Arithmetic cancellation candidate: the Dirichlet hyperbola bilinear decomposition of D_N

Date: 2026-09-20 (revised following independent review). Base commit: 271fc26. Prior work: FRONTIER_2026_09_12.md, SIGNED_MEAN_RENEWAL.md, FRONTIER_INDEPENDENT_REVIEW.md, RANK3_CONDUCTOR_SUM.md, FAREY_BASELINE_REPAIR.md, ARITHMETIC_CANCELLATION_REVIEW.md. Verification scripts: hunts/prime_pair_error/arithmetic_cancellation_candidate_check.py, hunts/prime_pair_error/independent_arithmetic_cancellation_check.py. Diagnostic records: hunts/prime_pair_error/results_arithmetic_cancellation_candidate.json, hunts/prime_pair_error/independent_arithmetic_cancellation_evidence.json.

1. Exact candidate and parameter ranges

Let $N \ge 4$ be an integer. Define \[ K = \lfloor\sqrt{N}\rfloor, \qquad Y = \frac{N}{K}. \] Then $K \le \sqrt{N} < K + 1$ and $\sqrt{N} \le Y$, with the valid upper bound $Y = N/K < N/(\sqrt{N} - 1) = \sqrt{N} + 1 + 1/(\sqrt{N} - 1)$ for $N \ge 4$ (corrected 2026-09-20: $Y < \sqrt{N} + 1$ is false; e.g. $N = 8$ gives $Y = 4 > \sqrt{8} + 1$, and every $N = m^2 - 1$ gives $Y = m + 1 > \sqrt{N} + 1$; 314 violations in $4 \le N \le 100000$, see results_factorization_diagnostic.json). The von Mangoldt function $\Lambda(n)$ is supported on all prime powers $n = p^m$ ($m \ge 1$), with $\psi(x) = \sum_{n \le x} \Lambda(n)$ and $R(x) = \psi(x) - x$. The target quantity isolated in SIGNED_MEAN_RENEWAL.md equation (17) is \[ D_N = N \int_{N/K}^\infty \frac{R(u)}{u^2}\,du - \sum_{k=2}^K R(N/k). \tag{1} \] The existing unconditional scale reduction proves $R(N) = D_N + O(\sqrt{N})$. The RH target is $|D_N| \ll_\epsilon N^{1/2+\epsilon}$ for every $\epsilon > 0$.

The candidate decomposition developed here is the exact algebraic partition \[ \boxed{\quad D_N = \mathcal{T}{\mathrm{bilinear}}(N, K) + \mathcal{T}{\mathrm{sawtooth}}(N, K) + \mathcal{R}_{\mathrm{boundary}}(N, K), \quad} \tag{2} \] where the constituent terms are defined explicitly as follows:

1. The centered bilinear hyperbolic sum:

\[ \mathcal{T}{\mathrm{bilinear}}(N, K) := -\sum{k=2}^K \sum_{Y < d \le N/k} \big(\Lambda(d) - 1\big). \tag{3} \] Because $\sum_{Y < d \le N/k} 1 = \lfloor N/k \rfloor - \lfloor Y \rfloor$ uses discrete integer floors while $R(N/k) - R(Y) = (\psi(N/k) - \psi(Y)) - (N/k - Y)$ uses continuous linear slopes, the relation to the prime-counting error $R$ is \[ \mathcal{T}{\mathrm{bilinear}}(N, K) = -\sum{k=2}^K \big( R(N/k) - R(Y) \big) - \mathcal{E}{\mathrm{frac}}(N, K), \tag{3'} \] where the exact rational discrepancy is \[ \mathcal{E}{\mathrm{frac}}(N, K) = \sum_{k=2}^K \left( \left\{ \frac{N}{k} \right\} - \{ Y \} \right). \] Indeed $\lfloor x \rfloor = x - \{x\}$, so $\mathcal{T}{\mathrm{bilinear}} + \sum{k=2}^K (R(N/k) - R(Y)) = \sum_{k=2}^K (\{Y\} - \{N/k\}) = -\mathcal{E}{\mathrm{frac}}$. (Corrected 2026-09-20: the repair draft had $+$ here; at $N = 49$, $\mathcal{E}{\mathrm{frac}} = \sum_{k=2}^7 \{49/k\} = 41/20$ while $\mathcal{T}{\mathrm{bilinear}} + \sum (R - R(Y)) = -41/20$, both sides checked independently in results_factorization_diagnostic.json.) Since $0 \le \{x\} < 1$, $|\mathcal{E}{\mathrm{frac}}(N, K)| < K - 1 < K \le \sqrt{N}$ is unconditionally bounded.

