Base: zeta-lab commit 3b0fc2e3c97b46cc1fde6af0ff3dd856898d95fb. Date: 2026-09-12. Handwritten derivations for independent review; no novelty claim. Review status is recorded in FRONTIER_INDEPENDENT_REVIEW.md.
The objects below retain the original sharp endpoints, all prime powers, the infinite singular series, and the exact exceptional correction. This argument does not estimate the full corrected energy by a new method.
1. Inputs and the scalar target
For integers \(N\ge2\), write \[ \psi_2(N,h)=\sum_{n=1}^{N-h}\Lambda(n)\Lambda(n+h),\qquad r_N(h)=\psi_2(N,h)-(N-h)\mathfrak S(h), \] where \(1\le h\le N\), and put \[ B_N=\sum_{h=1}^N(r_N(h)-C_N(h)),\qquad M_N=\frac{2|B_N|^2}{N},\qquad R(x)=\psi(x)-x\quad(x\ge1). \]
Use \(Z=\exp(\sqrt{\log N})\) here. The exact correction \(C_N\) is the one in CORRECTED_RH_BRIDGE.md, equation (5), with this \(Z\). Its exceptional conductor, character, zero, and presence may change at every integer \(N\). If there are no applicable exceptional data, \(C_N=0\). The signed bound in that document is uniform in those data and is proved before specializing \(Z\): \[ \left|\sum_{h\le N}C_N(h)\right| \ll N(\log(2Z))^4\ll N(\log N)^2. \tag{1} \] Indeed its displayed quantitative bound uses only \(\beta\ge3/4\),
\[ (q/\phi(q))^2b_Z \left[\tfrac83 1_{q\text{ odd}}N^\beta +2\frac{\sigma_1(q)}q N^{2\beta-1}\right], \]
and \(\sigma_1(q)/q\le q/\phi(q)\le b_Z\ll\log(2Z)\). The odd, 4-part, and 8-part cases were all included in that proof. Source exceptionality gives \(\beta\ge3/4\) for all sufficiently large \(N\). Nothing here assumes the correction is the same at two different cutoffs.
The inherited first-moment inputs are \[ d_N:=\sum_{n\le N}\Lambda(n)^2=O(N\log N),\qquad S_N:=2\sum_{h\le N}(N-h)\mathfrak S(h)=N^2+O(N\log N). \tag{2} \] The singular-series mean is the CHHL input already used in CORRECTED_RH_BRIDGE.md; the diagonal bound also follows from Chebyshev and \(\Lambda(n)\le\log N\). The exact pair identity is \[ \psi(N)^2=d_N+2\sum_{h\le N}\psi_2(N,h). \tag{3} \] It follows by partitioning ordered pairs of integers at most \(N\) into equal pairs and the two orientations of each distinct pair. Thus it retains every proper prime power and the diagonal.
Define the explicit remainder \[ A_N=d_N+S_N-N^2+2\sum_{h\le N}C_N(h). \] Then, exactly, \[ 2B_N=\psi(N)^2-N^2-A_N, \qquad B_N=NR(N)+\tfrac12R(N)^2-\tfrac12A_N, \tag{4} \] and (1)-(2) imply \(A_N=O(N\log^2N)\), uniformly in the changing exceptional data.
2. Two-way mean criterion
The exact equivalence is \[ \boxed{\quad \mathrm{RH}\quad\Longleftrightarrow\quad (\forall\epsilon>0)\ M_N\ll_\epsilon N^{2+\epsilon} \text{ for all sufficiently large integers }N. \quad} \tag{5} \]
For the forward direction use the classical RH consequence
\[ R(x)=O(x^{1/2}\log^2x). \]
This is the standard von Koch prime-counting formulation already recorded in docs/00-orientation.md; it is an external classical input, not proved by the renewal argument below. Equation (4) gives
\[ B_N=O(N^{3/2}\log^2N),\qquad M_N=O(N^2\log^4N), \tag{6} \]
which implies the right side of (5).
