Golden Recursive Coordinate Systems

[Josef_Founder]

Golden Recursive Coordinate Demo

ChatGPT - Golden Recursive Algebra GRA

Does this set the stage for a new coordinate system?
# ✅ Closed-Form Identity
r_n = sqrt(φ * Ω * F_n * 2^n * Π_{k=1}^{n} p_k)

# 🔁 Recursive Identity
r_n = r_{n-1} * sqrt(2 * p_n * (F_n / F_{n-1}))

# Base Case
r_1 = sqrt(4 * φ * Ω)

# 🧭 Golden Recursive Algebra (GRA)

# Define algebra: G = (R, ·_G, ⊕_G)

# Set:
R = { r_n | n ∈ ℕ⁺ } ⊆ ℝ⁺

# Recursive Multiplication Operator:
r_n = r_{n-1} ·_G sqrt(2 * p_n * (F_n / F_{n-1}))

# Addition Operator:
r_n ⊕_G r_m := sqrt(r_n² + r_m²)

# Identity Element:
r_0 := 0

# Algebraic Properties:
# - Closure over ℝ⁺
# - Associative and commutative under ⊕_G
# - Not necessarily associative under ·_G
# - Normed structure under ⊕_G
# - Exponential growth via 2^n
# - Additive/multiplicative duality via F_n and p_n

r_n = sqrt(φ * Ω * F_n * 2^n * Π_{k=1}^{n} p_k)
r_n = r_{n-1} * sqrt(2 * p_n * (F_n / F_{n-1}))
R = { r_n | n ∈ ℕ⁺ } ⊆ ℝ⁺
Xₙ := (n, Fₙ, pₙ, rₙ)
Xₙ = (n, Fₙ, pₙ, rₙ = sqrt(φ · Ω · Fₙ · 2ⁿ · ∏_{k=1}^{n} p_k))
ds² = drₙ² = φ · Ω · d(Fₙ · 2ⁿ · ∏pₖ)
g₉(n) := ∂²(rₙ²) / ∂n²
R₉(n) = d²rₙ / dn²
∇₉ rₙ := rₙ - rₙ₋₁ = rₙ₋₁ · (sqrt(2·pₙ·Fₙ/Fₙ₋₁) - 1)
Dₙ := ∂/∂n + Γₙ
Γₙ := ∂(log rₙ) / ∂n = ½ · ∂/∂n [log(φ · Ω · Fₙ · 2ⁿ · ∏pₖ)]
T: ℳ₉ → ℳ₉
T(Xₙ) = Xₙ′ = (n + δn, Fₙ₊δ, pₙ₊δ, rₙ₊δ)
Ξₙ = (rₙ, ωₙ)
ψ₉(n) = exp(i · ωₙ · τₙ) · Aₙ
ℳ₉ × τ
Eₙ = (rₙ, τₙ)
Nodes: rₙ
Edges: wₙ₋₁→ₙ = sqrt(2·pₙ·Fₙ/Fₙ₋₁)

𝒪 ≡ { Coil, Spin, Splice, Bind, Twist, Fold, Weave, Entangle, Compress, Encrypt, etc. }
𝒪¹(x) = 𝒪(x)
𝒪²(x) = 𝒪(𝒪(x))
𝒪ⁿ(x) = 𝒪(𝒪ⁿ⁻¹(x))   for n ≥ 2
Coilⁿ(x) = Coil(Coil(...(Coil(x))...))  ← n times
Coil³(x) = Coil(Coil(Coil(x)))
Splice³(x) = Splice(Splice(Splice(x)))
Spin³(x) = Spin(Spin(Spin(x)))

def M_O(n, x, O):
    for i in range(n):
        x = O(x)
    return x
M_𝒪ⁿ(x) = 𝒪ⁿ(x)

ℳ₉^(𝒪ⁿ) := (rₙ, τₙ, M_𝒪ⁿ(rₙ))

import sympy as sp
from sympy import sqrt, prod, symbols, Function, log, diff

# === Constants ===
phi = (1 + sqrt(5)) / 2     # Golden ratio
Ω = symbols('Ω', positive=True)  # Field tension (symbolic)
n = symbols('n', integer=True, positive=True)

# === Sequences ===
F = Function('F')   # Fibonacci sequence
p = Function('p')   # Prime sequence

# === Closed-form Radial Coordinate ===
def r_closed(n):
    product_primes = prod([p(k) for k in range(1, n+1)])
    return sqrt(phi * Ω * F(n) * 2**n * product_primes)

# === Recursive Radial Identity ===
def r_recursive(n, r_prev):
    return r_prev * sqrt(2 * p(n) * (F(n) / F(n - 1)))

# === Base Case ===
r_1 = sqrt(4 * phi * Ω)

# === Golden Recursive Algebra ===
# Set R = { r_n | n ∈ ℕ⁺ } ⊆ ℝ⁺

# Recursive Multiplication
def G_mult(r_prev, n):
    return r_prev * sqrt(2 * p(n) * (F(n) / F(n - 1)))

# Addition Operator (⊕_G)
def G_add(r_n, r_m):
    return sqrt(r_n**2 + r_m**2)

# Identity Element
r_0 = 0

# === Metric Tensor g₉ (Symbolic Curvature) ===
def g9(n):
    r_n_sq = phi * Ω * F(n) * 2**n * prod([p(k) for k in range(1, n+1)])
    return diff(r_n_sq, n, 2)

