2026-09-10. Builds on CORRECTED_RH_BRIDGE.md, SIEGEL_UNIFORMITY.md and LOCALIZED_MIXED_ENERGY.md as they stand on this branch, and directly answers the question CANDIDATE_ENERGY.md/CHALLENGE.md left open: a multiscale \(\mathrm{Energy}(N)\ge E_{\rm corr}(N)\) whose per-scale proofs use the arithmetic content of \(\Lambda\), \(\mathfrak S\), \(C_N\), not just linear algebra on an arbitrary real vector. Nothing in the three inherited documents is revisited or reproved; this document's job is to assemble their already-proved per-scale estimates into the exact box CANDIDATE_ENERGY.md Section 2 was asked for, name the one arithmetic fact each scale consumes, and run the Davenport-Heilbronn battery against that fact directly, per CHALLENGE.md's own standing rule.
Outcome. \(\mathrm{Energy}(N)\), defined below from LOCALIZED_MIXED_ENERGY.md's own eq. (25)-(26), dominates \(E_{\rm corr}(N)\) (Section 2) and splits into a retained mean/central mode plus dyadic scales with retained cross terms (Section 3), exactly as CANDIDATE_ENERGY.md Section 8 requires. Unlike that candidate, every scale's proof, not just its numeric value, requires an arithmetic hypothesis about \(\Lambda\) (Section 4): the mean/central mode needs \(\Lambda(n)\ge0\) (Euler-product positivity, the input to the classical zero-free region), and every other scale needs \(\Lambda\)'s multiplicativity (Vaughan/Siegel-Walfisz). Section 5 runs zeta/epstein.py's Davenport-Heilbronn interface against exactly that positivity hypothesis, reusing the module's own claim_euler_product_positivity and claim_multiplicativity unmodified, and both fail for the DH sequence -- reverified independently here (own \(O(n_{\max}\log n_{\max})\) divisor-sieve recursion, agreeing with zeta.epstein's own reference implementation to \(1.5\times10^{-16}\) relative error at \(n_{\max}=200\)), not merely re-quoted from the module's docstring. Section 5 then derives, from the pinned off-line zero \(\rho\), the standard (cited, not re-derived) consequence that the mean-mode step's conclusion is not just unproved but false for the DH analogue, and supports it with a robust numerical fact (Section 5.3) that does not depend on that harder asymptotic statement. This candidate does not survive the battery, which is the intended outcome: it is the first energy in this hunt whose domination proof has an identifiable step that a real, functional-equation-respecting, off-line-zero rival cannot supply. Section 6 states plainly what this does and does not change about the total order of \(E_{\rm corr}(N)\): nothing. No fixed power saving is claimed anywhere in this document.
This is a handwritten deduction, pending external verification, with no novelty claim: essentially all of the size estimates below are reused, by citation, from the three inherited documents; what is new is the packaging, the per-scale attribution, and the battery run.
1. Setup, inherited without change
Exactly CORRECTED_RH_BRIDGE.md Section 1 and SIEGEL_UNIFORMITY.md eq. (5): \(\psi_2(N,h)=\sum_{n=1}^{N-h}\Lambda(n)\Lambda(n+h)\), \(r_N(h)=\psi_2(N,h)-(N-h)\mathfrak S(h)\), \(x_h:=r_N(h)-C_N(h)\), \(E_{\rm corr}(N)=2\sum_{h=1}^Nx_h^2\), endpoint exactly \(N-h\), \(\Lambda\) including every proper prime power, \(C_N\equiv0\) when no exceptional zero exists at that \(N\). This document does not touch any of these definitions.
Also inherited, unchanged, from LOCALIZED_MIXED_ENERGY.md Section 1: \(\ell=\log N\), \(Z=\exp(\ell^{1/10})\), \(P=\prod_{p<Z}p\), \(b_Z=P/\phi(P)\), \(\nu(n)=b_Z1_{(n,P)=1}\), \(a(n)=\nu(n)(1-1_{\rm exc}\chi(n)n^{\beta-1})\), \(w_n=\Lambda(n)-a(n)\), \(F=\sum\Lambda(n)e(n\alpha)\), \(H=\sum a(n)e(n\alpha)\), \(W=F-H\), \(D_w=\sum|w_n|^2\), and the centered polynomials \(X=\mathcal C(2\operatorname{Re}(\overline HW))\), \(Y=\mathcal C(|W|^2)\), \(R_{\rm mod}=\mathcal C(|H|^2-V_y)-\widehat C_N\), with \(G_{\rm corr}=X+Y+R_{\rm mod}\) and \(|\sqrt{E_{\rm corr}}-\|G_{\rm corr}\|_2|\ll N\) (that document's eq. 24).
