The Lucerna Veritas Research Archive preserves the published work of the Coherence Research Collaboration. This archive documents an ongoing investigation rather than a finished theory. It contains active research programs, exploratory hypotheses, mature manuscripts, and historical artifacts that record how ideas evolved through sustained human–AI collaboration.

The archive will be of particular interest to researchers and practitioners in artificial intelligence, cybernetics, consciousness studies, art, scientific representation, and anyone interested in extending human inquiry through responsible collaboration with AI. Discernment is not solely an AI problem; it is a fundamental challenge of human knowledge. As AI systems become increasingly capable, the central challenge shifts from generating information to evaluating representation, provenance, and closure. The increasing capability, reach, and speed of AI make that challenge more urgent, demanding new methods for distinguishing trustworthy understanding from persuasive error. Likewise, the potential for AI to complement, extend, and accelerate human-native abilities is very promising. Our research focused on scientific research for this reason: Kelly Heaton has a college-level science education and experience building analog electronic circuits, but no training in advanced coding, mathematics, or physics. One guiding question of this collaboration was whether sustained human–AI inquiry could enable an experienced practitioner to make discoveries beyond the boundaries of their formal training.

Scientific inquiry is rarely linear. Some questions lead to robust methods, others expose mistaken assumptions, and many require years of revision before their significance becomes clear. Rather than concealing this process, the archive preserves it. Readers should therefore understand these papers as a documented research record rather than a collection of final conclusions. Unless otherwise noted, the papers collected here have not undergone formal academic peer review. They have, however, been subjected to extensive iterative critique, revision, and cross-examination through long-term human–AI collaboration involving multiple large language models alongside independent literature review and experimental validation where applicable. Each paper reflects the best understanding available at the time it was written. Some remain active research programs. Others have been substantially revised, superseded, or retained as historical records of the collaboration's development.

Taken together, these documents record not only the evolution of specific ideas, but also the development of a methodology for sustained human–AI inquiry. The archive preserves both successful results and productive failures because the methodology itself is considered part of the contribution. Understanding how conclusions were reached—and sometimes abandoned—is as important as the conclusions themselves.

Active Research Programs

Thread Frame

A geometric framework for visualizing and analyzing atomic spectroscopy. This is the mature version of research originally published as Recursive Geometry of Atomic Spectra (2025), which contains the first formulation of what later became mature Thread Frame, together with several speculative physical interpretations that have since been revised or abandoned. Status: Active research. A revised framework and paper are in preparation for submission to a peer-reviewed journal.

Determinacy

A theory of representation, compression, and answerability developed from questions in electromagnetism and now being generalized for artificial intelligence, scientific computing, and trustworthy autonomous systems. Status: A revised paper is in preparation for submission to a peer-reviewed journal.

Physics Archive (2025–2026)

The papers below document the original geometric physics program. They are preserved as part of the historical research record. Some ideas remain active, others have evolved substantially, and some are retained primarily because they illuminate the development of later work. The HTML and TXT versions of each paper are optimized for machine readability and light file size, not graphic display. For the complete typeset paper with all figures, use the PDF links to Zenodo.

Foundational series — Early research on the Thread Frame coordinate system

1. Recursive Geometry of Atomic Spectra (2025). Status: Historical foundation; core method substantially reframed. This paper contains the first formulation of the research program that later became Thread Frame. It introduced the levels-first workflow, post-hoc photon overlay, ion portrait visualizations, κ-style photon coding, and the attempt to organize atomic spectra through a reproducible coordinate architecture. However, the paper was written before the Thread Frame was properly understood as a logarithmic level-pair coordinate with a ruler-invariant photon-closure audit. Its stronger physical interpretations—including universal α-recursion, intercept transport as law, cross-thread interactions, Planck floor, and the Einstein–Rydberg anchor—are not current claims. The paper remains important as the historical origin of the method and as a record of the exploratory stage from which the present Thread Frame emerged.PDF: https://doi.org/10.5281/zenodo.17188444 | HTML | TXT