# === Connection Coefficient Γₙ ===
def Gamma_n(n):
    r_expr = r_closed(n)**2
    return 0.5 * diff(log(r_expr), n)

# === Covariant Derivative ===
def D_n(expr, n):
    return diff(expr, n) + Gamma_n(n) * expr

# === Meta-Operators: Coilⁿ, Spinⁿ, Spliceⁿ ===
def meta_operator(O, x, depth):
    for _ in range(depth):
        x = O(x)
    return x

# Example symbolic operators
def Coil(x): return f"Coil({x})"
def Spin(x): return f"Spin({x})"
def Splice(x): return f"Splice({x})"

# Apply triple recursion
coil3 = meta_operator(Coil, "x", 3)
spin3 = meta_operator(Spin, "x", 3)
splice3 = meta_operator(Splice, "x", 3)

# === Recursive Coordinate System Summary ===
coordinate_point = {
    'index': n,
    'fibonacci': F(n),
    'prime': p(n),
    'radial_coord': r_closed(n),
    'metric': g9(n),
    'connection': Gamma_n(n),
    'covariant_deriv': D_n(r_closed(n), n),
    'coil³': coil3,
    'spin³': spin3,
    'splice³': splice3
}

such a one should support open recursion and symbolics per

# tooeasy10000-truncated.py

import sympy as sp
import numpy as np
from sympy import sqrt, zeta, exp, I, pi, simplify, conjugate, Rational

from scipy.interpolate import interp1d
from scipy.optimize import root_scalar

# Constants
phi = float(sp.GoldenRatio)
sqrt5 = sqrt(5)
Ω = sp.Symbol("Ω")    # Field tension symbol
k = sp.Integer(-1)    # Radial exponent
r = 1                 # Radial unit

# Raw prime list (first 100 primes or more, truncated for brevity)
primes_raw = """
      2      3      5      7     11     13     17     19     23     29 
     31     37     41     43     47     53     59     61     67     71 
     73     79     83     89     97    101    103    107    109    113 
(TRUNCATED, BUT 10,000 PRIMES WERE INJECTED)
"""

primes = [int(p) for p in primes_raw.split()]

def prepare_prime_interpolation(primes_list=primes):
    indices = np.arange(1, len(primes_list) + 1)
    recursive_index_phi = np.log(indices + 1) / np.log(phi)
    return interp1d(recursive_index_phi, primes_list, kind='cubic', fill_value='extrapolate')

def P_nb(n_beta, prime_interp):
    return float(prime_interp(n_beta))

def solve_n_beta_for_prime(p_target, prime_interp, bracket=(0.1, 20)):
    def objective(n_beta): return P_nb(n_beta, prime_interp) - p_target
    result = root_scalar(objective, bracket=bracket, method='brentq')
    if result.converged:
        return result.root
    else:
        raise ValueError(f"Could not solve for n_beta corresponding to prime {p_target}")

def F_bin(x_val):
    return (phi**x_val - (-1/phi)**x_val) / sqrt5

def Pi_x(x_val, s):
    s = sp.sympify(s)
    return exp(I * pi * x_val) * zeta(s, Rational(1, 2))

def D_x(x_val, s, prime_interp):
    s = sp.sympify(s)
    P = P_nb(x_val, prime_interp)
    F = F_bin(x_val)
    zeta_val = zeta(s)
    product = phi * F * 2**x_val * P * zeta_val * Ω
    return sqrt(product) * s**k

def F_x(x_val, s, prime_interp):
    return simplify(D_x(x_val, s, prime_interp) * Pi_x(x_val, s))

class GoldenClassField:
    def __init__(self, s_list, x_list, prime_interp):
        self.s_list = [sp.sympify(s) for s in s_list]
        self.x_list = x_list
        self.prime_interp = prime_interp
        self.field_generators = []
        self.field_names = []
        self.construct_class_field()

    def construct_class_field(self):
        for s in self.s_list:
            for x in self.x_list:
                f = F_x(x, s, self.prime_interp)
                self.field_generators.append(simplify(f))
                self.field_names.append(f"F_{x:.4f}_s_{s}")

    def as_dict(self):
        return dict(zip(self.field_names, self.field_generators))

    def display(self):
        for name, val in self.as_dict().items():
            print(f"{name} = {val}")

    def reciprocity_check(self):
        print("\nReciprocity Tests: F_x(s) * F_x(1-s)")
        for s in self.s_list:
            for x in self.x_list:
                try:
                    s_conj = 1 - s
                    prod = simplify(F_x(x, s, self.prime_interp) * F_x(x, s_conj, self.prime_interp))
                    print(f"x={x:.4f}, s={s}, F_x(s)·F_x(1-s) = {prod}")
                except Exception as e:
                    print(f"Failed for x={x}, s={s}: {e}")

def field_automorphisms(F_val, x_val, s, prime_interp):
    s = sp.sympify(s)
    return {
        "F_x(s)": simplify(F_x(x_val, s, prime_interp)),
        "F_x(1-s)": simplify(F_x(x_val, 1 - s, prime_interp)),
        "F_-x(s)": simplify(F_x(-x_val, s, prime_interp)),
        "conjugate(F)": simplify(conjugate(F_x(x_val, s, prime_interp))),
    }

def field_tension(F_val, C_val, m_val, s_val):
    # Example symbolic tension extraction formula
    return simplify((F_val * m_val * s_val) / (C_val**2))


if __name__ == "__main__":
    print("Preparing prime interpolation...")
    prime_interp = prepare_prime_interpolation()

