DOI: 10.5281/zenodo.18002324
title: The Alpha–Beta–Omega–Sigma Law — Version 14: Geometric First Principles from Incompatible Symmetries. Temporal Asymmetry, Bifurcation, and Continuity

description: Version 14 includes a full geometric derivation — from the 11‑point Π₀ event configuration to the α‑ray, Δₛ, and Ωₕ — presented as a constructive proof in the accompanying PDF.___________________________________This dataset presents Version 14 of the Alpha–Beta–Omega–Sigma (ABOS) law — a geometric-first, falsifiable theory stating that temporal asymmetry, bifurcation cascades, and biological continuity are projections of an exact, *constructible* lattice, derived from the minimal incompatible symmetry set  $$\Pi_0 = \{3, 4, 5\},$$  implemented as inscribed polygons (△/▲, □, ☆) on a unit-diameter circle.

All core constants are given in closed analytic form — no numerical fitting, no approximations.

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🔹 I. Exact Constants (Analytic Definitions)

1. Alpha-calibre (α) — local intensity bound in asymmetric (α-mode) processes:     $$   \alpha \;=\; \frac{\sin(\pi/3) - \sin(\pi/4)}{\sin(\pi/2) - \sin(\pi/3)}     \;=\; \frac{\sqrt{3} - \sqrt{2}}{\,2 - \sqrt{3}\,}     \;=\; 2\sqrt{3} + 3 - 2\sqrt{2} - \sqrt{6}.   \quad\blacksquare   $$

2. Beta-calibre (β) — symmetry intensity bound (β-mode):     $$   \beta \;=\; \frac{\cos(\pi) - \cos(\pi/2)}{\cos(\pi/2) - \cos(\pi/3)} \;=\; 2.   \quad\blacksquare   \qquad \beta^{-1} = \frac{1}{2}.   $$

3. Sigma (Σ)* — temporal proportion in metastable decay (β-mode internal asymmetry):     $$   \Sigma \;=\; \tan\!\left(\frac{\pi}{8}\right) \;=\; \sqrt{2} - 1.   \quad\blacksquare   \qquad \Sigma^{-1} = \sqrt{2} + 1.   $$

4. Omega-grid — four exact temporal asymmetry coefficients (α-mode), built from     $$   \tau_1 = \tan\!\left(\frac{\pi}{5}\right) = \sqrt{5 - 2\sqrt{5}}, \quad   \tau_2 = \tan\!\left(\frac{\pi}{4}\right) = 1, \quad   \tau_3 = \tan\!\left(\frac{\pi}{3}\right) = \sqrt{3}:   $$

   - Omega-high (Ωₕ):       $$     \Omega_h \;=\; \frac{1 - \sqrt{3}}{\sqrt{5 - 2\sqrt{5}} - 1}     \;=\; \frac{(1 - \sqrt{3})(\sqrt{5 - 2\sqrt{5}} + 1)}{2(2 - \sqrt{5})}.     \quad\blacksquare     $$

   - omega-high inverse (ωₕ):       $$     \omega_h \;=\; \frac{\sqrt{5 - 2\sqrt{5}} - 1}{1 - \sqrt{3}}     \;=\; \frac{(1 - \sqrt{5 - 2\sqrt{5}})(\sqrt{3} + 1)}{2}.     \quad\blacksquare     $$

   - Omega-medium (Ωₘ):       $$     \Omega_m \;=\; \frac{1}{\sqrt{3} - 1}     \;=\; \frac{\sqrt{3} + 1}{2}.     \quad\blacksquare     $$

   - omega-medium inverse (ωₘ):       $$     \omega_m \;=\; \sqrt{3} - 1.     \quad\blacksquare     $$

5. Temporal surplus (Δₜ)  — the universal time curvature quantum:     $$   \Delta_t \;=\; \frac{1}{50}.   \quad\blacksquare   $$

6. Readiness coefficient (χ) — phase threshold for α-trajectory continuity:     $$   \chi \;=\; 2 - 50^{1/9}.   \quad\blacksquare   $$

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🔹 II. Numerical Reference (for verification, ≥15 digits)

