2026-09-10. Builds on CANDIDATE_ENERGY.md, CORRECTED_RH_BRIDGE.md, LOCALIZED_MIXED_ENERGY.md and ENDPOINT_BOUND.md as they stand on this branch. No base commit hash is cited: this pass does not use git. Nothing in those four documents is revisited or changed.
Outcome, stated first. The requested transition — bound the scale-\(k\) component of \(\mathrm{Energy}(N)\) by the scale-\((k-1)\) component plus an explicit arithmetic term, uniformly in \(k\), so the sum telescopes below the trivial order — fails, and it fails at an identifiable place: the only unconditional per-block arithmetic input available at short block lengths does not shrink as the block shrinks, while the number of blocks at scale \(k\) grows like \(N/2^k\); combining the two through the definition of \(\Delta_k\) produces a bound that is worse than the trivial \(N^3\) order at every scale this document can reach, including scale \(k=0\), where CANDIDATE_ENERGY.md's own tables show close to half of \(E_{\rm corr}(N)\) sits. The one place the transition does not blow up (the top \(O((\log N)^{3/5}(\log\log N)^{-1/5})\) rungs of a ladder of length \(K\sim\log_2N\)) is exactly where the measured energy share is under \(0.1\%\). No \(\theta\) beyond the inherited \(\theta=1/10\) of ENDPOINT_BOUND.md is produced by this route; that document's bound is not reproved here, only cited. This is a handwritten deduction, pending external verification, with no novelty claim.
1. Setup, inherited without change
Use CANDIDATE_ENERGY.md Sections 1-2 exactly: \(x_h=r_N(h)-C_N(h)\) for \(1\le h\le N\), extended by zero to \(M=2^K\ge N\), \(K=\lceil\log_2N\rceil\); \(P_k\) the block-average projection at block length \(2^k\); \(D_k=P_k-P_{k+1}\); \(\Delta_k(N)=\|D_kx\|2^2\); and the proved identity \[ E{\rm corr}(N)=\mathrm{Energy}(N) =2M\mu(N)^2+2\sum_{k=0}^{K-1}\Delta_k(N).\tag{$\star$} \] \((\star)\) is exact and carries no arithmetic content by itself (CANDIDATE_ENERGY.md Section 3): it does not, on its own, make any term smaller. A scale transition can only be useful here if it supplies new arithmetic information about each \(\Delta_k\) — something \((\star)\)'s linear algebra does not.
The target restated, so the known barrier is not rediscovered. A fixed power saving \(E_{\rm corr}(N)\ll N^{3-\delta}\) would invert CORRECTED_RH_BRIDGE.md equation (3) against Landau's \(\Omega\) theorem for \(\psi(N)-N\) and is not attempted here. The reachable target is a saving of shape \(N^3\exp(-c(\log N)^\theta)\), and the question is which \(\theta\), if any, a transition on \((\star)\) can reach, against the existing inherited bound \(E_{\rm corr}(N)\ll_\kappa N^3\exp(-c_\kappa(\log N)^\kappa)\) for every \(\kappa<1/10\) (CORRECTED_RH_BRIDGE.md equation (21)), and the proposed endpoint \(\theta=1/10\) (ENDPOINT_BOUND.md).
2. Two candidate transition mechanisms
Write \(\ell=\log N\). Two natural ways to try to bound \(\Delta_k(N)\) by \(\Delta_{k-1}(N)\) plus an arithmetic term were tried:
(a) A global block recursion. Bound the two block sums that make up each \(D_{k}\)-difference directly, using the strongest available unconditional, block-length-independent arithmetic input — the classical (Vinogradov-Korobov-strength) prime number theorem rate, already inherited in this project as \(|\psi(x)-x|\ll x\exp(-c\mathcal L(N))\), \(\mathcal L(N)=\ell^{3/5}(\log\ell)^{-1/5}\) (LOCALIZED_MIXED_ENERGY.md equations (18)-(19), citing TT Theorem 1.3(i)) — and propagate it through the definition of \(\psi_2(N,h)\) to every dyadic block, at every scale, by the same argument at every \(k\) (hence "uniformly in \(k\)"). Section 3 carries this out in full and shows it fails.
