V149_1 — Exact Decomposition of ζ′/ζ and the Limits of Cauchy-Packet Models

Description:
This paper gives an exact finite-packet decomposition of the logarithmic derivative of the zeta function and clarifies the limits of standard Cauchy-packet models. It shows that packet estimates imply analytic zero-exclusion only after the regular remainder and shifted-packet problem are also controlled.

🔹 Exact Decomposition of ζ′/ζ

Using the completed zeta function and Hadamard factorization, the logarithmic derivative is split into a finite singular zero-packet part and a regular remainder.

This gives an exact bridge between the analytic zeta-function energy and packet Gram forms.

🔹 Finite Zero-Packet Part

The finite zero-packet term contains the singular contributions from zeros up to a chosen height.

These are the terms that produce Gram-type packet energies on vertical lines.

🔹 Regular Remainder

The remainder contains compensation terms, omitted zero tails, the Hadamard constant, pole terms, and the gamma factor.

This part is not controlled by Cauchy-packet estimates and must be bounded separately.

🔹 Weighted Energy Identity

The vertical energy decomposes into packet energy, remainder energy, and a cross term.

Thus model packet estimates alone are not enough; the full analytic energy is controlled only when both packet and remainder energies are controlled.

🔹 Shifted Packet Structure

Standard Cauchy packets arise directly only for critical-line zeros.

For possible off-critical zeros, the exact packets are shifted horizontally.

Therefore a non-circular RH route must estimate shifted zero-packet Gram kernels without assuming all zeros are already on the critical line.

🔹 Finite-Strip Zero Exclusion

If both the packet energy and the regular remainder energy are finite on a finite strip, then the actual logarithmic-derivative energy is finite.

This excludes zeros on that finite vertical strip.

🔹 Logarithmic Low-Strip Consequence

On logarithmic low strips, the bridge gives logarithmic-cone zero exclusion.

This does not imply RH by itself because possible high-frequency off-critical zeros remain outside the cone.

🔹 Limits of Cauchy-Packet Models

The standard Cauchy-packet model should be interpreted as a critical-line packet model.

It cannot by itself give a non-circular proof of RH unless the shifted-packet problem and regular remainder are also addressed.

🔹 Conclusion
V149_1 identifies the exact analytic bridge and the exact limitation: Cauchy-packet Gram estimates are useful, but incomplete. A genuine RH route must next control shifted zero-packet Gram kernels, the regular remainder, omitted zero tails, and the high-frequency region.