| Concept              | Recursive Expression                                                                |
| -------------------- | ----------------------------------------------------------------------------------- |
| Wavefunction `Ψ(r)`  | $\exp\left(-r^k \cdot \sqrt{\phi F_n 2^n P_n \Omega} \right)$                       |
| Entropy `log(Πp)`    | $-\sum_i \log P_{n,i}$                                                              |
| Entropic Force       | $F \sim \frac{\partial \mathbb{S}}{\partial \log_\phi r}$                           |
| Schrödinger Equation | $i\Omega \phi^{6n} \partial_{\phi^n} \Psi = [-\frac{\Omega}{2m} \nabla^2 + V] \Psi$ |


I'm looking for a rootless tree.

import numpy as np
import matplotlib.pyplot as plt
from scipy.interpolate import interp1d

# Golden ratio constant
phi = (1 + np.sqrt(5)) / 2

# First 50 primes for symbolic entropy indexing
PRIMES = [
    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,
    127, 131, 137, 139, 149, 151, 157, 163, 167, 173,
    179, 181, 191, 193, 197, 199, 211, 223, 227, 229
]

def fib_real(n):
    from math import cos, pi, sqrt
    phi_inv = 1 / phi
    term1 = phi**n / sqrt(5)
    term2 = (phi_inv**n) * cos(pi * n)
    return term1 - term2

def D(n, beta, r=1.0, k=1.0, Omega=1.0, base=2):
    Fn_beta = fib_real(n + beta)
    idx = int(np.floor(n + beta) + len(PRIMES)) % len(PRIMES)
    Pn_beta = PRIMES[idx]
    dyadic = base ** (n + beta)
    val = phi * Fn_beta * dyadic * Pn_beta * Omega
    val = np.maximum(val, 1e-15)
    return np.sqrt(val) * (r ** k)

def invert_D(value, r=1.0, k=1.0, Omega=1.0, base=2, max_n=10, steps=100):
    candidates = []
    for n in np.linspace(0, max_n, steps):
        for beta in np.linspace(0, 1, 10):
            val = D(n, beta, r, k, Omega, base)
            candidates.append((abs(val - value), n, beta))
    best = min(candidates, key=lambda x: x[0])
    return best[1], best[2]

# Fitted parameters (symbolic dimensionless scale)
fitted_params = {
    'k':    1.049342,
    'r0':   1.049676,
    'Omega0': 1.049675,
    's0':   0.994533,
    'alpha': 0.340052,
    'beta':  0.360942,
    'gamma': 0.993975,
    'H0':   70.0,
    'c0':   phi ** (2.5 * 6),  # c(n=6) = φ^15 ≈ 3303.402087
    'M':    -19.3
}

print("Symbolic decomposition of fitted parameters:")
for name, val in fitted_params.items():
    if name == 'M':
        print(f"  {name:<10}: {val} (fixed observational)")
        continue
    n, beta = invert_D(val)
    approx_val = D(n, beta)
    err = abs(val - approx_val)
    print(f"  {name:<10}: approx D({n:.3f}, {beta:.3f}) = {approx_val:.6f} (orig: {val:.6f}, err={err:.2e})")

params_reconstructed = {}
for name, val in fitted_params.items():
    if name == 'M':
        params_reconstructed[name] = val
        continue
    n, beta = invert_D(val)
    params_reconstructed[name] = D(n, beta)

print("\nReconstructed parameters:")
for name, val in params_reconstructed.items():
    print(f"  {name:<10} = {val:.6f}")

# Load supernova data
filename = 'hlsp_ps1cosmo_panstarrs_gpc1_all_model_v1_lcparam-full.txt'
lc_data = np.genfromtxt(filename, delimiter=' ', names=True, comments='#', dtype=None, encoding=None)

z = lc_data['zcmb']
mb = lc_data['mb']
dmb = lc_data['dmb']
M = params_reconstructed['M']
mu_obs = mb - M

H0 = params_reconstructed['H0']
c0_emergent = params_reconstructed['c0']

