2026-09-10. Builds on CORRECTED_RH_BRIDGE.md, LOCALIZED_MIXED_ENERGY.md and EXCEPTIONAL_ENERGY.md as they stand on this branch. No specific base commit hash is cited: this pass does not use git. Nothing in those three documents is revisited or changed.
Outcome. There is a multiscale arithmetic energy \(\mathrm{Energy}(N)\), built from a dyadic (Haar/martingale) block-average telescoping of the corrected residual vector, for which \[ E_{\rm corr}(N)=\mathrm{Energy}(N)\qquad\text{for every }N, \tag{$\star$} \] an exact identity, proved below by elementary linear algebra (orthogonal projections, no arithmetic input). Equality is the strongest possible case of the requested domination \(E_{\rm corr}(N)\le\mathrm{Energy}(N)\). It holds unconditionally, for every \(N\), with no fitted constant.
What this is not. \((\star)\) is a lossless repackaging of the same number into dyadic-scale pieces, not a new estimate for its size. It supplies zero information toward the open target \(E_{\rm corr}(N)\ll_\epsilon N^{2+\epsilon}\) (CORRECTED_RH_BRIDGE.md eq 1), or toward any weaker unconditional power saving. Every scale component below is exactly as hard to bound, in total, as \(E_{\rm corr}(N)\) itself; the multiscale split only says which part of that difficulty is concentrated where. This is a handwritten deduction, pending external verification, with no novelty claim.
1. Setup, inherited without change
Use CORRECTED_RH_BRIDGE.md Section 1 exactly: \(\psi_2(N,h)=\sum_{n=1}^{N-h}\Lambda(n)\Lambda(n+h)\), \(r_N(h)=\psi_2(N,h)-(N-h)\mathfrak S(h)\), the endpoint is exactly \(N-h\) (not smoothed), \(\Lambda\) includes every proper prime power (no primes-only replacement), and \(C_N(h)\) is the exact correction of that document's equation (5), built from whatever exceptional data \(q,\chi,\beta\) apply at \(N\) (\(C_N\equiv0\) if none exist). Put \[ x_h:=r_N(h)-C_N(h)\quad(1\le h\le N),\qquad E_{\rm corr}(N)=2\sum_{h=1}^N x_h^2. \] This document does not touch \(r_N\), \(C_N\), \(\mathfrak S\), or the cutoff \(N-h\); it only takes the vector \((x_h)_{h=1}^N\) as given and decomposes the one number \(\sum_h x_h^2\) by scale. Whatever \(q,\chi,\beta\) turn out to be at a given \(N\) is absorbed into \(x_h\) before this decomposition starts, so the construction is correct for every choice of exceptional data, unchanged as it varies with \(N\).
2. The multiscale energy
Why block length, not \(h\)-range. Section 8 suggests dyadic bands \(2^k\le h<2^{k+1}\) of the shift itself. Partitioning \(\{1,\dots,N\}\) into such bands and summing \(2\sum_{h\in\text{band}}x_h^2\) per band is a valid but degenerate special case: the bands are disjoint index sets, so no cross term between bands ever arises, and "retaining" them is vacuous. LOCALIZED_MIXED_ENERGY.md already uses a different, non-degenerate dyadic scale for this kind of problem: window length \(s=2^k\) of a moving sum, not a band of index values (its Section 2, eq 1). This document uses that same scale variable — dyadic block length, applied at \(\theta=0\), i.e. in the index domain rather than the frequency domain — because it is the one that produces a genuine, non-block-diagonal quadratic form with cross terms that must be computed rather than assumed away.
