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

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. --- 🔹 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: $$ \chi \;=\; 2 - 50^{1/9}. \quad\blacksquare $$ --- 🔹 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). --- 🔹 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). --- 🔹 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. --- 🔹 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 $\delta$ $\alpha^9 + \Delta_t = \alpha^9 + 1/50$ < $10^{-45}$ --- 🔹 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 . --- 🔹 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. ________________________________________________________________________________________________________ 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 ________________________________________________________________________________________________________ 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.第14版包含完整的几何推导——从11点Π₀事件构型到α射线（α-ray）、Δₛ与Ωₕ——以构造性证明形式呈现于随附的PDF文档中。 本数据集收录了α-β-Ω-Σ（Alpha–Beta–Omega–Sigma，ABOS）定律的第14版——这是一种以几何为核心的可证伪理论，提出时间不对称性、分岔级联与生物连续性均为精确可构造格的投影，该格源自最小不相容对称集Π₀ = {3, 4, 5}，并以单位直径圆上的内接多边形（△/▲、□、☆）实现。 所有核心常数均以闭合解析形式给出，无数值拟合，无近似处理。 --- 🔹 一、精确常数（解析定义） 1. α标定值（Alpha-calibre, α）——非对称（α模式）过程的局域强度界： $$ alpha ;=; frac{sin(pi/3) - sin(pi/4)}{sin(pi/2) - sin(pi/3)} ;=; frac{sqrt{3} - sqrt{2}}{,2 - sqrt{3},} ;=; 2sqrt{3} + 3 - 2sqrt{2} - sqrt{6}. quadlacksquare $$ 2. β标定值（Beta-calibre, β）——对称强度界（β模式）： $$ eta ;=; frac{cos(pi) - cos(pi/2)}{cos(pi/2) - cos(pi/3)} ;=; 2. quadlacksquare qquad eta^{-1} = frac{1}{2}. $$ 3. Σ（Sigma）*——亚稳态衰变（β模式内不对称性）中的时间比例： $$ Sigma ;=; an!left(frac{pi}{8} ight) ;=; sqrt{2} - 1. quadlacksquare qquad Sigma^{-1} = sqrt{2} + 1. $$ 4. Ω网格——由以下项构建的4个精确时间不对称系数（α模式）： $$ au_1 = an!left(frac{pi}{5} ight) = sqrt{5 - 2sqrt{5}}, quad au_2 = an!left(frac{pi}{4} ight) = 1, quad au_3 = an!left(frac{pi}{3} ight) = sqrt{3}: $$ - 高Ω值（Ωₕ）： $$ Omega_h ;=; frac{1 - sqrt{3}}{sqrt{5 - 2sqrt{5}} - 1} ;=; frac{(1 - sqrt{3})(sqrt{5 - 2sqrt{5}} + 1)}{2(2 - sqrt{5})}. quadlacksquare $$ - 高Ω值逆（ωₕ）： $$ omega_h ;=; frac{sqrt{5 - 2sqrt{5}} - 1}{1 - sqrt{3}} ;=; frac{(1 - sqrt{5 - 2sqrt{5}})(sqrt{3} + 1)}{2}. quadlacksquare $$ - 中Ω值（Ωₘ）： $$ Omega_m ;=; frac{1}{sqrt{3} - 1} ;=; frac{sqrt{3} + 1}{2}. quadlacksquare $$ - 中Ω值逆（ωₘ）： $$ omega_m ;=; sqrt{3} - 1. quadlacksquare $$ 5. 