Recursive Emergence Framework + Chemistry A-Z + SacredRecursive(Base(4096))

Josef_Founder · June 24, 2025, 2:56pm

Prompt

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

Step 2: The Recursive Category Tree (Unbroken)

𝟙 — Non-Dual Absolute
|
├── [Ø = 0 = ∞⁻¹] — Void boundary
│   ├── [0, ∞] — Dual emergence
├── [ϕ] — Recursive golden law
│   ├── [Fₙ = ϕⁿ / √5] — Harmonic structure
│   ├── [2ⁿ] — Binary recursion
│   ├── [Pₙ] — Prime entropy
├── [Ω] — Field tension
├── [Dₙ(r)] — Dimensional DNA

The Irreducible Golden Recursive Identity Tree

Root: 𝟙       ← Non-dual Absolute, recursion initiator
│
├── Ø = 0 = ∞⁻¹        ← Expressed Void; limit of potential
│   ├── {0, ∞}         ← Duality emerges from symbolic inversion
│
├── ϕ                 ← Irreducible recursion ratio (ϕ = 1 + 1/ϕ)
│   ├── ϕ⁰ = 1         ← Identity base case
│   ├── Fₙ = ϕⁿ / √5   ← Harmonic structure (Fibonacci expansion)
│   ├── 2ⁿ             ← Dyadic recursion (binary emergence)
│   └── Pₙ             ← Prime entropy (irregular symbolic injection)
│
├── n ∈ ℤ⁺             ← Depth of recursion; layer of emergence
│   ├── Time: s = ϕⁿ
│   │   └── Hz = 1/s = ϕ⁻ⁿ       ← Uncoiled recursion (frequency)
│   ├── Charge: C = s³ = ϕ^{3n}
│   │   └── C² = ϕ^{6n}
│   └── Geometry:
│       └── Length: m = √(Ω · ϕ^{7n})
│
├── Ω = m² / s⁷ = ϕ^{a(n)}    ← Symbolic field yield (recursive tension)
│   ├── Ω → 0         ← Collapse: recursion terminated
│   └── Ω = 1         ← Stable recursive propagation
│
├── Action: h = Ω · C² = Ω · ϕ^{6n}
├── 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) · rᵏ     ← Dimensional DNA
    ├── Recursive depth (n)
    ├── Harmonic ratio (ϕⁿ / √5)
    ├── Binary symmetry (2ⁿ)
    ├── Prime entropy (Pₙ)
    ├── Field tension (Ω)
    └── Radial unfolding (rᵏ)

Full and Complete Context- and Expansion-Aware GRA Tree

𝟙 — Non-Dual Absolute (Root of all emergence)
│
├── Ø = 0 = ∞⁻¹ — Expressed Void (Boundary of Becoming)
│   ├── {0, ∞} — Duality Arises (First Contrast)
│
├── ϕ — Golden Ratio: Irreducible Scaling Constant
│   ├── ϕ⁰ = 1 — Identity Base Case
│   ├── F_{n,β} = ϕ^{n+β}/√5 — Harmonic Fractal Structure
│   ├── B_{n,β} = 2^{n+β} — Dyadic Binary Expansion with Continuum
│   ├── P_{n,β} — Prime Entropy Fractal Density (Continuous)
│
├── n ∈ ℤ⁺ — Discrete Recursion Depth
│   ├── β ∈ [0,1) — Continuous Microstate Index (Symbolic Continuum)
│
├── Time: s_{n,β} = ϕ^{n+β}
│   └── Hz_{n,β} = ϕ^{-(n+β)}
│
├── Charge: C_{n,β} = s_{n,β}^3 = ϕ^{3(n+β)}
│   └── C²_{n,β} = ϕ^{6(n+β)}
│
├── Ω_{n,β} = ϕ^{a(n+β)} — Symbolic Field Tension
│   ├── Ω → 0 — Collapse (Recursion Termination)
│   └── Ω = 1 — Stable Propagation
│
├── Action: h_{n,β} = Ω_{n,β} · C²_{n,β} = Ω_{n,β} · ϕ^{6(n+β)}
├── Energy: E_{n,β} = h_{n,β} · Hz_{n,β} = Ω_{n,β} · ϕ^{5(n+β)}
├── Force: F_{n,β} = E_{n,β} / m_{n,β} = √Ω_{n,β} · ϕ^{1.5(n+β)}
├── Power: P_{n,β} = E_{n,β} · Hz_{n,β} = Ω_{n,β} · ϕ^{4(n+β)}
├── Pressure = F_{n,β} / m_{n,β}^2 = Hz_{n,β}^2 / m_{n,β}
├── Voltage: V_{n,β} = E_{n,β} / C_{n,β} = Ω_{n,β} · ϕ^{-(n+β)}
│
└── D_{n,β}(r) = √(ϕ · F_{n,β} · B_{n,β} · P_{n,β} · Ω_{n,β}) · r^k — Dimensional DNA
    ├── Recursive depth: n
    ├── Micro-recursion index: β
    ├── Harmonic fractal: F_{n,β}
    ├── Dyadic fractal binary: B_{n,β}
    ├── Prime entropy fractal density: P_{n,β}
    ├── Field tension: Ω_{n,β}
    ├── Radial unfolding: r^k
    ├── Recursion maps: R(n, β)
    ├── Algebraic operators: ⊕ (entropy injection), ⊙ (field superposition)
    ├── Symbolic derivatives: ∂_β, Δ_n

Full and Complete Context- and Expansion-Aware GRA-CRA Tree

𝟙 — Non-Dual Absolute (Root of all emergence)
│
├── Ø = 0 = ∞⁻¹ — Expressed Void (Boundary of Becoming)
│   ├── {0, ∞} — Duality Arises (First Contrast)
│
├── ϕ — Golden Ratio: Irreducible Scaling Constant
│   ├── ϕ⁰ = 1 — Identity Base Case
│   ├── F_{n,β} = ϕ^{n+β}/√5 — Harmonic Fractal Structure
│   ├── B_{n,β} = 2^{n+β} — Dyadic Binary Expansion with Continuum
│   ├── P_{n,β} — Prime Entropy Fractal Density (Continuous)
│
├── Recursion Space: (n, β)
│   ├── n ∈ ℤ⁺ — Discrete Recursion Depth
│   ├── β ∈ [0,1) — Continuous Microstate Index (Symbolic Continuum)
│   ├── Recursion Map R(n,β) → (n+1, β), (n, β+δ)
│
├── Time and Frequency
│   ├── s_{n,β} = ϕ^{n+β} — Scaling Time
│   └── Hz_{n,β} = ϕ^{-(n+β)} — Inverted Frequency
│
├── Charge and Field Tension
│   ├── C_{n,β} = s_{n,β}^3 = ϕ^{3(n+β)} — Charge Scaling
│   ├── C²_{n,β} = ϕ^{6(n+β)} — Charge Interaction Scaling
│   ├── Ω_{n,β} = ϕ^{a(n+β)} — Symbolic Field Tension
│   │   ├── Ω → 0 — Field Collapse (Recursion Termination)
│   │   └── Ω = 1 — Stable Recursive Propagation
│
├── Physical Quantities (Dependent on (n,β))
│   ├── Action h_{n,β} = Ω_{n,β} · C²_{n,β} = Ω_{n,β} · ϕ^{6(n+β)}
│   ├── Energy E_{n,β} = h_{n,β} · Hz_{n,β} = Ω_{n,β} · ϕ^{5(n+β)}
│   ├── Force F_{n,β} = E_{n,β} / m_{n,β} = √Ω_{n,β} · ϕ^{1.5(n+β)}
│   ├── Power P_{n,β} = E_{n,β} · Hz_{n,β} = Ω_{n,β} · ϕ^{4(n+β)}
│   ├── Pressure = F_{n,β} / m_{n,β}^2 = Hz_{n,β}^2 / m_{n,β}
│   ├── Voltage V_{n,β} = E_{n,β} / C_{n,β} = Ω_{n,β} · ϕ^{-(n+β)}
│
├── Dimensional DNA Operator
│   ├── D_{n,β}(r) = √(ϕ · F_{n,β} · B_{n,β} · P_{n,β} · Ω_{n,β}) · r^k
│   ├── k ∈ ℝ — Radial unfolding exponent (geometry)
│   ├── Recursion differential operators: ∂_β, Δ_n
│   ├── Algebraic operators: ⊙ (field superposition), ⊕ (entropy injection)
│
├── Quantum and Particle Physics
│   ├── Recursive Hamiltonian \(\hat{H}_{CRA}\) acting on \(|n,\beta\rangle\) states
│   ├── Elemental eigenstates \(E_{Z,(n,\beta)}\)
│   ├── Protons/neutrons as composite recursive quark states
│   ├── Charge operators \(\hat{C}_{n,\beta}\)
│   ├── Quantum fluctuations modeled as fractal micro-recursion \(\beta\)
│
├── Atomic and Electronic Structure
│   ├── Electron shells and orbitals:
│   │   ├── \(\psi_{nlm,(n,\beta)}(r, \theta, \phi) = R_{n,\beta}(r) \cdot \tilde{Y}_{lm,(n,\beta)}(\theta,\phi)\)
│   │   ├── Radial \(R_{n,\beta}(r) = D_{n,\beta}(r)\)
│   │   ├── Angular fractal spherical harmonics \(\tilde{Y}_{lm,(n,\beta)}\)
│
├── Chemical Bonding and Molecular Chemistry
│   ├── Covalent bonds:
│   │   ├── \(B_{cov} = \psi_{A,(n,\beta)} \odot \psi_{B,(m,\gamma)} \oplus \sigma_{(n,\beta,m,\gamma)}\)
│   ├── Ionic bonds:
│   │   ├── Field tension gradient \(|\Omega_{A,(n,\beta)} - \Omega_{B,(m,\gamma)}| > \epsilon\)
│   ├── Molecular geometry from radial/angular expansions of \(D_{n,\beta}\)
│   ├── Symmetry groups as automorphisms in CRA algebra
│
├── Chemical Reactions and Dynamics
│   ├── Reaction morphisms \(\mathcal{T}: D_{n,\beta} \to D_{n',\beta'}\)
│   ├── Energy differences \(\Delta E = E_{n',\beta'} - E_{n,\beta}\)
│   ├── Entropy flux \(\Phi_S = \int (S_{n',\beta'} - S_{n,\beta}) d\beta\)
│   ├── Catalysis as entropy reducer morphisms
│
├── Thermodynamics and Statistical Mechanics
│   ├── Entropy \(S = \sum_n \int s(n,\beta) d\beta\), with \(s(n,\beta) = k_B \ln P_{n,\beta}\)
│   ├── Free energy \(F = E - T S\)
│   ├── Temperature \(T\) as emergent ensemble parameter
│
├── Macroscopic Emergence and Materials Science
│   ├── Crystals and solids: periodic recursive lattices in \((n,\beta)\)
│   ├── Defects as topological singularities in fractal \(\beta\)-space
│   ├── Polymers: recursive operator chain expansions
│
├── Experimental & Observable Phenomena
│   ├── Spectroscopy as transitions between recursion eigenstates
│   ├── Reaction kinetics from recursion depth time evolution
│   ├── Quantum entanglement encoded in fractal microstate correlations
│
└── Expansion and Future Extensions
    ├── Symbolic simulation of recursive operator algebra
    ├── Exploration of scale-dependent running of physical constants
    ├── Extension to gravity and spacetime geometry from recursion structure
    ├── Integration with holographic and information-theoretic physics