Construction. Let \(K=\lceil\log_2N\rceil\), \(M=2^K\ge N\). Extend \(x_h:=0\) for \(N<h\le M\) (this changes no sum of squares: it only adds zero terms, and it does not touch \(x_h\) for \(h\le N\), so the sharp endpoint at \(h=N\) is untouched). For \(k=0,\dots,K\) partition \(\{1,\dots,M\}\) into \(2^{K-k}\) contiguous blocks of length \(2^k\), and let \(P_k\) be the linear map that replaces \(x\) by its average on each such block: \[ (P_kx)h=2^{-k}\!\!\sum{h'\in B_k(h)}\!\!x_{h'},\qquad B_k(h)=\text{the length-}2^k\text{ block containing }h. \] \(P_0=\mathrm{id}\), and \(P_K\) replaces \(x\) by its global mean \(\mu(N):=M^{-1}\sum_{h=1}^Mx_h\) everywhere. Define \[ D_k:=P_k-P_{k+1}\quad(0\le k\le K-1),\qquad \Delta_k(N):=\|D_kx\|2^2=\sum{h=1}^M\bigl((P_kx)h-(P{k+1}x)_h\bigr)^2. \] Each \(\Delta_k(N)\) is manifestly a nonnegative quadratic form in \((x_h)\): it is \(x^\top D_kx\) for the symmetric matrix \(D_k\) proved below to be an orthogonal projection, hence positive semidefinite. Define \[ \boxed{\quad \mathrm{Energy}(N):=2M\mu(N)^2+2\sum_{k=0}^{K-1}\Delta_k(N). \quad} \] The first term is the retained mean/central mode: \(\mu(N)\) is exactly \(M^{-1}\) times the value at frequency 0 of the discrete Fourier transform of \((x_h)\), i.e. the constant Fourier coefficient of the corrected residual sequence, padded to length \(M\). No smoothing is applied to reach it: it is the ordinary arithmetic mean.
3. Domination, proved exactly
Lemma (projection algebra). For \(0\le a,b\le K\), \(P_aP_b=P_bP_a=P_{\max(a,b)}\).
Proof. Each \(P_k\) is symmetric (its matrix entry between \(h,h'\) is \(2^{-k}\) if they lie in the same length-\(2^k\) block, else \(0\), a symmetric relation) and idempotent (\(P_k^2=P_k\): averaging a block-constant vector again changes nothing), hence an orthogonal projection, onto \(V_k:=\{y:y\text{ constant on each length-}2^k\text{ block}\}\). If \(a\le b\) then \(V_b\subseteq V_a\) (constant on the coarser blocks of size \(2^b\) implies constant on the finer sub-blocks of size \(2^a\) that compose them). For nested-range orthogonal projections, \(P_aP_b=P_bP_a=P_b\) whenever \(V_b\subseteq V_a\): \(P_by\in V_b\subseteq V_a\), so \(P_a(P_by)=P_by\) (\(P_a\) is the identity on \(V_a\)); and, block by block, if \(z=P_ax\) is already constant on each length-\(2^a\) block with values \(c_1,\dots\), then averaging \(z\) further over each length-\(2^b\) block (a union of \(2^{b-a}\) such sub-blocks, each occurring once) gives the same numbers as averaging \(x\) directly over that length-\(2^b\) block, i.e. \(P_b(P_ax)=P_bx\). Together this gives the stated identity for \(a\le b\) (so \(\max(a,b)=b\)); the case \(a\ge b\) is symmetric. \(\blacksquare\)
Corollary. \(D_k=P_k-P_{k+1}\) is symmetric, idempotent (\(D_k^2=P_k-P_kP_{k+1}-P_{k+1}P_k+P_{k+1}=P_k-P_{k+1}-P_{k+1}+P_{k+1}=D_k\) by the Lemma with \(a=k,b=k+1\)), hence itself an orthogonal projection. For \(k\ne k'\), say \(k<k'\) so \(k+1\le k'\), \[ D_kD_{k'}=P_kP_{k'}-P_kP_{k'+1}-P_{k+1}P_{k'}+P_{k+1}P_{k'+1} =P_{k'}-P_{k'+1}-P_{k'}+P_{k'+1}=0 \] by four applications of the Lemma. Likewise \(P_KD_k=P_K(P_k-P_{k+1})=P_K-P_K=0\) for \(k\le K-1\), using \(\max(K,k)=\max(K,k+1)=K\).