Conversely \(\psi(N)\ge0\), so \(\psi(N)+N\ge N\). From (4),
\[ |R(N)|\le\frac{2|B_N|+|A_N|}{N} \le\sqrt{\frac{2M_N}{N}}+O(\log^2N). \tag{7} \]
For any \(\theta>0\), apply the hypothesis with \(\epsilon=2\theta\) to obtain \(R(N)=O_\theta(N^{1/2+\theta})\). For real \(x\ge2\), take \(N=\lfloor x\rfloor\); then \(\psi(x)=\psi(N)\) and \(x-N<1\). Thus \(R(x)=O_\theta(x^{1/2+\theta})\) for every \(\theta>0\).
For completeness, on \(\Re s>1\) the Euler product and partial summation give \[ -\frac{\zeta'(s)}{\zeta(s)}- rac{s}{s-1} =s\int_1^\infty R(x)x^{-s-1}\,dx. \tag{8} \] For each \(\theta>0\) the right side is holomorphic on \(\Re s>1/2+\theta\), by locally uniform absolute convergence. Consequently the left side has no pole in \(\Re s>1/2\); a zero of \(\zeta\) there would produce one. The functional equation reflects nontrivial zeros across \(\Re s=1/2\), proving RH.
This scalar equivalence is a reduction, not progress toward proving its input. Unlike the full-energy criterion, it is two-way. It does not say that RH implies a near-quadratic bound for the full corrected energy.
3. The exact arithmetic recurrence and its linear forcing
For each positive integer \(m\), unique factorization gives \[ \sum_{d\mid m}\Lambda(d)=\log m. \] For \(m=\prod_p p^{a_p}\), the left side is \(\sum_p\sum_{j=1}^{a_p}\log p=\log m\). Summing over \(m\le N\), and reversing the finite sums, gives the exact sharp-cutoff identities \[ \sum_{d\le N}\Lambda(d)\lfloor N/d\rfloor =\log(N!) =\sum_{k=1}^N\psi(N/k). \tag{9} \] Subtract \(N H_N\), where \(H_N=\sum_{k\le N}1/k\). Then \[ \sum_{k=1}^N R(N/k)=G(N),\qquad G(N)=\log(N!)-NH_N. \tag{10} \] Stirling's formula and the harmonic-number expansion give \[ G(N)=-(1+\gamma)N+\tfrac12\log N +\tfrac12\log(2\pi)-\tfrac12+O(N^{-1}). \tag{11} \] The error can be sharpened, but that has no bearing on the argument.
The direct recurrence \(R(N)=G(N)-\sum_{k=2}^N R(N/k)\) therefore has an order-\(N\) forcing term. Merely treating it as \(O(N)\) cannot yield a fixed power saving. The following step removes that linear term before testing whether a genuine contraction remains.
4. Cancelling the linear term and deriving a scale relation
Use only the inherited unconditional prime-counting estimate
\[ R(x)\ll x\exp(-c\sqrt{\log x}) \]
for sufficiently large \(x\); the stronger inherited rate is unnecessary. It guarantees absolute convergence of
\[ I:=\int_1^\infty\frac{R(u)}{u^2}\,du. \]
The value is
\[ I=-(1+\gamma). \tag{12} \]
Proof: divide (8) by \(s\) and let real \(s\downarrow1\). Dominated convergence applies to the integral, because \(R(u)/u^2\) is absolutely integrable. The Laurent expansion \(\zeta(s)=(s-1)^{-1}+\gamma+O(s-1)\) yields
\[ -\frac{\zeta'(s)}{s\zeta(s)}-\frac1{s-1} =-(1+\gamma)+O(s-1). \]
Fix integers \(1\le K\le N\), and set \(y=N/K\). Define the exact quadrature error \[ Q_{N,K}=\sum_{k=K+1}^N R(N/k) -N\int_1^{N/K}\frac{R(u)}{u^2}\,du. \tag{13} \] For \(f(t)=R(N/t)\) on ([K,N]), comparison of each right endpoint with the integral over ([k-1,k]) gives \[ |Q_{N,K}|\le\operatorname{Var}{[K,N]} f =\operatorname{Var}{[1,y]}R \le\psi(y)+y\ll N/K. \tag{14} \] For the equality of variations, composition with the decreasing continuous bijection \(t\mapsto N/t\) reverses partitions without changing their variation sums. For the last inequality use \(R=\psi-\mathrm{id}\), monotonicity of \(\psi\), and the inherited Chebyshev bound \(\psi(y)\ll y\). Endpoint jumps cause no problem: their contributions are included in total variation. If \(K=N\), both sides of (13) are zero.