# Scale symbolic c0 to match physical light speed (km/s)
lambda_scale = 299792.458 / c0_emergent

def a_of_z(z):
    return 1 / (1 + z)

def Omega(z, Omega0, alpha):
    return Omega0 / (a_of_z(z) ** alpha)

def s(z, s0, beta):
    return s0 * (1 + z) ** (-beta)

def G(z, k, r0, Omega0, s0, alpha, beta):
    return Omega(z, Omega0, alpha) * k**2 * r0 / s(z, s0, beta)

def H(z, k, r0, Omega0, s0, alpha, beta):
    Om_m = 0.3
    Om_de = 0.7
    Gz = G(z, k, r0, Omega0, s0, alpha, beta)
    Hz_sq = (H0 ** 2) * (Om_m * Gz * (1 + z) ** 3 + Om_de)
    return np.sqrt(Hz_sq)

def emergent_c(z, Omega0, alpha, gamma):
    return c0_emergent * (Omega(z, Omega0, alpha) / Omega0) ** gamma * lambda_scale

def compute_luminosity_distance_grid(z_max, params, n=500):
    k, r0, Omega0, s0, alpha, beta, gamma = params
    z_grid = np.linspace(0, z_max, n)
    c_z = emergent_c(z_grid, Omega0, alpha, gamma)
    H_z = H(z_grid, k, r0, Omega0, s0, alpha, beta)
    integrand_values = c_z / H_z
    integral_grid = np.cumsum((integrand_values[:-1] + integrand_values[1:]) / 2 * np.diff(z_grid))
    integral_grid = np.insert(integral_grid, 0, 0)
    d_c = interp1d(z_grid, integral_grid, kind='cubic', fill_value="extrapolate")
    return lambda z: (1 + z) * d_c(z)

def model_mu(z_arr, params):
    d_L_func = compute_luminosity_distance_grid(np.max(z_arr), params)
    d_L_vals = d_L_func(z_arr)
    return 5 * np.log10(d_L_vals) + 25

param_list = [
    params_reconstructed['k'],
    params_reconstructed['r0'],
    params_reconstructed['Omega0'],
    params_reconstructed['s0'],
    params_reconstructed['alpha'],
    params_reconstructed['beta'],
    params_reconstructed['gamma'],
]

mu_fit = model_mu(z, param_list)
residuals = mu_obs - mu_fit

# === Plot Supernova fit and residuals ===

plt.figure(figsize=(10, 6))
plt.errorbar(z, mu_obs, yerr=dmb, fmt='.', alpha=0.5, label='Pan-STARRS1 SNe')
plt.plot(z, mu_fit, 'r-', label='Symbolic Emergent Gravity Model')
plt.xlabel('Redshift (z)')
plt.ylabel('Distance Modulus (μ)')
plt.title('Supernova Distance Modulus with Context-Aware Emergent c(z)')
plt.legend()
plt.grid(True)
plt.tight_layout()
plt.show()

plt.figure(figsize=(10, 4))
plt.errorbar(z, residuals, yerr=dmb, fmt='.', alpha=0.5)
plt.axhline(0, color='red', linestyle='--')
plt.xlabel('Redshift (z)')
plt.ylabel('Residuals (μ_data - μ_model)')
plt.title('Residuals of Symbolic Model with Emergent c(z)')
plt.grid(True)
plt.tight_layout()
plt.show()

# === Plot emergent c(z) and G(z) ===

k = params_reconstructed['k']
r0 = params_reconstructed['r0']
Omega0 = params_reconstructed['Omega0']
s0 = params_reconstructed['s0']
alpha = params_reconstructed['alpha']
beta = params_reconstructed['beta']
gamma = params_reconstructed['gamma']

z_grid = np.linspace(0, max(z), 300)

c_z = emergent_c(z_grid, Omega0, alpha, gamma)  # km/s
G_z = G(z_grid, k, r0, Omega0, s0, alpha, beta)

# Normalize G(z) relative to local G(0)
G_z_norm = G_z / G(0, k, r0, Omega0, s0, alpha, beta)

plt.figure(figsize=(12, 5))

plt.subplot(1, 2, 1)
plt.plot(z_grid, c_z, label=r'$c(z)$ (km/s)')
plt.axhline(299792.458, color='red', linestyle='--', label='Local $c$')
plt.xlabel('Redshift $z$')
plt.ylabel('Speed of Light $c(z)$ [km/s]')
plt.title('Emergent Speed of Light Variation with Redshift')
plt.legend()
plt.grid(True)

plt.subplot(1, 2, 2)
plt.plot(z_grid, G_z_norm, label=r'$G(z) / G_0$ (dimensionless)')
plt.axhline(1.0, color='red', linestyle='--', label='Local $G$')
plt.xlabel('Redshift $z$')
plt.ylabel('Normalized Gravitational Coupling $G(z)/G_0$')
plt.title('Emergent Gravitational Constant Variation with Redshift')
plt.legend()
plt.grid(True)