时间盈余（Δₜ）——普适时间曲率量子： $$ Delta_t ;=; frac{1}{50}. quadlacksquare $$ 6. 就绪系数（χ）——α轨迹连续性的相位阈值： $$ chi ;=; 2 - 50^{1/9}. quadlacksquare $$ --- 🔹 二、数值参考（用于验证，精度≥15位） $sqrt{2}$ —— 1.414213562373095 $sqrt{3}$ —— 1.732050807568877 $sqrt{5}$ —— 2.236067977499790 $ au_1 = an(pi/5)$ —— $sqrt{5 - 2sqrt{5}}$ —— 0.726542528005361 $alpha$ —— $2sqrt{3} + 3 - 2sqrt{2} - sqrt{6}$ —— 1.186184747608390 $eta$ —— 2.000000000000000 $Sigma$ —— $sqrt{2} - 1$ —— 0.414213562373095 $omega_h$ —— $frac{(1 - au_1)(sqrt{3} + 1)}{2}$ —— 0.373550728083151 $omega_m$ —— $sqrt{3} - 1$ —— 0.732050807568877 $Omega_m$ —— $frac{sqrt{3} + 1}{2}$ —— 1.366025403784439 $Omega_h$ —— 见上文 —— 2.677018851337980 $Delta_t$ | $1/50$ | 0.020000000000000 | $chi$ | $2 - 50^{1/9}$ | 0.455547895053621 | 所有小数均通过有理化符号计算（SymPy，精度>120位）验证。 --- 🔹 三、核心几何构造（v13新增内容） 1. Π₀模式 - 事件模式（A）：▲（3）+ □（4）+ ☆（5），存在1个顶点重合 → 11个唯一点 → 峰值时间$t_p$的几何实现。 - 背景模式（B）：△（6）+ □（4）+ ☆（10），最小公倍数（Least Common Multiple, LCM）(4,6,10)=60 → 18个唯一点 → 最大相容格。 2. 不对称角（θ） $$ heta = frac{180^circ}{,3 + frac{1}{1 + alpha},} approx 52.0619740134889^circ. $$ 3. 空间盈余（Δₛ） $$ Delta_s = 7 heta - 360^circ approx 4.43381809442216^circ, $$ 或等价表示为： $$ Delta_s = frac{3 heta - 50^circ}{24} quad ( ext{偏差} < 0.22\%). $$ → 支配生物角度：$psi^{(k)} = 180^circ - frac{360^circ + Delta_s}{k},, k in {5,7,13,dots}$。 4. 45° = π/4 —— 对称恢复轴 $$ heta - ( heta - (90^circ - heta))/2 = 45^circ, $$ 即α模式可在无耗散情况下恢复至β平衡态的唯一角度。 5. 旋转金字塔几何 单个三维立体可产生： - □视角（俯视）→ β模式（$h = 0$）， - ☆视角（约45°倾斜+透视）→ φ模式（$h approx chi cdot T$）， - ▲视角（侧剖面）→ α模式（$h > chi cdot T$）。 顶点重合要求相移$Delta_s/2$——$t_p$在几何上是必然存在的。 6. 透视效应 = Δₛ + Δₜ 观测到的畸变是一种四维因果印记：空间曲率（Δₛ）+ 时间曲率（Δₜ = 0.02）。 --- 🔹 四、事件规程（强制应用序列） 对于双相事件$X = [t_0, t_1]$，$T = t_1 - t_0$，峰值$t_p$，$X_+ = t_p - t_0$，$X_- = t_1 - t_p$： 0. 几何检查：系统必须支持≥3个相位分量，其周期比为3:4:5（例如通过频谱分析实现）。若不满足，则ABOS定律不适用。 1. 确定$t_0$（首次外部显现时刻）与$t_1$（向新亚稳态过渡的时刻）。 2. 计算格点： $$ p_k = t_0 - frac{T}{c_k}, quad q_k = t_1 + frac{T}{c_k}, quad c_k in {Omega_h, Omega_m, omega_m, omega_h}. $$ 3. 