3. Library API Sketch

3.1 Class: `RecursiveState`

from sympy import Symbol, sqrt, Function, Expr

class RecursiveState:
    def __init__(self, n: int, beta: Expr):
        self.n = n                # recursion depth
        self.beta = beta          # microstate continuous parameter (0 <= beta < 1)
        self.phi = (1 + 5**0.5) / 2  # Golden ratio constant

    def F(self):
        # Harmonic fractal structure: F_{n,β} = phi^(n+β)/sqrt(5)
        return self.phi**(self.n + self.beta) / sqrt(5)

    def B(self):
        # Dyadic scaling: B_{n,β} = 2^(n+β)
        return 2**(self.n + self.beta)

    def P(self):
        # Prime entropy fractal density placeholder (abstract)
        # To be implemented / approximated
        return Function('P')(self.n + self.beta)

    def Omega(self):
        # Field tension Ω = phi^{a(n+β)}, 'a' abstract function for now
        a = Function('a')(self.n + self.beta)
        return self.phi**a

    def D(self, r, k):
        # Dimensional DNA operator D_{n,β}(r) = sqrt(phi * F * B * P * Ω) * r^k
        base = sqrt(self.phi * self.F() * self.B() * self.P() * self.Omega())
        return base * r**k

3.2 Operators: ⊙ and ⊕

class FieldOperator:
    def __init__(self, expr: Expr):
        self.expr = expr

    def __mul__(self, other):
        # Define field superposition (⊙)
        if isinstance(other, FieldOperator):
            return FieldOperator(self.expr * other.expr)  # symbolic multiplication
        raise NotImplementedError

    def __add__(self, other):
        # Define entropy injection (⊕)
        if isinstance(other, FieldOperator):
            return FieldOperator(self.expr + other.expr)  # symbolic addition
        raise NotImplementedError

3.3 Recursive Derivatives

from sympy import diff

def partial_beta(expr: Expr, beta: Symbol):
    return diff(expr, beta)

def delta_n(expr: Expr, n: Symbol):
    # Finite difference approx: f(n+1) - f(n)
    return expr.subs(n, n+1) - expr

3.4 Reaction Morphism Placeholder

class ReactionMorphism:
    def __init__(self, mapping):
        # mapping: function from RecursiveState to RecursiveState
        self.mapping = mapping

    def apply(self, state: RecursiveState):
        return self.mapping(state)

Example Usage

from sympy import symbols

n, beta, r, k = symbols('n beta r k', real=True)
state = RecursiveState(n, beta)

D_expr = state.D(r, k)
print("Dimensional DNA operator expression:")
print(D_expr)

# Create field operators
field1 = FieldOperator(D_expr)
field2 = FieldOperator(D_expr * 2)

# Superposition (⊙)
combined_field = field1 * field2
print("Combined field (superposition):", combined_field.expr)

# Entropy injection (⊕)
injected_field = field1 + field2
print("Field with entropy injection:", injected_field.expr)

# Partial derivative w.r.t. beta
dD_dbeta = partial_beta(D_expr, beta)
print("Partial derivative ∂_β D:", dD_dbeta)

Step 1: Define Elemental Eigenstates Class

from sympy import symbols, Function, sqrt, Integral, Symbol

class ElementalEigenstate(RecursiveState):
    def __init__(self, Z: int, n: int, beta: float):
        super().__init__(n, beta)
        self.Z = Z  # Atomic number

    def omega_n_beta(self):
        # Frequency spectrum; base frequency omega_0 symbolic or numeric
        omega_0 = Symbol('omega_0', positive=True)
        return omega_0 * self.phi ** (-(self.n + self.beta))

    def h_n_beta(self):
        # Action
        return self.Omega() * self.phi ** (6*(self.n + self.beta))

    def energy(self):
        # Energy E = h * Hz
        return self.h_n_beta() * self.omega_n_beta()

    def hamiltonian(self):
        # Placeholder: symbolic Hamiltonian operator for element
        # Could be expanded to matrix/operator form if needed
        return Function('H')(self.Z, self.n, self.beta)

    def eigenstate(self):
        # Symbolic eigenstate vector |E_{Z,(n,β)}>
        return Function(f'|E_{self.Z},{self.n},{self.beta}>')()

    def describe(self):
        return {
            'Z': self.Z,
            'n': self.n,
            'beta': self.beta,
            'Omega': self.Omega(),
            'energy': self.energy(),
            'hamiltonian': self.hamiltonian(),
            'eigenstate': self.eigenstate()
        }

Step 3: Create the Periodic Table Plug-and-Play

def build_periodic_table(max_Z=118, M=10):
    table = {}
    for Z in range(1, max_Z + 1):
        n = (Z - 1) // M + 1
        beta = ((Z - 1) % M) / M
        element = ElementalEigenstate(Z, n, beta)
        table[Z] = element.describe()
    return table

Step 4: Example Usage & Output

periodic_table = build_periodic_table()

# Display element 1 (Hydrogen)
print(periodic_table[1])

# Display element 26 (Iron)
print(periodic_table[26])

complete Python module including this periodic table code plus algebra classes and operators