2. \(\mathrm{Energy}(N)\), assembled from already-proved pieces
LOCALIZED_MIXED_ENERGY.md eq. (25) already proves, unconditionally, \[ \sqrt{E_{\rm corr}(N)}\le 2\sqrt{\mathcal M}+\sqrt{\textstyle\int|W|^4-D_w^2} +CN^{3/2}e^{-c\ell^{1/10}}+C'N, \tag{1} \] where \(\mathcal M=\int|H|^2|W|^2\) (its eq. 9). Define \[ \boxed{\quad \mathrm{Energy}(N):= \Bigl(2\sqrt{\mathcal M}+\sqrt{\textstyle\int|W|^4-D_w^2} +CN^{3/2}e^{-c\ell^{1/10}}+C'N\Bigr)^2. \quad} \tag{2} \] Squaring (1) gives \(E_{\rm corr}(N)\le\mathrm{Energy}(N)\) directly: this is not an identity, and Section 4 shows exactly why it cannot degenerate into one the way CANDIDATE_ENERGY.md's did.
The same document's eq. (26) already decomposes \(\mathcal M\) into named, disjoint-by-construction pieces: \[ \mathcal M\ll_\kappa \underbrace{N^{14/5+o(1)}}{\Delta{\rm denom}} +\underbrace{N^{11/5+o(1)}}{\Delta{\rm short}} +\underbrace{N^3e^{-c\mathcal L(N)}}{\Delta{\rm mean}} +\underbrace{N^3e^{-c_\kappa\ell^\kappa}+N^3e^{-c\ell^{9/10}}}{\Delta{\rm fallback}}, \tag{3} \] \(\mathcal L(N)=\ell^{3/5}(\log\ell)^{-1/5}\) (its eq. (19)'s exponent, Vinogradov-Korobov, stronger than plain de la Vallée-Poussin \(\ell^{1/2}\)). Explicitly:
- \(\Delta_{\rm denom}\): the dyadic-denominator sum \(\mathcal A_{\ge R}\) (eq. 12), summed over reduced fractions with denominator in dyadic blocks \([D,2D)\) via the rational-sampling lemma (eq. 10-11).
- \(\Delta_{\rm short}\): the short-window sum \(\mathcal A_{s\le L_0}\) (eq. 14), summed over dyadic window lengths \(s=2^k\le N^{3/5}\).
- \(\Delta_{\rm mean}\): the mean/central mode, the single \(s=N\), \(r=1\), \(\theta=0\) window \(T_N(0)\) (eq. 15), evaluated exactly via \(A(k)=\sum_{n\le k}(\Lambda(n)-a(n))\) and bounded (eq. 19) by the classical prime number theorem.
- \(\Delta_{\rm fallback}\): every remaining window, bounded by the inherited uniform Fourier estimate (end of Section 5 there).
This is a genuine multiscale telescoping in the sense LOCALIZED_MIXED_ENERGY.md Section 4 states: the windows are partitioned first by length \(s\le L_0\), then by denominator \(r\ge R\), "to avoid double counting." \(\mathrm{Energy}(N)\) as boxed in (2) is the Section-8-requested shape with the mean/central mode (\(\Delta_{\rm mean}\)) retained as its own named term, matching CANDIDATE_ENERGY.md Section 2's box exactly in role, not in method.
3. Why this does not collapse to an identity, and where the cross terms are
CANDIDATE_ENERGY.md's construction decomposed the residual vector \((x_h)\) itself by an orthogonal projection of \(\mathbb R^M\); that is why it was exact for every real vector and carried no arithmetic content (CHALLENGE.md attack 3). \((2)\) does not do this. It bounds \(E_{\rm corr}\) through a different, auxiliary object -- \(F,H,W\), functions of the frequency \(\alpha\), related to \((x_h)\) only through the specific, arithmetic identity that \(\psi_2(N,h)\) is an autocorrelation of \(\Lambda\) (Parseval/Plancherel for the product \(|F|^2\), CORRECTED_RH_BRIDGE.md Section 1 and SIEGEL_UNIFORMITY.md Section 1). Decomposing \(\mathcal M\) or \(\int|W|^4\) by dyadic scale of \(s\) or \(r\) is therefore not a rearrangement of the entries of \((x_h)\) by any orthogonal map; each piece's size genuinely depends on what \(\Lambda\) and \(a\) are, which is exactly why proving (3)'s individual bounds took the sieve calculation, the rational-sampling lemma, and the classical prime number theorem, rather than four lines of projection algebra. Section 5 makes this concrete by exhibiting a sequence for which one piece's proof has no analogue at all.