1. The signed sawtooth correlation:

\[ \mathcal{T}{\mathrm{sawtooth}}(N, K) := -\sum{d \le Y} \Lambda(d) \left( \left\{ \frac{N}{d} \right\} - \frac{1}{2} \right), \tag{4} \] where $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part.

1. The explicit boundary remainder:

\[ \mathcal{R}{\mathrm{boundary}}(N, K) := \frac{1}{2}\psi(Y) - \mathcal{B}{\mathrm{main}}(N, K) - \sum_{d \le Y} \Lambda(d) \left\lfloor \frac{N}{d} \right\rfloor + \mathcal{S}{\mathrm{smooth}}(N, K), \tag{5} \] in which \[ \mathcal{B}{\mathrm{main}}(N, K) = \sum_{k=2}^K \left( \left\lfloor \frac{N}{k} \right\rfloor - \lfloor Y \rfloor \right), \tag{6} \] and the smooth baseline is \[ \mathcal{S}_{\mathrm{smooth}}(N, K) = N \big( \log Y + H_K - 2 - \gamma \big), \qquad H_K = \sum_{k=1}^K \frac{1}{k}. \tag{7} \]

2. Complete derivation achieved

2.1. Exact reduction of the improper integral

The integral in (1) extends to infinity. In previous analyses, this feature appeared to prevent a discrete arithmetic attack. However, the unconditional PNT ensures absolute convergence of $\int_1^\infty R(u) u^{-2} du = -(1+\gamma)$. Therefore, the tail decomposes exactly as \[ \int_{N/K}^\infty \frac{R(u)}{u^2}\,du = -(1+\gamma) - \int_1^{N/K} \frac{R(u)}{u^2}\,du. \tag{8} \] On the finite interval $[1, Y]$ where $Y = N/K$, $\psi(u)$ is piecewise constant with jump discontinuities of magnitude $\Lambda(n)$ at prime powers $n \le Y$, so $R(u) = \psi(u) - u$ is piecewise linear with slope $-1$. Integrating by parts or summing over intervals between prime powers yields the identity: \[ \int_1^Y \frac{\psi(u)}{u^2}\,du = \sum_{d \le Y} \Lambda(d) \int_d^Y \frac{du}{u^2} = \sum_{d \le Y} \frac{\Lambda(d)}{d} - \frac{\psi(Y)}{Y}. \tag{9} \] Since $\int_1^Y u^{-1} du = \log Y$, we have the exact closed formula \[ \int_1^Y \frac{R(u)}{u^2}\,du = \sum_{d \le Y} \frac{\Lambda(d)}{d} - \frac{\psi(Y)}{Y} - \log Y. \tag{10} \] Multiplying by $N$ (and noting that $N/Y = K$): \[ N \int_{N/K}^\infty \frac{R(u)}{u^2}\,du = K \psi(Y) - N \sum_{d \le Y} \frac{\Lambda(d)}{d} + N \log Y - (1+\gamma)N. \tag{11} \] This removes the improper integral completely. Every term is now evaluated at scales $u \le Y \approx \sqrt{N}$.

2.2. Subtraction of the discrete sum

The discrete sum in (1) is \[ \sum_{k=2}^K R(N/k) = \sum_{k=2}^K \psi(N/k) - N \sum_{k=2}^K \frac{1}{k} = \sum_{k=2}^K \psi(N/k) - N (H_K - 1). \tag{12} \] Subtracting (12) from (11) gives the exact finite representation: \[ D_N = \left[ K \psi(Y) - \sum_{k=2}^K \psi(N/k) - N \sum_{d \le Y} \frac{\Lambda(d)}{d} \right] + \mathcal{S}{\mathrm{smooth}}(N, K), \tag{13} \] where $\mathcal{S}{\mathrm{smooth}}(N, K) = N (\log Y + H_K - 2 - \gamma)$ is given in (7).