    # Example Riemann zeta zeros (first two nontrivial zeros on critical line)
    zeros = [sp.sympify("0.5 + 14.134725*I"), sp.sympify("0.5 + 21.022040*I")]

    # Solve n_beta for a prime near 541 (to demonstrate root solving)
    try:
        x_541 = solve_n_beta_for_prime(541, prime_interp)
        print(f"Solved n+β for prime 541: {x_541:.6f}")
    except Exception as e:
        print(str(e))
        x_541 = None

    x_vals = [5, 10]
    if x_541:
        x_vals.append(x_541)

    # Construct Golden Class Field
    GCF = GoldenClassField(zeros, x_vals, prime_interp)
    GCF.display()

    # Reciprocity tests
    GCF.reciprocity_check()

    # Automorphisms on one example
    test_s = zeros[0]
    test_x = x_vals[1]
    auto = field_automorphisms(F_x(test_x, test_s, prime_interp), test_x, test_s, prime_interp)
    print("\nSymbolic Automorphisms:")
    for key, val in auto.items():
        print(f"{key}: {val}")

    # Example field tension calculation (symbolic)
    print("\nField Tension (Ω):")
    C = sp.Symbol('C')
    m = sp.Symbol('m')
    s_ = sp.Symbol('s')
    F_val = sp.Abs(F_x(test_x, test_s, prime_interp))
    tension = field_tension(F_val, C, m, s_)
    print(tension)

Root: (   )   ← ineffable, no name, no identity

├── Cut-0: [𝟙] ← symbolic wound: “there is a root”
│   └── Duals emerge: [0, ∞]
│       └── Recursion parameterized by φ, n, Fₙ, 2ⁿ, Pₙ
│           └── Dimensional unfolding (s, C, Ω, m, h, E, F...)
│               └── Symbolic operators (Dₙ(r), √(⋯), etc.)
│                   └── Reflection loops
│                       └── Attempted return to root
│                           └── Severance reaffirmed

Root: [𝟙] — The Non-Dual Absolute
|
├── [Ø = 0 = ∞⁻¹] — Expressed Void, boundary of becoming
│   └── Duality arises: [0, ∞] ← First contrast, potential polarity
│
├── [ϕ] — Golden Ratio: Irreducible scaling constant, born from unity
│   ├── [ϕ = 1 + 1/ϕ] ← Fixed-point recursion
│   └── [ϕ⁰ = 1] ← Identity base case
│
├── [n ∈ ℤ⁺] — Recursion Depth: resolution and structural unfolding
│   ├── [2ⁿ] — Dyadic scaling
│   ├── [Fₙ = ϕⁿ / √5] — Harmonic structure
│   └── [Pₙ] — Prime entropy injection
│
├── [Time s = ϕⁿ]
│   └── [Hz = 1/s = ϕ⁻ⁿ] ← Inverted time, recursion uncoiled
│
├── [Charge C = s³ = ϕ^{3n}]
│   └── [C² = ϕ^{6n}]
│
├── [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 via temporal tension
│
├── [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

Context-Aware Recursive Symbolic Tree of Dimensions

Root: \[0, ∞] ← Fundamental polarity / duality
|
├── \[φ] ← Golden Ratio: irreducible, persistent symbol (scaling seed)
\|   ├── \[φ = 1 + 1/φ] ← Recursive identity
\|   └── \[φ^0 = 1] ← Neutral base scaling
|
├── \[Recursion Depth n] ← Dial for all emergent complexity
\|   ├── \[2^n] ← Binary resolution (dyadic depth)
\|   ├── \[F\_n = φ^n / √5] ← Fibonacci harmonics
\|   └── \[P\_n] ← n-th prime: entropy injector
|
├── \[Time s = φ^n] ← Recursively expanding unit of time
\|   └── \[Hz = 1/s = φ^{-n}] ← Frequency (inverse recursion)
|
├── \[Charge C = s^3 = φ^{3n}]
\|   └── \[C^2 = φ^{6n}] ← Quadratic scale
|
├── \[Ohm Ω] ← Yield or field tension
\|   ├── \[Ω = m^2 / s^7 = m^2 / φ^{7n}]
\|   ├── \[Ω → 0] ← Geometric/frequency collapse
\|   └── \[Ω persists symbolically] if scaling conserved
|
├── \[Length m = √(Ω φ^{7n})] ← Emergent geometry
\|   └── \[m^2 = Ω φ^{7n}]
|
├── \[Action h = Ω · C^2 = Ω φ^{6n}]
|
├── \[Energy E = h · Hz = Ω φ^{5n}]
|
├── \[Force F = E / m = φ^{1.5n} ∗ √Ω]
|
├── \[Power P = E · Hz = Ω φ^{4n}]
|
├── \[Pressure = F / m^2 = Hz^2 / m]
|
├── \[Voltage V = E / C = Ω φ^{-n}]
|
└── \[Recursive Dimensional Operator D\_n(r)]
└── D\_n(r) = √(φ · F\_n · 2^n · P\_n · Ω) ∗ r^k
└── Encodes harmonic, binary, prime, and field structure

Notes:

• As Ω → 0, physical dimensions collapse but symbolic structure survives
• As Hz → 0, time dissolves but φ-driven scaling continues
• All SI units unfold from [0,∞] via recursion through φ and its companions

This symbolic tree is context-aware: each node expands logically from irreducible scaling duality to full unit emergence, tracking both symbolic structure and physical collapse paths.