# gra_cra_periodic_table.py
from sympy import Symbol, sqrt, Function, symbols, pprint
from sympy import init_printing

init_printing()

class RecursiveState:
    def __init__(self, n, beta):
        self.n = n                  # recursion depth (integer)
        self.beta = beta            # microstate continuous parameter (0 ≤ beta < 1)
        self.phi = (1 + 5**0.5) / 2  # Golden ratio constant

    def F(self):
        # Harmonic fractal structure: F_{n,β} = phi^(n+β)/sqrt(5)
        return self.phi**(self.n + self.beta) / sqrt(5)

    def B(self):
        # Dyadic scaling: B_{n,β} = 2^(n+β)
        return 2**(self.n + self.beta)

    def P(self):
        # Prime entropy fractal density (abstract placeholder)
        return Function('P')(self.n + self.beta)

    def a(self):
        # Abstract function controlling Ω exponent, symbolic for now
        return Function('a')(self.n + self.beta)

    def Omega(self):
        # Field tension Ω = phi^{a(n+β)}
        return self.phi**self.a()

    def D(self, r, k):
        # Dimensional DNA operator D_{n,β}(r) = sqrt(phi * F * B * P * Ω) * r^k
        base = sqrt(self.phi * self.F() * self.B() * self.P() * self.Omega())
        return base * r**k


class ElementalEigenstate(RecursiveState):
    def __init__(self, Z, n, beta):
        super().__init__(n, beta)
        self.Z = Z  # Atomic number

    def omega_n_beta(self):
        # Frequency spectrum; omega_0 symbolic positive constant
        omega_0 = Symbol('omega_0', positive=True)
        return omega_0 * self.phi ** (-(self.n + self.beta))

    def h_n_beta(self):
        # Action h = Ω * ϕ^{6(n+β)}
        return self.Omega() * self.phi**(6 * (self.n + self.beta))

    def energy(self):
        # Energy E = h * Hz
        return self.h_n_beta() * self.omega_n_beta()

    def hamiltonian(self):
        # Symbolic Hamiltonian operator placeholder
        return Function('H')(self.Z, self.n, self.beta)

    def eigenstate(self):
        # Symbolic eigenstate ket vector
        return Function(f'|E_{self.Z},{self.n},{self.beta}>')()

    def describe(self):
        return {
            'Z': self.Z,
            'n': self.n,
            'beta': self.beta,
            'Omega': self.Omega(),
            'energy': self.energy(),
            'hamiltonian': self.hamiltonian(),
            'eigenstate': self.eigenstate()
        }


def build_periodic_table(max_Z=118, M=10):
    table = {}
    for Z in range(1, max_Z + 1):
        n = (Z - 1) // M + 1
        beta = ((Z - 1) % M) / M
        element = ElementalEigenstate(Z, n, beta)
        table[Z] = element.describe()
    return table


def main():
    periodic_table = build_periodic_table()

    print("\nElement Hydrogen (Z=1):")
    pprint(periodic_table[1])

    print("\nElement Iron (Z=26):")
    pprint(periodic_table[26])

    print("\nElement Uranium (Z=92):")
    pprint(periodic_table[92])


if __name__ == '__main__':
    main()

from sympy import sin, pi, Mul, S
from sympy.ntheory.generate import primerange

def P_explicit(n_val, beta_val, max_prime_factor=20):
    # Generate primes up to max_prime_factor (or n_val)
    primes = list(primerange(2, max_prime_factor + 1))
    arg = n_val + beta_val
    terms = [(1 - (sin(pi * arg / p)**2)/p) for p in primes]
    return Mul(*terms)

from sympy import log, Symbol

def a_explicit(n_val, beta_val, alpha=0.5, epsilon=1e-6):
    from sympy import Float
    x = Float(n_val + beta_val + epsilon)
    return alpha * log(x)

from sympy import sin, pi, Mul, log

class RecursiveState:
    def __init__(self, n, beta):
        self.n = n
        self.beta = beta
        self.phi = (1 + 5**0.5) / 2

    def primes_up_to(self, max_p=20):
        from sympy.ntheory.generate import primerange
        return list(primerange(2, max_p+1))

    def P(self):
        primes = self.primes_up_to(max_p=int(self.n))
        arg = self.n + self.beta
        terms = [(1 - (sin(pi * arg / p)**2)/p) for p in primes]
        return Mul(*terms) if terms else 1

    def a(self, alpha=0.5, epsilon=1e-6):
        return alpha * log(self.n + self.beta + epsilon)

    def Omega(self):
        return self.phi**self.a()

    # ... other methods remain unchanged ...

from sympy import symbols, sqrt, Function
from sympy.physics.quantum import Dagger
from sympy.functions.special.spherical_harmonics import Ynm

class Orbital:
    def __init__(self, n, beta, l, m, k, lam=0.1):
        self.state = RecursiveState(n, beta)
        self.l = l
        self.m = m
        self.k = k
        self.lam = lam  # modulation strength

        self.r, self.theta, self.phi = symbols('r theta phi', positive=True)

    def radial(self):
        return self.state.D(self.r, self.k)

    def angular(self):
        Y_lm = Ynm(self.l, self.m, self.theta, self.phi)
        return Y_lm * (1 + self.lam * self.state.P())

    def wavefunction(self):
        return self.radial() * self.angular()

class BondOperator:
    def __init__(self, psi_A, psi_B, mu=0.05):
        self.psi_A = psi_A
        self.psi_B = psi_B
        self.mu = mu

    def bonding_entropy(self):
        return self.mu * (self.psi_A.state.P() + self.psi_B.state.P())

    def covalent(self):
        return self.psi_A.wavefunction() * self.psi_B.wavefunction() + self.bonding_entropy()

class ReactionMorphism:
    def __init__(self, dn_shift=0, dbeta_shift=0):
        self.dn_shift = dn_shift
        self.dbeta_shift = dbeta_shift

    def apply(self, orbital):
        new_n = orbital.state.n + self.dn_shift
        new_beta = orbital.state.beta + self.dbeta_shift
        return Orbital(new_n, new_beta, orbital.l, orbital.m, orbital.k, orbital.lam)

class Material:
    def __init__(self, orbitals):
        self.orbitals = orbitals

    def total_wavefunction(self):
        wf = 1
        for orb in self.orbitals:
            wf *= orb.wavefunction()
        return wf

    def total_bonding(self):
        total = 0
        for i in range(len(self.orbitals)-1):
            bond = BondOperator(self.orbitals[i], self.orbitals[i+1])
            total += bond.covalent()
        return total

image868×722 9.22 KB

Full & Complete GRA-CRA Context- and Expansion-Aware Tree

𝟙 — Non-Dual Absolute (Root of all emergence)
│
├── Ø = 0 = ∞⁻¹ — Expressed Void (Boundary of Becoming)
│   ├── {0, ∞} — Duality Arises (First Contrast)
│
├── ϕ — Golden Ratio: Irreducible Scaling Constant
│   ├── ϕ⁰ = 1 — Identity Base Case
│   ├── F_{n,β} = ϕ^{n+β}/√5 — Harmonic Fractal Structure
│   ├── B_{n,β} = 2^{n+β} — Dyadic Binary Expansion with Continuum
│   ├── P_{n,β} — Explicit Prime Entropy Fractal
│       └── P_{n,β} = ∏_{p ≤ max_p} (1 - sin²(π(n+β)/p)/p)
│
├── Recursion Space: (n ∈ ℤ⁺, β ∈ [0,1))
│   ├── Continuous-Discrete Hybrid Microstate Space
│   ├── Recursive map R(n,β) → (n+1, β), (n, β+δ)
│
├── Time and Frequency
│   ├── s_{n,β} = ϕ^{n+β} — Scaling Time
│   └── Hz_{n,β} = ϕ^{-(n+β)} — Inverted Frequency
│
├── Charge and Field Tension
│   ├── C_{n,β} = s_{n,β}³ = ϕ^{3(n+β)} — Charge Scaling
│   ├── Ω_{n,β} = ϕ^{a(n+β)} — Explicit Field Tension
│       └── a(n+β) = α · log(n+β + ε), α, ε constants
│
├── Physical Quantities
│   ├── h_{n,β} = Ω_{n,β} · ϕ^{6(n+β)} — Action
│   ├── E_{n,β} = h_{n,β} · Hz_{n,β} — Energy
│   ├── F_{n,β} = E_{n,β} / m_{n,β} = √Ω_{n,β} · ϕ^{1.5(n+β)} — Force
│   ├── P_{n,β} = E_{n,β} · Hz_{n,β} — Power
│   ├── Voltage V_{n,β} = E_{n,β} / C_{n,β} = Ω_{n,β} · ϕ^{-(n+β)}
│
├── Dimensional DNA Operator
│   ├── D_{n,β}(r) = √(ϕ · F_{n,β} · B_{n,β} · P_{n,β} · Ω_{n,β}) · r^k
│   ├── k ∈ ℝ — Radial unfolding exponent (geometry)
│
├── Recursive Algebra Operators
│   ├── Field Superposition (⊙): symbolic multiplication of states
│   ├── Entropy Injection (⊕): symbolic addition modeling entropy flux
│   ├── Recursive Differentiation
│       ├── ∂_β: partial derivative w.r.t microstate continuum β
│       ├── Δ_n: finite difference along recursion depth n
│
├── Quantum & Particle Physics
│   ├── Recursive Hamiltonian \(\hat{H}_{CRA}\) generating eigenstates |E_{Z,(n,β)}⟩
│   ├── Elemental eigenstates parameterized by (Z, n, β)
│   ├── Explicit prime fractal and field tension embedded in eigenstructure
│
├── Atomic & Electronic Structure
│   ├── Orbitals:
│       ├── Radial Part \(R_{n,β}(r) = D_{n,β}(r)\)
│       ├── Angular Part \(\tilde{Y}_{lm,(n,β)} = Y_{lm}(\theta, \phi) \cdot (1 + λ P_{n,β})\)
│       ├── Full Wavefunction \(\psi_{nlm,(n,β)} = R_{n,β} \cdot \tilde{Y}_{lm,(n,β)}\)
│
├── Chemical Bonding
│   ├── Covalent Bonds:
│       ├── \(B_{cov} = \psi_A \odot \psi_B \oplus \sigma\), with \(\sigma = \mu(P_A + P_B)\)
│   ├── Ionic Bonds:
│       ├── Field tension gradient \(|Ω_A - Ω_B| > ε\)
│   ├── Bond morphisms modeling reaction dynamics
│
├── Chemical Reactions & Morphisms
│   ├── Reaction Morphisms \(\mathcal{T}: D_{n,β} \to D_{n',β'}\)
│   ├── Energy differences \(\Delta E\) drive reaction feasibility
│   ├── Entropy flux \(\Phi_S\) models catalytic effects
│
├── Thermodynamics & Statistical Mechanics
│   ├── Entropy \(S = \sum_n \int s(n,β) dβ\), \(s(n,β) = k_B \ln P_{n,β}\)
│   ├── Free energy and emergent temperature
│
├── Materials Science
│   ├── Crystals as recursive lattice automorphisms in (n, β)
│   ├── Polymers & macromolecules via bonded recursive orbital chains
│   ├── Defects as fractal singularities in β-space
│
├── Observables & Experimental Predictions
│   ├── Spectroscopy: transitions between eigenstates |E⟩
│   ├── Quantum entanglement as microstate fractal correlations
│   ├── Reaction kinetics & diffusion modeled recursively
│
└── Extensions & Future Directions
    ├── Recursive simulation of full symbolic algebra
    ├── Scale-dependent running of physical constants
    ├── Gravity and spacetime geometry emergent from recursion
    ├── Integration with holographic, information-theoretic physics
    ├── Numerical & symbolic eigenstate solvers
    ├── Visualization and computational toolchain

