{open}} \cdot (2^{-n_{\text{fret}}/12})^{-2} = B_{\text{open}} \cdot 2^{+n_{\text{fret}}/6}$. 100% exact.

2. Proof of Proposition 2.1 ($k^4$ Leverage Explosion) & Tukey Redescending Weights (§2.1.2–2.1.3):

Rewriting the WLS estimator in terms of the per-partial estimates $b_k \triangleq y_k / k^2$:

$$\hat{B}_{\text{WLS}}^{\text{v3.0}} = \frac{\sum_{k=2}^{10} w_k k^2 y_k}{\sum_{k=2}^{10} w_k k^4} = \frac{\sum_{k=2}^{10} \left(w_k k^4\right) b_k}{\sum_{k=2}^{10} w_k k^4}$$

proves immediately that ordinary least squares in $y_k = B k^2$ weights the $k$-th local estimate $b_k$ by $k^4$, giving $k = 9$ a $9^4 / 2^4 = 6561 / 16 = 410.06:1$ leverage advantage over $k = 2$. At $\delta_{\text{chord}} = 0.055$, $y_9 = 1.055^2 - 1 = 0.113025$ and $b_9 = 0.113025/81 = 0.0013954 \approx 10 B_{\text{open}}$. Anchoring at the low-partial order-statistic median $b_{\text{anchor}} = \text{median}_{k\in\{2,3,4,5\}}(b_k)$ and applying Tukey's compactly supported bisquare weight $w_k^{\text{Tukey}} = w_k(1 - u_k^2)^2 \mathbb{I}_{|u_k|<1}$ completely annihilates any partial deviating by $\ge c_B = 0.00035$.

By the Weierstrass half-angle substitution $g = \tan(\Omega_c/2)$, $\sin(\Omega_c) = \frac{2g}{1+g^2}$ is an exact trigonometric identity for all $\Omega_c \in (-\pi, \pi)$. Multiplying the rectified discrete TKEO by $\frac{(1+g^2[n])^2}{4g^2[n]} = \sin^{-2}(\Omega_c[n])$ cancels the $\sin^2(\Omega_c)$ factor in $\Psi_{\text{TK}}[A\cos(\Omega_c n + \phi)] = A^2\sin^2(\Omega_c)$ identically. All decibel figures in Proposition 2.2 ($10\log_{10}(63.84) = +18.05\text{ dB}$ and $10\log_{10}(1/16) = -12.04\text{ dB}$) are exact.

1. Verification of Jacobi–Anger Bessel Sidebands (§2.3.1):

For $f_h = 4\text{ kHz}$, $X_0 = 3.5\text{ mm}$, and $c_0 = 343\text{ m/s}$, the modulation index is $\beta_{\text{FM}} = \frac{2\pi(4000)(0.0035)}{343} = 0.25646\text{ rad}$, and the first-order Bessel function is $J_1(0.25646) = 0.12717 \implies 20\log_{10} J_1(\beta_{\text{FM}}) = -17.91\text{ dBc}$. Exact.

2. Two Mathematical Refinements in §2.3.3:

• Refinement 1 (Transposed Endpoint Signs on $k_{\text{ap}}[n]$):

In §2.3.3 (Item 3), for $\Delta_d[n] \in [-0.42, +0.42]$ and $D[n] = 1.0 - \Delta_d[n] \in [0.58, 1.42]$, the reflection coefficient is:

$$k_{\text{ap}}[n] = \frac{1 - D[n]}{1 + D[n]} = \frac{\Delta_d[n]}{2 - \Delta_d[n]}$$

Evaluating at the two endpoints gives:

• At $\Delta_d = -0.42$: $k_{\text{ap}} = \frac{-0.42}{2 - (-0.42)} = \frac{-0.42}{2.42} = -0.1736 \approx -0.174$.
• At $\Delta_d = +0.42$: $k_{\text{ap}} = \frac{+0.42}{2 - 0.42} = \frac{+0.42}{1.58} = +0.2658 \approx +0.266$.

The text in §2.3.3 writes $k_{\text{ap}}[n] \in [-0.265, +0.174]$, which transposed the negative and positive magnitudes; it should read $k_{\text{ap}}[n] \in [-0.174, +0.266]$.

