Date: 2026-09-20. Runtime observed: Gemini 3.8 Flash high (provenance accurately recorded; not Opus, per coordinator notice). Scope: Bounded audit, repair, and final acceptance of package files under hunts/prime_pair_error (.). Base commit: 109c79808158252b7134c6a19543111bfdbb1e08.
1. Executive Disposition
Under ALIGNMENT.md Section 5, the mathematical status of this candidate is:
$$\mathbf{ATTEMPT\_UNRESOLVED}$$
ACCEPTED ONLY AS A PRESERVED UNRESOLVED ATTEMPT. No new cancellation estimate for $D_N$ has been obtained. Proposed finer analytic baselines (in particular the monotone-telescoping subexponential estimate for $\Sigma_2$) are not accepted; see section 4. No originality or novelty is claimed for standard identities ($\Lambda = \mu * \log$, Dirichlet hyperbola partition, Stirling/harmonic expansions); the package's contribution, if any, is only in their assembly and checking.
No new improved asymptotic bound on $D_N$ or $\psi_2(N, k)$ has been established. The scientific value of this package consists of:
- Exact, closed finite algebraic identities for the full functional $D_N$ without remainder leakage (proved by the written derivations plus inherited integral/PNT facts; finite-case machine checks per section 3 support, not replace, the proofs).
- Exact rational checks of stated finite cases: $\mathcal{E}_{\mathrm{frac}}$ values, prime-log coefficient matches at 15 cutoffs, the smooth/fractional split, and 4 planted lesion detections (section 3).
- Retraction of unsupported analytic sub-claims introduced in earlier drafts, as itemized in section 6.
- Elementary unconditional bound $\Sigma_2 \ll N \log^3 N$ retained as justified; the proposed subexponential baseline needs review.
- Sharply formulated remaining open joint analytic inequality.
Provenance honesty: the exact tests are worker-written code from this same campaign (a same-model worker, per the runtime record below), independent of the author checker but not a third-model review. Pending-review items are not complete.
2. Exact Finite Equations Accepted
The following exact algebraic equations are proved by the written derivations in the candidate and factorization notes together with the inherited integral/PNT facts; finite-case machine checks (section 3) support but do not replace those proofs, and no numerical oracle is invoked for the general case:
2.1. Candidate Bilinear Partition and Exact Integral Reduction
For integer $N \ge 4$, $K = \lfloor\sqrt{N}\rfloor$, and $Y = N/K$: $$D_N = \mathcal{T}{\mathrm{bilinear}}(N, K) + \mathcal{T}{\mathrm{sawtooth}}(N, K) + \mathcal{R}{\mathrm{boundary}}(N, K)$$ where the improper integral is evaluated in finite closed form: $$N \int{N/K}^\infty \frac{R(u)}{u^2} \, du = K \psi(Y) - N \sum_{d \le Y} \frac{\Lambda(d)}{d} + N \log Y - (1+\gamma)N$$ with $R(u) = \psi(u) - u$.
2.2. Exact Rational Discrepancy for Bilinear Sum
$$\mathcal{T}{\mathrm{bilinear}}(N, K) = -\sum{k=2}^K \big( R(N/k) - R(Y) \big) - \mathcal{E}{\mathrm{frac}}(N, K)$$ with confirmed minus sign and rational discrepancy: $$\mathcal{E}{\mathrm{frac}}(N, K) = \sum_{k=2}^K \left( \left\{ \frac{N}{k} \right\} - \{Y\} \right)$$ satisfying $|\mathcal{E}_{\mathrm{frac}}| < K \le \sqrt{N}$.
2.3. Exact Boundary Relation and Algebraic Coupling
With $A(N) = \sum_{d \le Y} \Lambda(d) \lfloor N/d \rfloor + \mathcal{B}{\mathrm{main}}(N, K)$: $$A(N) - \log(N!) = \mathcal{T}{\mathrm{bilinear}}(N, K) - \big(\psi(N) - \psi(Y)\big) \asymp -N$$ $$\mathcal{R}{\mathrm{boundary}}(N, K) = R(N) - \mathcal{T}{\mathrm{bilinear}}(N, K) + E_{\mathrm{det}}(N)$$ where $E_{\mathrm{det}}(N) = \mathcal{S}_{\mathrm{smooth}}(N, K) - \log(N!) + N - \frac{1}{2}\psi(Y) = -\frac{1}{2}R(Y) + O(\log N) = O(\sqrt{N})$. Consequently, the partition collapses to the exact renewal relation: $$D_N = R(N) + \mathcal{T}{\mathrm{sawtooth}}(N, K) + E{\mathrm{det}}(N) = R(N) + O(\sqrt{N})$$
2.4. Discrete Kernel Representation
$$D_N = \mathcal{S}{\mathrm{smooth}}(N, K) + \sum{m=1}^M w_N(m) \Lambda(m)$$ where $M = \lfloor N/2 \rfloor$, $\mathcal{S}_{\mathrm{smooth}}(N, K) = N (\log Y + H_K - 2 - \gamma)$, and: $$w_N(m) = \begin{cases} 1 - \frac{N}{m}, & 1 \le m \le Y \\ 1 - \lfloor \frac{N}{m} \rfloor, & Y < m \le M \end{cases}$$
2.5. Truncated Dirichlet Hyperbola Partition
Substituting $\Lambda = \mu * \log$: $$D_N = \mathcal{S}_{\mathrm{smooth}}(N, K) + \Sigma_1(N) + \Sigma_2(N)$$ with $U = \lfloor\sqrt{M}\rfloor$: $$\Sigma_1(N) = \sum_{a=1}^U \mu(a) \sum_{b=2}^{\lfloor M/a \rfloor} \log b \, w_N(a b)$$ $$\Sigma_2(N) = \sum_{b=2}^U \log b \sum_{a=U+1}^{\lfloor M/b \rfloor} \mu(a) w_N(a b)$$ Absence of remainder region: because $(U+1)^2 > M$, the region $\{a > U, b > U, ab \le M\}$ is strictly empty.
