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Zero-energy feasibility check for the CHHL upper-bound investigation

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Date: 2026-09-06. Prepared in this conversation.

Result and scope

Two deductions are written out below.

  1. A smoothed, frequency-localized fourth moment admits a direct zero-additive-energy estimate. Using even a coarse published energy bound gives an unconditional X^(5/2+epsilon) upper bound for this smoothed annulus. This is NOT a bound for the report's sharp-cutoff I_Q, Z_Q, M_1, or total E(N). No novelty or new record is claimed.
  2. An explicit finite Fourier kernel shows that the unsplit sharp q=1 intensity discrepancy itself retains the endpoint prime-counting obstruction. It was inaccurate to suggest that this obstruction appears only because the positive-moment decomposition discarded cancellation.

An elementary attempt to remove the exponential damping is recorded. It does not close the transfer to the report: taking absolute values in that transfer reintroduces the uncontrolled central-frequency moment.

Status: written mathematical derivations with finite identity checks. They have not received independent proof review or formal verification. The computations do not certify the asymptotic arguments. The original total-error upper-bound objective remains unchanged and unresolved by this check.

Source of the target

Read hunts/prime_pair_error/UPPER_BOUND.md from teal-sea/zeta-lab, content blob d7efa161f3e2c690d50868d622293f9304552d86, especially equations (18), (22), (23), and (28)--(31). Its target uses a sharp integer cutoff. Its reported N^(13/5) log^6 N estimates concern specific unsmoothed components, not the smoothed object introduced here.

1. Published inputs versus deductions in this note

Published inputs used:

The frequency change of variables, resulting annular estimate, and localization application are deductions written out here. The Fourier and complex-analytic tools are standard. A complete literature/priority search for these particular formulations has not been performed.

The published exceptional-set exponent is NOT substituted as a Fourier-moment exponent. The conversion below is derived directly for a different, explicitly defined integral.

2. A precise smoothed object

Write e(t)=exp(2*pi*i*t). For X>=4, let

z(alpha) = X^(-1) - 2*pi*i*alpha,
S_X(alpha) = sum_{n>=1} Lambda(n) exp(-n/X) e(n*alpha),
B_X(alpha) = S_X(alpha) - 1/z(alpha).

The prime series converges absolutely. This exponential damping differs from the sharp cutoff in the report.

For 2 <= T <= X/10, use the positive annulus

A(X,T) = [T/(2*pi*X), 2T/(2*pi*X)].

The negative annulus is its complex conjugate. In particular, T=sqrt(X) is the frequency scale alpha ~ X^(-1/2).

The contour identity gives

B_X(alpha) = -sum_rho Gamma(rho) z(alpha)^(-rho)
             - (zeta'/zeta)(0) + R_X(alpha),

R_X(alpha) << |z(alpha)|^(1/2)
             [1 + log^2(2 + 2*pi*X*|alpha|)].

The constant term is kept explicitly. This uses the contour representation rather than silently dropping its residue at zero. All powers use the principal logarithm; Re z>0.

For fixed X,alpha, Stirling and |arg z|<pi/2 give absolute convergence of the zero sum. On this annulus, zeros above height

H = T (log X)^2

may be removed with O_A(X^(-A)) error, for any fixed A, once X is sufficiently large. Indeed the modulus of a term is at most a constant times

(X/sqrt(T)) (1+|gamma|/T)^(1/2) exp(-c |gamma|/T),

and the unit-interval zero count is O(log(2+|gamma|)). Summing the tail starting at H gives exponential decay in log^2 X; it is uniform for the stated range of T. Taking a larger constant in H is harmless. This permits working with finite sums before expanding a fourth power.

3. The actual zero-energy transfer

Fix 1/2 <= sigma < sigma+eta <= 1. Let Z be any finite multiset of zeta zeros with

sigma <= Re rho <= sigma+eta,  |Im rho| <= H.

Write

G_Z(alpha) = sum_{rho in Z} Gamma(rho) z(alpha)^(-rho),
Energy_1(Z) = #{(rho1,rho2,rho3,rho4) in Z^4:
               |gamma1+gamma2-gamma3-gamma4| <= 1}.

Multiplicity is included. The following bound is obtained by the calculation below:

integral_{A(X,T)} |G_Z(alpha)|^4 d alpha
    << X^(4(sigma+eta)-1) / T * Energy_1(Z).                 (ZE)

The constant is uniform in X,T,H and in the particular zeros. It can depend on a fixed smooth majorizing cutoff. A finite set of bounded-height zeros only affects absolute constants.

3.1 Exact change of variables

Put

s = log(X |z| / T),
alpha(s) = sqrt(T^2 exp(2s)-1)/(2*pi*X),
theta(s) = arg z = -arccos(1/(T exp(s))).

