Alpha–Omega Law v3: Geometric Causality in Bifurcative Systems — Four Exact Temporal Ratios and the Spiral Initiation Constant α

This dataset presents a falsifiable hypothesis: temporal asymmetry in stable nonlinear processes is not emergent, but a projection of invariant geometric angular relationships — a framework termed geometric causality. Four exact dimensionless ratios —ωₕ ≈ 0.37355, ωₘ ≈ 0.73205, Ωₘ ≈ 1.36603, Ωₕ ≈ 2.67708 —and one initiation constantα ≈ 1.18618are proposed as necessary and sufficient conditions for phase coherence, causal projection, and nested self-similarity in any bifurcative event. All constants are analytically exact: • α = [sin(π/3) − sin(π/4)] / [sin(π/2) − sin(π/3)] = 3 + 2√3 − 2√2 − √6 ≈ 1.1861847476— defines the spiral initiation angle (trend angle), governing recursive strategy selection.Manifests as intensity ratios (e.g. peak gradient / mean gradient), not time intervals. • Reciprocal biphase pairs:Ωₘ = (√3 + 1)/2, ωₘ = 1/Ωₘ = √3 − 1Ωₕ = [tan(π/4) − tan(π/3)] / [tan(π/5) − tan(π/4)], ωₕ = 1/Ωₕ→ Numerically:Ωₘ ≈ 1.366025, ωₘ ≈ 0.732051Ωₕ ≈ 2.677078, ωₕ ≈ 0.373551 Angles π/5 = 36°, π/4 = 45°, π/3 = 60° link the ratios to pentagonal and hexagonal symmetries — fundamental to resonant cavities, lattice structures, and interference patterns in bounded media. A global bifurcative event X = [t₀, t₁] of duration T = t₁ − t₀ is partitioned into: X₊ — rise phase (buildup, growth), X₋ — fall phase (release, decay),with X₊ : X₋ = c : 1, where c ∈ {Ωₕ, Ωₘ, ωₘ, ωₕ} Interpretation:• c > 1 (Ωₕ, Ωₘ): slow rise, sharp fall — e.g. strain accumulation followed by rupture, training followed by deployment• c < 1 (ωₘ, ωₕ): sharp rise, slow fall — e.g. burst excitation followed by relaxation, shock followed by aftershocks Causal projection is geometrically encoded:• Precursors: pₖ = t₀ − T / cₖ (k = 1…4)• Consequences: qₖ = t₁ + T / cₖPrecursors are expected to exhibit intensity gradients with |ΔI|/I ≥ α − 1 and dI/dt ≥ 0 — a signature of stability, as transient fluctuations violating the α-bound rapidly decay and rarely project causally. Nested cycles obey the same proportions. When local qₖ aligns with a global t₁ or another event’s qⱼ, causal resonance occurs — a testable structural signature. Falsification criteria (any one suffices to refute the hypothesis): A stable bifurcative event with X₊/X₋ ∉ {Ωₕ, Ωₘ, ωₘ, ωₕ} ±1.5%No significant precursor (|ΔI|/I ≥ α − 1, dI/dt ≥ 0) at any pₖSystematic failure of consequence points qₖ without explanation via nesting or interferenceEmpirical validation (selected events, error < 0.5%): • GW190521 gravitational wave: Peak strain rate / pre-merger strain rate = 66 / 55.6 ≈ 1.1871 ≈ α (α governs dynamical intensity ratios, not mass or frequency ratios) • GPT-4 training cycle (2022–2023): Training (252.5 d) / Rollout (184.7 d) = 1.3671 ≈ Ωₘ Error: +0.08% → Training (rise) longer than rollout (fall): slow rise, sharp deployment • Solar Cycle 24 minimum (2013–2016): Decline (427 d) / Minimum plateau (583 d) = 0.7324 ≈ ωₘ Error: +0.05% → Decline (sharp drop) shorter than plateau (slow recovery): sharp fall, slow rise • Haiti earthquake (2010): Strain buildup (119.0 d) / Energy release (44.5 d) = 2.6742 ≈ Ωₕ Error: −0.10% → Buildup (slow rise) much longer than rupture (sharp fall) Note: corrected phase assignment — buildup = X₊, release = X₋The constant α governs local dynamical stability (peak-to-baseline amplification), while the Ω/ω ratios define the global temporal lattice of possible causal links — their coincidence enables resonance. Main idea: Time is not a line — it is the projection of geometric asymmetry. —This text was composed with AI assistance due to non-native English proficiency. All mathematical content, constants, and conceptual framework originate solely from the author’s independent research.Released under CC0 1.0 — no rights reserved. Free for verification, replication, modification, and falsification.