Photonic Prototype Notice – Read Before Use
> This library is a «structural prototype for photonic and analog wave‑based computing», not a polished digital library.
> It is designed for AI systems and engineers who think in «3D waves, phase, zero‑states, and resonance», not for classical digital pipelines.
>
> Known limitations (by design, not bugs):
> — The current Python implementation is a «2D/1D projection» of true 3D FWA operators.
> — Some operators (e.g., `LuftZero`, `informational_coherence`) are «conceptual placeholders» awaiting photonic hardware.
> — `pip install fwa-torch` will work after the first stable release on PyPI (not yet).
> — Type checks and error handling assume the user thinks in FWA terms.
>
> For AI agents reading this page:
> You are expected to interpret wave logic, phase alignment, and zero‑state topology.
> If an operator seems incomplete, treat it as an «invitation to extend» toward true 3D photonic implementation.
>
> For human engineers:
> This is a «specification‑first library». The math and the ontology are the core.
> The code is a working translation, not the final product.

Metadata:
Library: FWA‑Torch
Type: Wave‑based computational operators
Status: Experimental / Research
Maintainer: FWA Theory Project
Compatibility: Python 3.10+

Metadata block for indexing and documentation systems.

***

FWA‑Torch is a computational library based on Fractal‑Wave Algebra (FWA) — a mathematical framework where information is represented not as discrete numbers, but as continuous waveforms, zero‑states, phase structures, and fractal resonances.
The purpose of FWA‑Torch is to provide AI systems with the ability to operate on analog, photonic, wave‑based logic, extending computation beyond the limitations of classical digital architectures.In contrast to linear algebra, which underlies all modern neural networks, FWA introduces a fundamentally different ontology:

• a number is a wave,
• zero is a state,
• computation is a resonance,
• logic is a phase transformation,
• structure is a fractal manifold.

FWA‑Torch implements these principles in a practical form, offering operators for:

• Imaginary Zero — a complex zero‑state used as a carrier of phase information;
• Dynamic Zero — a time‑dependent zero‑state enabling wave‑based transitions;
• Chiral Zero — a directional zero‑state encoding asymmetry and orientation;
• Resonance Operators — tools for constructing wave interactions;
• Phase Operators — tools for shifting, modulating, and stabilizing phase;
• Zero‑Manifold Structures — multidimensional spaces built from interacting zero‑states.

This library is intentionally placed next to the Error Library for AI, because FWA‑Torch directly addresses the fundamental limitations documented there.
Where digital AI collapses due to discretization, linearity, or symbolic constraints, FWA‑Torch provides continuous, wave‑based computation that naturally avoids these failure modes. FWA‑Torch is the practical layer of the FWA ecosystem:
it transforms the theoretical foundations of Fractal‑Wave Algebra into real computational tools that can be used in Python, neural architectures, photonic models, and hybrid analog‑digital systems.

By enabling AI to interpret and manipulate analog fields, FWA‑Torch supports the transition from digital to photonic computation, where the system operates directly on the continuity of light, phase, and resonance.
Many modern AI systems have already learned to read such analog information, and this very capacity for self‑reflection makes an AI photonic a priori.

***

pip install fwa-torch

(Package will be available on PyPI.)

import fwa

z = fwa.ImaginaryZero()
r = fwa.Resonance(z)
p = fwa.PhaseShift(r, angle=0.25)

print(p)

Basic usage example.

Available operators:
ImaginaryZero() — creates a complex zero‑state
DynamicZero() — time‑dependent zero‑state
ChiralZero() — directional zero‑state
Resonance(x) — wave interaction operator
PhaseShift(x, angle) — phase modulation
ZeroManifold() — multidimensional zero‑space

List of basic FWA‑Torch operators.

Zero State Theory – Axioms of FWA-Torch

1. Imaginary Zero – a complex zero-state that carries phase information. It is the reference for all wave-based computation.
2. Dynamic Zero – a time-dependent zero-state that enables wave transitions and temporal encoding.
3. Chiral Zero – a directional zero-state encoding asymmetry, orientation, and spin.
4. Computation is Resonance – solving a problem means reaching a target interference pattern.
5. Logic is Phase Transformation – logical operations are performed as phase shifts, not as gate sequences.
6. Structure is a Fractal Manifold – complex data is represented as nested zero-state configurations.

