Below is a detailed analysis of the main errors and weaknesses in the text:
1. Major Inconsistency in Numerical Integration ("Marching Squares")
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Sections 3.3 & 4.3: The author claims to provide a certified proof using interval arithmetic via the inari library.
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The Flaw: Section 4.3 states that lemniscate length sampling relies on the standard double-precision floating-point (f64) marching squares algorithm, with a "conservative 5% grid error envelope absorbed into the interval arithmetic bounds".
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Why it is incorrect: The marching squares algorithm relies on spatial discretization and linear interpolation, which offer no rigorous bounds on the true arc length of the level set. Arbitrarily adding a 5% margin completely breaks the IEEE 1788 certification chain. A true rigorous proof would require bounding the curve length analytically or via rigorous interval integration techniques (e.g., Taylor model / Krawczyk methods).
2. Non-Rigorous Estimation of Lipschitz Constants
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Section 3.2: The certified upper bound on each box relies on the formula:
$$u_{i} \le L(p_{\text{center}}) + \text{Lip}(\nabla L) \cdot \text{diam}(B_i) + \epsilon_{\text{grid}}$$
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The Flaw: The text specifies that the Lipschitz constant $\text{Lip}(\nabla L)$ is "estimated via finite differences at multiple scales".
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Why it is incorrect: Estimating a Lipschitz constant using finite differences is a heuristic sampling technique, not a certified mathematical proof. It offers no guarantee that the function $L(p)$ does not exhibit steep derivatives or singularities between sample points. In a certified branch-and-bound proof, the Lipschitz constant must be analytically bounded or evaluated over the entire interval using automatic interval differentiation.
3. Flawed Hessian Analysis (Local Optimality)
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Sections 3.4 & 4.4: The author computes the Hessian matrix of $L(p)$ at $z^n - 1$ via central differences at various step sizes ($\epsilon \in \{10^{-4}, \dots, 10^{-7}\}$) and finds that all eigenvalues are strictly negative.
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Why it is incorrect: Computing the Hessian via central differences is a numerical approximation method. While it provides an indication of a local maximum, it does not constitute a certified proof.
4. Claim of a "New" Closed-Form Expression (Theorem 1)
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Section 2.2: The author presents the closed-form expression:
$$L(z^n - 1) = 2^{1/n} \sqrt{\pi} \frac{\Gamma(1/(2n))}{\Gamma(1/(2n) + 1/2)}$$
as a "new closed-form expression" (Contribution 2).
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The Reality: This exact formula for the arc length of the lemniscate $z^n - 1$ is a standard, well-known result in complex analysis and geometry (derived directly via polar coordinate substitution and reduction to the Beta/Gamma functions). Claiming it as a novel contribution reveals a lack of proper literature review.
5. Paradox in Complexity Jump ($n=10$ to $n=11$)
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Table 1: For $n=10$ (15 dimensions), the algorithm requires 32,768 evaluations and completes in 12 seconds.
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For $n=11$ (17 dimensions), the evaluation count surges to 10,223,616 taking 33 minutes.
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The Inconsistency: The author claims that "all non-extremizer boxes are eliminated at level 0 for $n \ge 5$". If boxes were truly eliminated at level 0 without further subdivision, the growth in evaluations would simply follow the increase in initial root boxes ($4^{n-2}$ or similar). The explosive ~300x jump between $n=10$ and $n=11$ indicates that deep box subdivisions occurred ($level > 0$), contradicting the claim that margins allow immediate elimination of the search space at level 0.
6 number of evaluations and computation time
Additional anomaly I found (not mentioned in the report): for n=4 (dim=3), the table shows 13,504 evaluations in 254 seconds, whereas 2³=8, and all other small values of n (3, 5, 6, 7...) resolve in under a second with evaluation counts matching 2^dim. Such a low dimension (3) justifies neither this number of evaluations nor this computation time — it's an isolated outlier in the table.