2.3. Hyperbola decomposition of the arithmetic bracket

We partition the term $K \psi(Y) - \sum_{k=2}^K \psi(N/k)$: \[ K \psi(Y) - \sum_{k=2}^K \psi(N/k) = \psi(Y) + \sum_{k=2}^K \big( \psi(Y) - \psi(N/k) \big) = \psi(Y) - \sum_{k=2}^K \sum_{Y < d \le N/k} \Lambda(d). \tag{14} \] Next, write $\Lambda(d) = 1 + (\Lambda(d) - 1)$. The double sum becomes \[ \sum_{k=2}^K \sum_{Y < d \le N/k} \Lambda(d) = \mathcal{B}{\mathrm{main}}(N, K) - \mathcal{T}{\mathrm{bilinear}}(N, K), \tag{15} \] with $\mathcal{B}{\mathrm{main}}(N, K)$ and $\mathcal{T}{\mathrm{bilinear}}(N, K)$ defined in (6) and (3). Similarly, decomposing $N/d = \lfloor N/d \rfloor + 1/2 + (\{N/d\} - 1/2)$, the divisor sum evaluates to \[ N \sum_{d \le Y} \frac{\Lambda(d)}{d} = \sum_{d \le Y} \Lambda(d) \left\lfloor \frac{N}{d} \right\rfloor + \frac{1}{2}\psi(Y) - \mathcal{T}{\mathrm{sawtooth}}(N, K), \tag{16} \] with $\mathcal{T}{\mathrm{sawtooth}}(N, K)$ defined in (4). Substituting (14), (15), and (16) into (13) proves the candidate identity (2) with zero defect: \[ D_N = \mathcal{T}{\mathrm{bilinear}}(N, K) + \mathcal{T}{\mathrm{sawtooth}}(N, K) + \mathcal{R}_{\mathrm{boundary}}(N, K). \] This algebraic match agrees independently with defect at most $6 \times 10^{-12}$ at $N \le 100000$ (independent checker, float-$\psi$ accumulation dominated) and $5 \times 10^{-16}$ at $N = 400$; the author's dps-40 checker reports defects below $10^{-35}$ on $N \le 400$. Finite agreement is diagnostic, not a proof of the exact identity; the proof is the algebra of (13)-(16).

3. Decisive arithmetic property

The candidate replaces the scalar definition of $D_N$ with three structured components:

1. Bilinear structure of $\mathcal{T}{\mathrm{bilinear}}$: The sum $\mathcal{T}{\mathrm{bilinear}}(N, K) = -\sum_{k=2}^K \sum_{Y < d \le N/k} (\Lambda(d) - 1)$ is an arithmetic bilinear form over the hyperbolic region $\{ (k, d) : 2 \le k \le K, Y < d \le N/k \}$. By Dirichlet convolution, $\Lambda(d) = \sum_{a b = d} \mu(a) \log b$. Therefore, $\mathcal{T}{\mathrm{bilinear}}$ expands into a ternary sum: \[ -\sum{k=2}^K \sum_{\substack{a b \le N/k \\ a b > Y}} \mu(a) \log b + \sum_{k=2}^K \left( \left\lfloor \frac{N}{k} \right\rfloor - \lfloor Y \rfloor \right). \] The decisive arithmetic property is the sign oscillation of the Möbius function $\mu(a)$ across coprime factors, paired with bilinear decoupling between the scale variable $k$ and the modulus $d$.
2. Equidistribution of fractional parts in $\mathcal{T}{\mathrm{sawtooth}}$: $\mathcal{T}{\mathrm{sawtooth}}(N, K) = -\sum_{d \le Y} \Lambda(d) \psi_0(N/d)$, where $\psi_0(x) = \{x\} - 1/2$. The decisive arithmetic property is the non-resonance of the prime-power sequence $d = p^m$ with the modular inverses or fractional parts $N/d$.
3. Exact boundary relation and algebraic coupling: Let $A(N) = \sum_{d \le Y} \Lambda(d) \lfloor N/d \rfloor + \mathcal{B}{\mathrm{main}}(N, K)$. Partitioning $\sum{d k \le N} \Lambda(d) = \log(N!)$ reveals the exact algebraic identity \[ A(N) - \log(N!) = \mathcal{T}{\mathrm{bilinear}}(N, K) - \big(\psi(N) - \psi(Y)\big). \] Because $\psi(N) - \psi(Y) = N - Y + R(N) - R(Y) = N + o(N)$, this discrepancy is order $N$, not $O(\sqrt{N})$. The complementary region $\{k=1, d > Y\}$ carrying the main term $N - Y$ accounts for this difference. Consequently, the boundary term satisfies the exact identity \[ \mathcal{R}{\mathrm{boundary}}(N, K) = R(N) - \mathcal{T}{\mathrm{bilinear}}(N, K) + E{\mathrm{det}}(N), \] where $E_{\mathrm{det}}(N) = \mathcal{S}{\mathrm{smooth}}(N, K) - \log(N!) + N - \frac{1}{2}\psi(Y) = -\frac{1}{2}R(Y) + O(\log N) = O(\sqrt{N})$. Hence bounding $\mathcal{R}{\mathrm{boundary}}$ is algebraically coupled to bounding $R(N) - \mathcal{T}{\mathrm{bilinear}}$. The form $E{\mathrm{det}}(N) = -\frac12 R(Y) + O(\log N)$ follows from Stirling $\log(N!) = N\log N - N + \frac12\log N + \frac12\log 2\pi + O(1/N)$ and $H_K = \log K + \gamma + 1/(2K) + O(1/K^2)$: since $YK = N$, $\mathcal{S}{\mathrm{smooth}} = N\log N - 2N + N/(2K) + O(1/K)$-times-$N$ with the $O(N/K^2) = O(Y/K) = O(1)$ remainder, and $N/(2K) - \psi(Y)/2 = -R(Y)/2$, giving $E{\mathrm{det}} = -R(Y)/2 - \frac12\log N
4. \frac12\log 2\pi + O(1)$. By Chebyshev $|R(Y)| \ll Y \ll \sqrt{N}$, $E_{\mathrm{det}} = O(\sqrt{N})$ unconditionally.

