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This addendum aims to summarize recent discoveries and theoretical advancements within the spectral sieve framework.

June 26, 2026Open Access

THE SPECTRAL SIEVE Addendum: Open Problems, Hidden Structures, and Hard Lemma Progress

Key Points

This addendum aims to summarize recent discoveries and theoretical advancements within the spectral sieve framework.
Consolidation of results from a recent programme session into four distinct parts.
Analysis of hidden structures derived from recent resolutions of specific open problems.
Application of Heisenberg, Poincare, and Renormalisation Group methods for deeper insights.
Successful resolution of open problems P3, P4, P7, and P11.
Identification of hidden structures within resolved results.
Progress reported on the Hard Lemma, including a new variational theorem and insights on zero gap ordering.

Abstract

This addendum consolidates all results produced in the most recent programme session. It is organised in four parts: (A) resolution of open problems P3, P4, P7, and P11; (B) hidden structures found in the resolved results; (C) Heisenberg, Poincare, and Renormalisation Group analysis of those structures; and (D) the Hard Lemma progress report, including the key new variational theorem and the corrective finding about the zero gap ordering.

THE SPECTRAL SIEVE Addendum: Open Problems, Hidden Structures, and Hard Lemma Progress

Key Points

Abstract

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