(b) A local geometric-decay recursion. Observe, empirically (Section 5), that \(\Delta_k(N)/\Delta_{k-1}(N)\approx1/2\) at the fine scales where the energy actually sits, and ask whether \(\Delta_k(N)\le(\tfrac12+\varepsilon)\Delta_{k-1}(N)+(\text{small arithmetic term})\) can be proved. Section 5 shows why this recursion, even if it held, would not close, and why proving it is not easier than the original problem.
Neither route produces a working transition; Sections 3-6 give why, with an explicit scale named in each case.
3. Mechanism (a): the global block recursion, worked out and shown to fail
Fix \(k\) and a length-\(2^{k+1}\) block, split into two adjacent length-\(2^k\) sub-blocks \(B_1=a,a+2^k)\), \(B_2=[a+2^k,a+2^{k+1})\) of the shift variable \(h\). Write \(S_i=\sum_{h\in B_i}x_h\) (\(i=1,2\)). From the block-average bookkeeping of [CANDIDATE_ENERGY.md Section 3 (the elementary Haar identity, not repeated here), \[ \Delta_k(N)=2^{-k-1}\sum_{\text{level-}(k+1)\text{ blocks}}(S_1-S_2)^2. \tag{1} \] This is exact (it is \((\star)\)'s bookkeeping specialized to one level); the arithmetic content must come from bounding \(S_1-S_2\).
Lemma (block-sum discrepancy, elementary). For \(2^k\le N^{1/4}\) and any block \(B=[a,a+2^k)\subseteq[1,N]\), \[ \sum_{n\le N}\Lambda(n)\!\!\sum_{\substack{h\in B\\n+h\le N}}\!\!\Lambda(n+h) =\sum_{n\le N}\Lambda(n)\,s(n)+O(N^2e^{-c\mathcal L(N)}), \tag{2} \] where \(s(n)=\#\{h\in B:n+h\le N\}\le2^k\).
Proof. For fixed \(n\), the inner sum is \(\psi(\min(N,n+a+2^k-1))-\psi(n+a-1)\). By the cited rate, each of these two values of \(\psi\) differs from its argument by \(O(Ne^{-c\mathcal L(N)})\) (using the elementary bound for arguments below \(\sqrt N\), absorbed since \(\sqrt N\log N=o(Ne^{-c\mathcal L(N)})\), exactly as in LOCALIZED_MIXED_ENERGY.md Section 5). Hence the inner sum is \(s(n)+O(Ne^{-c\mathcal L(N)})\), for every \(n\), with an implied constant independent of \(a\) and \(k\). Multiplying by \(\Lambda(n)\) and summing over \(n\le N\) (using \(\sum_{n\le N}\Lambda(n)\ll N\)) gives (2). \(\blacksquare\)
The left side of (2) is \(\sum_{h\in B}\psi_2(N,h)\), by exchanging the order of summation; this is exact bookkeeping, not an estimate. The right side splits \(\sum_{h\in B}x_h\) into the deterministic quantity \(\sum_n\Lambda(n)s(n)-\sum_{h\in B}\bigl[(N-h)\mathfrak S(h)+C_N(h)\bigr]\) plus the \(O(N^2e^{-c\mathcal L(N)})\) error of (2). Bounding that deterministic remainder for a general sub-block \(B\) (rather than the full range \([1,N]\)) needs a block-localized version of the first-moment computation CORRECTED_RH_BRIDGE.md carries out only for the complete range (its Section 4, and the signed estimate (2)/(15) for \(C_N\)); no such localized computation exists in the documents inherited here. We grant, generously and without proof, that this remainder costs no more than the same order \(O(N^2e^{-c\mathcal L(N)})\) — the most favorable assumption possible, since it is not established. Under that grant, \[ |S_i|=O(N^2e^{-c\mathcal L(N)})\qquad(i=1,2),\qquad |S_1-S_2|=O(N^2e^{-c\mathcal L(N)}), \tag{3} \] uniformly over blocks, for \(2^k\le N^{1/4}\).