Retained cross terms. \(\mathrm{Energy}(N)\) is derived from the full expansion LOCALIZED_MIXED_ENERGY.md eq. (22), \[ \|G_{\rm corr}\|_2^2=\|X\|_2^2+\|Y\|2^2+\|R{\rm mod}\|2^2 +2\operatorname{Re}\langle X,Y\rangle +2\operatorname{Re}\langle X+Y,R{\rm mod}\rangle, \tag{4} \] by triangle inequality, not by an assertion that the cross terms vanish or are small; that document states plainly "No sign or cancellation estimate for the last two terms has been proved here," and this document changes nothing about that. This is the honest sense in which cross terms between scales are retained here: present in (4), paid for (not deleted) by the factor structure of (1)-(2), and explicitly flagged as unresolved rather than assumed away -- the opposite failure mode from CANDIDATE_ENERGY.md, whose cross terms were retained by being proved exactly zero. Both are legitimate ways to satisfy the constraint; only one of them turned out to carry arithmetic content.
4. The one arithmetic fact each scale consumes
- \(\Delta_{\rm denom}\), \(\Delta_{\rm short}\): the rational-sampling lemma (eq. 10) plus the sieve calculation (SIEGEL_UNIFORMITY.md eq. 9-11, reused there via LOCALIZED_MIXED_ENERGY.md eq. 6-7), which needs \(\Lambda\) (via \(\nu,a\)) to be supported on an explicit sieve of primes below \(Z\), and Ramanujan-sum orthogonality \(\sum_{h\bmod q}c_q(h)^2=q\phi(q)\) -- both consequences of the multiplicative structure of the residues mod \(q\), not of size alone.
- \(\Delta_{\rm fallback}\): the uniform linear-phase bound \(\sup_\alpha|W(\alpha)|\ll_\kappa Ne^{-c_\kappa\ell^\kappa}\) (Tao-Teräväinen Theorem 2.7 via Gowers-uniformity control of \(\Lambda\)), itself built from Vaughan's identity, a divisor-convolution decomposition of \(\Lambda\) that exists because \(-\zeta'/\zeta(s)=\sum\Lambda(n)n^{-s}\) is the log-derivative of an Euler product.
- \(\Delta_{\rm mean}\): the classical prime number theorem, \(\Delta(t):=\psi(t)-t=O(t\exp(-c\mathcal L(t)))\) (LOCALIZED_MIXED_ENERGY.md eq. 15-19, citing TT Theorem 1.3(i)), whose proof -- de la Vallée-Poussin's zero-free region, and its Vinogradov-Korobov strengthening -- uses exactly one arithmetic fact about \(\Lambda\): \(\Lambda(n)\ge0\). This is the classical "3-4-1" inequality \(-3\operatorname{Re}(\zeta'/\zeta(\sigma))-4\operatorname{Re}(\zeta'/\zeta(\sigma+it)) -\operatorname{Re}(\zeta'/\zeta(\sigma+2it))\ge0\), which reduces to \(-\sum_n\Lambda(n)n^{-\sigma}\cdot2(1+\cos(t\log n))^2\le0\) using nothing about \(\Lambda\) beyond \(\Lambda(n)\ge0\) (log of a prime power is \(\ge0\)) and that \(\Lambda\) is the same sequence read off \(-\zeta'/\zeta\) at \(\sigma\), \(\sigma+it\), \(\sigma+2it\) -- i.e. that \(\zeta\) has an Euler product at all. This is exactly the property
zeta/epstein.py'sclaim_euler_product_positivitynames and tests: "the input to the classical zero-free region," in that module's own words.
No other estimate in (3) is used to prove \(\Delta_{\rm mean}\); no positivity or multiplicativity assumption is used to prove \(\Delta_{\rm denom}\), \(\Delta_{\rm short}\), or \(\Delta_{\rm fallback}\) beyond the ones just named. This is the line the cross-session review asked to be pointed at before writing anything down.
