Categorical Microcosm
About
What if quantum mechanics, particles, atoms, and chemistry were not postulates—but consequences of geometry?
Book IV turns the Panta Rhei program toward the microcosm: quantum mechanics, the particle spectrum, atomic structure, forces, and chemical bonding. Its core thesis is bold and sharply stated: microphysics emerges from the fiber τ² in the canonical fibration
τ³ = τ¹ ×₍τ²
with zero free parameters. The same calibration constant introduced earlier, ιτ = 2/(π + e), acts as the single numerical anchor.
The book develops a unified chain:
Stage → Quantum → Particles → Atoms → Forces → Molecules
Quantum mechanics from geometry
Quantum structure is derived from the topology and Fourier duality of the circle factors in T². In this framing:
- canonical commutation and uncertainty follow from sampling/duality on S¹,
- Schrödinger evolution is written in terms of a universal calibrated operator,
- measurement is reinterpreted as boundary sampling on the lemniscate screen—emphasizing inference over “collapse.”
Particles as character modes
Particles appear as character modes on the lemniscate boundary. A striking structural consequence is a topological three-generation pattern: distinct supports on the carrier distinguish three generations—and rule out a fourth by a capacity constraint.
Atoms, spectra, and a τ-periodic table
Atomic shells are described as winding modes (k₁, k₂) ∈ ℤ² on nested tori. From these winding constraints the book proposes:
- a τ-periodic table driven by topology rather than orbitals,
- selection rules and spectral lines as character transitions,
- fine-structure effects controlled by the same calibration.
Forces and couplings (derived)
Interactions are treated as intertwiners of character structure and holonomy on T², with gravity reserved for the base dynamics in Book V. The book presents derived coupling expressions—most notably a proposed derivation of the fine-structure constant from ιτ.
Chemistry as topology
Molecules are modeled as typed graphs whose geometry arises from holonomy minimization and winding exchange—offering a conceptual “chemistry without orbitals” viewpoint: reactions become graph transformations and barrier changes rather than classical trajectories.
Book IV closes by stating the Hermetic Principle of the series:
- As below (Book IV): physics in the fiber T²
- As above (Book V): physics along the base τ¹
- Together: a single framework intended to cover “all physics” across microcosm and macrocosm.
“Zero free parameters. One constant ιτ. All microcosm physics.”
Free reader downloads: To get a fast, high-signal overview of this volume, we provide two PDFs extracted from the original published pages of the book: the Table of Contents (see the full structure at a glance) and the Q&A Appendix (a reader’s guide to key ideas and common questions). Both are free to view and share for review/academic reference.
New to the series? Start with the Q&A Appendix; use the TOC to choose your entry points.