Yin-Yang Node internal geometry — pole separation r_yy = √(7/8)·ℓ_Planck and 7 dynamical processes

📖 This page documents the FOUNDATIONAL internal geometry of a Bagua Node. Sibling pages: Yin-Yang Z₂ symmetry · Glossary §3.

The single new principle: every Bagua Node has TWO internal poles (yin and yang) separated by

r_{y y} = \frac{7}{8} \cdot ℓ_{Planck} \approx 1.51 \times 1 0^{- 35} m

The 7/8 ratio is the SAME structural constant that produces the cascade slope d_0 = √7/4 in spt_sm_masses.py. From this single distance, ALL 7 dynamical processes (spin, flip, translation, lock, break, split, merge) inherit their length and energy scales — NO new parameters.

1. Foundational identity: r_yy = √(7/8)·ℓ_Planck

In the spt_sm_masses.py derivation of d_0 = √7/4, the dimensionless ratio r_eq² = 7/8 appears as the equilibrium yin-yang spacing in the weighted Laplacian L_w with edge weight w = 8/7 on the Q_6 hypercube. This ratio is now reinterpreted PHYSICALLY:

r_eq² = 7/8 (dimensionless, from L_w)

Equilibrium spacing in Bagua-natural units (lattice units = 1).

r_yy = √(7/8) · ℓ_Planck (physical units)

Yin-yang pole separation inside ONE Bagua Node, ≈ 1.51 × 10⁻³⁵ m (sub-Planckian).

Source

Same ratio that produces d_0 = √(7/16) = √7/4 in cascade slope (spt_sm_masses.py Stage 1).

SymPy verification

scripts/spt_yinyang_node.py Stage 1 — algebraic identity from r_eq² = 7/8.

2. Seven dynamical processes from one r_yy

#	Process	Formula	Match
1	Spin (xoay)	L = m_pole · r_yy · c = ℏ/2 quantization → m_pole = m_Planck · √(2/7) per pole; total node mass ≈ m_Planck.	✅ Tier-B (orbital quantization at v=c)
2	Flip (lật)	f_flip = c/a = 1/τ_Planck. E_flip = h·f_flip = 2π·E_Planck.	✅ Tier-B (membrane principle c=a/τ)
3	Translation (di chuyển)	Rigid-body propagation with v ≤ c (Klein-Gordon). r_yy invariant in rest frame.	✅ Tier-B (from spt_klein_gordon.py)
4	Phase-locking (khoá pha)	V(Δφ, R) = -E_lock(R)·cos(Δφ); E_lock(R) = E_Planck·(r_yy/R) (1/R Coulomb-like).	🟡 Heuristic (model-dependent)
5	Phase-breaking (vỡ pha)	E_crit = 2 · E_lock(R). Two break modes: classical (E > E_crit) or quantum tunneling.	🟡 Heuristic (textbook double-well)
6	Splitting (tách)	1 → 2 requires creating new yin-yang pair. E_split = 2·m_pole·c² ≈ 1.07·E_Planck. Drives Bagua subdivision 1→2→4→…	✅ Tier-B (Planck threshold)
7	Merging (hợp)	Time-reverse of splitting: 2 → 1 releases 2·m_pole·c². Hawking radiation IS this process at the BH horizon.	✅ Tier-B (matches Hawking energy budget)

All 7 dynamical processes inherit length scale r_yy = √(7/8)·ℓ_Planck and energy scale ~ m_pole·c² ≈ 0.5·E_Planck. NO new parameters.

3. Structural connections to known SPT principles

d_0 = √7/4

Same 7/8 ratio: d_0² = 7/16 = (7/8)/2, and r_eq²·d_0² = (7/8)·(7/16) = 49/128. The numerator 49 = 7² echoes 1/α_W(M_Pl) = 49 in spt_gauge_unification.py.

Cascade m_i = m_Pl·exp(−d_i/d_0)

Pole mass m_pole = m_Pl·√(2/7). Total node mass = 2·m_pole = m_Pl·√(8/7) ≈ 1.07·m_Pl. The √(8/7) factor connects directly to the weighted-Laplacian weight w = 8/7.

Bekenstein-Hawking S_BH = N/4

Each node carries 1 yin-yang bit on the horizon. With pole separation r_yy < ℓ_Planck inside each node, the discrete Hilbert space is 2^N at resolution a = ℓ_Planck — the BH unitarity argument (spt_bh_unitarity.py).

Splitting → cosmic expansion

The 1 → 2 → 4 → 8 → … Bagua subdivision is the geometric driver of cosmological expansion (see /theory/empty-space-is-tai-chi). Each subdivision costs 2·m_pole·c² ≈ E_Planck, and accumulated subdivisions produce the observed Hubble flow.

Yin-yang Z₂ symmetry

The Z₂ involution φ → −φ exchanges the two poles. Invariance under this exchange ⇒ θ_QCD ≡ 0 AND m_ν1 ≡ 0 (see Yin-Yang Z₂ page).

4. Download — verify offline

SymPy verify — download for offline testSYMPY ✓

Yin-Yang Node geometry — SymPy verification (10 May 2026 v3)

Ten stages: foundational identity, spin quantization, flip frequency, translation, phase-locking potential, phase-breaking critical energy, splitting energy, merging release, structural connections, falsifiability.

scripts/spt_yinyang_node.py

spt_yinyang_node.py — 7 dynamical processes — Stage 1: r_yy = √(7/8)·ℓ_Planck Tier-B EXACT. Stage 2: m_pole = m_Pl·√(2/7) from L=ℏ/2. Stage 3: E_flip = 2π·E_Planck. Stages 4-8: translation/lock/break/split/merge inherit r_yy and m_pole scales. Stage 9: structural connections to d_0, gauge unification, BH entropy.

320 LOCDownload

Reproduce in 30 seconds

pip install sympy numpy && python3 scripts/spt_yinyang_node.py

Or quick-verify with AI (Grok / Claude / ChatGPT)

Don't want to install Python? Paste the prompt straight into Grok / Claude / ChatGPT / Gemini — the AI fetches the public script URL below and independently verifies each assertion in ~30 s. Open grok.com or claude.ai , paste, send.

⚠️ AI can be wrong — running the Python above is the only 100% certain check. Full AI guide →

Inputs: Bagua integers + π/√ only — no CODATA, no PDG, no calibration (Tier B). SymPy-verified as exact fractions (not floating-point). See full context at /theory/sympy-breakthrough-2026.

5. Falsifiability claim FC-YY

FC-YY (yin-yang internal geometry): The pole separation r_yy = √(7/8)·ℓ_Planck is fixed by the same 7/8 ratio that gives d_0 = √7/4. Falsified if: (a) any sub-Planckian probe detects fermion substructure with length scale ≠ √(7/8)·ℓ_Planck at >5σ; OR (b) spin angular momentum quantization deviates from L = ℏ/2 in a way inconsistent with orbital model; OR (c) the 1 → 2 → 4 subdivision threshold is observed at energies different from 2·m_pole·c² ≈ E_Planck.

Bottom line. A single Bagua Node has internal yin-yang structure with pole separation r_yy = √(7/8)·ℓ_Planck. This single number, derived from the SAME 7/8 ratio that produces d_0 = √7/4, ENTIRELY determines the dynamics of the 7 processes (spin, flip, translation, phase-locking, phase-breaking, splitting, merging). NO new parameters. Closed-form Tier-B for stages 1, 2, 3, 6, 7 (foundational identity, spin, flip, splitting, merging); heuristic Tier-A for stages 4, 5 (lock, break) pending more rigorous derivation.