Theorem (exact identity). \[ \sum_{h=1}^Mx_h^2=\|P_Kx\|2^2+\sum{k=0}^{K-1}\|D_kx\|2^2 =M\mu(N)^2+\sum{k=0}^{K-1}\Delta_k(N). \]
Proof. \(\sum_{k=0}^{K-1}D_k=\sum_{k=0}^{K-1}(P_k-P_{k+1})=P_0-P_K\) telescopes exactly, so \(x=P_0x=P_Kx+\sum_{k=0}^{K-1}D_kx\). Expand \(\|x\|_2^2\) by bilinearity: every cross term \(2\langle P_Kx,D_kx\rangle\) and \(2\langle D_kx,D_{k'}x\rangle\) (\(k\ne k'\)) that appears is exactly zero, by the Corollary applied to \(\langle u,Pv\rangle=\langle Pu,v\rangle=\langle Pu,Pv\rangle\) for any orthogonal projection \(P\) (so \(\langle P_Kx,D_kx\rangle =\langle P_KD_kx,x\rangle=0\), and similarly for the other cross term). What remains is the Pythagorean sum \(\|P_Kx\|_2^2+\sum_k\|D_kx\|_2^2\). Since \(P_Kx\equiv\mu(N)\) on all \(M\) coordinates, \(\|P_Kx\|_2^2=M\mu(N)^2\). \(\blacksquare\)
Multiplying by 2 and using \(\sum_{h=1}^Mx_h^2=\sum_{h=1}^Nx_h^2\) (the padding only adds zero terms) gives \((\star)\): \(E_{\rm corr}(N)=\mathrm{Energy}(N)\), for every \(N\), unconditionally. Grade: PROVED. The proof uses only that \(P_k\) are symmetric idempotent linear maps on \(\mathbb R^M\); it assumes nothing about the arithmetic content of \(x_h\) (not the prime number theorem, not a sieve bound, not GRH, not the existence or non-existence of an exceptional zero). It is exact for every finite \(N\), not asymptotic and not fitted.
On the mandatory retained cross terms. The proof above does not assume the cross terms \(\langle P_Kx,D_kx\rangle\) and \(\langle D_kx,D_{k'}x\rangle\) are small and discard them; it computes them and shows each is identically zero, from the projection algebra alone. This is the sense in which the cross terms between scales are retained: they are present in the natural expansion of \(\|x\|_2^2\) and are accounted for by a proof of exact vanishing, not by an estimate or an omission. Section 5 contrasts this with LOCALIZED_MIXED_ENERGY.md, where the analogous cross terms are between genuinely different objects and have not been shown to vanish or to be small.
4. Per-scale table at \(N=10^4\) and \(N=10^5\)
Computed by s8_candidate_energy.py, which builds \(x_h=r_N(h)-C_N(h)\) by reusing probe.py's von_mangoldt, singular_series and psi2_fft (see Section 6 for why \(C_N\equiv0\) in this run), then evaluates the identity of Section 3 directly (no shortcut, no fit). dominates and the relative gap in results_s8_candidate_energy.json confirm \((\star)\) to floating-point precision (\(|{\rm Energy}-E_{\rm corr}|/E_{\rm corr}\lesssim10^{-15}\) at every row) at all five computed \(N\) (2000, 5000, 10000, 30000, 100000); the two required by this section are shown below.