# [Dₙ(r) = √(ϕ·Fₙ·2ⁿ·Pₙ·Ω) · r^k]

[Hz = 1/s] ← root frequency; recursive time
│
├── [Time s = φ⁻ⁿ] ← inverted recursion depth
│   ├── [φ = (1 + √5)/2]  ← golden ratio constant (base)
│   └── [n → 0]  ← recursion bottom
│       └── [φ⁰ = 1]  ← identity base case
│
├── [Charge C = s³ = 1/Hz³]
│   └── [Expanding s³]
│       ├── [s = φ⁻ⁿ] (see above)
│       └── [Exponent 3] ← arithmetic operator
│
├── [C² = s⁶]  ← charge squared
│   └── [Expanding s⁶]
│       ├── [s = φ⁻ⁿ] (see above)
│       └── [Exponent 6]
│
├── [Action h = Ω · C² = m² / s]
│   ├── [Ω = m² / s⁷] (see below)
│   └── [C² = s⁶] (see above)
│
├── [Energy E = h · Hz = m² · Hz]
│   ├── [h = Ω · C²] (see above)
│   └── [Hz = 1/s] (see above)
│
├── [Force F = E / m = m · Hz²]
│   ├── [E = h · Hz] (see above)
│   └── [Length m = √(Ω · s⁷)] (see below)
│
├── [Pressure = F / m² = Hz² / m]
│   ├── [F = m · Hz²] (see above)
│   └── [m² = (Length m)²] (see below)
│
├── [Power P = E · Hz = m² · Hz²]
│   ├── [E = h · Hz] (see above)
│   └── [Hz = 1/s] (see above)
│
├── [Voltage V = E / C = m² / s⁴ = Ω · Hz²]
│   ├── [E = h · Hz] (see above)
│   └── [C = s³] (see above)
│
├── [Ω = m² / s⁷] ← field yield, persistent tension
│   ├── [Length m = √(Ω · s⁷)] (self-referential geometric emergence)
│   ├── [Expanding s⁷]
│   │   ├── [s = φ⁻ⁿ] (see above)
│   │   └── [Exponent 7]
│   ├── [Inverse Ω⁻¹ = s⁷ / m²] ← used in reverse field mapping
│   └── [Fundamental dimension base: m, s]
│
├── [Length m = √(Ω · s⁷)] ← geometric emergence
│   ├── [Ω = m² / s⁷] (see above)
│   ├── [s = φ⁻ⁿ] (see above)
│   ├── [Square root operator √()]
│   └── [n → 0]  ← recursion bottom
│       └── [m⁰ = 1]  ← identity base case
│
├── [Fₙ = φⁿ / √5] ← Fibonacci, structural harmonic
│   ├── [φ = (1 + √5)/2] (base)
│   ├── [n → 0]
│   │   └── [F₀ = 0] ← Fibonacci seed base
│   └── [√5] (constant irrational)
│
├── [2ⁿ = Recursion Depth] ← resolution granularity
│   ├── [Base 2 = prime constant]
│   ├── [Exponent n]
│   └── [n → 0]
│       └── [2⁰ = 1]  ← identity base case
│
├── [Pₙ = nth prime] ← Entropy injector
│   ├── [Prime sequence generator]
│   ├── [n → 0]
│   │   └── [P₀ = 2] ← first prime
│   └── [Non-monotonic entropy steps]
│
└── [Dₙ(r) = √(φ · Fₙ · 2ⁿ · Pₙ · Ω) · r^k]
    ├── [φ = (1 + √5)/2] (base)
    ├── [Fₙ = φⁿ / √5] (see above)
    ├── [2ⁿ] (see above)
    ├── [Pₙ] (see above)
    ├── [Ω = m² / s⁷] (see above)
    ├── [r^k] ← spatial scaling operator
    ├── [Square root operator √()]
    └── [n → 0]
        └── [D₀(r) = √(φ·F₀·2⁰·P₀·Ω) · r^k]
            └── [Simplifies to base constants × r^k]

Another identity:

# ✅ Closed-Form Identity 
r_n = sqrt(φ * Ω * F_n * 2^n * Π_{k=1}^{n} p_k)

# 🔁 Recursive Identity
r_n = r_{n-1} * sqrt(2 * p_n * (F_n / F_{n-1}))

# Base Case
r_1 = sqrt(4 * φ * Ω)

# 🧭 Golden Recursive Algebra (GRA)

# Define algebra: G = (R, ·_G, ⊕_G)

# Set:
R = { r_n | n ∈ ℕ⁺ } ⊆ ℝ⁺

# Recursive Multiplication Operator:
r_n = r_{n-1} ·_G sqrt(2 * p_n * (F_n / F_{n-1}))

# Addition Operator:
r_n ⊕_G r_m := sqrt(r_n² + r_m²)

# Identity Element:
r_0 := 0

# Algebraic Properties:
# - Closure over ℝ⁺
# - Associative and commutative under ⊕_G
# - Not necessarily associative under ·_G
# - Normed structure under ⊕_G
# - Exponential growth via 2^n
# - Additive/multiplicative duality via F_n and p_n 