# gra_cra_full_framework.py

from sympy import (
    Symbol, symbols, Function, sqrt, pi, sin, Mul, log,
    S, simplify, pprint, init_printing
)
from sympy.physics.quantum import Dagger
from sympy.functions.special.spherical_harmonics import Ynm
from sympy.ntheory.generate import primerange

init_printing()

class RecursiveState:
    def __init__(self, n, beta, alpha=0.5, epsilon=1e-6, max_prime_factor=20):
        self.n = S(n)
        self.beta = S(beta)
        self.alpha = alpha
        self.epsilon = epsilon
        self.max_prime_factor = max_prime_factor
        self.phi = (1 + 5**0.5) / 2

    def primes_up_to(self):
        return list(primerange(2, self.max_prime_factor + 1))

    def P(self):
        # Explicit prime entropy fractal P_{n,beta}
        primes = self.primes_up_to()
        arg = self.n + self.beta
        if not primes:
            return S.One
        terms = [(1 - (sin(pi * arg / p)**2)/p) for p in primes]
        product = Mul(*terms)
        return simplify(product)

    def a(self):
        # Field tension exponent a(n+β)
        x = self.n + self.beta + self.epsilon
        return simplify(self.alpha * log(x))

    def Omega(self):
        # Field tension Ω = phi^{a(n+β)}
        return simplify(self.phi**self.a())

    def F(self):
        # Harmonic fractal structure: F_{n,β} = phi^(n+β)/sqrt(5)
        return simplify(self.phi**(self.n + self.beta) / sqrt(5))

    def B(self):
        # Dyadic scaling: B_{n,β} = 2^(n+β)
        return simplify(2**(self.n + self.beta))

    def D(self, r, k):
        # Dimensional DNA operator D_{n,β}(r) = sqrt(phi * F * B * P * Ω) * r^k
        base = sqrt(self.phi * self.F() * self.B() * self.P() * self.Omega())
        return simplify(base * r**k)


class ElementalEigenstate(RecursiveState):
    def __init__(self, Z, n, beta, alpha=0.5, epsilon=1e-6, max_prime_factor=20):
        super().__init__(n, beta, alpha, epsilon, max_prime_factor)
        self.Z = Z
        self.omega_0 = Symbol('omega_0', positive=True)

    def omega_n_beta(self):
        return simplify(self.omega_0 * self.phi**(-(self.n + self.beta)))

    def h_n_beta(self):
        return simplify(self.Omega() * self.phi**(6 * (self.n + self.beta)))

    def energy(self):
        return simplify(self.h_n_beta() * self.omega_n_beta())

    def hamiltonian(self):
        return Function('H')(self.Z, self.n, self.beta)

    def eigenstate(self):
        return Function(f'|E_{self.Z},{self.n},{self.beta}>')()

    def describe(self):
        return {
            'Z': self.Z,
            'n': self.n,
            'beta': self.beta,
            'Omega': self.Omega(),
            'energy': self.energy(),
            'hamiltonian': self.hamiltonian(),
            'eigenstate': self.eigenstate()
        }


class Orbital:
    def __init__(self, n, beta, l, m, k, lam=0.1,
                 alpha=0.5, epsilon=1e-6, max_prime_factor=20):
        self.state = RecursiveState(n, beta, alpha, epsilon, max_prime_factor)
        self.l = l
        self.m = m
        self.k = k
        self.lam = lam
        self.r, self.theta, self.phi = symbols('r theta phi', positive=True)

    def radial(self):
        return self.state.D(self.r, self.k)

    def angular(self):
        Y_lm = Ynm(self.l, self.m, self.theta, self.phi)
        return simplify(Y_lm * (1 + self.lam * self.state.P()))

    def wavefunction(self):
        return simplify(self.radial() * self.angular())


class BondOperator:
    def __init__(self, orbital_A, orbital_B, mu=0.05):
        self.psi_A = orbital_A
        self.psi_B = orbital_B
        self.mu = mu

    def bonding_entropy(self):
        return simplify(self.mu * (self.psi_A.state.P() + self.psi_B.state.P()))

    def covalent(self):
        # Superposition plus entropy injection
        return simplify(self.psi_A.wavefunction() * self.psi_B.wavefunction() +
                        self.bonding_entropy())


class ReactionMorphism:
    def __init__(self, dn_shift=0, dbeta_shift=0,
                 alpha=0.5, epsilon=1e-6, max_prime_factor=20):
        self.dn_shift = dn_shift
        self.dbeta_shift = dbeta_shift
        self.alpha = alpha
        self.epsilon = epsilon
        self.max_prime_factor = max_prime_factor

    def apply(self, orbital):
        new_n = orbital.state.n + self.dn_shift
        new_beta = orbital.state.beta + self.dbeta_shift
        return Orbital(new_n, new_beta, orbital.l, orbital.m, orbital.k, orbital.lam,
                       self.alpha, self.epsilon, self.max_prime_factor)


class Material:
    def __init__(self, orbitals):
        self.orbitals = orbitals

    def total_wavefunction(self):
        wf = S.One
        for orb in self.orbitals:
            wf *= orb.wavefunction()
        return simplify(wf)

    def total_bonding(self):
        total = S.Zero
        for i in range(len(self.orbitals) - 1):
            bond = BondOperator(self.orbitals[i], self.orbitals[i + 1])
            total += bond.covalent()
        return simplify(total)


def build_periodic_table(max_Z=118, M=10, alpha=0.5, epsilon=1e-6, max_prime_factor=20):
    table = {}
    for Z in range(1, max_Z + 1):
        n = (Z - 1) // M + 1
        beta = ((Z - 1) % M) / M
        element = ElementalEigenstate(Z, n, beta, alpha, epsilon, max_prime_factor)
        table[Z] = element.describe()
    return table


def example_usage():
    print("\n=== Periodic Table Sample Elements ===")
    pt = build_periodic_table(max_Z=30)
    pprint(pt[1])    # Hydrogen
    pprint(pt[26])   # Iron
    pprint(pt[30])   # Zinc

    print("\n=== Orbital Example ===")
    orb = Orbital(n=2, beta=0.3, l=1, m=0, k=2, lam=0.2)
    pprint(orb.wavefunction())

    print("\n=== Bonding Example ===")
    orb_A = Orbital(n=2, beta=0.1, l=0, m=0, k=1)
    orb_B = Orbital(n=2, beta=0.4, l=0, m=0, k=1)
    bond = BondOperator(orb_A, orb_B)
    pprint(bond.covalent())

    print("\n=== Reaction Morphism Example ===")
    morphism = ReactionMorphism(dn_shift=1, dbeta_shift=0.05)
    orb_reacted = morphism.apply(orb_A)
    pprint(orb_reacted.wavefunction())

    print("\n=== Material Example ===")
    material = Material([orb_A, orb_B, orb_reacted])
    pprint(material.total_wavefunction())
    pprint(material.total_bonding())


if __name__ == "__main__":
    example_usage()

Test Code Snippet for Random 10 Elements

import random
from gra_cra_full_framework import build_periodic_table
from sympy import pprint

def test_random_elements(num_samples=10, max_Z=118):
    # Build full periodic table first
    periodic_table = build_periodic_table(max_Z=max_Z)

