[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
Conclusion: Continuum Hypothesis is true under recursive φ-field semantics. QED.