Consequence. There are \(\asymp N/2^{k+1}\) level-\((k+1)\) blocks in (1). Using (3) in each term of (1), \[ \boxed{\quad \Delta_k(N)=O\!\left(\frac{N^5e^{-2c\mathcal L(N)}}{4^k}\right) \qquad(2^k\le N^{1/4}). \quad} \tag{4} \]
Where this fails. Compare (4) to the trivial order \(N^3\): \[ \frac{\Delta_k(N)_{\text{(4)}}}{N^3} =O\!\left(N^2e^{-2c\mathcal L(N)}/4^k\right). \] Since \(\mathcal L(N)=\ell^{3/5}(\log\ell)^{-1/5}=o(\ell)\), the quantity \(N^2e^{-2c\mathcal L(N)}\) grows like \(e^{2\ell-2c\mathcal L(N)}\to\infty\) far faster than any \(4^k\) with \(k\le K/4=\tfrac14\log_2N\) can compensate: at the top of the lemma's range of validity, \(4^{K/4}=N^{1/2}\), which is still \(\ll N^2e^{-2c\mathcal L(N)}\) for all large \(N\) (the exponent gap is \(3/2\log N\) against a term of size \(o(\log N)\) in the exponent of \(e\)). The bound (4) therefore exceeds the trivial \(N^3\) order at every scale \(k\) in its entire proved range \(0\le k\le\tfrac14\log_2N\), including \(k=0\), where it gives \(\Delta_0(N)=O(N^5e^{-2c\mathcal L(N)})\), worse than trivial by a factor \(N^2e^{-2c\mathcal L(N)}\to\infty\). This is not a borderline miss: the naive block recursion is not merely insufficient, it is strictly worse than doing nothing, throughout the only range where the Lemma above lets it be evaluated at all.
Solving \(4^{k^}=N^2e^{-2c\mathcal L(N)}\) for the threshold gives \(2^{k^}=Ne^{-c\mathcal L(N)}\), i.e. \(k^*=K-c\mathcal L(N)/\log2+O(1)\): only the top \(O(\mathcal L(N))=O(\ell^{3/5}(\log\ell)^{-1/5})\) rungs of the \(K\sim\log_2N\)-rung ladder are even candidates for this method not to blow up — a \((1-o(1))\)-fraction of all scales fail — and by CANDIDATE_ENERGY.md Section 4's measured tables, those top rungs (scale \(11\)-\(13\) at \(N=10^4\), scale \(14\)-\(16\) at \(N=10^5\)) each hold well under \(0.1\%\) of \(E_{\rm corr}(N)\). Grade: the Lemma and (4) are PROVED, conditional on the granted main-term-matching assumption stated above; the failure conclusion (exceeding trivial throughout the provable range) is unconditional and a fortiori — granting the assumption only makes the bound (4) as small as it could possibly be, and it still fails.
4. Why an aggregate (large-sieve-style) fix is not available either
The blow-up in Section 3 comes from bounding each of the \(\asymp N/2^k\) blocks' discrepancy separately and squaring the worst case, rather than bounding \(\sum(S_1-S_2)^2\) as a whole. A large-sieve-type inequality — bounding a sum of squares over many blocks by a global quantity without paying the block count as a separate factor — is exactly the kind of tool LOCALIZED_MIXED_ENERGY.md Section 4 uses, successfully, for a different aggregation: sums over reduced rationals \(a/r\) on the frequency circle (its equations (10)-(14)). That aggregation is on the frequency-denominator axis. \(\Delta_k\)'s blocks are on the shift-index axis at fixed block length; they are not the same objects, and no rational-denominator structure organizes them.