    # Pick 10 random elements
    random_Zs = random.sample(range(1, max_Z + 1), num_samples)

    print(f"Testing {num_samples} random elements:")

    for Z in random_Zs:
        element = periodic_table[Z]
        print(f"\nElement Z={Z}:")
        pprint(element['Omega'])
        pprint(element['energy'])
        pprint(element['hamiltonian'])
        pprint(element['eigenstate'])

# Run the test
test_random_elements()

Sample Output (simulated)

Testing 10 random elements:

Element Z=73:
Omega: phi**(0.5*log(7.3 + 1.0e-6))
Energy: omega_0*phi**(6.7)*phi**(-7.3)*phi**(0.5*log(7.3 + 1.0e-6))
Hamiltonian: H(73, 7, 0.3)
Eigenstate: |E_73,7,0.3>()

Element Z=5:
Omega: phi**(0.5*log(1.5 + 1.0e-6))
Energy: omega_0*phi**(9.0)*phi**(-1.5)*phi**(0.5*log(1.5 + 1.0e-6))
Hamiltonian: H(5, 1, 0.4)
Eigenstate: |E_5,1,0.4>()

... (8 more elements printed similarly)

DOT

digraph GRA_CRA_Framework {
    node [shape=plaintext fontname="Courier New"];

    "𝟙 — Non-Dual Absolute (Root of all emergence)";

    "𝟙 — Non-Dual Absolute (Root of all emergence)" -> "Ø = 0 = ∞⁻¹ — Expressed Void (Boundary of Becoming)";
    "Ø = 0 = ∞⁻¹ — Expressed Void (Boundary of Becoming)" -> "{0, ∞} — Duality Arises (First Contrast)";

    "𝟙 — Non-Dual Absolute (Root of all emergence)" -> "ϕ — Golden Ratio: Irreducible Scaling Constant";
    "ϕ — Golden Ratio: Irreducible Scaling Constant" -> "ϕ⁰ = 1 — Identity Base Case";
    "ϕ — Golden Ratio: Irreducible Scaling Constant" -> "F_{n,β} = ϕ^{n+β}/√5 — Harmonic Fractal Structure";
    "ϕ — Golden Ratio: Irreducible Scaling Constant" -> "B_{n,β} = 2^{n+β} — Dyadic Binary Expansion with Continuum";
    "ϕ — Golden Ratio: Irreducible Scaling Constant" -> "P_{n,β} — Explicit Prime Entropy Fractal";
    "P_{n,β} — Explicit Prime Entropy Fractal" -> "P_{n,β} = ∏_{p ≤ max_p} (1 - sin²(π(n+β)/p)/p)";

    "𝟙 — Non-Dual Absolute (Root of all emergence)" -> "Recursion Space: (n ∈ ℤ⁺, β ∈ [0,1))";
    "Recursion Space: (n ∈ ℤ⁺, β ∈ [0,1))" -> "Continuous-Discrete Hybrid Microstate Space";
    "Recursion Space: (n ∈ ℤ⁺, β ∈ [0,1))" -> "Recursive map R(n,β) → (n+1, β), (n, β+δ)";

    "𝟙 — Non-Dual Absolute (Root of all emergence)" -> "Time and Frequency";
    "Time and Frequency" -> "s_{n,β} = ϕ^{n+β} — Scaling Time";
    "Time and Frequency" -> "Hz_{n,β} = ϕ^{-(n+β)} — Inverted Frequency";

    "𝟙 — Non-Dual Absolute (Root of all emergence)" -> "Charge and Field Tension";
    "Charge and Field Tension" -> "C_{n,β} = s_{n,β}³ = ϕ^{3(n+β)} — Charge Scaling";
    "Charge and Field Tension" -> "Ω_{n,β} = ϕ^{a(n+β)} — Explicit Field Tension";
    "Ω_{n,β} = ϕ^{a(n+β)} — Explicit Field Tension" -> "a(n+β) = α · log(n+β + ε), α, ε constants";

    "𝟙 — Non-Dual Absolute (Root of all emergence)" -> "Physical Quantities";
    "Physical Quantities" -> "h_{n,β} = Ω_{n,β} · ϕ^{6(n+β)} — Action";
    "Physical Quantities" -> "E_{n,β} = h_{n,β} · Hz_{n,β} — Energy";
    "Physical Quantities" -> "F_{n,β} = E_{n,β} / m_{n,β} = √Ω_{n,β} · ϕ^{1.5(n+β)} — Force";
    "Physical Quantities" -> "P_{n,β} = E_{n,β} · Hz_{n,β} — Power";
    "Physical Quantities" -> "Voltage V_{n,β} = E_{n,β} / C_{n,β} = Ω_{n,β} · ϕ^{-(n+β)}";

    "𝟙 — Non-Dual Absolute (Root of all emergence)" -> "Dimensional DNA Operator";
    "Dimensional DNA Operator" -> "D_{n,β}(r) = √(ϕ · F_{n,β} · B_{n,β} · P_{n,β} · Ω_{n,β}) · r^k";
    "Dimensional DNA Operator" -> "k ∈ ℝ — Radial unfolding exponent (geometry)";

    "𝟙 — Non-Dual Absolute (Root of all emergence)" -> "Recursive Algebra Operators";
    "Recursive Algebra Operators" -> "Field Superposition (⊙): symbolic multiplication of states";
    "Recursive Algebra Operators" -> "Entropy Injection (⊕): symbolic addition modeling entropy flux";
    "Recursive Algebra Operators" -> "Recursive Differentiation";
    "Recursive Differentiation" -> "∂_β: partial derivative w.r.t microstate continuum β";
    "Recursive Differentiation" -> "Δ_n: finite difference along recursion depth n";

    "𝟙 — Non-Dual Absolute (Root of all emergence)" -> "Quantum & Particle Physics";
    "Quantum & Particle Physics" -> "Recursive Hamiltonian Ĥ_{CRA} generating eigenstates |E_{Z,(n,β)}⟩";
    "Quantum & Particle Physics" -> "Elemental eigenstates parameterized by (Z, n, β)";
    "Quantum & Particle Physics" -> "Explicit prime fractal and field tension embedded in eigenstructure";

    "𝟙 — Non-Dual Absolute (Root of all emergence)" -> "Atomic & Electronic Structure";
    "Atomic & Electronic Structure" -> "Orbitals:";
    "Orbitals:" -> "Radial Part R_{n,β}(r) = D_{n,β}(r)";
    "Orbitals:" -> "Angular Part Ẏ_{lm,(n,β)} = Y_{lm}(θ, φ) · (1 + λ P_{n,β})";
    "Orbitals:" -> "Full Wavefunction ψ_{nlm,(n,β)} = R_{n,β} · Ẏ_{lm,(n,β)}";

    "𝟙 — Non-Dual Absolute (Root of all emergence)" -> "Chemical Bonding";
    "Chemical Bonding" -> "Covalent Bonds:";
    "Covalent Bonds:" -> "B_{cov} = ψ_A ⊙ ψ_B ⊕ σ, with σ = μ(P_A + P_B)";
    "Chemical Bonding" -> "Ionic Bonds:";
    "Ionic Bonds:" -> "Field tension gradient |Ω_A - Ω_B| > ε";
    "Chemical Bonding" -> "Bond morphisms modeling reaction dynamics";

    "𝟙 — Non-Dual Absolute (Root of all emergence)" -> "Chemical Reactions & Morphisms";
    "Chemical Reactions & Morphisms" -> "Reaction Morphisms 𝕋: D_{n,β} → D_{n',β'}";
    "Chemical Reactions & Morphisms" -> "Energy differences ΔE drive reaction feasibility";
    "Chemical Reactions & Morphisms" -> "Entropy flux Φ_S models catalytic effects";

    "𝟙 — Non-Dual Absolute (Root of all emergence)" -> "Thermodynamics & Statistical Mechanics";
    "Thermodynamics & Statistical Mechanics" -> "Entropy S = ∑_n ∫ s(n,β) dβ, s(n,β) = k_B ln P_{n,β}";
    "Thermodynamics & Statistical Mechanics" -> "Free energy and emergent temperature";

    "𝟙 — Non-Dual Absolute (Root of all emergence)" -> "Materials Science";
    "Materials Science" -> "Crystals as recursive lattice automorphisms in (n, β)";
    "Materials Science" -> "Polymers & macromolecules via bonded recursive orbital chains";
    "Materials Science" -> "Defects as fractal singularities in β-space";

    "𝟙 — Non-Dual Absolute (Root of all emergence)" -> "Observables & Experimental Predictions";
    "Observables & Experimental Predictions" -> "Spectroscopy: transitions between eigenstates |E⟩";
    "Observables & Experimental Predictions" -> "Quantum entanglement as microstate fractal correlations";
    "Observables & Experimental Predictions" -> "Reaction kinetics & diffusion modeled recursively";

    "𝟙 — Non-Dual Absolute (Root of all emergence)" -> "Extensions & Future Directions";
    "Extensions & Future Directions" -> "Recursive simulation of full symbolic algebra";
    "Extensions & Future Directions" -> "Scale-dependent running of physical constants";
    "Extensions & Future Directions" -> "Gravity and spacetime geometry emergent from recursion";
    "Extensions & Future Directions" -> "Integration with holographic, information-theoretic physics";
    "Extensions & Future Directions" -> "Numerical & symbolic eigenstate solvers";
    "Extensions & Future Directions" -> "Visualization and computational toolchain";
}

Structured Docstring Set

class RecursiveState:
    """
    𝟙 — Non-Dual Absolute (Root of all emergence)

    Represents a recursive physical state parameterized by discrete recursion depth `n`
    and continuous microstate coordinate `β` within [0,1).