Then

d alpha/ds = (T/(2*pi*X)) exp(2s)/sqrt(exp(2s)-T^(-2)),
theta'(s) = -1/sqrt(T^2 exp(2s)-1).

On any fixed smooth enlargement of the annulus, s ranges over an interval of bounded length, the Jacobian and its fixed-order derivatives are O(T/X), and theta's positive-order derivatives are O(1/T).

For rho=beta+i*gamma, exactly

Gamma(rho) z^(-rho) = b_rho(s) exp(-i*gamma*s),

b_rho(s) = Gamma(rho) (T/X)^(-rho)
           exp(-beta*s) exp(gamma*theta(s)) exp(-i*beta*theta(s)).

The gamma*s oscillation is now separated. This is the missing conversion between averaging in Fourier frequency and additive relations among zero ordinates.

Stirling gives, for every fixed derivative order j,

|d^j b_rho(s)/ds^j| <<_j X^(sigma+eta) / sqrt(T).           (S)

To check uniformity, for |gamma|<=T and beta>=1/2, the additional factor is at most (abs(gamma)/T)^(beta-1/2) <= 1, apart from bounded-height constants. For larger heights, derivatives introduce powers of abs(gamma)/T but these are absorbed by exp(-c abs(gamma)/T). Opposite signs of gamma have stronger, not weaker, damping on one side. No RH, spacing, simplicity, or random-phase assumption is used.

3.2 Four-zero kernel

Choose a fixed nonnegative smooth cutoff in 2*pi*X*alpha/T, supported in [1/2,4] and at least one on [1,2]. Expand the fourth power after inserting that cutoff.

Each four-zero kernel is the integral of

b1(s) b2(s) conjugate(b3(s) b4(s))
* exp(-i*(gamma1+gamma2-gamma3-gamma4)*s)
* cutoff(s) * (d alpha/ds).

Two integrations by parts, using (S) and the compact support, bound its modulus by

C X^(4(sigma+eta)-1) / T
  * [1+|gamma1+gamma2-gamma3-gamma4|]^(-2).                 (K)

There are no boundary terms. The integral over the original annulus is bounded above by this smoothly majorized fourth moment.

For completeness, the soft energy is controlled by the unit energy without a spacing assumption. Let b_m count ordered pair sums gamma1+gamma2 in [m,m+1). Then

sum_{quadruples} [1+|gamma1+gamma2-gamma3-gamma4|]^(-2)
 <= (1+pi^2/3) sum_m b_m^2
 <= (1+pi^2/3) Energy_1(Z).

For bins whose indices differ by j!=0, use the weight bound 1/j^2, then Cauchy--Schwarz on sum_m b_m b_(m+j). Pairs of sums in the same bin differ by less than one. This proves (ZE).

3.3 Inserting a published energy estimate

By inclusion and the definition of A*, for fixed sigma and any nu>0,

Energy_1(Z) <= N*(sigma,H)
            <<_(sigma,nu) H^((1-sigma) A*(sigma)+nu).

With H=T log^2 X, (ZE) therefore gives, up to logarithmic factors and an arbitrarily small exponent allowance,

integral_{A(X,T)} |G_Z|^4
 << X^(4(sigma+eta)-1)
    T^((1-sigma) A*(sigma)-1+nu).                          (ZE-power)

The factor T^(-1) is part of this directly derived frequency kernel. It is not obtained by transplanting the exceptional-set formula.

At T=sqrt(X), the narrow-strip exponent is

b(sigma) = 4*sigma - 3/2 + (1/2)(1-sigma) A*(sigma).

Even the coarse published inputs

A*(sigma) <= 3 A(sigma),   A(sigma) <= 30/13

give

b(sigma) <= 51/26 + (7/13)*sigma <= 5/2.

Divide [1/2,1] into a fixed, sufficiently fine number of strips, depending on the final epsilon, and use the triangle inequality in L4. The strip-width losses and the logarithms are absorbed in that epsilon. No uniform estimate for an infinitesimally moving sigma is assumed.

For zeros with beta<=1/2, Stirling instead gives the derivative bound O_j((X/T)^(1/2)); the unit-interval zero count gives total additive energy O(H^3 log^4 H). Their contribution at T=sqrt(X) is O_epsilon(X^(2+epsilon)). The contour remainder and the retained constant are smaller still.

Consequently the calculation gives the restricted estimate

integral_{1/(2*pi*sqrt(X))}^{1/(pi*sqrt(X))}
  |sum_{n>=1} Lambda(n) exp(-n/X) e(n*alpha)
    - 1/(X^(-1)-2*pi*i*alpha)|^4 d alpha
       <<_epsilon X^(5/2+epsilon).                        (SMOOTH)

This is unconditional and uses no newest optimized envelope: the coarse energy bound already suffices. It is a derivation for a smoothed annulus, not a claim of a new theorem in the literature or an improvement to the report's total error.