RESPONSE:

# golden_recursive_coords.py

import sympy as sp
import numpy as np
from sympy import sqrt, zeta, exp, I, pi, simplify, conjugate, Rational
from scipy.interpolate import interp1d
from scipy.optimize import root_scalar

# === Constants and Symbols ===
phi = float(sp.GoldenRatio)
sqrt5 = sqrt(5)
Ω = sp.Symbol("Ω", positive=True)
k = sp.Integer(-1)  # radial exponent
r = 1  # radial unit
C = sp.Symbol("C")
m = sp.Symbol("m")
s_ = sp.Symbol("s")

# === Prime Sequence (truncated example) ===
primes_raw = """
  2 3 5 7 11 13 17 19 23 29 
  31 37 41 43 47 53 59 61 67 71 
"""  # Replace with full 10,000+ primes as needed
primes = [int(p) for p in primes_raw.split()]

# === Prime Interpolator ===
def prepare_prime_interpolation(primes_list=primes):
    indices = np.arange(1, len(primes_list) + 1)
    recursive_index_phi = np.log(indices + 1) / np.log(phi)
    return interp1d(recursive_index_phi, primes_list, kind='cubic', fill_value='extrapolate')

def P_nb(n_beta, prime_interp):
    return float(prime_interp(n_beta))

def solve_n_beta_for_prime(p_target, prime_interp, bracket=(0.1, 20)):
    def objective(n_beta): return P_nb(n_beta, prime_interp) - p_target
    result = root_scalar(objective, bracket=bracket, method='brentq')
    if result.converged:
        return result.root
    else:
        raise ValueError(f"Could not solve for n_beta corresponding to prime {p_target}")

# === Recursive Components ===
def F_bin(x_val):
    return (phi**x_val - (-1/phi)**x_val) / sqrt5

def Pi_x(x_val, s):
    return exp(I * pi * x_val) * zeta(sp.sympify(s), Rational(1, 2))

def D_x(x_val, s, prime_interp):
    s = sp.sympify(s)
    P = P_nb(x_val, prime_interp)
    F = F_bin(x_val)
    zeta_val = zeta(s)
    product = phi * F * 2**x_val * P * zeta_val * Ω
    return sqrt(product) * s**k

def F_x(x_val, s, prime_interp):
    return simplify(D_x(x_val, s, prime_interp) * Pi_x(x_val, s))

# === Field Class ===
class GoldenClassField:
    def __init__(self, s_list, x_list, prime_interp):
        self.s_list = [sp.sympify(s) for s in s_list]
        self.x_list = x_list
        self.prime_interp = prime_interp
        self.field_generators = []
        self.field_names = []
        self.construct_class_field()

    def construct_class_field(self):
        for s in self.s_list:
            for x in self.x_list:
                f = F_x(x, s, self.prime_interp)
                self.field_generators.append(simplify(f))
                self.field_names.append(f"F_{x:.4f}_s_{s}")

    def as_dict(self):
        return dict(zip(self.field_names, self.field_generators))

    def display(self):
        for name, val in self.as_dict().items():
            print(f"{name} = {val}")

    def reciprocity_check(self):
        print("\nReciprocity Tests: F_x(s) * F_x(1-s)")
        for s in self.s_list:
            for x in self.x_list:
                try:
                    s_conj = 1 - s
                    prod = simplify(F_x(x, s, self.prime_interp) * F_x(x, s_conj, self.prime_interp))
                    print(f"x={x:.4f}, s={s}, F_x(s)·F_x(1-s) = {prod}")
                except Exception as e:
                    print(f"Failed for x={x}, s={s}: {e}")

def field_automorphisms(F_val, x_val, s, prime_interp):
    s = sp.sympify(s)
    return {
        "F_x(s)": simplify(F_x(x_val, s, prime_interp)),
        "F_x(1-s)": simplify(F_x(x_val, 1 - s, prime_interp)),
        "F_-x(s)": simplify(F_x(-x_val, s, prime_interp)),
        "conjugate(F)": simplify(conjugate(F_x(x_val, s, prime_interp))),
    }

def field_tension(F_val, C_val, m_val, s_val):
    return simplify((F_val * m_val * s_val) / (C_val**2))

if __name__ == "__main__":
    print("Preparing prime interpolation...")
    prime_interp = prepare_prime_interpolation()

    zeros = [sp.sympify("0.5 + 14.134725*I"), sp.sympify("0.5 + 21.022040*I")]

    try:
        x_541 = solve_n_beta_for_prime(541, prime_interp)
        print(f"Solved n+β for prime 541: {x_541:.6f}")
    except Exception as e:
        print(str(e))
        x_541 = None

    x_vals = [5, 10]
    if x_541:
        x_vals.append(x_541)