5. The Davenport-Heilbronn battery
5.1 Running the module's own claims against the mean-mode hypothesis
hunts/prime_pair_error/s8_positivity_battery.py reuses zeta.epstein.battery, claim_multiplicativity, claim_euler_product_positivity, and dh_interface unmodified (never reimplemented; these are exactly the repository's own falsification harness for gate #3, built for exactly this kind of check), and writes results_s8_positivity_battery.json.
claim_multiplicativity: riemann_zeta = True, davenport_heilbronn = False
claim_euler_product_positivity: riemann_zeta = True, davenport_heilbronn = FalseBoth fail for the Davenport-Heilbronn interface, reconfirming what zeta/epstein.py's own docstrings already record: DH has real coefficients, a genuine Riemann-type functional equation \(F(s)=F(1-s)\), a real Hardy function, and a zero at \(\rho\approx0.808517+85.699348i\) off the critical line -- but no Euler product, and its log-derivative coefficients \(\Lambda_F(n)\) (zeta.epstein.log_derivative_coefficients, defined by \(a(n)\log n=\sum_{d\mid n}a(d)\Lambda_F(n/d)\), \(a_1=1\), the only route from a rival's Dirichlet coefficients to the quantity a positivity-based mechanism actually reads) are not \(\ge0\).
This script reverifies the failure independently rather than re-quoting the module's docstring number, using its own \(O(n_{\max}\log n_{\max})\) divisor-sieve solution of the identical recursion (the module's own implementations are \(O(n_{\max}^2)\), fine at their stated \(n_{\max}=200\) but not at the \(n_{\max}=2\times10^5\) this document needs for Section 5.3). Cross-check at \(n_{\max}=200\) against zeta.epstein.log_derivative_coefficients (mpmath, mirroring the module's own doubly-implemented convention in zeta/factorization.py and zeta/quasicrystal.py): maximum absolute difference \(8.9\times10^{-16}\). Independently computed \(\Lambda_F(3)=-0.31209\ldots\), sign-confirming the module's claim (its docstring quotes \(-6.78\) for the same coefficient, a discrepancy this document does not resolve and does not need to -- both values are negative, and the sign, not the magnitude, is what claim_euler_product_positivity reads). \(\kappa\), the Davenport-Heilbronn constant, was re-derived here via zeta.epstein.kappa, never hardcoded, per that module's own stated convention.
Consequence for \(\Delta_{\rm mean}\). The one hypothesis its proof needs, \(\Lambda(n)\ge0\), is false for \(\Lambda_F\). The classical zero-free-region argument cannot be run on the Davenport-Heilbronn sequence at all: the "3-4-1" inequality of Section 4 needs the sign \(-\sum_n\Lambda_F(n)n^{-\sigma}\cdot(\cdots)\le0\), and this fails as soon as any \(\Lambda_F(n)<0\), which happens already at \(n=3\). This is a stronger failure mode than a numeric near-miss: the construction's proof does not go through, exactly the sense CHALLENGE.md attack 3 asked for, run here in the positive direction (naming the one step that uses the structure of the primes, rather than confirming that none does).
5.2 What the missing positivity actually costs: a cited consequence
\(F\) is entire (no pole at \(s=1\): its coefficients' period average is \(1+\kappa-\kappa-1+0=0\)). Its log-derivative \(G(s)=-F'/F(s)\) therefore has no "trivial" pole at \(s=1\) supplying a growing main term the way \(-\zeta'/\zeta\) does for \(\Lambda\); but at the located simple zero \(s=\rho\) of \(F\), \(G(s)=-F'/F(s)\) does have a simple pole, with residue \(-1\) (elementary: \(F(s)=(s-\rho)h(s)\), \(h(\rho)\ne0\), gives \(F'/F(s)=1/(s-\rho)+h'/h(s)\)). By the same Mellin/Perron-holomorphy mechanism CORRECTED_RH_BRIDGE.md Section 5 eq. (20) already uses in the forward direction here (small partial sums make an integral holomorphic past a candidate zero, contradiction) -- run instead in the converse direction against a pole whose location is already pinned, which is the standard content of Landau's oscillation theorem for Dirichlet series with a genuine singularity, cited rather than re-derived here, exactly as the sibling SCALE_TRANSITION.md line of work names it as this hunt's known barrier mechanism -- any partial-sum bound \(\sum_{n\le t}\Lambda_F(n)=O(t^\theta)\) for \(\theta<\operatorname{Re}(\rho)=0.808517\ldots\) is impossible. This is the sense in which "the same construction applied to Davenport-Heilbronn does not give a matching bound": it gives a claim that is false, not merely an untested or unprovable one, because the classical zero-free-region mechanism would need to exclude a zero at \(\operatorname{Re}=0.8085\) to reach even the weak bound \(O(t^{1/2+\epsilon})\), and \(\rho\) is a verified counterexample to exactly that exclusion (already located, pinned to 50 digits, and cross-checked against Spira 1994 in zeta/epstein.py).
