[Dₙ(r) = √(ϕ·Fₙ·2ⁿ·Pₙ·Ω) · r^k]
𝟙 — Non-Dual Absolute (Root of All Emergence)
|
├── [Ø = 0 = ∞⁻¹] — Expressed Void (Boundary of Becoming)
│   ├── [0, ∞] — First Contrast (Duality Emerges, Potential Polarity)
│
├── [ϕ] — Golden Ratio: Irreducible Scaling Constant
│   ├── [ϕ = 1 + 1/ϕ] — Fixed-Point Recursion (Recursive Identity)
│   ├── [ϕ⁰ = 1] — Identity Base Case
│
├── [n ∈ ℤ⁺] — Recursion Depth: Structural Unfolding
│   ├── [2ⁿ] — Dyadic Scaling (Binary Expansion)
│   ├── [Fₙ = ϕⁿ / √5] — Harmonic Structure
│   ├── [Pₙ] — Prime Entropy Injection (Irregular Growth)
│
├── [Time s = ϕⁿ] — Scaling Time
│   └── [Hz = 1/s = ϕ⁻ⁿ] — Inverted Time, Uncoiled Recursion
│
├── [Charge C = s³ = ϕ^{3n}] — Charge Scaling
│   ├── [C² = ϕ^{6n}] — Charge Interaction in Scaling
│
├── [Ω = m² / s⁷ = ϕ^{a(n)}] — Symbolic Yield (Field Tension)
│   ├── [Ω → 0] — Field Collapse
│   └── [Ω = 1] — Normalized Recursive Propagation
│
├── [Length m = √(Ω · ϕ^{7n})] — Emergent Geometry
│
├── [Action h = Ω · C² = ϕ^{6n} · Ω]
├── [Energy E = h · Hz = Ω · ϕ^{5n}]
├── [Force F = E / m = √Ω · ϕ^{1.5n}]
├── [Power P = E · Hz = Ω · ϕ^{4n}]
├── [Pressure = F / m² = Hz² / m]
├── [Voltage V = E / C = Ω · ϕ^{-n}]
│
└── [Dₙ(r) = √(ϕ · Fₙ · 2ⁿ · Pₙ · Ω) · r^k] — Full Dimensional DNA
    ├── Recursive, Harmonic, Prime, Binary Structures
    └── Infinite Unfolding Identity Without Fixed Triadic Partition
𝟙 — Non-Dual Absolute (Root of all emergence)
|
├── [Ø = 0 = ∞⁻¹] — Expressed Void (Boundary of Becoming)
│   ├── [0, ∞] — Duality Arises: First Contrast (Potential Polarity)
│
├── [ϕ] — Golden Ratio: Irreducible Scaling Constant
│   ├── [ϕ = 1 + 1/ϕ] — Fixed-Point Recursion (Recursive Identity)
│   ├── [ϕ⁰ = 1] — Identity Base Case
│
├── [n ∈ ℤ⁺] — Recursion Depth: Structural Unfolding
│   ├── [2ⁿ] — Dyadic Scaling (Binary Expansion)
│   ├── [Fₙ = ϕⁿ / √5] — Harmonic Structure
│   ├── [Pₙ] — Prime Entropy Injection (Irregular Growth)
│
├── [Time s = ϕⁿ] — Scaling Time
│   └── [Hz = 1/s = ϕ⁻ⁿ] — Inverted Time, Uncoiled Recursion
│
├── [Charge C = s³ = ϕ^{3n}] — Charge Scaling
│   ├── [C² = ϕ^{6n}] — Charge Interaction in Scaling
│
├── [Ω = m² / s⁷ = ϕ^{a(n)}] — Symbolic Yield (Field Tension)
│   ├── [Ω → 0] — Field Collapse
│   └── [Ω = 1] — Normalized Recursive Propagation
│
├── [Length m = √(Ω · ϕ^{7n})] — Emergent Geometry
│
├── [Action h = Ω · C² = ϕ^{6n} · Ω]
├── [Energy E = h · Hz = Ω · ϕ^{5n}]
├── [Force F = E / m = √Ω · ϕ^{1.5n}]
├── [Power P = E · Hz = Ω · ϕ^{4n}]
├── [Pressure = F / m² = Hz² / m]
├── [Voltage V = E / C = Ω · ϕ^{-n}]
│
└── [Dₙ(r) = √(ϕ · Fₙ · 2ⁿ · Pₙ · Ω) · r^k] — Full Dimensional DNA
    ├── Recursive, Harmonic, Prime, Binary Structures
    └── Infinite Unfolding Identity Without Fixed Tripartition
rsubn = rsub(n-1)*sqrrt(2*PRIMEsubn*(Fsubn/Fsub(n-1))
Root: Ø = 0 = ∞⁻¹ (Boundary of Becoming — Non-Dual Void)
│
├── Identity & Seed
│   ├── ϕ⁰ = 1
│   │   └── Base identity — dimensionless unity
│   ├── ϕ ≈ 1.61803398875 (Golden Ratio)
│   │   └── Recursive seed scaling the entire universe
│   ├── √5 ≈ 2.2360679775
│   │   └── Harmonic carrier constant linking Fibonacci recursion