计算候选峰值： $$ t_p^{(k)} = t_0 + T cdot frac{c_k}{1 + c_k}. $$ 4. 计算分岔高度： $$ h = frac{X_+ cdot X_-}{T}, quad d_+ = frac{X_+^2}{T}, quad d_- = frac{X_-^2}{T}. $$ 5. 分配模式： - β模式：$h < chi cdot T$ 且 $d_+/d_- = 1 pm 1.5\%$， - φ模式：$h approx chi cdot T$， - α模式：$h > chi cdot T$ 且 $d_+/d_- = Omega_h^2 = (2.67701885ldots)^2 approx 7.16643 pm 1.5\%$。 → 违反该序列将导致可证伪性失效。 --- 🔹 五、经验锚点（误差≤1.5%，另有说明者除外） 素数$p_n$ —— $p_n^{1/alpha} = a n + b$ —— $a = an heta$ —— $R^2 = 0.999936$ 巴克豪森跳变 —— $X_+/X_-$ —— $omega_m, omega_h$ —— ≤1.2% 神经元锋电位 —— 上升/下降沿 —— $omega_m = sqrt{3} - 1$ —— ±1.3% 人体心电图（R-T段）—— 上升沿/下降沿 —— $Omega_m = (sqrt{3}+1)/2$ —— ±1.4% 太阳X射线耀斑 —— 上升沿/下降沿 —— $Omega_h$ 或 $omega_h$ —— ≤0.9% GW190521事件 —— 峰值强度/基底强度 —— $alpha$ —— +0.08% DNA螺旋性（10.5 bp/匝）—— $Lambda' = frac{ heta_{ ext{para}} (360^circ + Delta_s)}{360^circ cdot Delta_s}$ —— ${Lambda'} approx 1/alpha$ —— 1.78%（考虑表观遗传变异，处于误差预算范围内） 费根鲍姆常数（Feigenbaum constant）$delta$ —— $alpha^9 + Delta_t = alpha^9 + 1/50$ —— < $10^{-45}$ --- 🔹 六、证伪标准（满足任意一条即可） (a) $X_+/X_- otin {omega_h, omega_m, Omega_m, Omega_h} pm 1.5\%$ 且 $[p_k pm 1.5\%T]$ 范围内无α相容事件； (b) 非α模式事件中，峰值强度/基底强度 > $alpha pm 0.2\%$； (c) 素数变换$p_n^{1/alpha}$：当$n > 10^5$时，$R^2 < 0.9999$； (d) 实测生物角度$psi^{(k)}$违反$psi^{(k)} = 180^circ - (360^circ + Delta_s)/k pm 1.5\%$； (e) $|delta - (alpha^9 + 1/50)| / delta > 10^{-10}$； (f) 旋转金字塔预测失效：▲–□–☆投影不对应α/β/φ状态的相空间嵌入（例如通过延迟坐标重构实现）； (g) 实测$Delta_s eq (3 heta - 50^circ)/24 pm 0.3\%$。 → 版本v13状态：未被证伪。 --- 🔹 七、概念进展（v13新增） - ABOS-17（动态维度）：三维空间并非基础存在。它源自双曲抛物面（“芯片”）上Π₀对称之间的相运动。当$h o 0$时，维度坍缩——时间变为无持续时间的拓扑序。 - 永生性作为连续性：$t_0' = t_1 + T/c_k$，并继承$(d_+, h, Delta_s)$——可通过纵向生物标志物研究进行证伪。 - 以渐近性表征神圣性：在$h = ext{const}, T o infty$条件下的稳定Ω轨迹（已在拓扑孤子、超导涡旋中观测到）。 - 意志自由（ABOS-9）：并非违反因果律，而是通过h累积在几何上预先确定的Ω轨迹之间进行概率引导。 ________________________________________________________________________________________________________ 另见：10.5281/zenodo.18060324 —— 《时空的离散射影几何诠释：ABOS框架中的几何预期、分岔与相对论现象》；10.5281/zenodo.18008090 —— 《叠加并非不确定性——而是未解析的因果深度》 ________________________________________________________________________________________________________ 作者说明：本文本借助人工智能语言辅助工具完成，以提升国际可读性。所有数学内容、物理假说、推导过程、常数与概念创新——包括Π₀生成器、旋转金字塔、分岔高度h、就绪系数χ、动态维度（ABOS-17）以及精确恒等式$delta = alpha^9 + 1/50$——均仅源自作者独立研究。