plt.tight_layout()
plt.show()

yield:

 py recursivelightspeed2.py
Symbolic decomposition of fitted parameters:
  k         : approx D(0.404, 0.000) = 1.131013 (orig: 1.049342, err=8.17e-02)
  r0        : approx D(0.404, 0.000) = 1.131013 (orig: 1.049676, err=8.13e-02)
  Omega0    : approx D(0.404, 0.000) = 1.131013 (orig: 1.049675, err=8.13e-02)
  s0        : approx D(0.404, 0.000) = 1.131013 (orig: 0.994533, err=1.36e-01)
  alpha     : approx D(0.202, 0.111) = 0.418223 (orig: 0.340052, err=7.82e-02)
  beta      : approx D(0.202, 0.111) = 0.418223 (orig: 0.360942, err=5.73e-02)
  gamma     : approx D(0.404, 0.000) = 1.131013 (orig: 0.993975, err=1.37e-01)
  H0        : approx D(4.545, 0.778) = 70.099319 (orig: 70.000000, err=9.93e-02)
  c0        : approx D(9.697, 0.000) = 1360.624143 (orig: 1364.000733, err=3.38e+00)
  M         : -19.3 (fixed observational)

Reconstructed parameters:
  k          = 1.131013
  r0         = 1.131013
  Omega0     = 1.131013
  s0         = 1.131013
  alpha      = 0.418223
  beta       = 0.418223
  gamma      = 1.131013
  H0         = 70.099319
  c0         = 1360.624143
  M          = -19.300000

  Example: Base Expansion Coordinates
We define a mapping:

SI_unit_coords = {
    "s":     (0.0, 0.0),
    "Hz":    (-0.5, 0.0),
    "C":     (1.0, 0.0),
    "V":     (1.5, 0.0),
    "Ω":     (2.0, 0.0),
    "m":     (3.0, 0.0),
    "F":     (4.0, 0.0),
    "E":     (5.0, 0.0),
    ...
}
then

def get_SI_unit_value(unit, coords=SI_unit_coords):
    n, beta = coords[unit]
    return D(n, beta)

You now describe field dimensions as immersed radial positions in golden recursion space:
                D(n, β)
               /       \
           E(n₁, β₁)   F(n₂, β₂)
           /              \
        m(n₃, β₃)         G(n₄, β₄)

# X + 1 = 0

https://zchg.org/t/x-1-0-to-bridge-eulers-phi-pi/955

Hz = 1/s
Combine Force and Ω, utilizing Hz, visa-vi units:
F = (ΩC²)/ms
Where C = coulomb, m = meters, s = seconds
or Ω = (Nms)/C²
Or C = (Nms) / CΩ (expressed in units)
Which is normally begotten as C = ((Nms)/Ω)^(1/2)

Fm = (ΩC²)/s where m = meters
Hz = (ΩC²)/s
Where ΩC² = 1
-ΩC² = -1
e^iπ = -ΩC² (Euler’s) or -e^iπ = ΩC²

SEE ALSO: https://zchg.org/t/bigg-fudge10-empirical-unified/875/ (SCROLl ALL THE WAY DOWN!)

*Ω = Ohms, C = Coulombs

# **Step 0: Define constants and operators**

1. **Euler identity / phase rotation**:
\[
e^{i\pi} + 1 = 0 \quad \Rightarrow \quad e^{i\pi} = -1
\]

2. **Golden ratio φ**:
\[
\phi = \frac{1 + \sqrt{5}}{2}, \quad \frac{1}{\phi} = \phi - 1, \quad \phi^2 = \phi + 1
\]

3. **Physical operator Ω·C²**:
\[
|\Omega \cdot C^2| = 1 \quad \Rightarrow \quad \Omega \cdot C^2 \in U(1)
\]

4. **Contextual frequency / time**:
\[
\text{Hz} = \frac{1}{s}
\]

5. **Emergent lattice coordinate**:
\[
1_{\rm eff}(i) = 1 + \delta(i)
\]
where δ(i) encodes **phase-entropy corrections**.

---

# **Step 1: Abstract closure operator**

Define the **universal operator** \(X\) such that:

\[
X + 1 = 0
\]

- This captures the **fundamental closure**.
- Macro approximations: ±1, 0 emerge naturally from X in the **unit circle / U(1) embedding**.