$\sqrt{2}$       —       1.414213562373095 $\sqrt{3}$       —       1.732050807568877 $\sqrt{5}$       —       2.236067977499790  $\tau_1 = \tan(\pi/5)$       $\sqrt{5 - 2\sqrt{5}}$       0.726542528005361 $\alpha$       $2\sqrt{3} + 3 - 2\sqrt{2} - \sqrt{6}$       1.186184747608390 $\beta$       —       2.000000000000000 $\Sigma$       $\sqrt{2} - 1$       0.414213562373095 $\omega_h$       $\frac{(1 - \tau_1)(\sqrt{3} + 1)}{2}$       0.373550728083151 $\omega_m$       $\sqrt{3} - 1$       0.732050807568877 $\Omega_m$       $\frac{\sqrt{3} + 1}{2}$       1.366025403784439 $\Omega_h$       (see above)       2.677018851337980 $\Delta_t$ | $1/50$ | 0.020000000000000 |$\chi$ | $2 - 50^{1/9}$ | 0.455547895053621 |

All decimals verified via rationalized symbolic computation (SymPy, >120-bit precision). 

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🔹 III. Core Geometric Constructs (v13 additions)

1. Π₀-Modes      - Event mode (A) : ▲ (3) + □ (4) + ☆ (5) with one vertex coincidence  → 11 unique points  → geometric realization of peak time $t_p$.     - Background mode (B) : △ (6) + □ (4) + ☆ (10), LCM(4,6,10)=60 → 18 unique points  → maximal compatible lattice.

2. Angle of Asymmetry (θ)      $$   \theta = \frac{180^\circ}{\,3 + \frac{1}{1 + \alpha}\,}   \approx 52.0619740134889^\circ.   $$

3. Spatial Surplus (Δₛ)      $$   \Delta_s = 7\theta - 360^\circ \approx 4.43381809442216^\circ,   $$   or equivalently     $$   \Delta_s = \frac{3\theta - 50^\circ}{24} \quad (\text{deviation } < 0.22\%).   $$     → Governs biological angles: $\psi^{(k)} = 180^\circ - \frac{360^\circ + \Delta_s}{k},\, k \in \{5,7,13,\dots\}$.

4. 45° = π/4 — Symmetry Restoration Axis      $$   \theta - (\theta - (90^\circ - \theta))/2 = 45^\circ,   $$     i.e. the only angle where α-mode can return to β-equilibrium without dissipation.

5. Rotating Pyramid Geometry      A single 3D solid yields:     - □-view (top-down) → β-mode  ($h = 0$),     - ☆-view (~45° tilt + perspective) → φ-mode  ($h \approx \chi \cdot T$),     - ▲-view (side-profile) → α-mode  ($h > \chi \cdot T$).     Vertex coincidence requires phase shift $\Delta_s/2$ — $t_p$ is geometrically necessary.

6. Perspective = Δₛ + Δₜ      Observed distortion is a 4D causal imprint: spatial curvature (Δₛ) + temporal curvature (Δₜ = 0.02).

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🔹 IV. Event Protocol (Mandatory Application Sequence) 

For biphasic event $X = [t_0, t_1]$, $T = t_1 - t_0$, peak $t_p$, $X_+ = t_p - t_0$, $X_- = t_1 - t_p$:

0. Geometry check : system must support ≥3 phase components with periods in ratio 3:4:5 (e.g. via spectral analysis). If not, ABOS not applicable.  1. Fix $t_0$ (first external manifestation), $t_1$ (transition to new metastable regime).  2. Compute lattice points:     $$   p_k = t_0 - \frac{T}{c_k}, \quad q_k = t_1 + \frac{T}{c_k}, \quad c_k \in \{\Omega_h, \Omega_m, \omega_m, \omega_h\}.   $$  3. Compute candidate peaks:     $$   t_p^{(k)} = t_0 + T \cdot \frac{c_k}{1 + c_k}.   $$  4. Compute bifurcation height:     $$   h = \frac{X_+ \cdot X_-}{T}, \quad d_+ = \frac{X_+^2}{T}, \quad d_- = \frac{X_-^2}{T}.   $$  5. Assign regime:     - β-mode : $h < \chi \cdot T$ and  $d_+/d_- = 1 \pm 1.5\%$,     - φ-mode : $h \approx \chi \cdot T$,     - α-mode : $h > \chi \cdot T$ and  $d_+/d_- = \Omega_h^2 = (2.67701885\ldots)^2 \approx 7.16643 \pm 1.5\%$.