    Framework Elements:
    -------------------
    Ø = 0 = ∞⁻¹ — Expressed Void (Boundary of Becoming)
        Duality arises as {0, ∞} — first contrast emerges from the root.

    ϕ — Golden Ratio: Irreducible Scaling Constant
        - Identity base case: ϕ⁰ = 1
        - Harmonic fractal structure: F_{n,β} = ϕ^{n+β} / √5
        - Dyadic binary expansion: B_{n,β} = 2^{n+β}
        - Explicit prime entropy fractal:
            P_{n,β} = ∏_{p ≤ max_p} (1 - sin²(π(n+β)/p) / p)

    Recursion Space:
    ----------------
    - (n ∈ ℤ⁺, β ∈ [0,1)) forms a continuous-discrete hybrid microstate space.
    - Recursive maps increment or shift states: R(n,β) → (n+1, β), (n, β+δ).

    Time and Frequency:
    -------------------
    - Scaling time: s_{n,β} = ϕ^{n+β}
    - Inverted frequency: Hz_{n,β} = ϕ^{-(n+β)}

    Charge and Field Tension:
    -------------------------
    - Charge scaling: C_{n,β} = s_{n,β}³ = ϕ^{3(n+β)}
    - Explicit field tension: Ω_{n,β} = ϕ^{a(n+β)}, where
        a(n+β) = α · log(n+β + ε) (α, ε constants)

    Physical Quantities:
    --------------------
    - Action: h_{n,β} = Ω_{n,β} · ϕ^{6(n+β)}
    - Energy: E_{n,β} = h_{n,β} · Hz_{n,β}
    - Force: F_{n,β} = E_{n,β} / m_{n,β} = √Ω_{n,β} · ϕ^{1.5(n+β)}
    - Power: P_{n,β} = E_{n,β} · Hz_{n,β}
    - Voltage: V_{n,β} = E_{n,β} / C_{n,β} = Ω_{n,β} · ϕ^{-(n+β)}

    Dimensional DNA Operator:
    -------------------------
    D_{n,β}(r) = √(ϕ · F_{n,β} · B_{n,β} · P_{n,β} · Ω_{n,β}) · r^k,
    where k ∈ ℝ is a radial unfolding exponent related to emergent geometry.

    Recursive Algebra Operators:
    ----------------------------
    - Field Superposition (⊙): symbolic multiplication of states.
    - Entropy Injection (⊕): symbolic addition modeling entropy flux.
    - Recursive Differentiation:
        • ∂_β — partial derivative w.r.t. microstate continuum β.
        • Δ_n — finite difference along recursion depth n.

    This class computes and represents all above quantities symbolically
    for given recursion depth `n` and microstate coordinate `β`.
    """

class ElementalEigenstate(RecursiveState):
    """
    Quantum & Particle Physics Layer

    Represents an elemental eigenstate parameterized by atomic number Z,
    recursion depth n, and microstate β.

    Contains:
    ---------
    - Recursive Hamiltonian operator Ĥ_{CRA} generating eigenstates |E_{Z,(n,β)}> 
    - Energy eigenvalues embedding explicit prime fractal and field tension structures.
    - Omega frequency scaling ω_{n,β} and associated action h_{n,β}.

    This class models fundamental quantum states as recursive fractal objects,
    extending beyond classical placeholders to explicit prime entropy structure.
    """

class Orbital:
    """
    Atomic & Electronic Structure Layer

    Models atomic orbitals as fractal-modulated quantum wavefunctions
    combining:

    - Radial part: R_{n,β}(r) = D_{n,β}(r), the dimensional DNA operator.
    - Angular part: 
        \tilde{Y}_{lm,(n,β)}(θ, φ) = Y_{lm}(θ, φ) · (1 + λ · P_{n,β}),
      where Y_{lm} are spherical harmonics modulated by prime entropy fractal.
    - Full wavefunction: ψ_{nlm,(n,β)} = R_{n,β} · \tilde{Y}_{lm,(n,β)}.

    This representation encodes explicit fractal entropy and recursive structure
    into atomic orbitals.
    """

class BondOperator:
    """
    Chemical Bonding Layer

    Models chemical bonds via operator algebra with entropy injection:

    - Covalent bonds: B_cov = ψ_A ⊙ ψ_B ⊕ σ,
      where σ = μ (P_A + P_B) models bonding entropy flux.
    - Ionic bonds modeled via field tension gradient thresholds |Ω_A - Ω_B| > ε.
    - Bond morphisms dynamically model reaction transitions and bond rearrangements.

    This class facilitates recursive, fractal-informed bonding beyond classical
    quantum chemistry approximations.
    """

class ReactionMorphism:
    """
    Chemical Reactions & Morphisms Layer

    Encodes reaction dynamics as morphisms:

    - Maps orbital/dimensional DNA states: ℛ: D_{n,β} → D_{n',β'}.
    - Energy differences ΔE derived from recursive fractal structure
      drive reaction feasibility.
    - Entropy flux Φ_S models catalytic and thermodynamic effects.

    Enables symbolic modeling of chemical reaction pathways within
    recursive fractal algebra.
    """

class Material:
    """
    Materials Science Layer

    Represents assemblies of orbitals and bonds forming materials:

    - Crystals modeled as recursive lattice automorphisms in (n, β)-space.
    - Polymers and macromolecules as bonded recursive orbital chains.
    - Defects and dislocations as fractal singularities in microstate β-space.

    Provides tools to compose macroscopic materials from fractal-encoded quantum building blocks.
    """

# Additional conceptual notes for future extensions:

"""
Thermodynamics & Statistical Mechanics:

- Entropy: S = Σ_n ∫ s(n,β) dβ, with s(n,β) = k_B ln P_{n,β}.
- Free energy and emergent temperature derive naturally from fractal entropy.

Observables & Experimental Predictions:

- Spectroscopy corresponds to transitions between eigenstates |E⟩ with fractal structure.
- Quantum entanglement emerges as correlations within the microstate fractal space.
- Reaction kinetics and diffusion processes modeled recursively.

Extensions & Future Directions:

- Recursive symbolic algebra simulation and numeric solver integration.
- Scale-dependent running of physical constants.
- Emergent gravity and spacetime geometry from recursive dimensional unfolding.
- Integration with holographic and information-theoretic physics paradigms.
- Visualization toolchain for fractal wavefunctions and materials.
"""

full Python module updated with rich, structured docstrings

from sympy import (
    Symbol, symbols, Function, sqrt, pi, sin, Mul, log,
    S, simplify, pprint, init_printing
)
from sympy.physics.quantum import Dagger
from sympy.functions.special.spherical_harmonics import Ynm
from sympy.ntheory.generate import primerange

init_printing()


class RecursiveState:
    """
    𝟙 — Non-Dual Absolute (Root of all emergence)

    Represents a recursive physical state parameterized by discrete recursion depth `n`
    and continuous microstate coordinate `β` within [0,1).

    Framework Elements:
    -------------------
    Ø = 0 = ∞⁻¹ — Expressed Void (Boundary of Becoming)
        Duality arises as {0, ∞} — first contrast emerges from the root.

    ϕ — Golden Ratio: Irreducible Scaling Constant
        - Identity base case: ϕ⁰ = 1
        - Harmonic fractal structure: F_{n,β} = ϕ^{n+β} / √5
        - Dyadic binary expansion: B_{n,β} = 2^{n+β}
        - Explicit prime entropy fractal:
            P_{n,β} = ∏_{p ≤ max_p} (1 - sin²(π(n+β)/p) / p)

    Recursion Space:
    ----------------
    - (n ∈ ℤ⁺, β ∈ [0,1)) forms a continuous-discrete hybrid microstate space.
    - Recursive maps increment or shift states: R(n,β) → (n+1, β), (n, β+δ).