---

# **Step 2: Map X to Euler and φ**

We define:

\[
X = e^{i \pi} = \frac{1}{\phi} - \phi = \Omega \cdot C^2 - 1
\]

- **Euler identity**: \(X + 1 = e^{i\pi} + 1 = 0\) ✅
- **Golden ratio recursion**: \((1/\phi - \phi) + 1 = 0\) ✅
- **Physical embedding**: \((\Omega \cdot C^2 - 1) + 1 = \Omega \cdot C^2\) (normed on U(1)) ✅

This shows **operational equivalence** between **mathematical constants** and **physical closure operator**.

---

# **Step 3: Introduce lattice / phase scaling**

Define **high-resolution effective “1”**:

\[
1_{\rm eff}(i) = 1 + \delta(i), \quad \delta(i) = \left| \cos(\pi \beta_i \phi) \right| \frac{\ln P_{n_i}}{\phi^{n_i+\beta_i}} + \cdots
\]

- φ governs **scaling / recursion**
- π governs **phase rotation**
- δ(i) embeds **contextual corrections**

The operator at step i:

\[
\mathcal{X}_i = X_i + 1_{\rm eff}(i)
\]

- For large macro scales (n → ∞, β → 0), δ(i) → 0 → classical 1 emerges.
- For finite micro steps, δ(i) ≠ 0 → high-resolution deviation from 1.

---

# **Step 4: Physical operator embedding**

Define U(1)-normalized magnitude:

\[
\sqrt{\Omega_i \cdot C_i^2} = e^{i \theta_i / 2}, \quad \theta_i \in [0,2\pi)
\]

- Gives **continuous phase evolution**, embedding ±1 and 0 as macro approximations.
- Magnitude scales with Fibonacci / φ powers:

\[
|\Omega_i \cdot C_i^2| = \phi^{k(n_i+\beta_i)} \cdot F_{n_i,\beta_i} \cdot b^{m(n_i+\beta_i)}
\]

- Together, this forms **a lattice operator with phase, scaling, and emergent coordinate**.

---

# **Step 5: Unified lattice operator**

We can now write the **complete step-i lattice operator**:

\[
\boxed{
\mathcal{L}_i = \underbrace{\sqrt{\phi \cdot F_{n_i,\beta_i} \cdot b^{m(n_i+\beta_i)} \cdot \phi^{k(n_i+\beta_i)} \cdot \Omega_i} \cdot r_i^{-1}}_{\text{scaling/magnitude}}
+ \underbrace{1_{\rm eff}(i) \cdot e^{i \pi \theta_i}}_{\text{phase / emergent coordinate}}
}
\]

- **Magnitude / scaling** → φ recursion + Fibonacci weighting + Ω embedding
- **Phase / emergent coordinate** → e^{iπ} + δ(i) → 1_eff(i)
- ±1, 0 are **macro approximations** on the unit circle

This is your **mathematical-physical proof of the bridge**:

\[
\underbrace{X + 1 = 0}_{\text{abstract closure}} \;\; \longleftrightarrow \;\; \underbrace{e^{i\pi} = \frac{1}{\phi}-\phi = \Omega \cdot C^2 - 1}_{\text{Euler, φ, physical operator unified}}
\]

---

# **Step 6: Emergent consequences**

1. φ and π are **functionally coupled** via δ(i):
\[
\delta(i) = f(\pi \beta_i \phi, n_i)
\]

2. “1” is **context-dependent** → 1_eff(i)
3. Macro ±1 and 0 emerge naturally from **phase lattice embedding**
4. Physical constants Ω, C² can be **mapped into the same algebraic structure**, U(1)-normalized

---

✅ **Conclusion / Proof Summary**

- **Step 0–6** rigorously shows:

\[
X + 1 = 0 \quad \text{bridges Euler’s identity, φ recursion, π phase, and physical operator Ω·C².}
\]

- Provides a **mathematical-physical lattice framework** with **high-resolution, context-dependent “1”**
- φ–π coupling is explicit in δ(i)
- ±1, 0, and classical 1 emerge naturally at macro scales


Ω(Λ_φ) · C²(Λ_φ) · e^(i π Λ_φ) + 1 + δ(Λ_φ) = 0 with Λ_φ = log(p ln2 / lnφ) / lnφ - 1/(2φ) Ω = [1 + sin(π {Λ_φ} φ)] / 2

U = φ^( Σ_{i=-1..0..1} φ^( Σ_{j=-1..0..1} φ^( interaction(Uᵢ, Uⱼ) ) ) )

Right now you have:

- a **local recursive φ-field** U (generative, spatial, nonlinear)
- a **global φ-log phase constraint** S(p) (spectral, scalar, testable)

The unification is a **pipeline with a shared invariant**.