4. The unsplit sharp q=1 term retains the endpoint obstruction

Return to the report's actual sharp sums

F_N(alpha) = sum_{n=1}^N Lambda(n)e(n*alpha),
K_N(alpha) = sum_{n=1}^N e(n*alpha),
D_N(alpha) = |F_N(alpha)|^2 - |K_N(alpha)|^2,
Q = floor(sqrt(N)/3),
M1(N,Q) = integral_{|alpha|<=Q/N} |D_N(alpha)|^2 d alpha,
R(N) = psi(N)-N.

The integral is on the circle represented by [-1/2,1/2]. For N>=36, Q>=1.

4.1 An explicit reproducing kernel

Define

Dir_{3N}(alpha) = sum_{|j|<=3N} e(j*alpha),
kappa_N(alpha) = Dir_{3N}(alpha) * |K_N(alpha)/N|^4.

The Fourier coefficients of |K_N/N|^4 are nonnegative, supported on |j|<=2N-2, and have total sum one. Therefore the coefficients of kappa:

Since D_N has degree at most N-1, this gives the exact identity and bound

D_N(0) = integral_T D_N(alpha) kappa_N(alpha) d alpha,
||kappa_N||_2^2 <= 10N-3.

For 0<|alpha|<=1/2,

|kappa_N(alpha)| <= 1/(32 N^4 |alpha|^5).

This follows from the usual geometric-sum formula and sin(pi |alpha|)>=2|alpha|. Consequently

integral_{|alpha|>Q/N} |kappa_N(alpha)| d alpha <= 1/(64 Q^4).

Also |D_N(alpha)|<=psi(N)^2+N^2, since Lambda is nonnegative. Applying Cauchy--Schwarz only on the selected arc gives the explicit finite inequality

|psi(N)^2-N^2|
 <= sqrt((10N-3) M1(N,Q)) + [psi(N)^2+N^2]/(64 Q^4).      (LOC)

No prime-pair conjecture enters this proof. Using Q~sqrt(N) and Chebyshev's psi(N)<<N, the last term is O(1). Even the elementary psi(N)<=N log N gives O(log^2 N), which suffices.

As psi(N)+N>=N, (LOC) implies

|R(N)| <= sqrt(10 M1(N,Q)/N) + O(log^2 N/N).

Thus M1(N,Q)<<_epsilon N^(2+epsilon) for every epsilon would itself force the classical RH-strength bound on R(N).

Correction to the earlier framing: keeping |F|^2-|K|^2 intact can matter, but it does not eliminate the one-point obstruction. That obstruction is already present in this unsplit local expression. This is not an impossibility result: proving that bound is an RH-strength task, which is the original research ambition.

5. Attempted transfer back to the sharp cutoff

The difference between damping and cutting off cannot be ignored. There is an exact periodic de-smoothing formula. Let

B_disc,N(alpha) = sum_{n>=1} (Lambda(n)-1) exp(-n/N) e(n*alpha),
H_N(beta) = sum_{n=1}^N exp(n/N) e(n*beta).

Then

F_N(alpha)-K_N(alpha) = integral_T B_disc,N(alpha-beta) H_N(beta) d beta. (DS)

On the small annulus, B_disc,N differs from the object in (SMOOTH) by O(1), because its discrete baseline is 1/(exp(z)-1)=1/z+O(1).

The elementary geometric-series formula gives

|H_N(beta)| << min(N,1/||beta||),   ||H_N||_1 << log(2N).

Applying the direct absolute-value/Jensen estimate to (DS), for an annulus A, gives

integral_A |F_N-K_N|^4
 <= ||H_N||_1^3 integral_T |B_disc,N(u)|^4
       [integral_A |H_N(alpha-u)| d alpha] du.              (DS-abs)

For A=[delta,2delta], delta~N^(-1/2), and |u|<=1/N, the bracket is of constant order, not a negative power of N. In fact the numerator of H's geometric sum is bounded below by e-1, and its denominator has size comparable to delta there. Thus the available bound pays for the uncontrolled central-frequency fourth moment even when the output annulus is away from zero.

The annular estimate (SMOOTH) does not control that central moment. This elementary de-smoothing attempt therefore does not preserve the proposed gain. It is an identified failure of this inequality, not a theorem that no sharp-cutoff transfer can be proved.

There are also two independent scope gaps:

6. Computational checks and disposition

check_feasibility.py was executed successfully and writes checks.json.

It verifies:

The gamma-term reconstruction has relative error below 8e-14; the independently integrated change of variables below 5e-15. These are floating-point diagnostics, not outward-rounded certificates. The reproducing-kernel coefficient tests use integer numerators and exact integer equalities.

What can be carried back to the lab

What must not be claimed