    GCF = GoldenClassField(zeros, x_vals, prime_interp)
    GCF.display()
    GCF.reciprocity_check()

    test_s = zeros[0]
    test_x = x_vals[1]
    auto = field_automorphisms(F_x(test_x, test_s, prime_interp), test_x, test_s, prime_interp)
    print("\nSymbolic Automorphisms:")
    for key, val in auto.items():
        print(f"{key}: {val}")

    print("\nField Tension (Ω):")
    F_val = sp.Abs(F_x(test_x, test_s, prime_interp))
    tension = field_tension(F_val, C, m, s_)
    print(tension)

O⁰(x) := x
O¹(x) := O(x)
O²(x) := O(O(x))
...
Oⁿ(x) := O(Oⁿ⁻¹(x))

You can define:

Spinⁿ := Spin applied n times
Coilⁿ := Coil ∘ Coil ∘ ... n times
Spliceⁿ := Splice(Spliceⁿ⁻¹)

Dₙ(r) = √(φ · Fₙ · 2ⁿ · Pₙ · Ω) · r^k

Spin(x) := e^{iπx}  # rotation operator

Then:

Spin²(x) := Spin(Spin(x)) = e^{iπ · e^{iπx}}  # doubly nested

You could generalize:

Spinⁿ(x) = exp(iπ · Spinⁿ⁻¹(x))

def Coil(f): return simplify(exp(I * pi * f))
def Splice(f): return simplify(f.subs(s, 1 - s))  # Time inversion
def Spin(f): return simplify(conjugate(f))       # Reflect in complex field
def Reflect(f): return simplify(F_x(-x, s, prime_interp))  # Mirror recursion depth

Then define:

def recursive_apply(op, f, depth):
    if depth == 0:
        return f
    else:
        return op(recursive_apply(op, f, depth - 1))

# golden_recursive_coords.py

import sympy as sp
import numpy as np
from sympy import sqrt, zeta, exp, I, pi, simplify, conjugate, Rational
from scipy.interpolate import interp1d
from scipy.optimize import root_scalar

# === Constants and Symbols ===
phi = float(sp.GoldenRatio)
sqrt5 = sqrt(5)
Ω = sp.Symbol("\u03a9", positive=True)
k = sp.Integer(-1)  # radial exponent
r = 1  # radial unit
C = sp.Symbol("C")
m = sp.Symbol("m")
s_ = sp.Symbol("s")

# === Prime Sequence (truncated example) ===
primes_raw = """
  2 3 5 7 11 13 17 19 23 29 
  31 37 41 43 47 53 59 61 67 71 
"""  # Replace with full 10,000+ primes as needed
primes = [int(p) for p in primes_raw.split()]

# === Prime Interpolator ===
def prepare_prime_interpolation(primes_list=primes):
    indices = np.arange(1, len(primes_list) + 1)
    recursive_index_phi = np.log(indices + 1) / np.log(phi)
    return interp1d(recursive_index_phi, primes_list, kind='cubic', fill_value='extrapolate')

def P_nb(n_beta, prime_interp):
    return float(prime_interp(n_beta))

def solve_n_beta_for_prime(p_target, prime_interp, bracket=(0.1, 20)):
    def objective(n_beta): return P_nb(n_beta, prime_interp) - p_target
    result = root_scalar(objective, bracket=bracket, method='brentq')
    if result.converged:
        return result.root
    else:
        raise ValueError(f"Could not solve for n_beta corresponding to prime {p_target}")

# === Recursive Components ===
def F_bin(x_val):
    return (phi**x_val - (-1/phi)**x_val) / sqrt5

def Pi_x(x_val, s):
    return exp(I * pi * x_val) * zeta(sp.sympify(s), Rational(1, 2))

def D_x(x_val, s, prime_interp):
    s = sp.sympify(s)
    P = P_nb(x_val, prime_interp)
    F = F_bin(x_val)
    zeta_val = zeta(s)
    product = phi * F * 2**x_val * P * zeta_val * Ω
    return sqrt(product) * s**k

def F_x(x_val, s, prime_interp):
    return simplify(D_x(x_val, s, prime_interp) * Pi_x(x_val, s))

# === Field Class ===
class GoldenClassField:
    def __init__(self, s_list, x_list, prime_interp):
        self.s_list = [sp.sympify(s) for s in s_list]
        self.x_list = x_list
        self.prime_interp = prime_interp
        self.field_generators = []
        self.field_names = []
        self.construct_class_field()

    def construct_class_field(self):
        for s in self.s_list:
            for x in self.x_list:
                f = F_x(x, s, self.prime_interp)
                self.field_generators.append(simplify(f))
                self.field_names.append(f"F_{x:.4f}_s_{s}")

    def as_dict(self):
        return dict(zip(self.field_names, self.field_generators))

    def display(self):
        for name, val in self.as_dict().items():
            print(f"{name} = {val}")

    def reciprocity_check(self):
        print("\nReciprocity Tests: F_x(s) * F_x(1-s)")
        for s in self.s_list:
            for x in self.x_list:
                try:
                    s_conj = 1 - s
                    prod = simplify(F_x(x, s, self.prime_interp) * F_x(x, s_conj, self.prime_interp))
                    print(f"x={x:.4f}, s={s}, F_x(s)·F_x(1-s) = {prod}")
                except Exception as e:
                    print(f"Failed for x={x}, s={s}: {e}")