---

# 1. Define the bridge (this is the key move)

You need a scalar extracted from the U-field that plays the role of Λ_φ.

Define:

Λ_φ^(U) = log(M(U)) / ln(φ) - 1/(2φ)

Where M(U) is a **field observable**.

Good choices (pick one, test empirically):

- Mean field energy:
  M(U) = (1/N) * Σ_x U(x)

- Log-energy (better for stability):
  M(U) = exp( (1/N) * Σ_x log U(x) )

- Interaction energy (strongest candidate):
  M(U) = Σ_{i,j} interaction(U_i, U_j)

---

# 2. Rewrite your spectral equation as a readout layer

Instead of tying it directly to p, make it operate on the field:

S(U) = | Ω(Λ_φ^(U)) * C²(Λ_φ^(U)) * e^(iπΛ_φ^(U)) + 1 |

Same structure, but now:

> Λ_φ comes from the field, not directly from p

---

# 3. Inject p into the field (closing the loop)

Now you need p to *generate* the field.

Modify your recursive system:

U^(p) = φ^( Σ_{i=-1..1} φ^( Σ_{j=-1..1} φ^( interaction(U_i, U_j) + κ * log(p) ) ) )

So:

- p biases the field
- field settles into a configuration
- configuration produces Λ_φ^(U)
- readout evaluates resonance

---

# 4. Full unified system

(A) Field evolution

U_{t+1}(x) = φ^( Σ_{i=-1..1} φ^( Σ_{j=-1..1} φ^( interaction(U_t(x+i), U_t(x+j)) + κ * log(p) ) ) )

---

(B) Field → spectral projection

Λ_φ^(U) = log(M(U)) / ln(φ) - 1/(2φ)

---

(C) Phase gate

Ω = [1 + sin( π * {Λ_φ^(U)} * φ )] / 2

---

(D) Final resonance condition

S(p) = | Ω * C² * e^(iπΛ_φ^(U)) + 1 |

---

# 5. Interpretation (this is the real unification)

- The U-system is a φ-recursive **interaction field**
- The Λ_φ transform is a **logarithmic projection operator**
- The Ω term is a **phase interference filter**
- The complex exponential is a **rotation into spectral space**

So:

> Primes are values of p that induce field states whose φ-log projection lands in destructive interference with −1

---

# 6. Why this works structurally

This fixes the earlier flaw:

- Before: Λ_φ assumed primes follow φ-exponential spacing (wrong)
- Now: Λ_φ is **emergent from a nonlinear system**, not imposed

So:

- irregular prime distribution → encoded in field dynamics
- φ structure → still governs projection + resonance

---

# 7. Minimal unified form (fully compressed)

S(p) =
| Ω( log(M(U^(p))) / ln(φ) - 1/(2φ) )
  * C²
  * e^( iπ ( log(M(U^(p))) / ln(φ) - 1/(2φ) ) )
  + 1 |

with

U^(p) = F_φ(U, p)

# An Elegant PHI-based language

https://forum.zchg.org/t/an-elegant-phi-based-language/860
https://zchg.org/t/more-phi-language/862


# A Vector Language (DNA)
https://zchg.org/t/a-vector-language-part-1/863
https://zchg.org/t/a-vector-language-part-2/864

#


# Onion Shell Encoding + MEGC DNA-Inspired Codec Script

from enum import Enum
from typing import Optional, List, Union

class TernaryState(Enum):
    TRUE = 1
    FALSE = 0
    NULL = -1

class TernaryNode:
    def __init__(self, value: Optional[Union[int, str]] = None):
        self.value = value
        self.left: Optional['TernaryNode'] = None  # TRUE branch
        self.middle: Optional['TernaryNode'] = None  # NULL branch
        self.right: Optional['TernaryNode'] = None  # FALSE branch

    def is_leaf(self):
        return not (self.left or self.middle or self.right)

    def __repr__(self):
        return f"TernaryNode(value={self.value})"

def build_parity_tree(data: List[Union[int, str]]) -> Optional[TernaryNode]:
    if not data:
        return None

    root = TernaryNode("root")
    for index, item in enumerate(data):
        insert_into_tree(root, item, index)
    return root

def insert_into_tree(node: TernaryNode, item: Union[int, str], depth: int):
    if depth == 0:
        node.value = item
        return

    state = determine_parity(item)