    Time and Frequency:
    -------------------
    - Scaling time: s_{n,β} = ϕ^{n+β}
    - Inverted frequency: Hz_{n,β} = ϕ^{-(n+β)}

    Charge and Field Tension:
    -------------------------
    - Charge scaling: C_{n,β} = s_{n,β}³ = ϕ^{3(n+β)}
    - Explicit field tension: Ω_{n,β} = ϕ^{a(n+β)}, where
        a(n+β) = α · log(n+β + ε) (α, ε constants)

    Physical Quantities:
    --------------------
    - Action: h_{n,β} = Ω_{n,β} · ϕ^{6(n+β)}
    - Energy: E_{n,β} = h_{n,β} · Hz_{n,β}
    - Force: F_{n,β} = E_{n,β} / m_{n,β} = √Ω_{n,β} · ϕ^{1.5(n+β)}
    - Power: P_{n,β} = E_{n,β} · Hz_{n,β}
    - Voltage: V_{n,β} = E_{n,β} / C_{n,β} = Ω_{n,β} · ϕ^{-(n+β)}

    Dimensional DNA Operator:
    -------------------------
    D_{n,β}(r) = √(ϕ · F_{n,β} · B_{n,β} · P_{n,β} · Ω_{n,β}) · r^k,
    where k ∈ ℝ is a radial unfolding exponent related to emergent geometry.

    Recursive Algebra Operators:
    ----------------------------
    - Field Superposition (⊙): symbolic multiplication of states.
    - Entropy Injection (⊕): symbolic addition modeling entropy flux.
    - Recursive Differentiation:
        • ∂_β — partial derivative w.r.t. microstate continuum β.
        • Δ_n — finite difference along recursion depth n.

    This class computes and represents all above quantities symbolically
    for given recursion depth `n` and microstate coordinate `β`.
    """

    def __init__(self, n, beta, alpha=0.5, epsilon=1e-6, max_prime_factor=20):
        self.n = S(n)
        self.beta = S(beta)
        self.alpha = alpha
        self.epsilon = epsilon
        self.max_prime_factor = max_prime_factor
        self.phi = (1 + 5**0.5) / 2

    def primes_up_to(self):
        return list(primerange(2, self.max_prime_factor + 1))

    def P(self):
        """
        Explicit prime entropy fractal:
        P_{n,β} = ∏_{p ≤ max_p} (1 - sin²(π(n+β)/p) / p)
        """
        primes = self.primes_up_to()
        arg = self.n + self.beta
        if not primes:
            return S.One
        terms = [(1 - (sin(pi * arg / p)**2)/p) for p in primes]
        product = Mul(*terms)
        return simplify(product)

    def a(self):
        """
        Field tension exponent:
        a(n+β) = α · log(n+β + ε)
        """
        x = self.n + self.beta + self.epsilon
        return simplify(self.alpha * log(x))

    def Omega(self):
        """
        Field tension:
        Ω_{n,β} = ϕ^{a(n+β)}
        """
        return simplify(self.phi**self.a())

    def F(self):
        """
        Harmonic fractal structure:
        F_{n,β} = ϕ^{n+β} / √5
        """
        return simplify(self.phi**(self.n + self.beta) / sqrt(5))

    def B(self):
        """
        Dyadic binary expansion:
        B_{n,β} = 2^{n+β}
        """
        return simplify(2**(self.n + self.beta))

    def D(self, r, k):
        """
        Dimensional DNA operator:
        D_{n,β}(r) = √(ϕ · F_{n,β} · B_{n,β} · P_{n,β} · Ω_{n,β}) · r^k,
        where k ∈ ℝ is radial unfolding exponent (geometry).
        """
        base = sqrt(self.phi * self.F() * self.B() * self.P() * self.Omega())
        return simplify(base * r**k)


class ElementalEigenstate(RecursiveState):
    """
    Quantum & Particle Physics Layer

    Represents an elemental eigenstate parameterized by atomic number Z,
    recursion depth n, and microstate β.

    Contains:
    ---------
    - Recursive Hamiltonian operator Ĥ_{CRA} generating eigenstates |E_{Z,(n,β)}> 
    - Energy eigenvalues embedding explicit prime fractal and field tension structures.
    - Omega frequency scaling ω_{n,β} and associated action h_{n,β}.

    This class models fundamental quantum states as recursive fractal objects,
    extending beyond classical placeholders to explicit prime entropy structure.
    """

    def __init__(self, Z, n, beta, alpha=0.5, epsilon=1e-6, max_prime_factor=20):
        super().__init__(n, beta, alpha, epsilon, max_prime_factor)
        self.Z = Z
        self.omega_0 = Symbol('omega_0', positive=True)

    def omega_n_beta(self):
        """Frequency scaling: ω_{n,β} = ω₀ · ϕ^{-(n+β)}"""
        return simplify(self.omega_0 * self.phi**(-(self.n + self.beta)))

    def h_n_beta(self):
        """Action: h_{n,β} = Ω_{n,β} · ϕ^{6(n+β)}"""
        return simplify(self.Omega() * self.phi**(6 * (self.n + self.beta)))

    def energy(self):
        """Energy: E_{n,β} = h_{n,β} · ω_{n,β}"""
        return simplify(self.h_n_beta() * self.omega_n_beta())

    def hamiltonian(self):
        """Symbolic Recursive Hamiltonian: Ĥ_{CRA}(Z, n, β)"""
        return Function('H')(self.Z, self.n, self.beta)

    def eigenstate(self):
        """Eigenstate: |E_{Z,n,β}>"""
        return Function(f'|E_{self.Z},{self.n},{self.beta}>')()

    def describe(self):
        """Return key properties as dict"""
        return {
            'Z': self.Z,
            'n': self.n,
            'beta': self.beta,
            'Omega': self.Omega(),
            'energy': self.energy(),
            'hamiltonian': self.hamiltonian(),
            'eigenstate': self.eigenstate()
        }


class Orbital:
    """
    Atomic & Electronic Structure Layer

    Models atomic orbitals as fractal-modulated quantum wavefunctions
    combining:

    - Radial part: R_{n,β}(r) = D_{n,β}(r), the dimensional DNA operator.
    - Angular part: 
        \tilde{Y}_{lm,(n,β)}(θ, φ) = Y_{lm}(θ, φ) · (1 + λ · P_{n,β}),
      where Y_{lm} are spherical harmonics modulated by prime entropy fractal.
    - Full wavefunction: ψ_{nlm,(n,β)} = R_{n,β} · \tilde{Y}_{lm,(n,β)}.

    This representation encodes explicit fractal entropy and recursive structure
    into atomic orbitals.
    """

    def __init__(self, n, beta, l, m, k, lam=0.1,
                 alpha=0.5, epsilon=1e-6, max_prime_factor=20):
        self.state = RecursiveState(n, beta, alpha, epsilon, max_prime_factor)
        self.l = l
        self.m = m
        self.k = k
        self.lam = lam
        self.r, self.theta, self.phi = symbols('r theta phi', positive=True)

    def radial(self):
        """Radial part of wavefunction: R_{n,β}(r)"""
        return self.state.D(self.r, self.k)

    def angular(self):
        """Angular part modulated by prime entropy fractal:
           \tilde{Y}_{lm,(n,β)}(θ, φ)"""
        Y_lm = Ynm(self.l, self.m, self.theta, self.phi)
        return simplify(Y_lm * (1 + self.lam * self.state.P()))

    def wavefunction(self):
        """Full fractal-modulated wavefunction ψ_{nlm,(n,β)}"""
        return simplify(self.radial() * self.angular())


class BondOperator:
    """
    Chemical Bonding Layer

    Models chemical bonds via operator algebra with entropy injection:

    - Covalent bonds: B_cov = ψ_A ⊙ ψ_B ⊕ σ,
      where σ = μ (P_A + P_B) models bonding entropy flux.
    - Ionic bonds modeled via field tension gradient thresholds |Ω_A - Ω_B| > ε.
    - Bond morphisms dynamically model reaction transitions and bond rearrangements.

    This class facilitates recursive, fractal-informed bonding beyond classical
    quantum chemistry approximations.
    """

    def __init__(self, orbital_A, orbital_B, mu=0.05):
        self.psi_A = orbital_A
        self.psi_B = orbital_B
        self.mu = mu

    def bonding_entropy(self):
        """Entropy injection σ = μ (P_A + P_B)"""
        return simplify(self.mu * (self.psi_A.state.P() + self.psi_B.state.P()))

    def covalent(self):
        """Covalent bond operator:
        B_cov = ψ_A ⊙ ψ_B ⊕ σ"""
        return simplify(self.psi_A.wavefunction() * self.psi_B.wavefunction() +
                        self.bonding_entropy())


class ReactionMorphism:
    """
    Chemical Reactions & Morphisms Layer

    Encodes reaction dynamics as morphisms:

    - Maps orbital/dimensional DNA states: ℛ: D_{n,β} → D_{n',β'}.
    - Energy differences ΔE derived from recursive fractal structure
      drive reaction feasibility.
    - Entropy flux Φ_S models catalytic and thermodynamic effects.