def field_automorphisms(F_val, x_val, s, prime_interp):
    s = sp.sympify(s)
    return {
        "F_x(s)": simplify(F_x(x_val, s, prime_interp)),
        "F_x(1-s)": simplify(F_x(x_val, 1 - s, prime_interp)),
        "F_-x(s)": simplify(F_x(-x_val, s, prime_interp)),
        "conjugate(F)": simplify(conjugate(F_x(x_val, s, prime_interp))),
    }

def field_tension(F_val, C_val, m_val, s_val):
    return simplify((F_val * m_val * s_val) / (C_val**2))

if __name__ == "__main__":
    print("Preparing prime interpolation...")
    prime_interp = prepare_prime_interpolation()

    zeros = [sp.sympify("0.5 + 14.134725*I"), sp.sympify("0.5 + 21.022040*I")]

    try:
        x_541 = solve_n_beta_for_prime(541, prime_interp)
        print(f"Solved n+β for prime 541: {x_541:.6f}")
    except Exception as e:
        print(str(e))
        x_541 = None

    x_vals = [5, 10]
    if x_541:
        x_vals.append(x_541)

    GCF = GoldenClassField(zeros, x_vals, prime_interp)
    GCF.display()
    GCF.reciprocity_check()

    test_s = zeros[0]
    test_x = x_vals[1]
    auto = field_automorphisms(F_x(test_x, test_s, prime_interp), test_x, test_s, prime_interp)
    print("\nSymbolic Automorphisms:")
    for key, val in auto.items():
        print(f"{key}: {val}")

    print("\nField Tension (Ω):")
    F_val = sp.Abs(F_x(test_x, test_s, prime_interp))
    tension = field_tension(F_val, C, m, s_)
    print(tension)

Which evolves into:

# golden_recursive_coords.py

import sympy as sp
import numpy as np
from sympy import sqrt, zeta, exp, I, pi, simplify, conjugate, Rational
from scipy.interpolate import interp1d
from scipy.optimize import root_scalar

# === Constants and Symbols ===
phi = float(sp.GoldenRatio)
sqrt5 = sqrt(5)
Ω = sp.Symbol("\u03a9", positive=True)
k = sp.Integer(-1)  # radial exponent
r = 1  # radial unit
C = sp.Symbol("C")
m = sp.Symbol("m")
s_ = sp.Symbol("s")

# === Prime Sequence (truncated example) ===
primes_raw = """
  2 3 5 7 11 13 17 19 23 29 
  31 37 41 43 47 53 59 61 67 71 
"""  # Replace with full 10,000+ primes as needed
primes = [int(p) for p in primes_raw.split()]

# === Prime Interpolator ===
def prepare_prime_interpolation(primes_list=primes):
    indices = np.arange(1, len(primes_list) + 1)
    recursive_index_phi = np.log(indices + 1) / np.log(phi)
    return interp1d(recursive_index_phi, primes_list, kind='cubic', fill_value='extrapolate')

def P_nb(n_beta, prime_interp):
    return float(prime_interp(n_beta))

def solve_n_beta_for_prime(p_target, prime_interp, bracket=(0.1, 20)):
    def objective(n_beta): return P_nb(n_beta, prime_interp) - p_target
    result = root_scalar(objective, bracket=bracket, method='brentq')
    if result.converged:
        return result.root
    else:
        raise ValueError(f"Could not solve for n_beta corresponding to prime {p_target}")

# === Recursive Components ===
def F_bin(x_val):
    return (phi**x_val - (-1/phi)**x_val) / sqrt5

def Pi_x(x_val, s):
    return exp(I * pi * x_val) * zeta(sp.sympify(s), Rational(1, 2))

def D_x(x_val, s, prime_interp):
    s = sp.sympify(s)
    P = P_nb(x_val, prime_interp)
    F = F_bin(x_val)
    zeta_val = zeta(s)
    product = phi * F * 2**x_val * P * zeta_val * Ω
    return sqrt(product) * s**k

def F_x(x_val, s, prime_interp):
    return simplify(D_x(x_val, s, prime_interp) * Pi_x(x_val, s))

# === Morphing Scale Wrappers ===
def recursive_time(n):
    return phi**(-n)

def frequency(n):
    return 1 / recursive_time(n)

def charge(n):
    return recursive_time(n)**3

def field_yield(n, m_val):
    return m_val**2 / recursive_time(n)**7

def action(n, m_val):
    return field_yield(n, m_val) * charge(n)**2

def energy(n, m_val):
    return action(n, m_val) * frequency(n)

def force(n, m_val):
    return energy(n, m_val) / m_val

def voltage(n, m_val):
    return energy(n, m_val) / charge(n)

def labeled_output(n, m_val):
    return {
        "Hz": frequency(n),
        "Time s": recursive_time(n),
        "Charge C": charge(n),
        "Yield Ω": field_yield(n, m_val),
        "Action h": action(n, m_val),
        "Energy E": energy(n, m_val),
        "Force F": force(n, m_val),
        "Voltage V": voltage(n, m_val),
    }