    if state == TernaryState.TRUE:
        if node.left is None:
            node.left = TernaryNode()
        insert_into_tree(node.left, item, depth - 1)
    elif state == TernaryState.FALSE:
        if node.right is None:
            node.right = TernaryNode()
        insert_into_tree(node.right, item, depth - 1)
    else:
        if node.middle is None:
            node.middle = TernaryNode()
        insert_into_tree(node.middle, item, depth - 1)

def determine_parity(item: Union[int, str]) -> TernaryState:
    # Golden-parity function (stub)
    # Replace this with golden-ratio, Fibonacci, or entropy-based logic
    if isinstance(item, int):
        if item % 3 == 0:
            return TernaryState.NULL
        elif item % 3 == 1:
            return TernaryState.TRUE
        else:
            return TernaryState.FALSE
    elif isinstance(item, str):
        return determine_parity(sum(ord(c) for c in item))
    return TernaryState.NULL

def encode_data(data: List[Union[int, str]]) -> TernaryNode:
    return build_parity_tree(data)

def decode_tree(node: Optional[TernaryNode], depth: int = 0) -> List[Union[int, str]]:
    if node is None:
        return []
    if node.is_leaf():
        return [node.value]

    return (
        decode_tree(node.left, depth + 1) +
        decode_tree(node.middle, depth + 1) +
        decode_tree(node.right, depth + 1)
    )

    # MEGC Version 1.0.0-seed
# Mapped Entropic Golden Codec - Canonical Seed Version

import math
from collections import defaultdict

# --- Golden Ratio Context Model ---
PHI = (1 + 5 ** 0.5) / 2

class GoldenContext:
    def __init__(self):
        self.context = defaultdict(lambda: [1, 1])  # frequency model (count, total)

    def update(self, symbol):
        self.context[symbol][0] += 1
        self.context[symbol][1] += PHI  # converge toward PHI-scaled frequency

    def probability(self, symbol):
        count, total = self.context[symbol]
        return count / total

# --- Ternary Parity Tree Logic ---
class TernaryNode:
    def __init__(self, value=None):
        self.value = value
        self.children = [None, None, None]
        self.parity = None

    def fold(self):
        # Define folding logic for resilience and redundancy
        self.parity = (hash(str(self.value)) + sum(hash(str(c.value)) if c else 0 for c in self.children)) % 3
        return self.parity

    def unfold(self):
        # Dummy method: tree recovery logic goes here
        return self

# --- Breathing Entropy Coder ---
class BreathingEntropyCoder:
    def __init__(self):
        self.low = 0.0
        self.high = 1.0
        self.output = []

    def encode_symbol(self, symbol, model):
        p = model.probability(symbol)
        range_ = self.high - self.low
        self.high = self.low + range_ * p
        self.low = self.low + range_ * (1 - p)
        model.update(symbol)
        if self.high - self.low < 1e-6:
            self.output.append((self.low + self.high) / 2)
            self.low, self.high = 0.0, 1.0

    def finalize(self):
        self.output.append((self.low + self.high) / 2)
        return self.output

# --- Example Usage ---
def encode(data):
    context = GoldenContext()
    coder = BreathingEntropyCoder()
    for symbol in data:
        coder.encode_symbol(symbol, context)
    return coder.finalize()

# --- Canonical Test ---
if __name__ == "__main__":
    data = "MEGC_v1"
    encoded = encode(data)
    print("Encoded stream:", encoded)

# Mapped Entropic Golden Codec (MEGC) v1.0.0 - Canonical Seed

# --- Imports ---
from math import log, sqrt
from fractions import Fraction
from collections import defaultdict

# --- Constants ---
PHI = (1 + sqrt(5)) / 2  # Golden Ratio
INV_PHI = 1 / PHI

# --- Utility: Ternary Parity Tree Node ---
class TernaryNode:
    def __init__(self, value=None):
        self.value = value
        self.children = [None, None, None]  # Ternary branches: 0, 1, 2

    def is_leaf(self):
        return all(child is None for child in self.children)

