Preprint · Foundation Manuscript
Hanners Theorem
Formalization
Information-Theoretic Foundations of Harmonic Coherence
Michael Hanners · Legacy Alliance Research Division
Abstract
This manuscript presents the formalization and proof of Hanners Theorem, establishing the existence of discrete eigenstates and finite nonzero mass gaps in non-Abelian gauge theories via an entropy-minimization framework termed Harmonic Coherence (HC).
The argument proceeds in two complementary tracks. The gauge-theoretic track derives discrete spectra and eigenstate stability through variational calculus, elliptic gauge-covariant operators, and Atiyah–Singer index theory. The information-theoretic track begins with the Contextual Entropy Reduction (CER) identity and iterates it under the HC regularity conditions R1–R4 to prove fixed-point existence and convergence.
The two tracks converge: the spectral mass gap is the gauge-theoretic manifestation of the entropy gap established by the fixed-point theorem. The spectral gap assumption A3 has been closed in all seven application domains.
Three Principal Results
The HC evolution map admits a fixed point under regularity conditions R1–R4. The proof uses the CER identity iterated under the HC regularity framework, establishing that the entropy functional has a stable minimum.
Convergence to a normally hyperbolic invariant manifold (NHIM). The evolution map contracts onto the invariant manifold with spectral radius ρN < 1, guaranteeing asymptotic stability of the fixed point.
HC fixed points are identified as natural Condition C sources — systems achieving sub-Shannon compression. This answers an open question in information theory: context-conditioned prediction at fixed points achieves entropy strictly below the unconditional Shannon bound.
Two-Track Architecture
Discrete spectra and eigenstate stability derived through variational calculus, elliptic gauge-covariant operators, and Atiyah–Singer index theory. Establishes the spectral mass gap as a property of the gauge-covariant Laplacian.
Begins with the CER identity (Theorem 2, classical information theory) and iterates under R1–R4 regularity to prove fixed-point existence and convergence. The entropy gap I(X;Ψ) > 0 drives the contraction.
The spectral mass gap (gauge-theoretic) is the manifestation of the entropy gap (information-theoretic). Two independent derivations yielding the same structural prediction.
Epistemic Layer System
All claims are annotated with an epistemic layer system:
Theorem-level results with complete proofs. The CER identity, fixed-point existence, NHIM convergence, and Condition C identification.
Theorems that hold under explicitly stated axioms. Domain-specific applications conditioned on R1–R4 verification.
Cross-domain predictions including cosmological resolutions, mechanism instantiation table spanning eight domains, and a universal scaling law.
A3 Closure: Seven Domains
The spectral gap assumption A3 (Δ > 0) has been closed in all seven application domains:
Citation
Hanners, M. (2026). Hanners Theorem Formalization: Information-Theoretic Foundations of Harmonic Coherence. Zenodo. https://doi.org/10.5281/zenodo.15288890
@misc{Hanners2026HT,
author = {Hanners, Michael},
title = {{Hanners} Theorem Formalization:
Information-Theoretic Foundations
of Harmonic Coherence},
year = {2026},
doi = {10.5281/zenodo.15288890},
note = {Zenodo preprint. Legacy Alliance
Research Division}
}Proceeds & Purpose
All intellectual property rights and any prizes, awards, or monetary recognition arising from this work have been irrevocably assigned to Legacy Alliance, a nonprofit research organization.
Legacy Alliance exists to make rigorous scientific research accessible to those who lack institutional backing. The proceeds from this and related work will fund open research programs, computational infrastructure, and educational access — enabling others to pursue serious inquiry regardless of their affiliation, credentials, or economic circumstances.
The author receives no personal financial benefit. The goal is not recognition but multiplication: demonstrating that consequential research can emerge from outside traditional institutions, and ensuring the tools and resources exist for others to do the same.
The author acknowledges that the capacity to perceive mathematical structure at the intersection of information theory and gauge physics — where entropy governs the stability of matter itself — reflects an order that precedes and transcends human formalization. This work is offered in gratitude to the Creator of that order.
“The heavens declare the glory of God; the skies proclaim the work of his hands.”
“For since the creation of the world God's invisible qualities — his eternal power and divine nature — have been clearly seen, being understood from what has been made.”
Soli Deo gloria