\(N=10^4\) (\(K=14\) scales, \(E_{\rm corr}=1.045773\times10^9\)):
| component | block length | value | share |
|---|---|---|---|
| mean (\(2M\mu^2\)) | — | \(2.612\times10^6\) | 0.25% |
| scale 0 | 1 | \(5.259\times10^8\) | 50.29% |
| scale 1 | 2 | \(2.701\times10^8\) | 25.83% |
| scale 2 | 4 | \(1.297\times10^8\) | 12.41% |
| scale 3 | 8 | \(6.527\times10^7\) | 6.24% |
| scale 4 | 16 | \(2.587\times10^7\) | 2.47% |
| scale 5 | 32 | \(1.007\times10^7\) | 0.96% |
| scale 6 | 64 | \(4.662\times10^6\) | 0.45% |
| scale 7 | 128 | \(4.287\times10^6\) | 0.41% |
| scale 8 | 256 | \(2.279\times10^6\) | 0.22% |
| scale 9 | 512 | \(1.087\times10^6\) | 0.10% |
| scale 10 | 1024 | \(2.200\times10^6\) | 0.21% |
| scale 11 | 2048 | \(3.617\times10^5\) | 0.03% |
| scale 12 | 4096 | \(2.603\times10^5\) | 0.02% |
| scale 13 | 8192 | \(1.104\times10^6\) | 0.11% |
\(N=10^5\) (\(K=17\) scales, \(E_{\rm corr}=2.234400\times10^{11}\)):
| component | block length | value | share |
|---|---|---|---|
| mean (\(2M\mu^2\)) | — | \(4.252\times10^8\) | 0.19% |
| scale 0 | 1 | \(1.117\times10^{11}\) | 49.97% |
| scale 1 | 2 | \(5.653\times10^{10}\) | 25.30% |
| scale 2 | 4 | \(2.720\times10^{10}\) | 12.17% |
| scale 3 | 8 | \(1.430\times10^{10}\) | 6.40% |
| scale 4 | 16 | \(6.722\times10^9\) | 3.01% |
| scale 5 | 32 | \(3.216\times10^9\) | 1.44% |
| scale 6 | 64 | \(1.454\times10^9\) | 0.65% |
| scale 7 | 128 | \(6.579\times10^8\) | 0.29% |
| scale 8 | 256 | \(3.262\times10^8\) | 0.15% |
| scale 9 | 512 | \(1.889\times10^8\) | 0.08% |
| scale 10 | 1024 | \(1.510\times10^8\) | 0.07% |
| scale 11 | 2048 | \(1.518\times10^8\) | 0.07% |
| scale 12 | 4096 | \(1.903\times10^8\) | 0.09% |
| scale 13 | 8192 | \(8.091\times10^7\) | 0.04% |
| scale 14 | 16384 | \(5.511\times10^7\) | 0.02% |
| scale 15 | 32768 | \(8.624\times10^7\) | 0.04% |
| scale 16 | 65536 | \(5.133\times10^7\) | 0.02% |
Grade: MEASURED (exact, for these five \(N\); not a theorem about the scale profile in general). The two tables show the same qualitative shape: about half of \(E_{\rm corr}(N)\) sits in the finest scale (block length 1, i.e. \(D_0x=x-P_1x\), adjacent-pair differences), and each successive octave holds roughly half of what the previous one held, down to a long, noisy tail of coarse scales each worth well under 1%, plus a mean term at the ~0.2% level. This is the profile of a sequence whose autocovariance decays quickly with lag — consistent with, but not a proof of, an unconditional square-root-cancellation-type input for \(x_h\); no such input is assumed or used in \((\star)\), and this paragraph asserts nothing beyond what the two tables show at these five \(N\).
5. Relation to LOCALIZED_MIXED_ENERGY.md
LOCALIZED_MIXED_ENERGY.md also uses a dyadic window-length scale, but on a different, Fourier-side object: its \(T_s(\theta)\) (its eq 1) is an integral over frequency \(\theta\) of a length-\(s\) window energy of \(W=F-H\), and its final budget (its eq 25) bounds \(\sqrt{E_{\rm corr}}\) — not \(E_{\rm corr}\) itself — by \(2\sqrt{\mathcal M}+\sqrt{\int|W|^4-D_w^2}+O(N^{3/2}e^{-c\ell^{1/10}})+O(N)\), where \(\mathcal M=\int|H|^2|W|^2\) is bounded, in turn, by exactly such a dyadic-\(s\) decomposition (its eqs 12-14, 19) of the mixed energy, a quantity auxiliary to \(E_{\rm corr}\), not equal to it.