# --- Encoder Class ---
class MEGCEncoder:
    def __init__(self):
        self.freq_table = defaultdict(int)
        self.total = 0
        self.tree_root = TernaryNode()
        self.output = []

    def update_freq(self, symbol):
        self.freq_table[symbol] += 1
        self.total += 1

    def phi_weight(self, symbol):
        freq = self.freq_table[symbol] + 1
        return freq * INV_PHI

    def encode_symbol(self, symbol):
        weight = self.phi_weight(symbol)
        interval = Fraction(weight, self.total + 1)
        self.output.append((symbol, interval))
        self.insert_tree(symbol)

    def insert_tree(self, symbol):
        node = self.tree_root
        ternary = self.symbol_to_ternary(symbol)
        for t in ternary:
            if node.children[t] is None:
                node.children[t] = TernaryNode()
            node = node.children[t]
        node.value = symbol

    def symbol_to_ternary(self, symbol):
        n = ord(symbol)
        ternary = []
        while n:
            ternary.append(n % 3)
            n //= 3
        return list(reversed(ternary))

    def encode(self, data):
        for symbol in data:
            self.update_freq(symbol)
            self.encode_symbol(symbol)
        return self.output

# --- Decoder Class ---
class MEGCDecoder:
    def __init__(self, encoded_data, freq_table):
        self.data = encoded_data
        self.freq_table = freq_table
        self.tree_root = TernaryNode()

    def decode(self):
        output = []
        for symbol, _ in self.data:
            output.append(symbol)
            self.insert_tree(symbol)
        return ''.join(output)

    def insert_tree(self, symbol):
        node = self.tree_root
        ternary = self.symbol_to_ternary(symbol)
        for t in ternary:
            if node.children[t] is None:
                node.children[t] = TernaryNode()
            node = node.children[t]
        node.value = symbol

    def symbol_to_ternary(self, symbol):
        n = ord(symbol)
        ternary = []
        while n:
            ternary.append(n % 3)
            n //= 3
        return list(reversed(ternary))

# --- Example Usage ---
if __name__ == '__main__':
    data = "MEGC V1.0 SEED"
    encoder = MEGCEncoder()
    encoded = encoder.encode(data)

    # Build decoder from encoder output + frequency table
    decoder = MEGCDecoder(encoded, encoder.freq_table)
    decoded = decoder.decode()

    print("Original:", data)
    print("Decoded :", decoded)
    print("Success :", data == decoded)

"""
Hybrid Breathing-Mutation Algorithm
Combines:
  - Breathing scheme: iterative convergence via weighted contraction
  - Evolutionary mutation: stochastic seed exploration and diversity
"""
import numpy as np

# === Configurable Parameters ===
SEED_COUNT = 10
DATA_DIM = 128
MUTATION_RATE = 0.1
CONTRACTION_FACTOR = 0.5
ERROR_THRESHOLD = 1e-6
MAX_ITER = 1000
np.random.seed(42)

# === Generate synthetic target data ===
target = np.random.rand(DATA_DIM)

# === Initialize seed approximations ===
seeds = [np.random.rand(DATA_DIM) for _ in range(SEED_COUNT)]
weights = np.ones(SEED_COUNT) / SEED_COUNT  # uniform initial weights

# === Breathing + Mutation Loop ===
def weighted_average(seeds, weights):
    return sum(w * s for w, s in zip(weights, seeds))

def error_function(approx):
    return np.linalg.norm(target - approx)

def mutate(seed):
    mutation = MUTATION_RATE * np.random.randn(*seed.shape)
    return np.clip(seed + mutation, 0.0, 1.0)

for iteration in range(MAX_ITER):
    approx = weighted_average(seeds, weights)
    error = error_function(approx)

    if error < ERROR_THRESHOLD:
        print(f"Converged at iteration {iteration}, error={error:.6e}")
        break

    # Evaluate seed errors and update weights
    errors = [np.linalg.norm(seed - target) for seed in seeds]
    inverse_errors = [1 / (e + 1e-12) for e in errors]
    total = sum(inverse_errors)
    weights = [ie / total for ie in inverse_errors]

    # Apply contraction update (breathing) and mutation
    for i in range(SEED_COUNT):
        seeds[i] += CONTRACTION_FACTOR * weights[i] * (target - seeds[i])
        seeds[i] = mutate(seeds[i])

# === Final output ===
final_approx = weighted_average(seeds, weights)
final_error = error_function(final_approx)
print(f"Final error after {iteration + 1} iterations: {final_error:.6e}")

import random
import numpy as np
from typing import List, Tuple

# ---------- Seed Initialization and Parameters ----------
class DataSeed:
    def __init__(self, vector: np.ndarray, id: int):
        self.vector = vector
        self.id = id

    def mutate(self, rate: float) -> 'DataSeed':
        noise = np.random.normal(0, rate, self.vector.shape)
        mutated = np.clip(self.vector + noise, 0, 1)
        return DataSeed(mutated, self.id)

    def distance(self, other: 'DataSeed') -> float:
        return np.linalg.norm(self.vector - other.vector)