Summary
The paper attempts to blend formal verification terminology (IEEE 1788, Rust inari, branch-and-bound) with non-rigorous heuristic approximations (marching squares, finite-difference Lipschitz estimates, approximated Hessians). The certificate chain is broken, meaning the document does not constitute a valid mathematical verification of the conjecture for $n \in \{3, \dots, 12\}$.
errors_Mendoza_EHP_n3-12_March2026.pdf
Below is a detailed analysis of the main errors and weaknesses in the text:
1. Major Inconsistency in Numerical Integration ("Marching Squares")
Sections 3.3 & 4.3: The author claims to provide a certified proof using interval arithmetic via the
inarilibrary.The Flaw: Section 4.3 states that lemniscate length sampling relies on the standard double-precision floating-point (
f64) marching squares algorithm, with a "conservative 5% grid error envelope absorbed into the interval arithmetic bounds".Why it is incorrect: The marching squares algorithm relies on spatial discretization and linear interpolation, which offer no rigorous bounds on the true arc length of the level set. Arbitrarily adding a 5% margin completely breaks the IEEE 1788 certification chain. A true rigorous proof would require bounding the curve length analytically or via rigorous interval integration techniques (e.g., Taylor model / Krawczyk methods).
2. Non-Rigorous Estimation of Lipschitz Constants
Section 3.2: The certified upper bound on each box relies on the formula:
The Flaw: The text specifies that the Lipschitz constant$\text{Lip}(\nabla L)$ is "estimated via finite differences at multiple scales".
Why it is incorrect: Estimating a Lipschitz constant using finite differences is a heuristic sampling technique, not a certified mathematical proof. It offers no guarantee that the function$L(p)$ does not exhibit steep derivatives or singularities between sample points. In a certified branch-and-bound proof, the Lipschitz constant must be analytically bounded or evaluated over the entire interval using automatic interval differentiation.
3. Flawed Hessian Analysis (Local Optimality)
Sections 3.4 & 4.4: The author computes the Hessian matrix of$L(p)$ at $z^n - 1$ via central differences at various step sizes ($\epsilon \in \{10^{-4}, \dots, 10^{-7}\}$ ) and finds that all eigenvalues are strictly negative.
Why it is incorrect: Computing the Hessian via central differences is a numerical approximation method. While it provides an indication of a local maximum, it does not constitute a certified proof.
4. Claim of a "New" Closed-Form Expression (Theorem 1)
Section 2.2: The author presents the closed-form expression:
as a "new closed-form expression" (Contribution 2).
The Reality: This exact formula for the arc length of the lemniscate$z^n - 1$ is a standard, well-known result in complex analysis and geometry (derived directly via polar coordinate substitution and reduction to the Beta/Gamma functions). Claiming it as a novel contribution reveals a lack of proper literature review.
5. Paradox in Complexity Jump ($n=10$ to $n=11$ )
Table 1: For$n=10$ (15 dimensions), the algorithm requires 32,768 evaluations and completes in 12 seconds.
For$n=11$ (17 dimensions), the evaluation count surges to 10,223,616 taking 33 minutes.
The Inconsistency: The author claims that "all non-extremizer boxes are eliminated at level 0 for$n \ge 5$ ". If boxes were truly eliminated at level 0 without further subdivision, the growth in evaluations would simply follow the increase in initial root boxes ($4^{n-2}$ or similar). The explosive ~300x jump between $n=10$ and $n=11$ indicates that deep box subdivisions occurred ($level > 0$ ), contradicting the claim that margins allow immediate elimination of the search space at level 0.
6 number of evaluations and computation time
Additional anomaly I found (not mentioned in the report): for n=4 (dim=3), the table shows 13,504 evaluations in 254 seconds, whereas 2³=8, and all other small values of n (3, 5, 6, 7...) resolve in under a second with evaluation counts matching 2^dim. Such a low dimension (3) justifies neither this number of evaluations nor this computation time — it's an isolated outlier in the table.
Summary
The paper attempts to blend formal verification terminology (IEEE 1788, Rust$n \in \{3, \dots, 12\}$ .
inari, branch-and-bound) with non-rigorous heuristic approximations (marching squares, finite-difference Lipschitz estimates, approximated Hessians). The certificate chain is broken, meaning the document does not constitute a valid mathematical verification of the conjecture forerrors_Mendoza_EHP_n3-12_March2026.pdf