Unified Physics, E₈ Geometry, and Foundational Logic
A unified research program exploring mathematical self-consistency in physical law — from the Fixed-Point Paradox in logic to E₈ projection geometry in quantum field theory.
This research establishes the epistemological ground for investigating self-consistent systems: what fundamental limits does logical structure impose on a system's self-knowledge?
A rigorous formal theorem demonstrating the axiomatic incoherence of counterfactual freedom (CFF) in temporally consistent systems. Using Lawvere's fixed-point theorem and modal epistemic logic, we show that an embedded agent cannot possess both infallible knowledge of a future action and the metaphysical ability to alter it without contradiction.
An extended thought experiment proving that even an omnipotent, hypercomputational being (Q) fails the FPP. This establishes the Axiomatic Opacity Constraint: the logical incoherence of CFF is principled, not contingent on human limitations.
Interactive φ-spiral tracer for exploring frequency, weight, and phase projections in complex systems. This visualizes the self-similar, recursive constraints generated by golden ratio scaling in dynamical systems.
The Fixed-Point Paradox demonstrates the deep power of self-consistency constraints in abstract systems. Golden Geometric Coherence (GGC) investigates whether E₈ projection geometry represents the unique physical structure that satisfies a stringent self-consistency criterion in the context of spacetime and quantum fields.
Derived purely from E₈ geometry — no free parameters, no input data.
The geometric framework predicts α⁻¹ = 137.036051635 from E₈ root structure and golden ratio scaling.
Experimental value: 137.035999084 — 0.38 parts per million accuracy
This result is a mathematical derivation from first principles that reproduces one of nature's most precisely measured constants to extraordinary accuracy, prior to any phenomenological fitting.
Golden Geometric Coherence (GGC) proposes that fundamental physics emerges from the projection of the 8-dimensional E₈ exceptional Lie algebra onto 4-dimensional spacetime through four golden ratio projection operators:
Manifold: Position space M (observable universe)
Weight: φ⁻ⁿ — decreasing positive series
Content: Quarks, leptons, all visible matter
Time flow: Forward (t → +∞)
Manifold: Momentum space M̄ (quantum hidden sector)
Weight: φⁿ — increasing positive series
Content: Quantum structure, uncertainty, entanglement
Dual to: Matter sector via canonical commutation
Manifold: Anti-position M̃ (anti-universe)
Weight: (−1)ⁿφ⁻ⁿ — alternating signs
Content: Antiquarks, antileptons
Time flow: Backward (relative to M)
Manifold: Anti-momentum M̄̃ (doubly hidden)
Weight: (−1)ⁿφⁿ — alternating, increasing
Content: Anti-quantum structure
CPT conjugate: To momentum sector
The alternating sign (−1)ⁿ in the antimatter sectors creates a predictable geometric asymmetry leading to **destructive interference** when the infinite projection series is bounded by the 8D boundary. This geometric asymmetry naturally generates the matter-antimatter imbalance we observe in the universe.
The framework derives fundamental constants and particle properties from the E₈ projection geometry:
Generation indices κ ∈ {3, 5, 8} emerge from Fibonacci constraint 1 < F_n < 8. Fourth generation (κ = 13) is geometrically forbidden by the 8D boundary closure.
The three observed generations are a consequence of geometric necessity from E₈ dimension.