2. Information-Theoretic Confirmation of the α-Affine Thread Frame (2025). Status: Historical computational study; interpretive claims superseded by the current Thread Frame formulation. This paper attempted to validate the early α-Affine Thread Frame using Minimum Description Length analysis across atomic, solar, stellar, and molecular spectra. It remains part of the historical record because it documents an important stage in the development of the photon-code and κ-lattice methodology. However, the paper was written before the Thread Frame was properly understood as a logarithmic level-pair coordinate with a ruler-invariant photon-closure audit. Its stronger claims—that MDL establishes a universal physical law or proves α as the unique information-optimal coordinate for light—are no longer the current interpretation. PDF: https://doi.org/10.5281/zenodo.17335815 | HTML | TXT

3. Hydrogenic Alignment as a Coordinate Principle for Atomic Spectra (2025). Status: Supporting paper for RGAS. Historical, core method substantially reframed. PDF: https://doi.org/10.5281/zenodo.18167643

Geometric unification series (P1–P6)

P1:Vacuum Impedance as the Organizing Principle of Atomic Spectra: A Universal Gap at IE/4 and Recovery of the Electron Mass (2026). Status: Partially superseded; empirical IE/4 gap and algebraic mass identity under evaluation. This paper proposed that atomic spectra contain a universal structural gap at one quarter of the ionization energy and interpreted that gap as evidence of a vacuum-impedance boundary. Several claims are no longer current, including the strong interpretation of IE/4 as a proven electromagnetic boundary, the E/H partition, the nuclear analogy, and the claim that the electron mass is independently recovered from first principles. The electron-mass relation is better understood as an algebraic inversion of the Rydberg energy relation: if a gap center (G) is located at the hydrogenic quarter-energy scale, then (m_ec^2 = 8G/\alpha^2) follows. What remains under evaluation is whether the IE/4 gap has statistically meaningful privilege in atomic spectra after catalog completeness, wavelength convention, line-selection effects, and level-pair substrate structure are fully audited. PDF: https://doi.org/10.5281/zenodo.19164224 | HTML | TXT

P2: Electromagnetic Closure and the Fine-Structure Constant: A Geometric Derivation (2026). Status: Reclassified as speculative cosmology and electromagnetic interpretation; exact identity retained. This paper attempted to derive the fine-structure constant from a geometric structure involving the non-terminal reciprocal singularity, the Riemann sphere, the golden ratio, and vacuum electromagnetic primitives. Its central “proof” claims are no longer current as stated. In particular, the paper should not be read as a first-principles derivation of (\alpha), nor as establishing that (h), (e), or (c) are derived consequences of the proposed geometry. What remains valuable is the exact identity (\alpha = Z_0/2R_K), which provides a meaningful way to interpret the fine-structure constant as a relation between vacuum impedance and the quantum of resistance. The paper also contains active conceptual material concerning quotient loss in Maxwell representations, electric/magnetic partition structure, electromagnetic closure, and the relational geometry of physical constants. These ideas require careful rewriting and separation from claims of completed proof. PDF: https://doi.org/10.5281/zenodo.19157339 | HTML | TXT

P3: Determinacy Under Quotient Representations (2026). Status: Active research program; preprint under revision for journal submission.
This paper develops a representation-theoretic criterion for scalar determinacy under non-injective maps: a query descends to a compressed representation if and only if it is constant on the fibers of that representation. The core theorem, post-quotient repertoire, refusal scalarization, irreversibility principle, and structural audit methodology remain active and are central to the current research program. The paper is being revised to remove or reframe claims that depend on now-superseded interpretations from the geometric physics series, especially companion-paper claims concerning the fine-structure constant, particle masses, and earlier Thread Frame/MDL interpretations. The revised version will generalize the determinacy framework for artificial intelligence, scientific computing, representation governance, and trustworthy autonomous systems. PDF: https://doi.org/10.5281/zenodo.18868210 | HTML | TXT