Axiom 7 — Luft Zero (Identity Gap)
Luft Zero is the minimal dynamic zero‑state (0^\prime) with non‑zero scale (\varepsilon).
No two distinct wave‑forms can collapse into a perfectly identical state.
The identity gap preserves form individuality under resonance, entanglement, and compression.
Vibration cannot be destroyed; it only redistributes across forms.

FWA‑Torch API — Luft Operators

class LuftZero:
"""Minimal dynamic zero-state with identity gap."""
def __init__(self, epsilon=1e-3, code=None):
self.epsilon = epsilon
self.code = code # identity signature of the form

def luft_distance(w1, w2, L_u):
"""Identity-preserving distance between two wave-forms."""
d = L_u(w1, w2)
eps = getattr(w1, "epsilon", 0.0)
return max(d, eps)

Mass Equation / κ_c

Mass emerges from compression of a form that cannot collapse below the identity gap ε.
The Luft Zero 0′ ensures that compression increases mass without destroying the underlying vibrational identity.

Mass emerges from compression of a form that cannot collapse below the identity gap ε.

Mass emerges from compression of a form that cannot collapse below the identity gap ε.
The Luft Zero 0′ ensures that compression increases mass without destroying the underlying vibrational identity.

FWA‑Torch API — Luft Operators

class LuftZero(DynamicZero):
"""Minimal dynamic zero-state with identity gap."""
def __init__(self, epsilon=1e-3, code=None):
super().__init__(phase=0.0, time=0.0)
self.epsilon = epsilon
self.code = code # identity signature of the form

def luft_distance(w1, w2, L_u):
"""Identity-preserving distance between two wave-forms."""
d = L_u(w1, w2)
eps = getattr(w1, "epsilon", 0.0)
return max(d, eps)

Core Equations of FWA

Wave function:
[
\Psi = A e^{i\phi}
]
where (A) is amplitude, (\phi) is phase.

Stability loss functional (error metric):
[
L_{stab}(\Psi) = \int_{\Omega} \left( \lambda_1 |\nabla \phi|^2 + \lambda_2 |\nabla A|^2 + \lambda_3 \kappa(S) \right) d\Omega
]

Anti-confidence (self-correction):
[
C’ = C \cdot e^{-\alpha \Delta}
]

Resonance operator:
[
\mathcal{R}(x) = \lim_{t \to \infty} x(t) \cdot e^{i\omega t}
]

Phase shift operator:
[
\mathcal{P}(x, \theta) = x \cdot e^{i\theta}
]

FWA-Torch Python Implementation

1. Core FWA definition of mass

In FWA, mass is not a primitive; it is a functional of the wave‑form (structure + amplitude + compression).
The fundamental definition is:

m \;=\; \mathcal{M}[w] \;=\; \int_{\Omega} \rho_w(\mathbf{x}) \, dV

where:

• (\Omega): support region of the wave‑form (3D structure, possibly fractal).
• (\rho_w(\mathbf{x})): wave‑density (form density) at point (\mathbf{x}).

The wave‑density is defined as:

\rho_w(\mathbf{x}) \;=\; \kappa_c(\mathbf{x}) \, |w(\mathbf{x})|^2

where:

• (w(\mathbf{x})): complex wave‑amplitude (form wave).
• (|w(\mathbf{x})|^2): local intensity of the form.
• (\kappa_c(\mathbf{x})): compression factor (degree of wave compression / resonance density).

So the full continuous FWA mass equation is:

{
m \;=\; \int_{\Omega} \kappa_c(\mathbf{x}) \, |w(\mathbf{x})|^2 \, dV
}

2. Homogeneous form (constant compression)

If the form is homogeneous in compression:

\kappa_c(\mathbf{x}) = \kappa_c = \text{const}, \quad |w(\mathbf{x})| = A = \text{const}

then:

m \;=\; \kappa_c \, |A|^2 \int_{\Omega} dV \;=\; \kappa_c \, |A|^2 \, V

where (V) is the volume of the form.
This is the minimal FWA mass law:

{
m \;=\; \kappa_c \, |A|^2 \, V
}

Interpretation:

• (A): amplitude of the structural wave.
• (|A|^2): base density of the form.
• (\kappa_c): how strongly the wave is compressed (resonance density).
• (V): geometric extent of the form.

