The Quasi-Riemann Hypothesis

The proposed argument

The paper tries to contradict the existence of a rightmost possible zero in an entire Hecke L-function family. It constructs a completed cubic-theta sum with two exact representations: reflection gives a small direct bound, while Poisson summation exposes a principal Mellin signal containing the reciprocal of the target L-function. Bounds for the other rows would continue that reciprocal across the possible zeros. A balanced first stage claims a zero-free half-plane to the right of 11/12; prime compensation and two additional moment inductions claim 7/8. The full Hecke family is needed because Poisson introduces twists, even when the final target is zeta. These are the manuscript’s proposed steps, not a completed verification of its difficult inputs.

Reading order

1. 01

   §§1–2 — continuation

   Replace nonvanishing with a uniform reciprocal-continuation problem: a direct bound and a common power saving approximation force every reciprocal through the rightmost family zeros.

2. 02

   §§3–6 — base reflection

   Build one completed cubic-theta probe and bound it through its reflected form; balanced scales give the low Z^(1/4+epsilon) estimate.

3. 03

   §§7–11 — first high

   Poisson-expand the same probe, isolate its principal reciprocal signal, count troublesome nonprincipal rows through saturated inverse witnesses and the sextic sieve, then close 11/12.

4. 04

   §§12–16 — compensated low

   Insert independent selected prime slots and asymmetric scales; obtain a new reflected low bound and an exact holomorphic local correction with controlled errors.

5. 05

   §17 — marked inverse

   Prove the selected prime inverse moment by a canonical invariant, terminal reflected energy and two masked Poisson transforms; initialize it and amplify the no-slot estimate.

6. 06

   §18 — plain fourth

   Prove the plain-witness fourth moment by decreasing effective width; center equal-product-scale comparisons so fixed inducing-character main terms cancel.

7. 07

   §19 — refined counts

   Turn the two simultaneous witnesses and independent short prime slots into complementary inverse/plain row counts, with explicit slot capacity and supply margins.

8. 08

   §20 — final closure

   Balance the complementary counts, certify a small positive endpoint margin, reserve all losses before choosing target dependent heights, and apply continuation plus quadratic transfer.