4. Strongest justified bound per term

Baseline bounds details:

• $\mathcal{T}{\mathrm{sawtooth}}$: Because $|\{N/d\} - 1/2| \le 1/2$ for all real arguments, \[ |\mathcal{T}{\mathrm{sawtooth}}(N, K)| \le \frac{1}{2} \sum_{d \le Y} \Lambda(d) = \frac{1}{2} \psi(Y). \] With the inherited Chebyshev bound $\psi(Y) \ll Y$ and $Y \ll \sqrt{N}$, $|\mathcal{T}_{\mathrm{sawtooth}}| \le \frac12\psi(Y) \ll \sqrt{N}$. (Corrected 2026-09-20: the draft cited $Y \le \sqrt{N} + 1$ and the constants $1.04$, $0.52$ without valid ranges; both are removed and no explicit constant is claimed.) This is already within the target unconditionally.
• $\mathcal{R}{\mathrm{boundary}}$: Because $\mathcal{R}{\mathrm{boundary}} = R(N) - \mathcal{T}{\mathrm{bilinear}} + E{\mathrm{det}}(N)$ with $E_{\mathrm{det}}(N) = -\frac{1}{2}R(Y) + O(\log N) \ll \sqrt{N}$, unconditional bounds on $R(N)$ and $\mathcal{T}{\mathrm{bilinear}}$ transfer directly to $\mathcal{R}{\mathrm{boundary}}$. An elementary $O(\sqrt{N})$ bound for $\mathcal{R}{\mathrm{boundary}}$ is not established independently of $R(N) - \mathcal{T}{\mathrm{bilinear}}$.

5. Exact remaining obstacle

The exact remaining obstacle to proving $|D_N| \ll_\epsilon N^{1/2+\epsilon}$ is: \[ \boxed{\quad \left| \mathcal{T}{\mathrm{bilinear}}(N, K) \right| \ll\epsilon N^{1/2+\epsilon} \quad \text{and} \quad \left| R(N) - \mathcal{T}{\mathrm{bilinear}}(N, K) \right| \ll\epsilon N^{1/2+\epsilon}. \quad} \tag{17} \] Notice the crucial analytical features:

