2026-09-12. Continuation from science commit 3b0fc2e3 and the separately recorded board/history at 88b5d1a5. Earlier work is preserved. The current proofs and their independent challenge are in SIGNED_MEAN_RENEWAL.md, FAREY_BASELINE_REPAIR.md, and FRONTIER_INDEPENDENT_REVIEW.md.
Objective and current mathematical status
RH has not been proved. This continuation establishes no new fixed power saving for the prime-counting error or prime-pair energy. The new material is a scalar reduction, an arithmetic scale identity, a precise obstruction to the attempted induction, and corrections to the previous route closure. No novelty claim or formal proof claim is made.
The strongest inherited complete corrected-energy estimate remains \[ E_{\mathrm{corr}}^{(Z)}(N) \ll N^3\exp(-c''\sqrt{\log N}),\qquad Z=\exp(\sqrt{\log N}),\quad c''<2c_0, \] under the sufficiently small constant conditions (H'') in SHARP_EXPONENT.md. This is still \(N^{3-o(1)}\). Its proof in sections 2 and 3 does not use the defective AP-variance calculation of section 1.2. The conditional lower-bound discussion is not an unconditional assertion that the true exponent equals \(2c_0\), and does not exhibit a Siegel zero.
The actual RH-bearing quantity
Retain all prime powers, the infinite singular series and the exact exceptional correction. Define \[ B_N=\sum_{h=1}^N(r_N(h)-C_N(h)),\qquad M_N=\frac{2|B_N|^2}{N},\qquad R(x)=\psi(x)-x. \] The signed correction is uniformly \(O(N\log^2N)\) at the current \(Z\), even when the exceptional data change with \(N\). The exact pair identity therefore gives \[ B_N=NR(N)+\tfrac12R(N)^2+O(N\log^2N). \] Consequently \[ \mathrm{RH}\quad\Longleftrightarrow\quad (\forall\epsilon>0)\ M_N\ll_\epsilon N^{2+\epsilon}. \] This is a direct consequence of the already established bridge, isolated as a two-way scalar criterion. The mean projection was already present in the generic Haar energy decomposition. The advance here is testing an arithmetic estimate for that projection, not claiming that a new name for it controls the primes. A full corrected-energy bound remains sufficient; its converse under RH is not asserted.
Best new attack and what happened after it failed
Unique factorization supplies an exact arithmetic identity, \[ \sum_{k\le N}\psi(N/k)=\log(N!). \] Its direct recurrence for \(R\) has a linear forcing term. Cancelling that term with the absolutely convergent integral \(\int_1^\infty R(u)u^{-2}du=-1-\gamma\) gives, uniformly in \(1\le K\le N\), \[ \sum_{k\le K}R(N/k) -N\int_{N/K}^{\infty}\frac{R(u)}{u^2}\,du =O(N/K+\log N). \] At \(K=\lfloor\sqrt N\rfloor\) the error is \(O(\sqrt N)\). The remaining signed combination is exactly \[ D_N=N\int_{N/K}^{\infty}\frac{R(u)}{u^2}\,du -\sum_{k=2}^K R(N/k),\qquad R(N)=D_N+O(\sqrt N). \] The experiment then took three successive steps rather than treating the small forcing as a bound for \(R\).
- Absolute-value induction. A putative envelope \(|R(u)|\le Au^\theta\) gives an induction coefficient growing like \(2K^{1-\theta}/(1-\theta)\), not a contraction.
- Retain signs and positivity. The transfer multiplier on \(u^\rho\) is \(\zeta(\rho)\), with remainder \(O_\rho((N/K)^{\Re\rho})\). Thus a hypothetical off-critical zero contribution survives at every scale. An explicit smooth perturbation preserves nonnegative monotonicity, the integral normalization and the PNT envelope. It is a conditional diagnostic model, not the von Mangoldt sequence or an unconditional counterexample to RH.
- Retain the exact forcing. Mellin transformation gives \(\widehat G(s)=\zeta(s)\widehat R(s)\). Dividing by \(\zeta(s)\) leaves residue \(-m/\rho\) at a zero of multiplicity \(m\). The exact forcing does not cancel that pole. This identifies the missing inverse estimate; it supplies no zero exclusion.
The exact remaining estimate for this route is \[ |D_N|\ll_\epsilon N^{1/2+\epsilon} \quad\text{for every }\epsilon>0. \] No such bound, or even a new bound \(D_N\ll N^{1-\delta}\) for fixed \(\delta>0\), was obtained. Improving the quadrature constant or choosing another fixed split does not address the multiplier.
Recomputed disposition of the older routes
| Route | Current disposition |
|---|---|
| Arc transfer and composite Gauss cross terms | Earlier fictitious transfer and primitivity obstacles were already removed. The remaining low-denominator mixed term contains the principal PNT error. |
| BDH/AP variance | The principal term and the small-conductor repetitions cannot be discarded. The corrected estimates retain their large remainder. |
| Haar energy and measured scale decay | Exact for generic sequences, including the existing DH control. No arithmetic fine-scale bound follows. |
| Endpoint/model refinements | The later arc split already removed the minor-arc divisor approximant from the complete bound. Repeating its endpoint tuning does not attack the current obstacle. |
| Farey/Gallagher at the smaller cutoff | Section 1.2's displayed proof of closure is withdrawn. Bounded overlap and Parseval give a valid \(N\log N\) second moment and Vaughan gives \(N^3\log^9N/Q\) for \(Q\le N^{2/5}\). No stronger estimate or general no-go is obtained. |
| Constructive Theorem-B sequence | Its claimed absolute zero expansion loses a numerator. That construction needs truncation or smoothing; the settled nonconstructive Theorem B is unaffected. |
The broad statement that no density estimate could improve the Farey route is not supported by its former calculation. Reopening a faulty closure does not, by itself, make that route a better RH experiment than the signed mean. The old upper bound is preserved without importing that conclusion.
What would justify the next mathematical experiment
A continuation should start from the signed mean or an equally explicit RH implication, and propose an estimate using arithmetic information that the smooth zero-mode diagnostic does not possess: prime-power support, multiplicative identities, or a concrete bilinear cancellation mechanism. Its first test is whether that additional information actually bounds the signed combination, rather than replacing it with the old positive norm.
The current material contains no defensible further estimate to test at this point. A fresh model after its limit resets may supply one. It should receive this proof and obstruction, not a request to retune the old exponential constant or replay the stale proposal queue. Any genuinely stronger arithmetic estimate needs a separate adversarial review before integration. No future run has been scheduled by this document.