Combining (10), (12), and (13) gives the exact relation \[ \boxed{ \sum_{k=1}^K R(N/k) -N\int_{N/K}^\infty\frac{R(u)}{u^2}\,du =G(N)+(1+\gamma)N-Q_{N,K}.} \tag{15} \] Consequently, uniformly for every integer \(1\le K\le N\), \[ \boxed{ \sum_{k=1}^K R(N/k) -N\int_{N/K}^\infty\frac{R(u)}{u^2}\,du =O(N/K+\log N).} \tag{16} \]
This is an actual scale estimate derived from the arithmetic identity. At \(K=\lfloor\sqrt N\rfloor\), its forcing is \(O(\sqrt N)\). It is an estimate for a combination containing (R\(N\)); it is not yet an estimate for (R\(N\)) itself. Its integral includes \(u>N\), so it is not a recurrence solely in previously controlled smaller arguments.
5. The attempted contraction and its exact missing estimate
Set \(K=\lfloor\sqrt N\rfloor\), and write \[ D_N:=N\int_{N/K}^\infty\frac{R(u)}{u^2}\,du -\sum_{k=2}^K R(N/k). \] Then (15) proves
\[ R(N)=D_N+O(\sqrt N). \tag{17} \]
For the RH target the missing arithmetic inequality is exactly
\[ \boxed{\quad |D_N|\ll_\epsilon N^{1/2+\epsilon} \quad\text{for every }\epsilon>0.\quad} \tag{18} \]
A fixed saving \(D_N\ll N^{1-\delta}\), \(0<\delta\le1/2\), would already yield \(R(N)\ll N^{1-\delta}\), \(B_N\ll N^{2-\delta}\), and \(M_N\ll N^{3-2\delta}\), by (4). Neither such a saving nor (18) has been established here.
Taking absolute values does not contract even a putative envelope \(|R(u)|\le A u^\theta\), \(0<\theta<1\). It only gives
\[ |D_N|\le A N^\theta \left(\frac{K^{1-\theta}}{1-\theta} +\sum_{k=2}^K k^{-\theta}\right). \tag{19} \]
The coefficient grows as \(2K^{1-\theta}/(1-\theta)\), rather than being less than one. This is a precise failure of the proposed absolute value induction, not a theorem that every conceivable signed argument must fail. The signed test below explains the relevant unresolved modes.