# === Field Class ===
class GoldenClassField:
    def __init__(self, s_list, x_list, prime_interp):
        self.s_list = [sp.sympify(s) for s in s_list]
        self.x_list = x_list
        self.prime_interp = prime_interp
        self.field_generators = []
        self.field_names = []
        self.construct_class_field()

    def construct_class_field(self):
        for s in self.s_list:
            for x in self.x_list:
                f = F_x(x, s, self.prime_interp)
                self.field_generators.append(simplify(f))
                self.field_names.append(f"F_{x:.4f}_s_{s}")

    def as_dict(self):
        return dict(zip(self.field_names, self.field_generators))

    def display(self):
        for name, val in self.as_dict().items():
            print(f"{name} = {val}")

    def reciprocity_check(self):
        print("\nReciprocity Tests: F_x(s) * F_x(1-s)")
        for s in self.s_list:
            for x in self.x_list:
                try:
                    s_conj = 1 - s
                    prod = simplify(F_x(x, s, self.prime_interp) * F_x(x, s_conj, self.prime_interp))
                    print(f"x={x:.4f}, s={s}, F_x(s)·F_x(1-s) = {prod}")
                except Exception as e:
                    print(f"Failed for x={x}, s={s}: {e}")

def field_automorphisms(F_val, x_val, s, prime_interp):
    s = sp.sympify(s)
    return {
        "F_x(s)": simplify(F_x(x_val, s, prime_interp)),
        "F_x(1-s)": simplify(F_x(x_val, 1 - s, prime_interp)),
        "F_-x(s)": simplify(F_x(-x_val, s, prime_interp)),
        "conjugate(F)": simplify(conjugate(F_x(x_val, s, prime_interp))),
    }

def field_tension(F_val, C_val, m_val, s_val):
    return simplify((F_val * m_val * s_val) / (C_val**2))

if __name__ == "__main__":
    print("Preparing prime interpolation...")
    prime_interp = prepare_prime_interpolation()

    zeros = [sp.sympify("0.5 + 14.134725*I"), sp.sympify("0.5 + 21.022040*I")]

    try:
        x_541 = solve_n_beta_for_prime(541, prime_interp)
        print(f"Solved n+β for prime 541: {x_541:.6f}")
    except Exception as e:
        print(str(e))
        x_541 = None

    x_vals = [5, 10]
    if x_541:
        x_vals.append(x_541)

    GCF = GoldenClassField(zeros, x_vals, prime_interp)
    GCF.display()
    GCF.reciprocity_check()

    test_s = zeros[0]
    test_x = x_vals[1]
    auto = field_automorphisms(F_x(test_x, test_s, prime_interp), test_x, test_s, prime_interp)
    print("\nSymbolic Automorphisms:")
    for key, val in auto.items():
        print(f"{key}: {val}")

    print("\nField Tension (Ω):")
    F_val = sp.Abs(F_x(test_x, test_s, prime_interp))
    tension = field_tension(F_val, C, m, s_)
    print(tension)

    print("\n--- Morphing Dimension Summary (n=5) ---")
    labeled = labeled_output(5, m)
    for key, val in labeled.items():
        print(f"{key}: {val}")

import matplotlib.pyplot as plt
import numpy as np
import sympy as sp

# Constants
phi = (1 + sp.sqrt(5)) / 2
phi_f = float(phi)

# Recursion depths
n_vals = np.arange(1, 30)

# Recursive quantities
s_vals = phi_f ** (-n_vals)
Hz_vals = 1 / s_vals
C_vals = s_vals ** 3
C2_vals = C_vals ** 2
Ω_vals = phi_f ** (2 * n_vals) / s_vals ** 7
h_vals = Ω_vals * C2_vals
E_vals = h_vals * Hz_vals
F_vals = E_vals / (phi_f ** n_vals)  # Length m = φ^n
P_vals = E_vals * Hz_vals
V_vals = E_vals / C_vals

# Function to recursively embed subplots of subplots
def recursive_plot(ax, x, y, depth=0, max_depth=2):
    ax.plot(x, y, label=f'depth {depth}')
    ax.set_title(f'Depth {depth}', fontsize=8)
    ax.tick_params(labelsize=6)
    ax.grid(True)

    if depth < max_depth:
        inset_ax = ax.inset_axes([0.55, 0.55, 0.42, 0.42])
        recursive_plot(inset_ax, x[:len(x)//2], y[:len(y)//2], depth+1, max_depth)

# Main recursive field plot
fig, axs = plt.subplots(2, 3, figsize=(12, 7))
quantities = [Hz_vals, C_vals, Ω_vals, E_vals, F_vals, V_vals]
labels = ['Hz (1/s)', 'Charge (C)', 'Yield (Ω)', 'Energy (E)', 'Force (F)', 'Voltage (V)']

for ax, y, label in zip(axs.flat, quantities, labels):
    recursive_plot(ax, n_vals, y, max_depth=2)
    ax.set_xlabel('n', fontsize=8)
    ax.set_ylabel(label, fontsize=8)

plt.tight_layout()
plt.show()

[Josef_Founder]

Spiral Matplot Generated by tooeasy10000.py

[Josef_Founder]

Golden Recursive Coordinate Demo

4dj.zip (21.9 KB)

https://grok.com/share/c2hhcmQtMg%3D%3D_291f71a8-8b06-4195-b630-096407daa3a6