1. Absolute value norm loses the target: If one takes absolute values inside the sum, then even assuming the Riemann Hypothesis $|R(x)| \le C x^{1/2} \log^2 x$, partial summation gives the upper bound \[ \sum_{k=2}^K |R(N/k)| \ll N^{1/2} \log^2 N \sum_{k=2}^{\sqrt{N}} k^{-1/2} \ll N^{3/4} \log^2 N. \] This upper bound leaves an $N^{1/4}$ gap above $N^{1/2}$. Retaining signed cancellation across scales $k$ is mandatory.
2. Spectral representation: Under the formal explicit formula $R(x) \approx -\sum_\rho \frac{x^\rho}{\rho}$, the mode sum is \[ \sum_{k=2}^K R(N/k) \approx -\sum_\rho \frac{N^\rho}{\rho} \sum_{k=2}^K k^{-\rho}. \] By Euler summation, $\sum_{k=1}^K k^{-\rho} = \frac{K^{1-\rho}}{1-\rho} + \zeta(\rho) + \text{tail}_K$, where $\text{tail}K = O(K^{-\beta})$. Crucially, $\sum{k=2}^K k^{-\rho} = \sum_{k=1}^K k^{-\rho} - 1$. When testing the pure power mode $g(u) = u^\rho$ for fixed $\rho$ with $0 < \Re\rho < 1$ in the functional $D_N$, we find \[ D_N[u^\rho] = N \int_{N/K}^\infty u^{\rho-2}\,du - \sum_{k=2}^K (N/k)^\rho = N^\rho B(\rho), \] where \[ B(\rho) = \frac{K^{1-\rho}}{1-\rho} - \sum_{k=2}^K k^{-\rho} = 1 - \zeta(\rho) - \text{tail}_K. \] At any zeta zero, $\zeta(\rho) = 0$, so $B(\rho) = 1 - \text{tail}_K \to 1$ as $K \to \infty$. Therefore, for a zero mode: \[ D_N[u^\rho] \sim N^\rho. \] On the critical line ($\Re\rho = 1/2$), this remainder has magnitude $|D_N[u^\rho]| = N^{1/2}|B(\rho)| \sim N^{1/2}$, which is borderline at the target scale, not $N^{1/4}$. (The $1/|\rho|$ factor belongs to the explicit-formula coefficient $N^\rho/\rho$ of $R$ itself, not to the pure mode tested here.) Off-critical with $\Re\rho = \beta > 1/2$, the mode produces $|D_N[u^\rho]| \sim N^\beta \gg \sqrt{N}$. For non-real zeros $\rho = \beta + i\gamma$, the mode $N^\rho = N^\beta e^{i\gamma\log N}$ oscillates; the bound is an envelope $|N^\rho| = N^\beta$, not monotonic growth at every integer. For a real exponent $\rho = \beta$ the mode $N^\beta$ would instead grow monotonically; no real zero is asserted to exist. This reconciles Section 5 with Section 7.

6. Quantitative implication for $D_N$ if resolved

Adding the exact boundary identity $\mathcal{R}{\mathrm{boundary}} = R(N) - \mathcal{T}{\mathrm{bilinear}} + E_{\mathrm{det}}(N)$ to the decomposition (2) yields the exact relation \[ D_N = R(N) + \mathcal{T}{\mathrm{sawtooth}}(N, K) + E{\mathrm{det}}(N). \tag{18} \] Since $|\mathcal{T}{\mathrm{sawtooth}}| \le \frac12\psi(Y) \ll \sqrt{N}$ and $|E{\mathrm{det}}| = |-\frac{1}{2}R(Y) + O(\log N)| \ll \sqrt{N}$ (by Chebyshev $|R(Y)| \le \psi(Y) + Y \ll Y \ll \sqrt{N}$) are unconditionally bounded by $O(\sqrt{N})$, we recover $D_N = R(N) + O(\sqrt{N})$. Therefore:

1. Bounding $|D_N| \ll_\epsilon N^{1/2+\epsilon}$ is equivalent to bounding $|R(N)| \ll_\epsilon N^{1/2+\epsilon}$.
2. In decomposition (2), establishing $|\mathcal{T}{\mathrm{bilinear}}| \ll\epsilon N^{1/2+\epsilon}$ and $|\mathcal{R}{\mathrm{boundary}}| \ll\epsilon N^{1/2+\epsilon}$ jointly proves $|D_N| \ll_\epsilon N^{1/2+\epsilon}$, which by the two-way mean criterion (SIGNED_MEAN_RENEWAL.md equation (5)) is equivalent to RH.