2.6. Signed Smooth and Fractional Splitting of $\Sigma_2$
Because $1 - \lfloor x \rfloor = 1 - x + \{x\}$ (plus sign for the fractional part): $$\mathcal{M}b(N) = \sum{U < a \le \lfloor M/b \rfloor} \mu(a) \left( 1 - \frac{N}{ab} \right) + \sum_{U < a \le \lfloor M/b \rfloor} \mu(a) \left\{ \frac{N}{ab} \right\} =: \mathcal{M}{b,\mathrm{smooth}}(N) + \mathcal{M}{b,\mathrm{frac}}(N)$$ $$\Sigma_2(N) = \Sigma_{2,\mathrm{smooth}}(N) + \Sigma_{2,\mathrm{frac}}(N)$$ verified independently in exact rational arithmetic with zero tolerance.
2.7. Guarded Level-Set Mertens Expansion
For $hi = \min(\lfloor M/b \rfloor, \lfloor N/(bk) \rfloor)$ and $lo = \max(U, \lfloor N/(b(k+1)) \rfloor)$: $$\mathcal{M}b(N) = [M(\lfloor M/b \rfloor) - M(U)] - \sum{k=2}^K k [M(hi) - M(lo)] \cdot \mathbf{1}_{hi > lo}$$ where the guard $hi > lo$ is required to prevent wrong-signed nonzero contributions on empty cells.
2.8. Spectral Mode Normalization
For fixed $\rho$ with $0 < \Re\rho < 1$ and $g(u) = u^\rho$: $$D_N[u^\rho] = N^\rho B(\rho), \qquad B(\rho) = 1 - \zeta(\rho) - \text{tail}_K$$ At any zeta zero $\zeta(\rho) = 0$, $B(\rho) \to 1$. For a critical zero $\rho = 1/2 + i\gamma$, $|D_N[u^\rho]| = N^{1/2}|B(\rho)| \sim N^{1/2}$, 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.) Since $|u^\rho| = u^\beta$ with $\beta = \Re\rho$: the cosine $\cos(\gamma \log N)$ describes only the real part $\Re(N^\rho)$, whose oscillation is not monotonic growth at every integer.
3. Evidence Classes
The mathematical assertions in this package are grounded in three distinct evidence classes:
| Class | Method | Scope | Key Results | ||||
|---|---|---|---|---|---|---|---|
| Class A | Exact rational Fraction arithmetic (zero tolerance) | Stated finite cases only | Rational residual $\mathcal{E}_{\mathrm{frac}}$ values at 13 cutoffs including N = 49 (both signs discriminated); prime-log coefficient matches at 15 finite cutoffs (N = 12..400 per results_factorization_diagnostic.json); smooth/fractional split $M_b = M_{b,\mathrm{smooth}} + M_{b,\mathrm{frac}}$ at checked cells; 4 planted lesion detections. These prove the stated finite cases only. | ||||
| Class B | High-precision transcendental evaluation (mpmath dps=80) | Transcendentals ($\log, \gamma, \zeta$) at checked cutoffs | Candidate identity (2) defect $\le 5.3 \times 10^{-12}$; integral reduction formula defect $\le 6.0 \times 10^{-17}$; remainder identities defects $\le 2.5 \times 10^{-10}$. Finite-cutoff agreement only. | ||||
| Class C | Measured floating-point diagnostics (double precision) | Envelopes, ratios, and diagnostics | Measured defect $ | D_N - (\mathcal{S}_{\mathrm{smooth}} + \Sigma_1 + \Sigma_2) | $ at float64 level; $(A - \log(N!))/N \approx -1$; spectral bracket values tracking $ | 1 - \zeta(\rho) | $; $Y$-range sweep violations count (314). Diagnostic only. |
General algebraic identities rest on the written derivations plus the inherited integral/PNT facts, not on any numerical oracle. Kernel and partition identities were checked in floats (Class C), not in exact arithmetic: there is no exact kernel verification. All test code is campaign-worker code (same-model family), independent of the author checker but not a third-model review.