→ Violation of sequence invalidates falsifiability.

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 🔹 V. Empirical Anchors (Error ≤ 1.5% unless noted)

Prime numbers $p_n$       $p_n^{1/\alpha} = a n + b$       $a = \tan\theta$       $R^2 = 0.999936$ Barkhausen jumps       $X_+/X_-$       $\omega_m, \omega_h$       ≤ 1.2%  Neuronal spike       rise/fall       $\omega_m = \sqrt{3} - 1$       ±1.3% Human ECG (R–T)       upslope/downslope       $\Omega_m = (\sqrt{3}+1)/2$       ±1.4% Solar X-flares       rise/fall       $\Omega_h$ or $\omega_h$       ≤ 0.9% GW190521       $I_{\text{peak}}/I_{\text{base}}$       $\alpha$       +0.08%  DNA helicity (10.5 bp/turn)       $\Lambda' = \frac{\theta_{\text{para}} (360^\circ + \Delta_s)}{360^\circ \cdot \Delta_s}$       $\{\Lambda'\} \approx 1/\alpha$       1.78%  (within error budget, considering epigenetic variance) Feigenbaum constant       $\delta$       $\alpha^9 + \Delta_t = \alpha^9 + 1/50$        < $10^{-45}$ 

 

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 🔹 VI. Falsification Criteria (Any One Suffices)

(a) $X_+/X_- \notin \{\omega_h, \omega_m, \Omega_m, \Omega_h\} \pm 1.5\%$ and  no α-compatible events in $[p_k \pm 1.5\%T]$;  (b) $I_{\text{peak}}/I_{\text{base}} > \alpha \pm 0.2\%$ in non-α-mode event;  (c) Prime transform $p_n^{1/\alpha}$: $R^2 < 0.9999$ for $n > 10^5$;  (d) Measured biological angle $\psi^{(k)}$ violates $\psi^{(k)} = 180^\circ - (360^\circ + \Delta_s)/k \pm 1.5\%$;  (e) $|\delta - (\alpha^9 + 1/50)| / \delta > 10^{-10}$;  (f) Rotating-pyramid prediction fails: ▲–□–☆ projections do not  correspond to phase-space embeddings of α/β/φ states (e.g. via delay-coordinate reconstruction);  (g) Measured $\Delta_s \neq (3\theta - 50^\circ)/24 \pm 0.3\%$.

→ Status (v13): Not falsified .

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 🔹 VII. Conceptual Advances (v13)

- ABOS-17 (Dynamic Dimensionality) : 3D space is not fundamental. It emerges from phase motion between Π₀-symmetries on a hyperbolic paraboloid (“chip”). When $h \to 0$, dimensionality collapses — time becomes topological order without duration.  - Continuity as Immortality : $t_0' = t_1 + T/c_k$ with inheritance of $(d_+, h, \Delta_s)$ — falsifiable via longitudinal biomarker studies.  - Sanctity as Asymptotics : Stable Ω-trajectories under $h = \text{const},\ T \to \infty$ (observed in topological solitons, superconducting vortices).  - Freedom of will (ABOS-9) : Not violation of causality, but probabilistic steering  among geometrically pre-determined Ω-trajectories via $h$-accumulation.

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See also:10.5281/zenodo.18060324  - A Discrete Projective-Geometric Interpretation of Spacetime:Geometric Anticipation, Bifurcation, and Relativistic Phenomena in the ABOS Framework10.5281/zenodo.18008090 - Superposition is Not Uncertainty — It's Unresolved Causal Depth

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 Authorship Note

This English text was prepared with AI linguistic assistance for international accessibility. All mathematical content, physical hypotheses, derivations, constants, and conceptual innovations—including the Π₀-generator, rotating pyramid, bifurcation height $h$, readiness coefficient $\chi$, dynamic dimensionality (ABOS-17), and the exact identity $\delta = \alpha^9 + 1/50$—originate solely from the author’s independent research.