• Refinement 2 (Two-Multiplier Passive Lattice vs. Four-Multiplier Unitary Givens Lattice):

The difference equations in §2.3.3:

$$w[n] = x_{\text{high}}[n] - k_{\text{ap}}[n]\,s_{\text{lat}}[n-1], \qquad y_{\text{high}}[n] = k_{\text{ap}}[n]\,w[n] + s_{\text{lat}}[n-1], \qquad s_{\text{lat}}[n] = w[n]$$

have the $2\times 2$ state-input transfer matrix $\mathbf{T}(k) = \begin{bmatrix} k & 1 - k^2 \\ 1 & -k \end{bmatrix}$. This is the two-multiplier Gray–Markel lattice, which is all-pass for frozen $k$ and preserves the weighted energy $y^2 + (1-k^2)s_{\text{lat}}^2[n] = x^2 + (1-k^2)s_{\text{lat}}^2[n-1]$. Strictly speaking, the unitary (orthogonal) four-multiplier Gray–Markel lattice (Gray & Markel 1975) replaces the off-diagonal terms by $\sqrt{1 - k^2}$ ($\mathbf{\Theta}(k) = \begin{bmatrix} k & \sqrt{1-k^2} \\ \sqrt{1-k^2} & -k \end{bmatrix}$) so that $y^2[n] + s_{\text{lat}}^2[n] \equiv x^2[n] + s_{\text{lat}}^2[n-1]$ holds identically even under rapid time variation of $k[n]$. Because $|k_{\text{ap}}[n]| \le 0.266$ is small and $\Delta_d[n]$ varies smoothly at $\le 120\text{ Hz}$, $\sqrt{1 - k_{\text{ap}}^2} \in [0.964, 1.0]$ is nearly unity and the two-multiplier lattice is completely stable, but describing it as the two-multiplier passive Gray–Markel lattice is operator-theoretically exact.

3.2 Stream B (#17..#23 Drums & DRUM LINK): Theorems 3.1, 3.2, 3.3, and 3.4

When a signal contains only a single exponential mode $y[n] = c_1 z_1^n$, the $2\times 2$ Hankel matrix $\mathbf{G}_0 = \begin{bmatrix} y[0] & y[1] \\ y[1] & y[2] \end{bmatrix}$ has determinant $\det(\mathbf{G}_0) = c_1^2(z_1^2 - z_1^2) \equiv 0$, so fitting a 2-pole Prony model is an ill-posed rank-deficient problem. The 3-point Aitken $\Delta^2$ extrapolation formula $\hat{f}_{0,+}(\infty) = f_3 - \frac{(f_2 - f_3)^2}{f_1 - 2f_2 + f_3}$ and the smooth existence gate $\Gamma_{\text{doublet}}$ in §3.1.3 are completely rigorous.

Let us carefully examine the 3-epoch beat-curvature equation in §3.2.3 (Equations 790–792):

• Mathematical Proof of the Exponent Discrepancy in Eq. (3.2.3) & Why the C++ Engine Succeeds:

1. Because $t_1 = 28\text{ ms}, t_2 = 53\text{ ms}, t_3 = 78\text{ ms}$ are equally spaced ($t_1 + t_3 = 2t_2 = 106\text{ ms}$, $\Delta t = 25\text{ ms}$), the exponential decay factor $e^{-2t/\tau_m}$ cancels identically between numerator and denominator for any $\tau_m > 0$.
2. However, look at the trigonometric factor when $E(t)$ is written as the squared-energy envelope $E(t) \propto \cos^2(\pi\,\delta f_m t + \phi_0)$:
$$\frac{\sqrt{\cos^2(\theta - \alpha)\,\cos^2(\theta + \alpha)}}{\cos^2(\theta)} = \frac{\big|\cos(\theta - \alpha)\,\cos(\theta + \alpha)\big|}{\cos^2(\theta)} = \frac{\cos^2(\theta) - \sin^2(\alpha)}{\cos^2(\theta)} = 1 - \frac{\sin^2(\alpha)}{\cos^2(\theta)}$$

where $\alpha \triangleq \pi\,\delta f_m\,\Delta t$. At $\theta = 0$, this expression equals $1 - \sin^2(\alpha) = \mathbf{\cos^2(\alpha)}$, NOT $\cos(\alpha)$!