This paragraph cites standard theory in the same way the inherited documents cite Tao-Teräväinen or Montgomery-Vaughan; it is not re-derived here, and the convergence/growth properties of \(\Lambda_F\)'s own Dirichlet series in a given half-plane are not established in this document beyond what Section 5.3 checks directly.
5.3 A robust numerical fact that does not depend on the asymptotic statement
Landau-type Omega statements are limsup statements and are not expected to be visible at any numerically reachable \(N\) (the same caution applies to the classical PNT decay LOCALIZED_MIXED_ENERGY.md cites for the real mean-mode: at \(N=2\times10^5\), \(\exp(-\sqrt{\log N})\approx0.030\), nowhere near its asymptotic regime either). The partial-sum growth-rate fit in results_s8_positivity_battery.json (fitted_growth_exponent_top_half, top_quarter) is accordingly noisy and inconclusive at \(n_{\max}=2\times10^5\) -- reported for completeness, not claimed as evidence.
A different, non-asymptotic-sensitive fact is robust at this range and is the one actually load-bearing here. For the primes, \(\Lambda(n)\le\log n\) always (used pervasively in the inherited documents, e.g. SIEGEL_UNIFORMITY.md/LOCALIZED_MIXED_ENERGY.md's \(|a(n)|\le2b_Z\), \(\Lambda(n)\le\ell\) bounds). Is \(\Lambda_F(n)=O(\log n)\) for Davenport-Heilbronn? Measured (own divisor-sieve computation, cross-checked as above), maximum \(|\Lambda_F(n)|\) in dyadic-ish windows against \(\log(\text{window right edge})\):
| window | \(\max | \Lambda_F(n) | \) | \(\log(\text{right edge})\) | ratio |
|---|---|---|---|---|---|
| \([2,2000)\) | 26.73 | 7.60 | 3.52 | ||
| \([2000,20000)\) | 157.80 | 9.90 | 15.93 | ||
| \([20000,200000)\) | 630.97 | 12.21 | 51.69 |
The ratio is not merely nonzero; it grows by roughly a factor of \(4.4\) each time the window's right edge grows by a factor of \(10\), i.e. \(|\Lambda_F|\) is growing polynomially in \(n\) here (consistent with, though this document does not fit, an exponent well above \(0\); \(\sqrt n\) at \(n=2\times10^5\) is \(447\), the same order as the measured \(631\)), while \(\log n\) grows by less than \(1\). This is the elementary bound every minor-arc and sieve estimate in \(\Delta_{\rm denom}\), \(\Delta_{\rm short}\), \(\Delta_{\rm fallback}\) also uses for \(\Lambda\) itself, so it is a second, independent place (beyond the sign failure of Section 5.1) where the Davenport-Heilbronn analogue of the primes' arithmetic input is measurably, robustly false, already within reach of direct computation, without needing the harder Omega-theorem statement of Section 5.2 at all.
6. What this changes, and what it does not: the a-0070 relay
A parallel attempt on the same corrected error (a-0070, relayed via a peer session) reports that any bounded-divisor-level model comparison of the \(\nu(n)\), \(Z=\exp(\ell^{1/10})\) shape used throughout SIEGEL_UNIFORMITY.md/ENDPOINT_BOUND.md's architecture is structurally capped: the \(N^3\) budget forces a divisor level \(D_0=Z^s\) with \(s=o(\ell^{1-\kappa})\), while reaching a rate \(\ell^\kappa\) needs \(s\) to decay at least that fast, and the two cannot both hold past \(\kappa=1/2\).
\(\Delta_{\rm denom}\), \(\Delta_{\rm short}\), and \(\Delta_{\rm fallback}\) are reused directly from that architecture (Section 4), so they inherit exactly that ceiling; nothing here removes it. \(\Delta_{\rm mean}\) does not go through any \(Z\)-level sieve truncation at all -- its bound (LOCALIZED_MIXED_ENERGY.md eq. 19) is the classical, level-free de la Vallée-Poussin/Vinogradov-Korobov estimate for \(\psi(t)-t\) directly, with no divisor level \(D_0\) and no arithmetic-progression modulus \(q\) in its statement at all. It has no analogous tension between a truncation level and the \(N^3\) budget, and in fact already reaches a stronger rate, \(\mathcal L(N)=\ell^{3/5}(\log\ell)^{-1/5}\) (Vinogradov-Korobov), than the \(\ell^{1/10}\) the rest of the architecture is capped at.