7. Analysis of the smooth zero-mode diagnostic

In SIGNED_MEAN_RENEWAL.md section 8, a smooth perturbation $R_\eta(u)$ was constructed with $R_\eta(u) = -a + \eta \Re u^\rho$ for $u \ge 4$, where $\zeta(\rho) = 0$ with $\beta = \Re\rho > 1/2$. That diagnostic satisfies:

• Nonnegativity $\Psi_\eta \ge 0$ and monotonicity $\Psi_\eta' \ge 1/2$;
• The integral normalization $\int_1^\infty R_\eta(u) u^{-2} du = -(1+\gamma)$;
• The Chebyshev total variation bound $\operatorname{Var}{[1, y]} R\eta \ll y$;
• The scale relation $(L_K R_\eta)(N) = O(N/K) = O(\sqrt{N})$.

Why $R_\eta$ fails to bound $D_N$:

1. $D_N[R_\eta]$ is large: By definition, $D_N[f] = f(N) - (L_K f)(N)$. For the smooth model $R_\eta$, $(L_K R_\eta)(N) = O(\sqrt{N})$, but $R_\eta(N) \asymp \eta N^\beta$. Therefore, \[ D_N[R_\eta] \asymp \eta N^\beta \gg \sqrt{N}. \] Thus, $R_\eta$ does not satisfy $|D_N| \ll \sqrt{N}$. It honestly demonstrates that $L_K f(N) = O(\sqrt{N})$ does not by itself force $f(N) = O(\sqrt{N})$ for general smooth functions.
2. $R_\eta$ lacks the decisive arithmetic hypotheses:
3. No prime-power support: $R_\eta$ is $C^1$ smooth; its derivative has no delta masses.
4. Violates exact divisor recurrence: For the true von Mangoldt function, $\sum_{k \le N} R(N/k) = G(N) = \log(N!) - N H_N$. For $R_\eta$, $\sum_{k \le N} R_\eta(N/k) = -(1+\gamma)N + O(1)$, which misses the exact arithmetic forcing by $\frac{1}{2}\log N + O(1)$.
5. No Dirichlet convolution structure: $R_\eta$ cannot be decomposed into a bilinear form with Möbius signs. The smooth model is a valid diagnostic for the transfer operator $L_K$, not a counterexample to RH.

8. Falsifiable small-case diagnostic for Muse

Independent values agreeing with the author's JSON records (results_arithmetic_cancellation_candidate.json) to the displayed decimals (diagnostic agreement, not a proof):

9. Distinction from equivalent rewritings of R

A trivial manipulation would express $D_N = R(N) - (R(N) - D_N) = R(N) + O(\sqrt{N})$. Candidate decomposition (2):

1. Syntactic elimination: The terms $\mathcal{T}{\mathrm{bilinear}}$, $\mathcal{T}{\mathrm{sawtooth}}$, and $\mathcal{R}_{\mathrm{boundary}}$ are defined without invoking $R(N)$ or $\psi(N)$; the largest argument is $N/2$.
2. Algebraic coupling: As revealed by the exact identity $\mathcal{R}{\mathrm{boundary}} = R(N) - \mathcal{T}{\mathrm{bilinear}} + E_{\mathrm{det}}(N)$, the decomposition does not eliminate $R(N)$ semantically. Rather, it partitions $D_N$ into a short-range sawtooth term $\mathcal{T}{\mathrm{sawtooth}}$ and the coupled difference $R(N) - \mathcal{T}{\mathrm{bilinear}}$.
3. Identification of structure: It separates the short-range fractional correlation $\mathcal{T}{\mathrm{sawtooth}}$ (bounded by $\frac12\psi(Y) \ll \sqrt{N}$) from the multi-scale hyperbolic remainder $\mathcal{T}{\mathrm{bilinear}}$.

10. Conclusion and disposition

Under ALIGNMENT.md Section 5:

• Disposition: Attempt unresolved.
• The arithmetic bilinear decomposition (2) is exact and verified independently.
• The sawtooth term $\mathcal{T}_{\mathrm{sawtooth}}$ is unconditionally bounded by $O(\sqrt{N})$.
• The boundary term $\mathcal{R}{\mathrm{boundary}}$ and bilinear term $\mathcal{T}{\mathrm{bilinear}}$ remain unresolved and algebraically coupled.
• The spectral remainder at a zero mode is of order $N^\beta$ (borderline $N^{1/2}$ on the critical line), not $N^{1/4}$.
• No claim of having proved RH is made.