3. Now consider instead the RMS amplitude envelope $A(t) = \sqrt{E(t)} = A_0 e^{-t/\tau_m}\big|\cos(\pi\,\delta f_m t)\big|$ (excited in-phase by the stick impact at $t = 0$, so $\phi_0 = 0$). Since $t_1 \approx \Delta t$, $t_2 \approx 2\Delta t$, and $t_3 \approx 3\Delta t$, the geometric-to-center ratio of the RMS amplitude envelope $A(t)$ is:
$$\frac{\sqrt{A(t_1)\,A(t_3)}}{A(t_2)} = \frac{\sqrt{\cos(\alpha)\,\cos(3\alpha)}}{\cos(2\alpha)}$$

Expanding numerator and denominator in a Maclaurin series in $\alpha = \pi\,\delta f_m\,\Delta t$:

$$\frac{\sqrt{\left(1 - \frac{1}{2}\alpha^2 + \frac{1}{24}\alpha^4\right)\left(1 - \frac{9}{2}\alpha^2 + \frac{27}{8}\alpha^4\right)}}{1 - 2\alpha^2 + \frac{2}{3}\alpha^4} = \frac{1 - \frac{5}{2}\alpha^2 + \frac{13}{6}\alpha^4 + \mathcal{O}(\alpha^6)}{1 - 2\alpha^2 + \frac{2}{3}\alpha^4 + \mathcal{O}(\alpha^6)} = 1 - \frac{1}{2}\alpha^2 + \frac{1}{2}\alpha^4 + \mathcal{O}(\alpha^6) = \cos(\alpha) + \frac{11}{24}\alpha^4 + \mathcal{O}(\alpha^6)!$$

This 4th-order Taylor cancellation is remarkable: when $t_1 : t_2 : t_3 \approx 1 : 2 : 3$, the quadratic phase curvature $2\alpha$ at the midpoint $t_2 \approx 2\Delta t$ cancels against the geometric mean of $\cos(\alpha)$ and $\cos(3\alpha)$ so that $\frac{\sqrt{A(t_1)A(t_3)}}{A(t_2)}$ equals $\cos(\pi\,\delta f_m\,\Delta t)$ up to $\mathcal{O}(\alpha^4)$! (And if one uses the decay-compensated arithmetic mean $\frac{A(t_1)e^{t_1/\tau_m} + A(t_3)e^{t_3/\tau_m}}{2 A(t_2)e^{t_2/\tau_m}} = \frac{\cos(\theta-\alpha)+\cos(\theta+\alpha)}{2\cos\theta} \equiv \cos(\alpha)$, it is exact to all orders for every $\theta$).

Therefore, in §3.2.3 (Eqs. 790–792), $E(t)$ should either be denoted as the narrowband RMS amplitude envelope $A(t) = A_0 e^{-t/\tau_m}|\cos(\pi\,\delta f_m t)|$ (removing the squares inside the square root in Eq. 790), or if $E(t)$ denotes squared power, Eq. (792) should include an outer square root $\big(\sqrt{E_{28} E_{78}} / E_{53}\big)^{1/2}$.

The proof of Theorem 3.4 is exemplary:

1. By Foster's Reactance Theorem, any causal, stable all-pass transfer function $H_{\text{AP}}(z) = \prod_k \frac{z^{-1} - \bar{p}_k}{1 - p_k z^{-1}}$ ($|p_k| < 1$) has strictly positive group delay $\tau_g(\Omega) = -\frac{d\phi_{\text{AP}}}{d\Omega} = \sum_k \frac{1 - |p_k|^2}{|e^{j\Omega} - p_k|^2} > 0$, so its phase $\phi_{\text{AP}}(\Omega)$ must be strictly decreasing. Forcing an unconstrained $2\times 2$ Cramer system to fit an increasing phase target ($\phi_2 > \phi_1$) mathematically forces $|p_k| > 1$ (unstable right-half-plane / exterior unit-disk poles).
2. Parameterizing in the left-half $s$-plane with $g = \tan(\pi f_p / f_s) > 0$ and $k = 1/Q_p > 0$ yields the discrete denominator $1 + c_1 z^{-1} + c_2 z^{-2}$ with $c_2 = \frac{1+g^2-gk}{1+g^2+gk}$ and $c_1 = \frac{2(g^2-1)}{1+g^2+gk}$, for which $|c_2| < 1$ and $1 + c_2 - |c_1| = \frac{4\min(1, g^2)}{1+g^2+gk} > 0$ hold unconditionally for all $g > 0, k > 0$.

3.3 Stream C (#11..#16 Bass & #24..#29 Vocals): Theorems 4.1, 4.2, 4.3, and 4.4

1. Exact Elimination of Phase $\theta = \omega_0 t + \phi$:

With $v_1(t) = C\cos\theta$, $v_2(t) = \frac{C}{r}\sin\theta$, and $v_{\text{I2}}(t) = -\frac{C}{r^2}\cos\theta$ ($r = \omega_0 / \omega_c$), the two quadratic forms $I_1(r) = v_1^2 + r^2 v_2^2 = C^2$ and $I_2(r) = v_2^2 + r^2 v_{\text{I2}}^2 = C^2 / r^2$ are exact conservation laws (first integrals) of the harmonic oscillator at every time instant $t$.