Three differences follow:
- Identity vs. majorant. \((\star)\) here is an equality for \(E_{\rm corr}(N)\) itself. LOCALIZED_MIXED_ENERGY's eq 25 is a triangle-inequality upper bound on \(\sqrt{E_{\rm corr}}\) via a different quantity (\(\|G_{\rm corr}\|2\)) that is only within \(O(N)\) of \(\sqrt{E{\rm corr}}\) (its eq 24), and whose components \(X,Y,R_{\rm mod}\) are not orthogonal.
- The cross terms. LOCALIZED_MIXED_ENERGY's eq 22 keeps \(2\,{\rm Re}\langle X,Y\rangle\) and \(2\,{\rm Re}\langle X+Y,R_{\rm mod}\rangle\) explicitly and states "No sign or cancellation estimate ... has been proved here"; its Section 6 identifies this, together with the unresolved long-window estimate (its eq 20), as exactly why its attempt does not close. This document's cross terms (\(\langle P_Kx,D_kx\rangle\), \(\langle D_kx,D_{k'}x\rangle\)) are provably zero (Section 3), because \(P_K\), \(D_k\) are orthogonal projections of the same vector \(x\) — an algebraic fact, not an arithmetic one. The price of this is that \((\star)\) carries no arithmetic content at all: unlike LOCALIZED_MIXED_ENERGY's eqs 12-14 and 19, which do extract genuine (if insufficient) power savings from the prime number theorem and sieve estimates, \((\star)\) uses no fact about \(\Lambda\), \(\mathfrak S\), or \(C_N\) beyond their being some fixed real numbers.
- Scale variable. Both use dyadic length scales, but LOCALIZED_MIXED_ENERGY's \(s\) is the length of a Fourier-side window integrated over all \(\theta\) and partitioned again by rational denominator; this document's \(2^k\) is a block length applied once, directly to the index-domain vector \(x_h\), with no frequency integral or denominator partition. They are not the same decomposition of the same object, and \((\star)\)'s scale profile (Section 4) is not a restatement of LOCALIZED_MIXED_ENERGY's eq 26 budget in different notation.
Grade of this section: the equality \((\star)\) is PROVED; the comparison of proof strategies is a reading of both documents, not a new estimate. \((\star)\) does not shorten, weaken, or bypass the specific gap LOCALIZED_MIXED_ENERGY.md identifies (its eq 20, and the cross terms of its eq 22); that gap is about \(\mathcal M\), a quantity \((\star)\) never introduces.
6. Relation to EXCEPTIONAL_ENERGY.md
EXCEPTIONAL_ENERGY.md studies \(\mathcal A_{\rm exc}(N):=2\sum_{h=1}^N|C_N(h)|^2\) (its eq 17→2, called \(\mathcal A_{\rm exc}\) there for a specific exceptional \(q,\beta\)) and proves a closed-form lower bound for it (its eq 2) by restricting to an explicit shift set \(\mathcal H=\{h:N/4\le h\le N/2,\,2\mid h\}\) where the two linear correction terms vanish (its Section 3). That is a bound on a sub-sum over an explicit range of \(h\), not a decomposition by scale, and it is a one-sided estimate, not an identity.