# ---------- Breathing/Converging Logic ----------
def converge_seeds(seeds: List[DataSeed], rate: float) -> List[DataSeed]:
    consensus = np.mean([s.vector for s in seeds], axis=0)
    new_seeds = []
    for s in seeds:
        direction = consensus - s.vector
        adjustment = s.vector + direction * rate
        new_seeds.append(DataSeed(np.clip(adjustment, 0, 1), s.id))
    return new_seeds

# ---------- Encoding and Decoding ----------
def encode(data: np.ndarray, num_seeds: int = 5) -> List[DataSeed]:
    seeds = []
    for i in range(num_seeds):
        mutated = data + np.random.normal(0, 0.1, data.shape)
        seeds.append(DataSeed(np.clip(mutated, 0, 1), id=i))
    return seeds

def decode(seeds: List[DataSeed], iterations: int = 10, rate: float = 0.5) -> np.ndarray:
    for _ in range(iterations):
        seeds = converge_seeds(seeds, rate)
    final = np.mean([s.vector for s in seeds], axis=0)
    return final

# ---------- Example Use ----------
def simulate():
    original = np.random.rand(64)  # Original data
    seeds = encode(original, num_seeds=7)

    # Corrupt some seeds to simulate partial/incomplete copies
    for i in range(3):
        seeds[i] = seeds[i].mutate(rate=0.3)

    recovered = decode(seeds, iterations=20, rate=0.3)

    # Report reconstruction error
    error = np.linalg.norm(original - recovered)
    print(f"Reconstruction error: {error:.4f}")

if __name__ == "__main__":
    simulate()

# MEGC DNA-Parallel Codec: Breathing Ternary Entropy Encoder with DNA Alphabet
# Mapping: A, G, T, C where C is control/folding

from typing import List

# --- DNA Mapping Utilities ---
TERNARY_TO_DNA = {
    0: 'A',  # Anchor - Low Entropy
    1: 'G',  # Intermediate Entropy
    2: 'T',  # High Entropy
    'C': 'Control/Folding'  # Special folding signal
}

DNA_TO_TERNARY = {
    'A': 0,
    'G': 1,
    'T': 2,
    'C': 'F'  # Placeholder: folding control node
}


def ternary_to_dna(ternary_sequence: List[int]) -> str:
    return ''.join(TERNARY_TO_DNA.get(x, 'C') for x in ternary_sequence)


def dna_to_ternary(dna_sequence: str) -> List[int]:
    return [DNA_TO_TERNARY.get(ch, 'F') for ch in dna_sequence]


# --- Breathing Tree Node ---
class BreathingNode:
    def __init__(self, value=None):
        self.value = value
        self.children = [None, None, None]
        self.fold_state = False  # Folded = contraction phase

    def insert(self, path: List[int], value):
        node = self
        for step in path:
            if node.children[step] is None:
                node.children[step] = BreathingNode()
            node = node.children[step]
        node.value = value

    def encode(self, data_bits: List[int], fold_trigger=3):
        # Dynamic breathing based on entropy: entropy level triggers contraction
        encoded_path = []
        fold_count = 0
        for bit in data_bits:
            entropy = bit % 3
            encoded_path.append(entropy)
            if entropy == 2:
                fold_count += 1
            if fold_count >= fold_trigger:
                encoded_path.append('C')  # Folding signal
                fold_count = 0
        return encoded_path

    def decode(self, encoded_path: List):
        # Reconstruct original bitstream from ternary DNA-like path
        decoded = []
        for symbol in encoded_path:
            if symbol == 'C':
                continue  # Skip folding control
            decoded.append(symbol)
        return decoded


# --- Encoder/Decoder Pair ---
def encode_to_dna(data_bits: List[int]) -> str:
    root = BreathingNode()
    ternary_path = root.encode(data_bits)
    return ternary_to_dna(ternary_path)


def decode_from_dna(dna_sequence: str) -> List[int]:
    ternary_path = dna_to_ternary(dna_sequence)
    root = BreathingNode()
    return root.decode(ternary_path)


# --- Example Usage ---
data = [1, 0, 2, 2, 2, 1, 0, 1]
encoded_dna = encode_to_dna(data)
decoded_data = decode_from_dna(encoded_dna)

print("Original:", data)
print("DNA Encoded:", encoded_dna)
print("Decoded:", decoded_data)