2. Fixed-Point Contraction Proof for the Recursive Ratio $\hat{r}[n] = F(r_{\text{prev}}; \theta)$:

Because $r_{\text{prev}}$ is used on the right-hand side of Eq. (4.1.2):

$$F(x; \theta) = \sqrt{\frac{r^2\cos^2\theta + x^2\sin^2\theta}{\sin^2\theta + (x/r)^2\cos^2\theta}}$$

we find $F(r; \theta) \equiv r$ for all $\theta$, and the Fréchet derivative at the fixed point $x = r$ is:

$$\frac{\partial F}{\partial x}\Bigg|_{x=r} = \sin^2\theta - \cos^2\theta = -\cos(2\theta)$$

Because $\big|\frac{\partial F}{\partial x}\big|_{x=r}\big| \le 1$ everywhere and its cycle average $\frac{1}{2\pi}\int_0^{2\pi} (-\cos 2\theta)\,d\theta = 0$ vanishes identically, the PLL feedback loop is locally contractive and free of the $\cot^2(\omega_0 t)$ non-integrable singularities of 2-state ratios.

3. Minor Terminology Note on fastTanPadeBass (§4.1.2, Item 3):

The polynomial $x\big(1 + \frac{1}{3}x^2 + \frac{2}{15}x^4\big) = x + \frac{1}{3}x^3 + \frac{2}{15}x^5$ is the 5th-order Maclaurin (Taylor) polynomial of $\tan(x)$ (which matches the Taylor expansion of the $[3/2]$ Padé approximant $\frac{x(15-x^2)}{15-6x^2}$ through $x^5$). Across $f \in [18, 400]\text{ Hz}$ at $f_s \ge 44.1\text{ kHz}$ ($x = \pi f / f_s \le 0.0285\text{ rad}$), the relative error is $\frac{17}{315}x^6 \le 2.9 \times 10^{-11}$ ($2.9 \times 10^{-9}\%$), far better than the claimed $< 0.001\%$.

• Theorem 4.2: The cancellation of $L(n_{\text{fret}})$ in $f_{\text{null},m} = \left(\frac{m L(n_{\text{fret}})}{x_p}\right)\left(\frac{c_s}{2 L(n_{\text{fret}})}\right) = \frac{m}{\beta_{0,\text{bridge}}}f_{0,\text{open},s}$ is physically and algebraically exact.
• Theorem 4.3: Evaluating the Fant–Hanson resonator gain $L_{F_1}(f) = 10\log_{10}\!\left(\frac{F_1^4 + F_1^2 B_1^2}{(f^2 - F_1^2)^2 + f^2 B_1^2}\right)$ at $f_0 = 523.25\text{ Hz}$ (C5), $F_1 = 523.25\text{–}530\text{ Hz}$, $B_1 = 120\text{ Hz}$ gives $L_{F_1}(f_0) = +13.1\text{ dB}$, $L_{F_1}(2f_0) = -9.6\text{ dB}$, and $\Delta L_{F_1} = +22.7\text{ dB}$ ($+28.6\text{ dB}$ at $B_1 = 60\text{ Hz}$), confirming Proposition 4.3.

3.4 Stream D & Collective Cochlear Control Core: Theorems 5.1, 5.2, and 5.3

1. Verification of Proposition 5.1 ($+7.22\text{ dB}$ Track-Splitting Error):

Splitting a $75\text{ dB SPL}$ bus across $M = 8$ equal tracks reduces each track's level by $10\log_{10}(8) = 9.0309\text{ dB}$, which steepens Zwicker's upper masking slope $S_u(L) = 24 + \frac{0.23}{f_{\text{kHz}}} - 0.2(L - 40)_+$ by $\Delta S_u = 0.2 \times 9.0309 = 1.80618\text{ dB/Bark}$. Over $\Delta z = +4\text{ Bark}$, the cumulative error is $4 \times 1.80618 = \mathbf{+7.2247\text{ dB}}$. Exact.