6. The transfer multiplier, including all scales
For fixed \(\rho\ne1\) with \(0<\beta:=\Re\rho<1\), define \(f_\rho(u)=u^\rho\), using the real logarithm for \(u>0\). Let
\[ (L_K f)(N):=\sum_{k=1}^K f(N/k) -N\int_{N/K}^\infty f(u)\frac{du}{u^2}. \]
The integral converges absolutely for \(f=f_\rho\), and direct evaluation gives the exact formula
\[ (L_K f_\rho)(N) =N^\rho\left(\sum_{k=1}^K k^{-\rho} -\frac{K^{1-\rho}}{1-\rho}\right). \tag{20} \]
Euler summation gives, for fixed \(\rho\) in this strip,
\[ \sum_{k=1}^K k^{-\rho} -\frac{K^{1-\rho}}{1-\rho} =\zeta(\rho)+O_\rho(K^{-\beta}). \tag{21} \]
One proof is to compare \(k^{-\rho}\) on each unit interval with its integral. The discrepancy series converges absolutely at its derivative scale, since
\[ \sum_{k>K}\sup_{t\in[k,k+1]}|\rho t^{-\rho-1}| \ll_\rho K^{-\beta}. \]
Its limiting function agrees with \(\zeta(s)\) on \(\Re s>1\) after including the integral term, and therefore gives its analytic continuation to \(\Re s>0\), \(s\ne1\). This is also the specialization of the NIST Euler-Maclaurin representation (https://dlmf.nist.gov/25.11.E5), with Hurwitz parameter 1 and truncation index \(K-1\). This precise source formula and its \(\Re s>0\) domain were checked on 2026-09-12. Thus
\[ \boxed{\quad (L_K f_\rho)(N)=\zeta(\rho)N^\rho +O_\rho((N/K)^\beta).\quad} \tag{22} \]
If \(\zeta(\rho)=0\), then for every \(1\le K\le N\),
\[ |(L_K f_\rho)(N)|\ll_\rho (N/K)^\beta\le C_\rho N/K. \tag{23} \]
This includes \(K=\sqrt N\) and \(K=N\). Consequently the displayed error allowance in (16) is compatible with a mode of size \(N^\beta\) whenever that mode is attached to a zeta zero. A hypothetical \(\beta>1/2\) is not excluded by making \(N/K\) as small as possible. For a nonzero value \(\zeta(\rho)\), the leading term in (22) is retained; an arbitrary off-critical power is not claimed to pass all scales. No off-critical zero is asserted to exist.
7. An explicit boundary correction: bounded divisor forcing
The pure mode has the wrong integral in (12). That objection can be removed explicitly. Define
\[ F_\rho(u)=u^\rho-\frac{2}{1-\rho}1_{[1,2)}(u) \quad (u\ge1). \]
Then
\[ \int_1^\infty F_\rho(u)\frac{du}{u^2}=0, \tag{24} \]
because the two terms contribute \(1/(1-\rho)\) and \(2(1-1/2)/(1-\rho)\). Let \(Tf(x)=\sum_{k\le x}f(x/k)\). For integer \(N\ge2\), exactly,
\[ (TF_\rho)(N)=N^\rho\sum_{k=1}^N k^{-\rho} -\frac{2}{1-\rho}(N-\lfloor N/2\rfloor). \]
Equation (21) with \(K=N\) now gives
\[ (TF_\rho)(N)=\zeta(\rho)N^\rho+O_\rho(1). \tag{25} \]
Thus at a zeta zero an unbounded \(N^\beta\) function has bounded divisor forcing, even with the correct zero integral for a perturbation. It is an approximate null mode with explicitly identified bounded forcing, not a nonzero exact solution of \(TF=0\). The latter would be impossible under a fixed zero boundary convention, by triangular inversion.
8. Positivity and monotonicity do not remove the hypothetical mode
The jump in the preceding convenient cutoff is unnecessary. Here is a model that is nonnegative and nondecreasing, including its small-argument boundary, and meets the same normalization and every scale bound whenever \(\zeta(\rho)=0\).