4. Conditional and Unconditional Baselines
The table below summarizes the strongest justified baselines on record:
| Quantity | Justified Baseline | Status / Derivation | ||||
|---|---|---|---|---|---|---|
| $\mathcal{T}_{\mathrm{sawtooth}}(N, K)$ | $O(\sqrt{N})$ | Established unconditionally. $ | \psi_0(x) | \le 1/2 \implies | \mathcal{T}_{\mathrm{sawtooth}} | \le \frac{1}{2}\psi(Y) \ll \sqrt{N}$. |
| $E_{\mathrm{det}}(N)$ | $O(\sqrt{N})$ | Established unconditionally. $E_{\mathrm{det}} = -\frac{1}{2}R(Y) + O(\log N) \ll \sqrt{N}$ via Chebyshev. | ||||
| $\mathcal{T}_{\mathrm{bilinear}}(N, K)$ | $O(N \log N \exp(-c\sqrt{\log N}))$ | Unconditional. Inherited from PNT for $R(x)$ across $k \le K$. Under RH: $\sum_{k \le K} | R(N/k) | \ll N^{3/4}\log^2 N$. | ||
| $\mathcal{R}_{\mathrm{boundary}}(N, K)$ | $O(N \log N \exp(-c\sqrt{\log N}))$ | Unconditional (algebraically coupled). Coupled to $R(N) - \mathcal{T}_{\mathrm{bilinear}} + O(\sqrt{N})$. | ||||
| $\Sigma_2(N)$ (trivial) | $O(N \log^3 N)$ | Justified. Absolute values $ | \mu | \le 1$ without sign cancellation. Honestly labeled not best known. | ||
| $\Sigma_2(N)$ (monotone telescoping) | $O(N \exp(-c''\sqrt{\log N}))$ | Proposed baseline, needs review — not accepted. The written derivation does not rigorously handle active jump ranges, changing exponential constants, $V$ endpoints, and small $N$. | ||||
| Joint target $D_N$ (under RH for $\zeta$) | $O(N^{1/2}\log^2 N)$ | Proved conditionally. Follows from $D_N = R(N) + O(\sqrt{N})$ and von Koch $ | R(x) | \ll x^{1/2}\log^2 x$. |
5. The Remaining Open Joint Inequality
The remaining obstacle to establishing $|D_N| \ll_\epsilon N^{1/2+\epsilon}$ without assuming RH is the joint cancellation:
$$\boxed{\quad \left| \mathcal{S}_{\mathrm{smooth}}(N, K) + \Sigma_1(N) + \Sigma_2(N) \right| \ll_\epsilon N^{1/2+\epsilon} \quad}$$
What this audit does and does not say:
- The estimates attempted here do not establish the required joint bound. No lower bound for $\Sigma_2$ was proved, and nothing here shows that separate bounds saturate, that $N^{1/2+\epsilon}$ cannot be achieved piece by piece, or that joint cancellation is required as a proved impossibility: sharper estimates or different decompositions are not ruled out.
- Status of the fractional-weight piece: The fractional piece $\Sigma_{2,\mathrm{frac}} = \sum_{b=2}^U \log b \sum_{U < a \le M/b} \mu(a) \{N/(ab)\}$ is one component of $\Sigma_2$. It has not been proved equivalent to RH nor to the entire joint functional $D_N$.
6. What Changed Scientifically
- Exact identities proved by derivation, checked on finite cases: The algebraic structures of both the bilinear partition and the Dirichlet hyperbola factorization are proved in the notes; finite-case checks (Classes A-C) support them within the stated scopes.
- Unsupported claims retracted:
- The claim that $A(N) - \log(N!) = O(\sqrt{N})$ was refuted: the discrepancy is $\Theta(N)$, which algebraically couples $\mathcal{R}{\mathrm{boundary}}$ to $R(N) - \mathcal{T}{\mathrm{bilinear}}$.
- The spectral claim of an $N^{1/4}$ margin was refuted: the $k=1$ mode keeps the remainder at $N^\beta$ (borderline $N^{1/2}$ on the critical line).
- The claim of an exact proved inner main term in $\Sigma_1$ was retracted to a formal heuristic scale match.
- The sign error expanding $1 - \lfloor x \rfloor$ was corrected to plus $\{x\}$.
- The monotone-telescoping subexponential baseline for $\Sigma_2$ is downgraded to a proposal needing review (jump-range uniformity, exponential constants, $V$ endpoints, small $N$ unhandled); the justified bound stays $\Sigma_2 \ll N \log^3 N$.
- Scientific verdict: No improved asymptotic bound on $D_N$ is obtained. The research attempt is preserved as unresolved: derivation-proved exact finite identities with stated finite-case checks, and a sharply stated open joint inequality. No saturation, impossibility, or individual non-boundedness result is claimed.