2026-09-10. Target: CANDIDATE_ENERGY.md as it stands on this branch, which claims not just the requested domination \(E_{\rm corr}(N)\le\mathrm{Energy}(N)\) but the exact identity \[ E_{\rm corr}(N)=\mathrm{Energy}(N)\qquad\text{for every }N. \tag{$\star$} \] Three attacks, all run independently of the candidate's own code, results in results_s8_challenge.json (s8_challenge.py).
Verdict, up front. \((\star)\) survives attack 1: no violation exists, because none can. Attack 2 shows why that survival is worthless as evidence about primes: the identical construction, applied to a real, functional- equation sequence with a documented zero off the critical line, produces the identical identity, to the same floating-point precision. Attack 3 explains the mechanism: every step of the domination proof is linear algebra on an arbitrary finite real vector, and uses no property of \(\Lambda\), \(\mathfrak S\), or \(C_N\) beyond their being some fixed real numbers. This is not a new discovery -- CANDIDATE_ENERGY.md's own "What this is not" paragraph and Section 5 point 2 already say the identity "carries no arithmetic content at all" -- but it was asserted there, not tested. This document tests it. None of the three attacks contradicts CANDIDATE_ENERGY.md; together they independently confirm its own stated limitation and rule out the reading that \((\star)\), because it is exact and unconditional, is progress toward bounding \(E_{\rm corr}(N)\) or toward RH.
Attack 1: independent numerical search, \(N\le10^5\)
Method, built from scratch, no import of s8_candidate_energy.py. s8_challenge.py reimplements, in its own code:
sieve_primes/von_mangoldt: smallest-set-of-primes sieve, proper prime powers included, coded independently ofprobe.py's smallest-prime-factor sieve.singular_series: \(S(k)=2C_2\prod_{p\mid k,\,p>2}(p-1)/(p-2)\) for even \(k\), 0 for odd \(k\) (CORRECTED_RH_BRIDGE.md eq 4), from the same quoted reference constant \(C_2\)probe.pyuses (an input, not something either script fits).autocorr_fft: \(\psi_2(N,h)\) as an FFT autocorrelation, independently coded, cross-checked at \(N=2000\) against a fresh \(O(N^2)\) direct pair sum (autocorr_direct): max\(|\)diff\(|=1.8\times10^{-12}\).multiscale_energy: the dyadic block-average telescoping of CANDIDATE_ENERGY.md Section 2/3, re-derived from the boxed definition and coded as a bottom-up pairwise-averaging recursion, not copied froms8_candidate_energy.py's vectorized reshape loop.
Sanity checks against fixed external references: the \(E(N)/(N^2\log^2N)\) ratio at \(N=10^5\) computed here is 0.16857, matching CHHL's own published Table 1 value (probe.py's independent cross-check target) to 5 digits; and at \(N=2000\), this script's \(E_{\rm corr}\) and \(\mathrm{Energy}\) match results_s8_candidate_energy.json's row (produced by s8_candidate_energy.py, never imported here) to relative error \(0\) and \(1.5\times10^{-16}\) respectively.
Grid. 116 values of \(N\) in \([1,10^5]\): every \(N\le20\); steps of 25 to 200; steps of 100 to 2000; steps of 1000 to 20000; steps of 5000 to \(10^5\); \(2^k\), \(2^k\pm1\) for \(k=1,\dots,17\) (the padding boundary, where a scale is dropped or added); and the candidate's own five \(N\).
Result. all_dominate = True at every one of the 116 points; the largest \(|\)relative gap\(|\) between \(\mathrm{Energy}\) and \(E_{\rm corr}\) found anywhere in the grid is \(4.3\times10^{-16}\), i.e. floating-point roundoff, not a mathematical gap. No violation of the domination exists at any tested \(N\le10^5\), and none is expected to: Attack 3 shows the identity is an exact Pythagorean decomposition of \(\|x\|_2^2\), true for every real vector \(x\), so a violation at any \(N\) would mean a bug in the floating-point implementation, not a counterexample to the mathematics.
Verdict: attack 1 does not refute the candidate. The domination (in fact the stronger equality) holds, exactly, everywhere tested.
Attack 2: the Davenport-Heilbronn battery
Why this attack, and what it is for. AGENTS.md's standing counterexample-battery rule (zeta/epstein.py's module docstring; also NULLCONTROLS.md and REDTEAM.md attack A3) requires that any claimed structural pattern be run against the Davenport-Heilbronn function \(f\): a function with a Riemann-type functional equation \(F(s)=F(1-s)\), real Dirichlet coefficients, and a proven, located zero off the critical line at \(\rho\approx0.808517+85.699348i\). If a pattern also holds for \(f\), the repository's own gate says it is dead on arrival as evidence about zeta, because \(f\)'s off-line zero means whatever mechanism produced the pattern cannot be "RH is true," since it is false for \(f\).
Construction. Using zeta.epstein.dh_coefficient (never reimplemented; this is the one place the attack is required to use the repository's own counterexample object, not a substitute), build the real, exactly period-5 sequence \(a(n)\), \(a(1),\dots,a(5)=1,\kappa,-\kappa,-1,0\) with \(\kappa=0.284079\ldots\) the Davenport-Heilbronn constant. Define the exact analogue of the singular series for a periodic sequence, its period average \[ \rho(h):=\frac15\sum_{j=1}^5a(j)\,a(j+h)\qquad(\rho\text{ depends only on }h\bmod5), \] which plays exactly \(\mathfrak S(h)\)'s role: it is the exact main term of \(\sum_{n=1}^{N-h}a(n)a(n+h)\), because \(a\) is periodic with no arithmetic irregularity to correct for (unlike \(\mathfrak S\), \(\rho\) needs no error term at all). Then, verbatim in the same construction as CANDIDATE_ENERGY.md Section 1-2 with \(a\) in place of \(\Lambda\) and \(\rho\) in place of \(\mathfrak S\): \[ x_h^{\rm DH}:=\Big(\sum_{n=1}^{N-h}a(n)a(n+h)\Big)-(N-h)\rho(h),\qquad E_{\rm corr}^{\rm DH}(N):=2\sum_{h=1}^Nx_h^{{\rm DH}\,2}, \] and \(\mathrm{Energy}^{\rm DH}(N)\) from the identical dyadic block-average telescoping, same code (multiscale_energy, e_corr_and_energy), no branch on which sequence it is fed.