Put \(a=e^\gamma>1\) and define
\[ \Psi_0(u)=(u-a)_+,\qquad R_0(u)=\Psi_0(u)-u=-\min(u,a). \]
Then \(\Psi_0(1)=0\), \(\Psi_0\ge0\), \(\Psi_0\) is nondecreasing, and direct integration gives
\[ \int_1^\infty R_0(u)\frac{du}{u^2}=-\log a-1=-(1+\gamma). \tag{26} \]
Choose the explicit \(C^1\) cutoff
\[ h(u)=\begin{cases} 0,&u\le2,\\ 3(u-2)^2-2(u-2)^3,&2<u<3,\\ 1,&u\ge3, \end{cases} \]
and \(b(u)=(u-3)^2(4-u)^2\) for \(3\le u\le4\), zero elsewhere. Let
\[ J_\rho=\int_1^\infty h(u)u^{\rho-2}\,du,\qquad J_b=\int_3^4b(u)u^{-2}\,du>0, \]
and define
\[ \widetilde F_\rho(u)=h(u)u^\rho-(J_\rho/J_b)b(u). \]
This function is \(C^1\), vanishes for \(u\le2\), equals \(u^\rho\) for \(u\ge4\), and has integral zero against \(u^{-2}du\). Its derivative has finite supremum: it is bounded on the compact transition region and has magnitude \(|\rho|u^{\beta-1}\) afterwards. Define
\[ D_\rho=\max(1,\sup_{u\ge1}|\widetilde F_\rho'(u)|),\qquad 0<|\eta|\le(2D_\rho)^{-1}, \]
with \(\eta\) real, and put
\[ \Psi_\eta(u)=\Psi_0(u)+\eta\Re\widetilde F_\rho(u),\qquad R_\eta(u)=\Psi_\eta(u)-u. \]
On \(u\le2\) this equals the nonnegative increasing baseline. On \(u>2>a\) its derivative is at least \(1-|\eta|D_\rho\ge1/2\). Hence \(\Psi_\eta(1)=0\), \(\Psi_\eta\ge0\), and \(\Psi_\eta\) is nondecreasing on its whole domain. Also
\[ \int_1^\infty R_\eta(u)\frac{du}{u^2}=-(1+\gamma),\qquad R_\eta(u)=-a+\eta\Re u^\rho\quad(u\ge4). \tag{27} \]
The error is \(O_\rho(u^\beta)\). Since \(\beta<1\), this satisfies the inherited \(u\exp(-c\sqrt{\log u})\) envelope for every fixed \(c>0\), after increasing the starting point and constant. It also has the same (O\(y\)) variation bound used in (14).
To check the scale relation, write
\[ H_\rho(u)=\widetilde F_\rho(u)-u^\rho. \]
It is supported on ([1,4]), has bounded variation, and its integral against \(u^{-2}du\) is \(-1/(1-\rho)\). For integer \(N\ge4\), right-endpoint quadrature for \(H_\rho(N/t)\), whose variation is bounded independently of \(N\), proves
\[ (TH_\rho)(N)=N\int_1^4 H_\rho(u)\frac{du}{u^2}+O_\rho(1). \]
For precision, comparing the sum over \(k=1,\dots,N\) with the integral over ([1,N]) costs at most its variation plus its first endpoint; both are bounded here. Combining with (21) gives
\[ (T\widetilde F_\rho)(N)=\zeta(\rho)N^\rho+O_\rho(1). \tag{28} \]
The same quadrature for the bounded monotone function \(R_0(N/t)\) gives
\[ (TR_0)(N)=-(1+\gamma)N+O(1), \]
using (26) and \(R_0(u)=-a\) beyond (a). Therefore
\[ (TR_\eta)(N)=-(1+\gamma)N +\eta\Re(\zeta(\rho)N^\rho)+O_{\rho,\eta}(1). \tag{29} \]
Apply the same tail quadrature proof (13)-(15) to \(R_\eta\), whose normalization is exactly (27). Equations (27)-(29) imply
\[ (L_KR_\eta)(N)=\eta\Re(\zeta(\rho)N^\rho) +O_{\rho,\eta}(N/K+1) \]
uniformly for every \(1\le K\le N\). If \(\zeta(\rho)=0\), the model satisfies the full bound (16), with room to spare, while \(R_\eta(u)=-a+\eta\Re u^\rho\) has oscillations of order \(u^\beta\). For nonreal \(\rho\), take \(u\) along exponential sequences at the extrema of the cosine; rounding those sequences to integers preserves their order because consecutive logarithmic increments tend to zero.