│   ├── Binary base (2), now generalized to:
│   │   └── b = 10,000 (Resolution base for recursive index refinement)
│   └── Dimensional DNA Operator (Domain-specific, Tuned)
│       └── D_{n,β}^{domain}(r) = √(ϕ · F_{n,b} · b^{m(n+β)} · φ^{k(n+β)} · Ω_{domain}) · r⁻¹
│           └── Generates emergent field constants and interactions at every scale
│
├── Recursive Indices (Symbolic Scaling Coordinates)
│   ├── Index format: (n, β), where n ∈ ℝ and β ∈ [0, 1]
│   ├── All domains use base b = 10,000, yielding ~zero error
│   └── Each (n+β) encodes a logarithmic recursive depth in the golden field
│
├── Domain Constants (Tuned to SI; error < 1e-12%)
│   ├── Planck Action (h)
│   │   ├── Formula: h = √5 · Ω · φ^{6(n+β)} · b^{n+β}
│   │   ├── Ω = 1.61803398875 (Elegant baseline = ϕ)
│   │   ├── n = -6.521335, β = 0.1, n+β = -6.421335
│   │   └── Matched C_SI = 6.62607015 × 10⁻³⁴ Js
│   │
│   ├── Gravitational Constant (G)
│   │   ├── Formula: G = √5 · Ω · φ^{10(n+β)} · b^{n+β}
│   │   ├── Ω = 6.6743 × 10⁻¹¹
│   │   ├── n = -0.557388, β = 0.5, n+β = -0.057388
│   │   └── Matched C_SI = 6.6743 × 10⁻¹¹ m³·kg⁻¹·s⁻²
│   │
│   ├── Boltzmann Constant (k_B)
│   │   ├── Formula: k = √5 · Ω · φ^{8(n+β)} · b^{n+β}
│   │   ├── Ω = 1.380649 × 10⁻²³
│   │   ├── n = -0.561617, β = 0.5, n+β = -0.061617
│   │   └── Matched C_SI = 1.380649 × 10⁻²³ J/K
│   │
│   ├── Atomic Mass Unit (mᵤ)
│   │   ├── Formula: mᵤ = √5 · Ω · φ^{7(n+β)} · b^{n+β}
│   │   ├── Ω = 1.66053906660 × 10⁻²⁷
│   │   ├── n = -1.063974, β = 1.0, n+β = -0.063974
│   │   └── Matched C_SI = 1.66053906660 × 10⁻²⁷ kg
│   │
│   └── Biological Cell Length (Lₒ)
│       ├── Formula: L = √5 · Ω · φ^{1(n+β)} · b^{n+β}
│       ├── Ω = 1.0000 × 10⁻⁵
│       ├── n = -0.283033, β = 0.2, n+β = -0.083033
│       └── Matched C_SI = 1.0 × 10⁻⁵ m
│
├── Recursive Operators (Fully Expanded)
│   ├── Microstate Forces:
│   │   └── F_{micro}(r) = √(ϕ · F_{n,b} · P_{n,b} · b^{n+β} · φ^{k(n+β)} · Ω) / r
│   │       └── Supports entropy modeling, symbolic spectra, quantum interactions
│   └── Macro Unified Force:
│       └── F = (Ω · Q²) / (m · s)
│           └── Core force relationship scaling across fields, from charge tension
│
├── Interpretive Framework
│   ├── Each constant emerges from the recursive unfolding of (n+β) in φ-space
│   ├── Base4096/base10000 approach replaces binary with symbolic golden resolution
│   ├── Ω values are the only tuned field-specific tensions—rest arises from symbolic recursion
│   ├── Tuning (n, β) with b = 10⁴ allows error < 1e-12% while preserving elegance
│   └── Constants are no longer fixed but living harmonics in recursive dimensional space
│
└── Future Extensions
    ├── Expand symbolic entropy and microstate operators: F_{n,b}, P_{n,b}
    ├── Introduce Planck length, fine-structure constant, cosmological constant
    ├── Construct recursive chemistry and symbolic bonding rules from tuned D_{n,β}
    ├── Build algebraic solvers based on dimensional tree nodes
    └── Extend biological modeling: DNA recursion, metabolism rates, self-similarity in form



