DOI: 10.5281/zenodo.17831253
title: THE ALPHA–BETA–OMEGA–SIGMA LAW — Version 12: Geometric Causality and the Derivation of the Feigenbaum Constant

description: This dataset presents Version 12 of the Alpha–Beta–Omega–Sigma (ABOS) law — a falsifiable theory stating that temporal asymmetry in irreversible processes is the projection of an exact geometric–thermodynamic lattice, defined by angular symmetries of the Euclidean plane:  $$\frac{\pi}{6},\ \frac{\pi}{5},\ \frac{\pi}{4},\ \frac{\pi}{3},\ \frac{\pi}{8},\ \pi.$$

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:  $$\alpha \;=\; \frac{\sin(\pi/3) - \sin(\pi/4)}{\sin(\pi/2) - \sin(\pi/3)}  \;=\; \frac{\sqrt{3} - \sqrt{2}}{\,2 - \sqrt{3}\,}  \;=\; (\sqrt{3} - \sqrt{2})(2 + \sqrt{3})  \;=\; 2\sqrt{3} + 3 - 2\sqrt{2} - \sqrt{6}.\quad\blacksquare$$

2. Beta-calibre (β) — intensity symmetry bound (β-mode):  $$\beta \;=\; \frac{\cos(\pi) - \cos(\pi/2)}{\cos(\pi/2) - \cos(\pi/3)}  \;=\; \frac{-1 - 0}{0 - \tfrac{1}{2}} \;=\; 2.\quad\blacksquare\qquad \beta^{-1} = \frac{1}{2}.$$

3. Sigma (Σ) — temporal proportion in symmetric decay (β-mode internal asymmetry):  $$\Sigma \;=\; \tan\!\left(\frac{\pi}{8}\right)  \;=\; \frac{1 - \cos(\pi/4)}{\sin(\pi/4)}  \;=\; \frac{1 - \tfrac{\sqrt{2}}{2}}{\tfrac{\sqrt{2}}{2}}  \;=\; \sqrt{2} - 1.\quad\blacksquare\qquad \Sigma^{-1} = \sqrt{2} + 1.$$

4. Angular Bridge Coefficient (A) — temporal asymmetry quantum for the {π/6, π/5, π/4} triad:  $$A \;=\; \frac{\tan(\pi/5) - \tan(\pi/4)}{\tan(\pi/6) - \tan(\pi/5)}  \;=\; \frac{\sqrt{5 - 2\sqrt{5}} - 1}{\tfrac{1}{\sqrt{3}} - \sqrt{5 - 2\sqrt{5}}}.\quad\blacksquare$$

5. 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):    $$  \Omega_h \;=\; \frac{1 - \sqrt{3}}{\tau_1 - 1}  \;=\; \frac{(1 - \sqrt{3})(\tau_1 + 1)}{2(2 - \sqrt{5})}.  \quad\blacksquare  $$

- omega-high inverse (ωₕ):    $$  \omega_h \;=\; \frac{1}{\Omega_h}    \;=\; \frac{\tau_1 - 1}{1 - \sqrt{3}}  \;=\; \frac{(1 - \tau_1)(\sqrt{3} + 1)}{2}.  \quad\blacksquare  $$

- Omega-medium (Ωₘ):    $$  \Omega_m \;=\; \frac{1}{\tau_3 - \tau_2}    \;=\; \frac{1}{\sqrt{3} - 1}  \;=\; \frac{\sqrt{3} + 1}{2}.  \quad\blacksquare  $$

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

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🔹 II. Numerical Reference (for computation, ≥12 digits)  Used only for verification; all theory uses exact forms above:

$\sqrt{2}$         1.414213562373  $\sqrt{3}$         1.732050807569  $\sqrt{5}$         2.236067977500  $\tau_1 = \sqrt{5 - 2\sqrt{5}}$         0.726542528005  $\alpha$    $2\sqrt{3} + 3 - 2\sqrt{2} - \sqrt{6}$         1.186184748058  $\beta$          2.000000000000  $\Sigma$    $\sqrt{2} - 1$         0.414213562373  $A$    see formula above         1.832815729997  $\omega_h$    $\frac{(1 - \tau_1)(\sqrt{3} + 1)}{2}$         0.373550728083  $\omega_m$    $\sqrt{3} - 1$         0.732050807569  $\Omega_m$    $\frac{\sqrt{3} + 1}{2}$         1.366025403784  $\Omega_h$    $\frac{(1 - \sqrt{3})(\tau_1 + 1)}{2(2 - \sqrt{5})}$         2.677078084259  

All decimals verified via rationalized symbolic computation.