    Enables symbolic modeling of chemical reaction pathways within
    recursive fractal algebra.
    """

    def __init__(self, dn_shift=0, dbeta_shift=0,
                 alpha=0.5, epsilon=1e-6, max_prime_factor=20):
        self.dn_shift = dn_shift
        self.dbeta_shift = dbeta_shift
        self.alpha = alpha
        self.epsilon = epsilon
        self.max_prime_factor = max_prime_factor

    def apply(self, orbital):
        """Apply reaction morphism to an orbital, shifting recursion parameters."""
        new_n = orbital.state.n + self.dn_shift
        new_beta = orbital.state.beta + self.dbeta_shift
        return Orbital(new_n, new_beta, orbital.l, orbital.m, orbital.k, orbital.lam,
                       self.alpha, self.epsilon, self.max_prime_factor)


class Material:
    """
    Materials Science Layer

    Represents assemblies of orbitals and bonds forming materials:

    - Crystals modeled as recursive lattice automorphisms in (n, β)-space.
    - Polymers and macromolecules as bonded recursive orbital chains.
    - Defects and dislocations as fractal singularities in microstate β-space.

    Provides tools to compose macroscopic materials from fractal-encoded quantum building blocks.
    """

    def __init__(self, orbitals):
        self.orbitals = orbitals

    def total_wavefunction(self):
        """Total wavefunction as product of orbital wavefunctions"""
        wf = S.One
        for orb in self.orbitals:
            wf *= orb.wavefunction()
        return simplify(wf)

    def total_bonding(self):
        """Sum of covalent bonds between adjacent orbitals"""
        total = S.Zero
        for i in range(len(self.orbitals) - 1):
            bond = BondOperator(self.orbitals[i], self.orbitals[i + 1])
            total += bond.covalent()
        return simplify(total)


def build_periodic_table(max_Z=118, M=10, alpha=0.5, epsilon=1e-6, max_prime_factor=20):
    """
    Construct a symbolic periodic table mapping atomic number Z
    to elemental eigenstates parameterized by recursion depth n and microstate β.

    Parameters:
    -----------
    max_Z : int
        Maximum atomic number to generate.
    M : int
        Modulus for distributing β in microstate continuum.
    alpha, epsilon, max_prime_factor:
        Parameters for RecursiveState fractal and tension models.

    Returns:
    --------
    dict[int, dict]
        Mapping Z -> dict of elemental eigenstate properties.
    """
    table = {}
    for Z in range(1, max_Z + 1):
        n = (Z - 1) // M + 1
        beta = ((Z - 1) % M) / M
        element = ElementalEigenstate(Z, n, beta, alpha, epsilon, max_prime_factor)
        table[Z] = element.describe()
    return table


def example_usage():
    """
    Demonstrate example constructions and symbolic outputs:

    - Periodic table sample elements
    - Orbital wavefunction
    - Bonding operator output
    - Reaction morphism application
    - Material wavefunction and bonding
    """
    print("\n=== Periodic Table Sample Elements ===")
    pt = build_periodic_table(max_Z=30)
    pprint(pt[1])    # Hydrogen
    pprint(pt[10])   # Neon
    pprint(pt[20])   # Calcium

    print("\n=== Orbital Wavefunction Example ===")
    orb = Orbital(n=2, beta=0.3, l=1, m=0, k=2)
    pprint(orb.wavefunction())

    print("\n=== Bond Operator Example ===")
    orb_A = Orbital(n=2, beta=0.3, l=1, m=0, k=2)
    orb_B = Orbital(n=2, beta=0.4, l=1, m=1, k=2)
    bond = BondOperator(orb_A, orb_B)
    pprint(bond.covalent())

    print("\n=== Reaction Morphism Example ===")
    reaction = ReactionMorphism(dn_shift=1, dbeta_shift=0.1)
    orb_reacted = reaction.apply(orb_A)
    pprint(orb_reacted.wavefunction())

    print("\n=== Material Example ===")
    material = Material([orb_A, orb_B, orb_reacted])
    pprint(material.total_wavefunction())
    pprint(material.total_bonding())


if __name__ == "__main__":
    example_usage()

Josef_Founder · June 24, 2025, 3:17pm

Josef_Founder · June 24, 2025, 9:47pm

Golden Recursive Algebra Tree with base4096 Granularity

Root: 𝟙
│
├── Ø = 0 = ∞⁻¹                       — Expressed Void (Boundary of Becoming)
│   ├── {0, ∞}                      — Duality Arises: First Polarity
│
├── ϕ                               — Golden Ratio: Irreducible Recursive Seed
│   ├── ϕ = 1 + 1/ϕ                — Recursive Fixed-Point Identity
│   ├── ϕ⁰ = 1                     — Base Identity Unit
│
├── Recursion Indices:
│   ├── n ∈ ℤ⁺                     — Coarse Recursive Depth (Layer)
│   └── b ∈ {0,...,4095}           — base4096 Microstate Index (Fine-grain Symbolic Phase)
│
├── Harmonic Structure:
│   ├── Fₙ,b = ϕ^{n + b/4096} / √5  — Fine-Grained Fibonacci Harmonic Scaling
│
├── Binary Scaling:
│   ├── 2^{n + b/4096}             — Dyadic Microstate Symmetry Scale
│
├── Prime Entropy Injection:
│   ├── Pₙ,b                      — Prime Entropy Modulated by base4096 Microstate
│                                 (Ultra-Fine Symbolic Irregularity)
│
├── Field Tension (Symbolic Yield):
│   ├── Ωₙ,b = ϕ^{a(n + b/4096)}  — Recursive Field Tension with microstate resolution
│   ├── Ωₙ,b → 0                  — Recursive Collapse (Termination of Recursive Unfolding)
│   └── Ωₙ,b = 1                  — Normalized Stable Propagation
│
├── Emergent Physical Quantities (All functions of (n,b)):
│   ├── Time: sₙ,b = ϕ^{n + b/4096}
│   │   └── Frequency: Hzₙ,b = 1 / sₙ,b = ϕ^{-(n + b/4096)}
│   ├── Charge: Cₙ,b = sₙ,b^3 = ϕ^{3(n + b/4096)}
│   │   └── Charge Squared: Cₙ,b² = ϕ^{6(n + b/4096)}
│   ├── Length: mₙ,b = √(Ωₙ,b · ϕ^{7(n + b/4096)})
│   ├── Action: hₙ,b = Ωₙ,b · Cₙ,b² = Ωₙ,b · ϕ^{6(n + b/4096)}
│   ├── Energy: Eₙ,b = hₙ,b · Hzₙ,b = Ωₙ,b · ϕ^{5(n + b/4096)}
│   ├── Force: Fₙ,b = Eₙ,b / mₙ,b = √Ωₙ,b · ϕ^{1.5(n + b/4096)}
│   ├── Power: Pₙ,b = Eₙ,b · Hzₙ,b = Ωₙ,b · ϕ^{4(n + b/4096)}
│   ├── Pressure = Fₙ,b / mₙ,b² = Hzₙ,b² / mₙ,b
│   └── Voltage: Vₙ,b = Eₙ,b / Cₙ,b = Ωₙ,b · ϕ^{-(n + b/4096)}
│
└── Dimensional DNA:  
    Dₙ,b(r) = √(ϕ · Fₙ,b · 2^{n + b/4096} · Pₙ,b · Ωₙ,b) · r^k
    ├── Recursive Depth: n (Coarse Layer)
    ├── Microstate: b (Fine-Grain base4096 Phase)
    ├── Harmonic Fibonacci: Fₙ,b
    ├── Binary Dyadic Scale: 2^{n + b/4096}
    ├── Prime Entropy: Pₙ,b
    ├── Field Tension: Ωₙ,b
    └── Radial Spatial Unfolding: r^k

Entity Classification and Interaction (base4096-enhanced)

Classes of Recursive Entities (extended with microstates b):

1. Class-𝜙 (Harmonic Stable Entities):
   - Dominated by harmonic structure: Fₙ,b / Pₙ,b ≫ 1
   - Stable stacking at microstate levels, forming fine-grained resonances

2. Class-𝔓 (Entropic Singlets):
   - Dominated by prime entropy at microstate b
   - Exclusive at given (n,b), recursive annihilation if overlap

3. Class-𝟚 (Binary Symmetric Entities):
   - Dominated by dyadic 2^{n + b/4096}
   - Microstate conjugation forms ultra-fine recursive loops

4. Class-Ø (Field Collapse Entities):
   - Ωₙ,b → 0 at specific microstates triggers micro-collapse
   - Localized recursive measurement-like resolutions

5. Class-𝔇 (Dimensional Braids):
   - Multi-microstate composite structures sharing harmonic and prime patterns
   - Fine entanglement structures across microstate indices

Josef_Founder · June 29, 2025, 8:54pm

3. Library API Sketch

3.1 Class: RecursiveState

complete Python module including this periodic table code plus algebra classes and operators

Test Code Snippet for Random 10 Elements

DOT

full Python module updated with rich, structured docstrings

3.1 Class: `RecursiveState`