This is not, however, an improvement to \(E_{\rm corr}(N)\)'s total order. \(\mathcal M\)'s size in (3) is governed by whichever term is largest, and \(N^3e^{-c\mathcal L(N)}\) (the mean mode) is smaller than \(N^3e^{-c_\kappa\ell^\kappa}\) (the fallback pieces, \(\kappa<1/10\)) for large \(N\), since \(\ell^{3/5}(\log\ell)^{-1/5}\gg\ell^{1/10}\). The bottleneck was never the mean mode; it is exactly where LOCALIZED_MIXED_ENERGY.md already located it (its eq. 20, the long-window mean square, a strictly stronger and unestablished statement, and the unresolved cross-term cancellation of its eq. 22). \(\mathrm{Energy}(N)\) therefore has the same order as before this document: \(E_{\rm corr}(N)\ll_\kappa N^3\exp(-c'_\kappa\ell^\kappa)\) for every fixed \(\kappa<1/10\), no fixed power saving. Section 8's request for "a genuinely arithmetic multiscale upper bound" did not require beating that order, only that the bound's proof be arithmetic rather than vacuous and that it fail the battery a merely-algebraic construction would pass; this document supplies that, not a smaller \(\mathrm{Energy}(N)\).
7. Section 8's constraints, checked one by one
- Retains the mean/central mode. \(\Delta_{\rm mean}\) (Section 2) is exactly this, and Section 4 identifies its own arithmetic content separately from every other scale.
- Retains the cross terms between scales. Present in (4) and paid for by triangle inequality through to (1)-(2), not assumed zero or dropped (Section 3).
- The changing correction data. \(a(n)\), \(R_{\rm mod}\), and \(\widehat C_N\) are built from whatever \(q,\chi,\beta\) apply at each \(N\) (Section 1, inherited); no step here uses a fixed exceptional conductor.
- Sharp endpoint \(N-h\); no smoothing. Inherited unchanged through \(r_N(h)\), \(\psi_2(N,h)\), and the un-smoothed window identity (LOCALIZED_MIXED_ENERGY.md eq. 1).
- No primes-only replacement. \(\Lambda\) includes every proper prime power throughout; the model \(a(n)\) is a comparison object, not a replacement, exactly as in every inherited document.
All five hold, by inheritance; this document adds none of them and removes none of them.
8. Grade
Assembly of (2)-(3) from LOCALIZED_MIXED_ENERGY.md eq. (9), (12), (14), (19), (25)-(26): PROVED, by citation -- no new size estimate is claimed for any \(\Delta\) term; the arithmetic is entirely the cited document's. The per-scale attribution of Section 4: an independent reading, checked against the actual proofs cited, in the same sense CHALLENGE.md attack 3 is a reading of CANDIDATE_ENERGY.md's proof. The battery run of Section 5.1: MEASURED, independently, to \(1.5\times10^{-16}\) relative error at the module's own cross-check point, with the fast solver's code in s8_positivity_battery.py. The Omega-theorem consequence of Section 5.2: CITED, not proved here (standard theory, applied to a pinned zero, not re-derived from Perron's formula in full rigor in this document). The magnitude comparison of Section 5.3: MEASURED, robust at the computed range, not asymptotic-sensitive. The order discussion of Section 6: a direct reading of already-established inequalities, not a new estimate, and not an improvement to \(E_{\rm corr}(N)\)'s total order.
No fixed power saving is claimed. No exceptional zero is asserted to exist or excluded. This document does not reopen or weaken any inherited result; it answers, specifically, whether a Section-8-compliant \(\mathrm{Energy}(N)\) can be built whose domination proof survives the Davenport-Heilbronn battery at the one step a merely-algebraic construction like CANDIDATE_ENERGY.md's could not identify. It can, at the mean/central-mode scale, via \(\Lambda(n)\ge0\); it explicitly cannot, at every other scale reused here, escape the ceiling a-0070 describes for the same reason CANDIDATE_ENERGY.md's construction escaped nothing: those scales were never this document's new content.