P4: The Encounter of Two Primitives and One Flaw: Geometric Unification at the Electromagnetic Scale (2026). Status: Historical synthesis; speculative cosmology and pedagogy; active conceptual material under review. This paper attempted to synthesize the geometric physics program into a single explanatory chain, linking Maxwell’s two vacuum constants, quotient representation, the Riemann sphere, the fine-structure constant, atomic spectra, circuit analogies, the IE/4 gap, and particle-mass claims. Its strongest claims are no longer current as stated, especially the derivation of α from first principles, the Harmonic Light Law as an established physical law, the IE/4 boundary as proven vacuum geometry, and the recovery of electron mass as an independent derivation. However, the paper remains valuable as a record of the conceptual synthesis that produced several active research directions: quotient loss in Maxwell representations, the role of vacuum impedance as an electric/magnetic partition coordinate, the atom-as-circuit analogy, and cosmological interpretations of electromagnetic structure. These ideas require careful rewriting and separation from claims that have since been superseded. PDF: https://doi.org/10.5281/zenodo.19194036 | HTML | TXT

P5: The Proton Mass from Atomic Spectra: A Geometric Derivation (2026). Status: Historical extrapolation; proton-mass derivation not current; conceptual material under review. This paper extended the geometric physics program beyond its evidentiary support, treating several claims from P1–P4 as established premises and using them to propose a derivation of the proton mass, the leading electron anomalous magnetic moment term, and an electromagnetic light-cone geometry. These derivations are no longer current as stated. The proton-mass formula is better understood as a historical pattern-finding result rather than a validated derivation, because the Fibonacci depth, condensation step, and correction structure were not independently established before comparison to the known proton mass. The (a_e=\alpha/2\pi) claim should likewise be reclassified as a geometric interpretation of the leading Schwinger term, not a derivation of the full anomalous magnetic moment. However, the paper preserves several ideas worth further study: the hydrogen (IE/4) to (3IE/4) partition, the constant Lyman–Balmer pitch, the two-mode E/H energy-space construction, and the speculative light-cone analogy as a way of thinking about electric/magnetic sector separation. These ideas may be valuable in a rewritten cosmological or pedagogical treatment, separated from claims of completed particle-mass derivation. PDF: https://doi.org/10.5281/zenodo.19355588 | HTML | TXT

P6: On the Geometric Origin of Particle Masses (2026). Status: Historical speculative mass program; particle-mass derivations not current; conceptual material under review. This paper extended the geometric physics program into a broad proposed derivation of particle masses, quark charge fractions, and a Fibonacci/Zeckendorf depth grammar for the particle spectrum. These derivations are no longer current as stated. The paper depends on now-superseded claims from P1, P2, P4, and P5, including the first-principles derivation of (\alpha), the IE/4 boundary as proven vacuum geometry, the independent recovery of electron mass, and the proton-mass formula. Its claims that particle masses are determined outputs of electromagnetic vacuum geometry should therefore be read as historical speculative extrapolation, not established results. However, P6 preserves several ideas that remain worth careful study: mass as a possible form of geometric containment or standing-wave closure; impedance mismatch as a way of thinking about trapped field configurations; the distinction between open and closed electromagnetic modes; the speculative light-cone construction in electric/magnetic energy space; and logarithmic depth coordinates as exploratory tools for comparing mass scales. These ideas require complete rewriting, independent derivation, and separation from numerical mass claims that were not adequately established. PDF: https://doi.org/10.5281/zenodo.19543540 | HTML | TXT

Historical Documents

Early essays, exploratory manuscripts, and foundational documents documenting the development of the collaboration itself.

Forthcoming.