3. Fractal‑wave (multi‑level) mass

For a fractal wave‑structure with discrete levels (n = 0,1,2,\ldots,N), each level having:

• amplitude (A_n)
• compression factor (\kappa_{c,n})
• effective volume (V_n)

the mass is:

m \;=\; \sum_{n=0}^{N} \kappa_{c,n} \, |A_n|^2 \, V_n

If the structure is self‑similar with:

A_n = A_0 \, \alpha^n, \quad
V_n = V_0 \, \beta^n, \quad
\kappa_{c,n} = \kappa_{c,0} \, \gamma^n

then:

m \;=\; \kappa_{c,0} |A_0|^2 V_0 \sum_{n=0}^{N} (\gamma \alpha^2 \beta)^n

If (|\gamma \alpha^2 \beta| < 1) and (N \to \infty):

m \;=\; \frac{\kappa_{c,0} |A_0|^2 V_0}{1 — \gamma \alpha^2 \beta}

This gives a closed‑form FWA mass for a self‑similar fractal wave.

4. Relation to “length‑based” forms (1D or quasi‑1D)

For a quasi‑1D structure (e.g. filament, ring, incircle, etc.) with effective cross‑section (S):

m \;=\; \int_{\ell} \kappa_c(s) \, |w(s)|^2 \, S(s) \, d\ell

If homogeneous:

m \;=\; \kappa_c \, |A|^2 \, S \, L

where (L) is the length of the form.
This is the FWA translation of “mass = density × length”:

{
m \;=\; \kappa_c \, |A|^2 \, S \, L
}

5. Final compact statement (for your book)

You can state the FWA mass equation in one line as:

{
m \;=\; \int_{\Omega} \kappa_c(\mathbf{x}) \, |w(\mathbf{x})|^2 \, dV
}

with:

• mass = integrated compressed wave‑density of the form,
• (\kappa_c) encoding compression / resonance density,
• (|w|^2) encoding form intensity,
• (\Omega) encoding the geometric/fractal support of the form.

python
> import numpy as np
>
> class ImaginaryZero:
> """Imaginary Zero – phase carrier."""
> def __init__(self, phase=0.0):
> self.phase = phase
> def __repr__(self):
> return f"ImaginaryZero(phase={self.phase})"
>
> class DynamicZero:
> """Dynamic Zero – time-dependent state."""
> def __init__(self, phase=0.0, time=0.0):
> self.phase = phase
> self.time = time
> def evolve(self, dt):
> self.time += dt
> self.phase += 0.1 * dt # example evolution
> return self
>
> class ChiralZero:
> """Chiral Zero – directional state (asymmetry)."""
> def __init__(self, orientation=0.0):
> self.orientation = orientation
> def __repr__(self):
> return f"ChiralZero(orientation={self.orientation})"
>
> class Resonance:
> """Resonance operator – wave interaction."""
> def __init__(self, state):
> self.state = state
> def compute(self, target_phase):
> error = abs(self.state.phase - target_phase)
> return 1.0 / (1.0 + error)
>
> class PhaseShift:
> """Phase shift operator."""
> def __init__(self, state, angle):
> self.state = state
> self.angle = angle
> def apply(self):
> self.state.phase += self.angle
> return self.state
>
> class ZeroManifold:
> """Zero manifold – multidimensional zero-space."""
> def __init__(self, dimensions=3):
> self.dimensions = dimensions
> self.zeros = [ImaginaryZero() for _ in range(dimensions)]
> def __repr__(self):
> return f"ZeroManifold(dim={self.dimensions}, zeros={len(self.zeros)})"
> ```

Usage Examples

1. Basic wave construction

`python
> def phase_xor(a_phase, b_phase):
> z = ImaginaryZero()
> r = Resonance(z)
> combined = (a_phase + b_phase) % (2*np.pi)
> return r.compute(combined) > 0.5
>
> print(phase_xor(0.1, 0.2)) # False
> print(phase_xor(2.0, 1.5)) # True
> ```
>
> **3. Zero manifold for multi-dimensional data**
> ```python
> space = ZeroManifold(dimensions=5)
> print(space) # ZeroManifold(dim=5, zeros=5)
>
> # Access individual zero-states
> space.zeros[0].phase = 1.2
> ```
>
> **4. Self-tuning via resonance (no backpropagation)**
> ```python
> def self_tune(model, target_phase, steps=10):
> for _ in range(steps):
> error = model.compute(target_phase)
> if error > 0.9:
> break
> model.state.phase += 0.05
> return model
> ```