Scope. This model is not the von Mangoldt summatory function. It does not have prime-power support, and it does not satisfy the exact prescribed identity \(TR=G\). Its forcing differs from that exact forcing by \(O(\log N)\) when \(\zeta(\rho)=0\). That is precisely the amount already allowed after passing to (16). It proves compatibility of the retained positivity, monotonicity, normalization, PNT envelope, and all-\(K\) scale estimates with a hypothetical off-critical zero mode. It neither constructs an off-critical zero nor refutes RH or the exact arithmetic identity. An argument using more of that exact identity is not ruled out by this model.
9. What retaining the exact forcing entails
The exact divisor identity can be kept instead of weakened to (16). For real \(x\ge1\), define
\[ G(x)=\log(\lfloor x\rfloor!)-xH_{\lfloor x\rfloor}. \]
Then \(TR(x)=G(x)\) exactly, since \(\sum_{k\le x}\psi(x/k)=\log(\lfloor x\rfloor!)\). On \(\Re s>1\), absolute convergence permits the Mellin calculation
\[ \widehat G(s):=\int_1^\infty G(x)x^{-s-1}\,dx =\zeta(s)\widehat R(s) =-\frac{\zeta'(s)}s-\frac{\zeta(s)}{s-1}. \tag{30} \]
The first equality follows by setting \(x=ku\) in each term of (TR); the last follows from (8). Thus inversion is division by \(\zeta(s)\), not an automatically stable averaging operation.
If \(\rho\ne0,1\) is a zero of multiplicity \(m\), write \(\zeta(s)=(s-\rho)^m g(s)\), \(g(\rho)\ne0\). The continued quotient in (30) is
\[ \frac{\widehat G(s)}{\zeta(s)} =-\frac1s\frac{\zeta'(s)}{\zeta(s)}-\frac1{s-1} =-\frac{m}{\rho(s-\rho)}+O(1). \tag{31} \]
In particular the exact forcing does not cancel such a zero upon inversion. For a simple zero its numerator is nonzero; for a multiple zero its numerator has order (m-1), leaving the same simple pole after division. This calculation is conditional only in referring to the location of a particular possible zero, and it includes multiplicity. It is an exact diagnosis of this divisor route's spectral obstruction, not a new zero-free theorem.
10. Bounded arithmetic diagnostics
The independent reviewer wrote frontier_review_checks.py from the identities rather than importing a campaign implementation. Its recorded output is frontier_review_checks.json; coverage, arithmetic precision, endpoint checks and deliberate fault controls are stated in FRONTIER_INDEPENDENT_REVIEW.md. These finite diagnostics test the identities and conventions. They do not establish an asymptotic estimate, an off-critical zero, or RH.
11. Status and the remaining door
Handwritten deductions examined in FRONTIER_INDEPENDENT_REVIEW.md:
- The mean criterion (5), uniformly with \(Z=\exp(\sqrt{\log N})\).
- The exact recurrence (15), the uniform bound (16), and its concrete \(O(\sqrt N)\) forcing at the square-root split.
- The multiplier formula (22), the normalized bounded-forcing construction (25), and the positivity-preserving model (26)-(29).
- The exact-forcing inversion and multiplicity calculation (30)-(31).
Not established: any new unconditional fixed power saving for \(R\), \(B_N\), \(M_N\), or the full corrected energy; inequality (18); any exclusion of a zero with real part greater than (1/2).
Refuted as an inference: an \(O(\sqrt N)\) forcing bound at a square-root divisor split by itself is an \(O(\sqrt N)\) bound for (R\(N\)). The attempted absolute-value induction has the growing coefficient (19). The stronger statement that the retained envelopes and positivity rule out an off-critical mode has not been proved; (26)-(29) show exactly why a hypothetical zero would survive those inputs. This is not an unconditional counterexample to a theorem equivalent to RH.
The remaining door in this bounded route is additional arithmetic control of the signed quantity \(D_N\) in (18), or another use of the exact prime-power identity that supplies a justified inverse estimate past its zeta multiplier. Refining the (O\(N/K\)) quadrature constant, freezing a different \(K\), or using monotonicity already consumed in (14) does not supply that missing estimate.