A genuine fix would need a second-moment (variance-of-correlations) estimate for \(\sum_{h\in B}x_h\) across many blocks \(B\) simultaneously, which is a quartic-in-\(\Lambda\) statement (each \(x_h\) is already quadratic in \(\Lambda\)). The one place in the inherited documents where a quartic-in-\(\Lambda\) long-range quantity of comparable shape is handled is LOCALIZED_MIXED_ENERGY.md's controlling long window \(T_N(0)\) (its equations (15)-(19)), which is a variance of prefix sums \(A(k)=\psi(k)-k+O(\cdots)\) around the global mean, i.e. its own \(k=K\)-type (single, coarsest, global) object — it proves \(T_N(0)\ll N^3e^{-c\mathcal L(N)}\), Vinogradov-Korobov strength (\(\theta=3/5\)), stronger than \(1/10\), but that document is explicit (its equation (20) and the surrounding discussion) that this bound on the raw second moment does not imply the needed mean-square around the block-local mean at shorter lengths, which is precisely a shorter version of what Section 3 above would need at every \(k<K\). The gap identified there is not resolved by relabeling it in the index domain: it is the same open estimate, restated. No unconditional aggregate input beating Section 3's per-block bound is available in the inherited documents. Grade: OBSERVATION (a comparison of what is and is not proved elsewhere; it does not itself prove non-existence of such a tool).
5. Mechanism (b): the empirical geometric-decay recursion
At the scales where the energy actually sits, CANDIDATE_ENERGY.md's own results_s8_candidate_energy.json shows a near-constant ratio between consecutive fine scales:
| \(N\) | \(\Delta_1/\Delta_0\) | \(\Delta_2/\Delta_1\) | \(\Delta_3/\Delta_2\) | \(\Delta_4/\Delta_3\) | \(\Delta_5/\Delta_4\) | \(\Delta_6/\Delta_5\) |
|---|---|---|---|---|---|---|
| \(2000\) | 0.449 | 0.544 | 0.350 | 0.731 | 0.527 | 0.178 |
| \(5000\) | 0.474 | 0.477 | 0.648 | 0.346 | 0.576 | 0.615 |
| \(10000\) | 0.514 | 0.480 | 0.503 | 0.396 | 0.389 | 0.463 |
| \(30000\) | 0.474 | 0.519 | 0.544 | 0.483 | 0.401 | 0.490 |
| \(100000\) | 0.506 | 0.481 | 0.526 | 0.470 | 0.478 | 0.452 |
(computed directly from the existing per_scale fields already in results_s8_candidate_energy.json; no new script was written, none was needed to read an existing file.) The ratios cluster around \(0.45\)-\(0.55\) at \(N\ge10^4\), suggestive of an approximate \(\Delta_k(N)\approx\Delta_{k-1}(N)/2\) law at fine-to-medium scales — consistent with, but visibly not exactly, a fixed halving (the range \(0.35\)-\(0.65\) at smaller \(N\) or larger \(k\) shows real scatter, and CANDIDATE_ENERGY.md Section 4 already calls the coarse tail "noisy"). This is measured, finite-\(N\) data, not a law derived from any theorem here.