2. Verification of Partition of Unity & $M$-Independent Stability:

Because the responsibility weights $w_{j\to m}[b]$ are non-negative and satisfy $\sum_{j \notin \text{Tier1}}\sum_{b=0}^{11} w_{j\to m}[b] = \frac{E_{\text{nonTier1}}[m]}{E_{\text{nonTier1}}[m] + \epsilon} < 1$, the collective feedback operator has spectral radius strictly bounded independent of track count $M \in [1, 64]$.

• Theorem 5.2: Extending the discrete Legendre basis to degree $4$ ($\mathbf{p}_1, \mathbf{p}_2, \mathbf{p}_3, \mathbf{p}_4 \in \mathbb{R}^{12}$) while gating $(c_3, c_4)$ by the full-bandwidth occupancy factor $\gamma_{\text{full}} \in [0, 1]$ prevents discrete Runge boundary oscillation when low/high ERB bands sit at the noise floor.
• Theorem 5.3: For a $50\%$-duty-cycle sine-phased square wave $x_{\text{sq}}(t) = \frac{4}{\pi}\sum_{m \text{ odd}}\frac{1}{m}\cos(m\omega_0 t - \pi/2)$, $\phi_1 = \phi_3 = -\frac{\pi}{2} \implies \Delta\Phi_{3,1} = \phi_3 - 3\phi_1 = +\pi \implies \mathcal{B}_{3,1} = \cos(\pi) = -1.000$, which is identical to the biphase of a symmetrically clipped sinusoid. Pairing squareWaveImmunity01 with sign-opposed $+4.5\text{ dB}$-clamped Cubic Hermite interpolation ($10^{4.5/20} = 1.6788$) is mathematically rigorous.

4. Actionable List of Mathematical & Editorial Refinements for v3.5

To make the Taxonomy38 v3.5 archival monograph $100\%$ bulletproof, we recommend four targeted refinements:

1. Theorem 3.2 (§3.2.3, Eqs. 790–792) — Clarify RMS Amplitude vs. Squared-Energy Envelope in the 3-Epoch Beat Ratio:

• Replace $E(t) = E_0 e^{-2t/\tau_m}\cos^2(\pi\,\delta f_m t + \phi_0)$ with the RMS amplitude envelope $A(t) = A_0 e^{-t/\tau_m}\big|\cos(\pi\,\delta f_m t)\big|$ (so $\frac{\sqrt{A(t_1)A(t_3)}}{A(t_2)} = \frac{\sqrt{\cos(\alpha)\cos(3\alpha)}}{\cos(2\alpha)} = \cos(\pi\,\delta f_m\,\Delta t) + \mathcal{O}(\alpha^4)$), OR if $E(t)$ denotes squared power, write $\left(\frac{\sqrt{E_{28} E_{78}}}{E_{53}}\right)^{1/2}$ inside the $\arccos(\cdot)$ argument so the exponent of $\cos(\pi\,\delta f_m\,\Delta t)$ is $1$ rather than $2$.

1. Theorem 2.3 (§2.3.3, Item 3) — Fix the Transposed Sign Bounds on $k_{\text{ap}}[n]$ & Specify Two-Multiplier Gray–Markel Lattice:

• Change $k_{\text{ap}}[n] = \frac{\Delta_d[n]}{2 - \Delta_d[n]} \in [-0.265, +0.174]$ to $k_{\text{ap}}[n] \in [-0.174, +0.266]$ for $\Delta_d[n] \in [-0.42, +0.42]$.
• Note that Eq. (739) is the two-multiplier passive Gray–Markel all-pass lattice (with $\sqrt{1 - k_{\text{ap}}^2} \in [0.964, 1.0]$ near unity because $|k_{\text{ap}}| \le 0.266$).

1. Theorem 4.1 (§4.1.2, Item 3) — Maclaurin/Taylor Terminology for fastTanPadeBass:

• Note that $x\big(1 + \frac{1}{3}x^2 + \frac{2}{15}x^4\big)$ is the 5th-order Maclaurin (Taylor) polynomial of $\tan(x)$ (matching the Taylor expansion of the $[3/2]$ Padé approximant through $x^5$, with $< 3\times 10^{-9}\%$ error at $400\text{ Hz}$).

1. HTML Anchor IDs & KaTeX Superscript Braces in taxonomy38-v35-research-paper.html:

• In the Hero callout (line 530), change $\sin^-2(\Omega_c)$ to $\sin^{-2}(\Omega_c)$.
• In the sidebar Table of Contents (lines 572–593), synchronize the href="#..." anchor slugs with the 48-character <h3 id="..."> attributes in the article body so every TOC sub-link scrolls directly to its target section.