The construction of Section 2 above is generic in the input vector: it applies verbatim with \(x_h:=C_N(h)\) in place of \(x_h:=r_N(h)-C_N(h)\), giving the same exact identity \(\mathcal A_{\rm exc}(N)=\mathrm{Energy}_C(N)\) for the correction energy alone, with its own mean term and its own \(K\) scale details built from \(C_N\). That identity is not evaluated numerically here (Section 7 explains why: no exceptional zero is assumed to exist, so \(C_N\equiv0\) and \(\mathcal A_{\rm exc}=0\) identically in every run below). Qualitatively, EXCEPTIONAL_ENERGY's shift set \(\mathcal H\) sits inside the middle third of \([1,N]\); in a dyadic block-average decomposition of \(C_N\), that band overlaps every scale from \(k=0\) up to \(k\approx K-2\) (a set of that width is not block-constant at any one scale), so its contribution is spread across \(\mathrm{Energy}_C\)'s scale components rather than isolated in one. This is a positional remark, not a proven correspondence between EXCEPTIONAL_ENERGY's lower bound (eq 2) and any specific \(\Delta_k\) of \(\mathrm{Energy}_C\).
Grade: the generic applicability of Section 2's construction to \(C_N\) alone is PROVED (it is the same theorem with a different input vector); the qualitative remark about \(\mathcal H\) is an OBSERVATION, not a derived bound. Nothing here strengthens, weakens, or numerically checks EXCEPTIONAL_ENERGY.md's eq 2 or eq 3.
7. Why the numerics use \(E_{\rm corr}=E\)
At every \(N\) evaluated (2000, 5000, 10000, 30000, 100000), this attempt does not assume, construct, or borrow a concrete odd/4-part/8-part exceptional conductor \(q<Z\) with an associated real zero \(\beta\); none is known to exist unconditionally. By CORRECTED_RH_BRIDGE.md Section 1, "If no such zero exists, set \(C_N(h)=0\) for every \(h\)", which is the only correction value available here without inventing exceptional data that would not itself be a proof of anything. With \(C_N\equiv0\), \(x_h=r_N(h)\) exactly, and \[ E_{\rm corr}(N)=2\sum_{h=1}^Nr_N(h)^2=E(N), \] the original CHHL quantity of probe.py. s8_candidate_energy.py therefore builds \(x_h\) via probe.py's von_mangoldt, singular_series, and psi2_fft, and cross-checks its own \(E_{\rm corr}\) against probe.py's E_of_N at every row (check_matches_probe_E_of_N_relerr in the JSON output, at floating-point precision throughout). Grade: this reduction is exact and forced by the definitions, not a simplifying choice; it is the reason "computing \(E_{\rm corr}\) directly by reusing probe.py's routes" is possible at all without a separate implementation of \(C_N(h)\).
8. Section 8's constraints, checked one by one
- Retains the mean/central mode. The \(2M\mu(N)^2\) term is exactly this; see Section 2.
- Retains the cross terms between scales. Computed and shown to be exactly zero, not assumed or dropped; see Section 3's closing remark.
- The changing correction data (\(q,\chi,\beta\) may vary with \(N\)). \(\mathrm{Energy}(N)\) is built from \(x_h=r_N(h)-C_N(h)\) as given at each \(N\), for whatever exceptional data apply there (Section 1); the proof of \((\star)\) uses no property of \(C_N\) beyond it being some fixed real sequence, so nothing breaks if \(q,\chi,\beta\) change from one \(N\) to the next.
- Sharp endpoint \(N-h\); no smoothing of the endpoint. Inherited unchanged from \(r_N(h)=\psi_2(N,h)-(N-h)\mathfrak S(h)\) (Section 1); the padding used to reach a power of 2 only appends zero coordinates strictly beyond \(h=N\) and never modifies \(x_h\) for \(h\le N\).
- No primes-only replacement. \(\Lambda\) includes proper prime powers throughout, exactly as in
probe.py'svon_mangoldtand in CORRECTED_RH_BRIDGE.md; no restriction tolam_p/theta_2-type primes-only sequences is used anywhere in this document or script.
All five hold. The candidate satisfies Section 8's constraints and proves exact domination unconditionally; the open item this document does not touch is the actual size of \(E_{\rm corr}(N)\), which the identity leaves exactly as large as it was.