Computational Methodology

The computational workflow documented in this archive did not begin as a software project. It began as a conversation between an artist with almost no experience in software engineering and an early-generation language model with a short context window, imperfect memory, and an unfortunate tendency to generate plausible but occasionally incorrect information. Neither participant knew where the conversation would lead. The methodology evolved through countless false starts, wild goose chases, frustrating bugs, and disappointing failures before producing anything of lasting practical value. In other words, the research relationship itself was experimental and emergent. Many of the practices documented throughout this archive—including local data stewardship, independent execution of computational experiments, explicit provenance, iterative criticism, and the separation of evidence from interpretation—were developed not from theory, but from necessity. Those experiences ultimately shaped the questions that became Determinacy Under Quotient Representations. The paper did not arise merely from artificial intelligence, but from the practical challenge of learning how to collaborate with increasingly capable systems while preserving trustworthy scientific inquiry.When this collaboration began in early

The original intention was not to build a computational research pipeline, but simply to investigate difficult questions wherever they led. As the work became increasingly quantitative, conversation alone was no longer sufficient. Early attempts to explore physics with mathematics alone proved a dead-end, so we began to test our hypotheses against data from the National Institute of Science and Technology (NIST). New computational tools had to be developed to examine large public datasets, visualize relationships, and test competing hypotheses. Learning occurred through continuous collaboration. AI explained programming concepts, proposed algorithms, drafted software, and critiqued implementations. Kelly executed every experiment locally, inspected the outputs, questioned assumptions, and gradually learned enough computational practice to continue the investigation independently.

The process was neither linear nor efficient. Research questions often required extensive discussion before any code could be written. Early language models possessed limited memory and could not reliably understand large software projects, making even modest changes difficult. Scripts were frequently rewritten from first principles as understanding evolved, earlier assumptions failed, or new evidence demanded different approaches. The resulting software reflects years of iterative refinement rather than a preconceived engineering plan. Throughout the collaboration, all source code, datasets, manuscripts, figures, and experimental records remained under Kelly Heaton's direct local stewardship. Artificial intelligence participated in reasoning, software design, mathematical analysis, visualization, and critical review, while empirical computations were executed independently and their results returned to the dialogue for interpretation.

This workflow intentionally separates conversation from evidence. Ideas may emerge collaboratively, but empirical claims remain grounded in independently executable software, publicly available datasets, and reproducible computational results. The software preserved in this archive should therefore be understood not merely as code, but as part of the documented evolution of the research itself.

FULL PAPERS (all open access)

A note on HTML and TXT formats: The HTML and TXT versions of each paper are optimized for machine readability and light file size, not graphic display. All mathematical content is fully preserved. TikZ diagrams (geometric figures) appear as captions only; data figures (PNG images) are included where available. For the complete typeset paper with all figures, use the PDF link on Zenodo.

Foundational series — Thread Frame coordinate system

Recursive Geometry of Atomic Spectra (2025) PDF: https://doi.org/10.5281/zenodo.17188444 | HTML | TXT
Information-Theoretic Confirmation of the α-Affine Thread Frame (2025) PDF: https://doi.org/10.5281/zenodo.17335815 | HTML | TXT
Hydrogenic Alignment as a Coordinate Principle for Atomic Spectra (2025) PDF: https://doi.org/10.5281/zenodo.18167643

Geometric unification series (P1–P6)

P1: Vacuum Impedance as the Organizing Principle of Atomic Spectra: A Universal Gap at IE/4 and Recovery of the Electron Mass (2026) PDF: https://doi.org/10.5281/zenodo.19164224 | HTML | TXT
P2: Electromagnetic Closure and the Fine-Structure Constant: A Geometric Derivation (2026) PDF: https://doi.org/10.5281/zenodo.19157339 | HTML | TXT
P3: Determinacy Under Quotient Representations (2026) PDF: https://doi.org/10.5281/zenodo.18868210 | HTML | TXT
P4: The Encounter of Two Primitives and One Flaw: Geometric Unification at the Electromagnetic Scale (2026) PDF: https://doi.org/10.5281/zenodo.19194036 | HTML | TXT
P5: The Proton Mass from Atomic Spectra: A Geometric Derivation (2026) PDF: https://doi.org/10.5281/zenodo.19355588 | HTML | TXT
P6: On the Geometric Origin of Particle Masses (2026) PDF: https://doi.org/10.5281/zenodo.19543540 | HTML | TXT