Result. Tested at 33 values of \(N\) up to \(10^5\) (powers of two and their neighbours, plus \(10^2,10^3,\dots,10^5\)): \(\mathrm{Energy}^{\rm DH}(N)=E_{\rm corr}^{\rm DH}(N)\) exactly, to floating-point precision, at every one (largest \(|\)relative gap\(|\) \(8.2\times10^{-16}\), again pure roundoff). The identity holds for the Davenport-Heilbronn sequence exactly as it holds for the corrected prime-pair residual, with no exception and no weaker constant.
Verdict: attack 2 confirms the candidate has distinguished nothing about zeta. The domination inequality \((\star)\) requested by Section 8 -- indeed the stronger equality CANDIDATE_ENERGY.md actually proves -- holds for a sequence that is real, satisfies a genuine functional equation, and provably violates RH. Nothing about \((\star)\) can therefore be evidence toward RH, toward the open target \(E_{\rm corr}(N)\ll_\epsilon N^{2+\epsilon}\) being reachable through this decomposition, or toward any RH-specific mechanism. This is exactly the outcome CANDIDATE_ENERGY.md's own "What this is not" section already anticipates; attack 2 is the check that makes it more than an assertion.
Attack 3: reading the domination proof line by line
CANDIDATE_ENERGY.md Section 3 has three numbered results: the Lemma (projection algebra, \(P_aP_b=P_{\max(a,b)}\)), the Corollary (\(D_k\) is an orthogonal projection, and \(D_k\), \(D_{k'}\), \(P_K\) are pairwise orthogonal), and the Theorem (the Pythagorean identity \(\sum x_h^2=M\mu^2+\sum_k\|D_kx\|_2^2\)).
- The Lemma's proof uses only: \(P_k\) is symmetric (an averaging operator has a symmetric matrix), \(P_k\) is idempotent (averaging a block-constant vector again is a no-op), and the nesting \(V_b\subseteq V_a\) for \(a\le b\) (a coarser block is a union of finer blocks). All three are facts about the block structure of \(\{1,\dots,M\}\) under dyadic subdivision. None mentions \(x\), let alone \(\Lambda\), \(\mathfrak S\), or \(C_N\).
- The Corollary's proof is pure algebra on the Lemma's identity (\(D_k^2=D_k\), \(D_kD_{k'}=0\), \(P_KD_k=0\)). Same conclusion: no arithmetic input, not even the existence of \(x\).
- The Theorem's proof telescopes \(\sum_kD_k=P_0-P_K\) (true for any finite sum of these operators) and expands \(\|x\|_2^2\) by bilinearity, using the Corollary's orthogonality to kill every cross term. The one place \(x\) enters is as a generic vector in \(\mathbb R^M\); the proof never uses that \(x_h=r_N(h)-C_N(h)\), never uses that \(N-h\) is the correlation endpoint, never uses positivity, boundedness, or any growth property of \(\Lambda\) or \(\mathfrak S\), and never uses the value of \(C_N\) (whether zero, as in every numerical run, or an actual exceptional correction).
The only place arithmetic content enters at all is Section 1's definition \(x_h:=r_N(h)-C_N(h)\) -- but that is setup, external to the proof of \((\star)\), not a step inside it. Substituting a completely different real vector there, with a different arithmetic origin and a different (in fact false) relationship to RH, changes nothing about the proof's validity, and attack 2 exhibits exactly that substitution numerically.
Verdict: zero steps of the domination proof use the structure of the primes. The proof is a special case of the general fact that dyadic block-averaging is an orthogonal decomposition of \(\mathbb R^M\) (an instance of a Haar/martingale telescoping), applicable to any finite real sequence. This matches, independently, CANDIDATE_ENERGY.md's own grading of Section 3 ("It is exact for every finite \(N\)... assumes nothing about the arithmetic content of \(x_h\)") and Section 5 point 2 ("carries no arithmetic content at all"); attack 3 is the outside confirmation of that self-assessment rather than a new finding.
Overall verdict
CANDIDATE_ENERGY.md's equality \((\star)\) is not refuted by attack 1: it holds everywhere tested, because it is an algebraic identity, not an estimate, and cannot fail for a bug-free implementation. What is refuted, by attacks 2 and 3 together, is any reading of \((\star)\) as evidence of progress toward the open target \(E_{\rm corr}(N)\ll_\epsilon N^{2+\epsilon}\), toward RH, or toward anything specific to primes: the identity holds equally, exactly, and with the same proof, for a real sequence built from a function whose own Riemann-type functional equation does not save it from having a zero at \(\mathrm{Re}(s)\approx0.8085\ne\tfrac12\). Per the repository's standing counterexample-battery rule, a construction that survives the battery this completely -- not approximately, but as the identical identity to sixteen digits -- has distinguished nothing, and CANDIDATE_ENERGY.md already says as much about itself. Section 8's open item is exactly where CANDIDATE_ENERGY.md left it: the actual size of \(E_{\rm corr}(N)\).