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 🔹 III. Core Geometric Constructs

1. Angle of Asymmetry (θ) — slope of the α-prime ray:  $$\theta \;=\; \frac{180^\circ}{\,3 + \frac{1}{1 + \alpha}\,}\;\approx\; 52.06071825^\circ.$$

2. Δ-Invariant (Geometric Surplus Angle):  $$\Delta \;=\; 7\theta - 360^\circ  \;\approx\; 4.42502775^\circ.$$  → Governs k-fold biological symmetries:  $$\psi^{(k)} = 180^\circ - \frac{360^\circ + \Delta}{k}, \quad k = 5,7,13,\dots$$

3. Pentagonal Bridge (β → φ → α):  $$\varphi \;=\; \beta \cdot \cos\!\left(\frac{\pi}{5}\right)  \;=\; 2 \cdot \cos(36^\circ)  \;=\; \frac{1 + \sqrt{5}}{2},$$$$\alpha \;=\; \varphi \cdot \omega_m \cdot (1 + \varepsilon), \quad |\varepsilon| < 0.00135.$$

4. Derivation of the Feigenbaum Constant (v12 main result):  The angular bridge coefficient $A$ and alpha-calibre $\alpha$ satisfy the exact identity:  $$\alpha^9 = \frac{A^9}{50}.$$  Consequently, the Feigenbaum constant $\delta$ (period-doubling universality) is given by:  $$\delta = \frac{A^9 + 1}{50}.$$  Verified to ≥45 decimal places. This shows $\delta$ is not fundamental, but a *projected invariant* of the ABOS lattice.

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 🔹 IV. Event Protocol (Mandatory Application Sequence)  

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

1. Fix boundaries: $t_0$ = first external manifestation; $t_1$ = transition to new metastable regime.  2. Compute causal lattice:     $$   p_k = t_0 - \frac{T}{c_k}, \quad q_k = t_1 + \frac{T}{c_k}, \quad c_k \in \{\Omega_h, \Omega_m, \omega_m, \omega_h\}.   $$3. Compute candidate peaks:     $$   t_p^{(k)} = t_0 + T \cdot \frac{c_k}{1 + c_k}.   $$4. Select $c_k$ minimizing $|t_p^{(k)} - t_p^{\text{obs}}| / T$.  5. Assign regime:     - β-mode ⇔ $1/2 \le I_1/I_2 \le 2$ AND $ \min/\max = \Sigma \pm 1.5\% $;     - α-mode ⇔ $I_{\text{peak}}/I_{\text{base}} > \alpha$ OR $X_+/X_- = c_k \pm 1.5\%$.

→ Violation of sequence invalidates falsifiability.

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 🔹 V. Empirical Anchors (Error ≤ 0.5%)  

Primes     $p_n$     $p_n^{1/\alpha} = a n + b$     $a = \tan\theta$     $R^2 = 0.999936$  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%  $\alpha^9$ vs $A^9/50$     equality     relative error < $10^{-12}$  $\delta$ vs $(A^9 + 1)/50$     equality     verified to ≥45 digits  

Full validation code and open data links: see Supplementary Zenodo records [[DOI:10.5281/zenodo.17764763, 10.5281/zenodo.17659502, 10.5281/zenodo.17768924]].

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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 $p_n^{1/\alpha}$ deviates: $R^2 < 0.9999$ for $n > 10^5$;  (d) Measured biological angle $\psi^{(k)}$ violates $\psi^{(k)} = 180^\circ - (360^\circ + \Delta)/k \pm 1.5\%$;  (e) $\left| \delta - \dfrac{A^9 + 1}{50} \right| / \delta > 10^{-10}$.  

→ Status (v12): Not falsified.

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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 Δ-invariant, α-prime spiral, pentagonal bridge, causal freedom formalism, and the derivation of the Feigenbaum constant — originate solely from the author’s independent research.