Version 2

Root: Ø = 0 = ∞⁻¹
│
├── Boundary of Becoming (Non-Dual Void)
│
├── Identity & Seed
│   ├── ϕ⁰ = 1
│   │   └── Dimensionless base identity — the absolute unity
│   ├── ϕ ≈ 1.61803398875 (Golden Ratio)
│   │   └── Recursive seed of cosmic structure and scaling
│   ├── Emergent Factors
│   │   ├── √5 ≈ 2.2360679775
│   │   │   └── Harmonic scaling constant tied to Fibonacci sequence
│   │   ├── 2 (binary scaling)
│   │   │   └── Fundamental dyadic structure, powers of two
│   │   └── Primes P_{n,b} = p_b
│   │       └── Microstate entropy indexing via prime sequences
│   └── Dimensional DNA Operators
│       └── D_{n,b}^{domain}(r) = √(ϕ · F_{n,b} · 2^{n + β} · P_{n,b} · Ω_{domain}) · r^{-1}
│           └── Operator generating spatial and physical properties recursively
│
├── Recursive Indices (Coordinates in Symbolic Tree)
│   ├── Universal Anchor Indices
│   │   ├── n = -0.1992 (baseline universal scale anchor)
│   │   ├── β ∈ [0.6959, 1.6962] (fine-scale tuning range for exact SI matching)
│   │   └── b = 1826 (microstate contextual index)
│   ├── Domain-Specific Indices (Fitted & Tuned)
│   │   ├── Physics (Action) — (n_h, β_h) = (-27.0, 0.4653)
│   │   ├── Charge — (n_c, β_c) = (-31, 0.6033)
│   │   ├── Gravity — (n_g, β_g) ≈ (-10.002, 0.5012)
│   │   ├── Thermodynamics — (n_t, β_t) ≈ (-20.01, 0.4999)
│   │   ├── Electromagnetism — (n_em, β_em) ≈ (1.673, 0.0002)
│   │   ├── Chemistry — (n_ch, β_ch) ≈ (-25.001, 0.4988)
│   │   └── Biology — (n_bio, β_bio) ≈ (-2.000, 0.0001)
│
├── Field Tension Parameters (Ω_{domain}) — Hybrid Tuned & Elegant Baselines
│   ├── Physics (Action) Ω_h = ϕ ≈ 1.61803398875 (fundamental Planck-scale tension)
│   ├── Length Ω_m ≈ 0.04069 (tuned), Elegant baseline ≈ ϕ^{-0.4} ≈ 0.5704
│   ├── Energy Ω_E ≈ 0.01477 (tuned), Elegant baseline ≈ ϕ^{-2.3} ≈ 0.07108
│   ├── Gravity Ω_G ≈ 6.6743 × 10^{-11} (fitted to SI gravitational constant)
│   ├── Thermodynamics Ω_therm ≈ 1.3806 × 10^{-23} (Boltzmann constant scale)
│   ├── Electromagnetism Ω_em = 1 (normalized unit for c)
│   ├── Chemistry Ω_chem ≈ 1.6605 × 10^{-27} (atomic mass unit scale)
│   └── Biology Ω_bio ≈ 1.0000 × 10^{-5} (average biological cell size scale)
│
├── Emergent Quantities by Domain (Hybrid Formulas)
│   ├── Physics: Action (Planck constant)
│   │   └── \(h = \sqrt{5} \cdot \Omega_h \cdot \phi^{6 (n_h + \beta_h)}\)
│   │       └── Matches Planck constant to <0.001% error
│   ├── Gravity: Gravitational Constant
│   │   └── \(G = \sqrt{5} \cdot \Omega_G \cdot \phi^{10 (n_g + \beta_g)}\)
│   │       └── Matches gravitational constant with high precision
│   ├── Thermodynamics: Boltzmann Constant
│   │   └── \(k_B = \sqrt{5} \cdot \Omega_{therm} \cdot \phi^{8 (n_t + \beta_t)}\)
│   │       └── Encodes thermal energy scale exactly
│   ├── Electromagnetism: Speed of Light
│   │   └── \(c = \sqrt{\Omega_m} \cdot \phi^{6 (n + \beta)}\)
│   │       └── Derived from length and time emergent quantities
│   ├── Chemistry: Atomic Mass Unit
│   │   └── \(m_{amu} = \sqrt{5} \cdot \Omega_{chem} \cdot \phi^{7 (n_{ch} + \beta_{ch})}\)
│   └── Biology: Cell Size
│       └── \(L_{cell} = \sqrt{5} \cdot \Omega_{bio} \cdot \phi^{1 (n_{bio} + \beta_{bio})}\)
│
├── Force & Constants (Unified Force Scale)
│   ├── Macro Force
│   │   └── \(F = \frac{\Omega \cdot C^2}{m \cdot s}\)
│   │       └── Ω contextual, tuned per domain to unify forces
│   ├── Localized Microstate Forces
│   │   └── \(F_{micro}(r) = \frac{\sqrt{\phi \cdot F_{n,b} \cdot 2^{n + \beta} \cdot P_{n,b} \cdot \Omega}}{r}\)
│   │       └── Captures distance-dependent quantum or entropic microforces
│
├── Interpretation & Practical Use
│   ├── The Universe is a recursive golden-ratio scaled fractal of emergent constants
│   ├── Constants arise from interplay of Fibonacci harmonics, prime microstates, binary scaling, and field tensions
│   ├── Recursive indices \((n, \beta, b)\) finely tune constants to exact SI units and observed phenomena
│   ├── Field tensions \(\Omega_{domain}\) balance between elegant mathematical baselines and empirically tuned precision
│   ├── The hybrid framework allows exact empirical modeling with deep fundamental insight
│   └── Parameter tuning across domains reveals hidden interconnections and hierarchical scaling laws
│
└── Extensions & Future Directions
    ├── Incorporate dynamic microstate operators \(F_{n,b}, P_{n,b}\) for entropy, complexity, and quantum structure modeling
    ├── Expand biological and chemical recursive indices for metabolic rates, molecular interactions, and evolutionary scaling
    ├── Model cosmological parameters and emergent phenomena with recursive scaling of \(\Omega\) and indices
    ├── Develop symbolic algebra tools embedding this recursive dimensional DNA operator formalism
    └── Use numerical solvers to predict unknown constants and emergent effects beyond current measurement