Why this cannot be turned into the requested transition, even granting the pattern. Suppose, optimistically, that \(\Delta_k(N)\le(\tfrac12+\varepsilon)\Delta_{k-1}(N)+R_k(N)\) could be proved for some small remainder \(R_k\). Telescoping downward from \(k=K-1\) to \(k=1\) gives \(\Delta_k(N)\lesssim2^{-(k)}\Delta_0(N)+ \sum R_j\)-type accumulation — i.e. every scale's bound would still be anchored to \(\Delta_0(N)\), the single finest scale, exactly the term CANDIDATE_ENERGY.md's own "what this is not" paragraph already identifies as "exactly as hard to bound... as \(E_{\rm corr}(N)\) itself." A recursion running the other way (upward, bounding \(\Delta_{k}\) in terms of \(\Delta_{k-1}\) starting from \(k=1\)) has the same anchor at the bottom. Either direction, the base case is the finest scale, and nothing above supplies a bound on \(\Delta_0(N)\) beyond the global \(N^3e^{-c_\kappa\ell^\kappa}\), \(\kappa<1/10\), already known without any scale decomposition. Proving the ratio bound itself is no easier: \(\Delta_k\) and \(\Delta_{k-1}\) differ only by which pairs of blocks are compared, both are quartic-in-\(\Lambda\) quantities of the shift-block type Section 4 identified as unresolved, and there is no reduction here to something weaker. Grade: the ratio table is MEASURED (exact, for the five listed \(N\)); the proposed recursion is neither proved nor reducible to a weaker open statement — it is exactly as open as \(\Delta_0(N)\) itself, which is exactly as open as \(E_{\rm corr}(N)\).
6. Synthesis: the exact failure point and the resulting \(\theta\)
Reading the dyadic ladder from the coarse end (\(k=K-1\), closest to the global mean \(\mu(N)\), where CORRECTED_RH_BRIDGE.md's signed first-moment estimate (equations (2), (15)-(16)) already gives good control) down toward the fine end (\(k=0\), where CANDIDATE_ENERGY.md's tables put roughly half the energy): the transition fails immediately below the top \(O(\ell^{3/5}(\log\ell)^{-1/5})\) rungs, i.e. after descending only a vanishing fraction of the \(K\sim\log_2N\) rungs available, by the exact margin computed in Section 3 (equation (4) and the threshold \(k^=K-c\mathcal L(N)/\log2+O(1)\)). Below \(k^\) the only tried mechanism (a) is strictly worse than the trivial bound, and mechanism (b) never leaves the status of an empirical pattern requiring a proof as hard as \(\Delta_0(N)\) itself (Section 5). No mechanism reaches the scales carrying the energy.
\[ \boxed{\quad \text{The scale transition on }(\star)\text{ yields no }\theta.\quad \text{It neither reaches nor beats }\theta=1/10. \quad} \]
The only bound available for \(E_{\rm corr}(N)=\mathrm{Energy}(N)\) at the end of this attempt is the one already established without any scale decomposition: \(E_{\rm corr}(N)\ll_\kappa N^3\exp(-c_\kappa(\log N)^\kappa)\) for every \(\kappa<1/10\) (CORRECTED_RH_BRIDGE.md equation (21)), and the separately proposed \(\theta=1/10\) endpoint of ENDPOINT_BOUND.md, neither of which this document reproves, weakens, or improves. The exact identity \((\star)\) is unaffected; it remains true and arithmetically inert, exactly as characterized in CANDIDATE_ENERGY.md.
7. What is and is not claimed
Section 3's Lemma and the bound (4) are proved, conditional on one explicitly granted (not proved) main-term-matching assumption; the conclusion that this route fails throughout its provable range holds regardless of that assumption's truth, since granting it is the most favorable case. Section 4's comparison to LOCALIZED_MIXED_ENERGY.md is a reading of that document, not a new non-existence proof for every conceivable aggregate estimate. Section 5's ratio table is exact, finite-\(N\) measurement; the impossibility argument built on it is a reduction to an already-identified open quantity (\(\Delta_0(N)\)), not a proof that no future argument can succeed. No claim is made that the scale transition is impossible in principle — only that the two mechanisms tried here, built directly from the tools inherited from CORRECTED_RH_BRIDGE.md, LOCALIZED_MIXED_ENERGY.md and ENDPOINT_BOUND.md, both fail, and fail at the specific scales named above. This does not touch, weaken, or improve the RH-sufficiency bridge, the exceptional lower bound, or the proposed \(\theta=1/10\) endpoint; none of those are reproved or re-derived here.