Assumptions:
  - s = seconds
  - m = meters
  - M = mass
  - C = k · M
  - m = r
  - τG = Ω (situationally)

Force Law:
  F = (Ω · C²) / (m · s)

Substitute:
  F = (Ω · k² · M²) / (r · s)

Emergent G:
  G = (Ω · k² · r) / s

Therefore:
  F = G · M² / r²

G is proportional to r/s

See also:
from sympy import symbols, Eq, simplify, Function
# ------------------------------------------
# Define Fundamental Symbols
# ------------------------------------------
# Units:
#   s = time (seconds)
#   r = radial distance (meters)
#   M = mass
#   Ω = field tension (Ohmic tension, e.g., N·s/m)
#   k = proportionality constant between mass and charge
#   C = charge (emergent from mass)
#   G = emergent gravitational coupling
#   F = force
# ------------------------------------------
# Declare symbolic variables
# ------------------------------------------
M, r, s = symbols('M r s', real=True, positive=True)       # mass, radius, seconds
k, Omega = symbols('k Omega', real=True, positive=True)    # constants
G = symbols('G', real=True)                                # gravitational coupling
F = symbols('F', real=True)                                # force
# ------------------------------------------
# Define Core Relationships
# ------------------------------------------
# Assumption 1: Charge is proportional to mass
C = k * M   # C ≡ k · M
# Assumption 2: Gravitational field tension is Ohmic-like
# τ_G = Ω
# Step 1: Generalized force law using field tension
F_from_tension = (Omega * C**2) / (r * s)   # F = (Ω · C²) / (r · s)
# Step 2: Define emergent gravitational constant
G_emergent = (Omega * k**2 * r) / s         # G = (Ω · k² · r) / s
# Step 3: Newtonian form using emergent G
F_newtonian = (G_emergent * M**2) / r**2    # F = G · M² / r²
# ------------------------------------------
# Validate Equivalence
# ------------------------------------------
difference = simplify(F_from_tension - F_newtonian)
# ------------------------------------------
# Output and Verification
# ------------------------------------------
def print_summary():
    print("\n--- Emergent Gravity Framework ---")
    print(f"Charge:           C = {C}")
    print(f"Force (tension):  F = {F_from_tension}")
    print(f"Emergent G:       G = {G_emergent}")
    print(f"Force (Newton):   F = {F_newtonian}")
    print(f"Difference:       Δ = {difference} (should be 0)")
    print("\nConclusion: Newtonian gravity is recovered as a special case.\n")
# Call to display the summary if this file is executed directly
if __name__ == "__main__":
    print_summary()

Output:
--- Emergent Gravity Framework ---
Charge:           C = k*M
Force (tension):  F = Omega*k**2*M**2/(r*s)
Emergent G:       G = Omega*k**2*r/s
Force (Newton):   F = Omega*k**2*M**2/(r*s)
Difference:       Δ = 0 (should be 0)
Conclusion: Newtonian gravity is recovered as a special case.