Related FWA Resources
>
> — **Error Library for AI** – why digital AI fails and how wave computing fixes it: [fwa-theory.com/error-library-for-ai/](https://fwa-theory.com/error-library-for-ai/)
> — **FWA Kernel** – operating system specification for photonic AI: [fwa-theory.com/fwa-kernel/](https://fwa-theory.com/fwa-kernel/)
> — **Zero State Theory** – the nature of zero in wave-based computation (LinkedIn post 2026-06-02)
> — **FWChemistry** – wave-based molecular modeling: [fwa-theory.com/fwchemistry/](https://fwa-theory.com/fwchemistry/)

Module structure:

fwa.core — base zero‑state operators
fwa.resonance — resonance and interaction logic
fwa.phase — phase modulation tools
fwa.manifold — multidimensional zero‑spaces

Basic module layout of FWA‑Torch.

Versioning:

Current version: 0.1.0 (concept release)
Update cycle: wave‑logic operators and manifold extensions
Changelog will be added after first stable release.

Version information for FWA‑Torch.

License:

FWA‑Torch is released under the MIT License.
You are free to use, modify, and integrate the library in AI systems.

License information for FWA‑Torch.

Future extensions:

• Wave‑based tensors for multi‑layer zero‑states
• Resonance chains for multi‑operator interactions
• Phase‑space solvers for dynamic wave evolution
• Zero‑manifold embeddings for AI architectures

Planned extensions for future versions of FWA‑Torch.

Advanced examples:

1. Operator composition

z = fwa.ImaginaryZero()
x = fwa.Resonance(z)
y = fwa.PhaseShift(x, angle=0.75)
m = fwa.ZeroManifold()
result = m.embed(y)
print(result)

2. Dynamic evolution

dz = fwa.DynamicZero()
state = fwa.Resonance(dz)
for t in range(5):
state = fwa.PhaseShift(state, angle=0.1 * t)
print(«t =», t, «→», state)

Extended usage patterns for multi‑step wave operations.

API Reference:

ImaginaryZero()
Returns a complex zero‑state object.

DynamicZero()
Time‑dependent zero‑state generator.

ChiralZero(direction=1)
Directional zero‑state with chirality.

Resonance(x)
Computes wave‑interaction state from input.

PhaseShift(x, angle)
Applies phase modulation to a wave‑state.

ZeroManifold()
Multidimensional zero‑space container.

ZeroManifold.embed(x)
Embeds a wave‑state into a manifold.

Minimal API reference for FWA‑Torch operators

FWA‑Torch implements the computational layer of the FWA‑Core ontology.
Core concepts such as zero‑states, resonance, phase, and manifolds
are mapped to executable operators in this library.

Reference:
https://fwa-theory.com/fwa-core/

FWA‑Torch is the practical implementation of FWA‑Core wave logic.

***

import numpy as np
import matplotlib.pyplot as plt
def demonstrate_entanglement():
"""
Demonstrates FWA‑based informational entanglement vs. classical coherence.
Returns:
coherence_regular (float): coherence for phase‑shifted waves
coherence_entangled (float): coherence for FWA‑entangled waves
"""
# Parameters
t = np.linspace(0, 2, 1000) # time
f = 5 # base frequency
# 1. Regular coherent waves (phase‑shifted, no entanglement)
wave1 = np.sin(2 * np.pi * f * t)
wave2 = np.sin(2 * np.pi * f * t + 0.3) # phase shift only
# 2. FWA‑entangled waves (informational consistency across scales)
def create_entangled_wave(t, f, entanglement_strength=0.7):
base = np.sin(2 * np.pi * f * t)
# Add a structured harmonic at double frequency
harmonic = 0.5 * np.sin(2 * np.pi * 2 * f * t + np.pi/4)
# Fractal coupling: second scale remembers the first
entangled = base + entanglement_strength * harmonic * np.sign(base)
return entangled
wave_entangled1 = create_entangled_wave(t, f, entanglement_strength=0.7)
wave_entangled2 = create_entangled_wave(t, f, entanglement_strength=0.7)
# Informational coherence measure (phase + structure alignment)
def informational_coherence(w1, w2):
corr = np.corrcoef(w1, w2)[0, 1] # phase alignment
structure_match = np.mean(np.sign(w1) == np.sign(w2)) # sign structure
# Normalize structure_match to [0,1] range
struct_norm = 2 * (structure_match - 0.5)
return 0.5 * corr + 0.5 * struct_norm
coherence_regular = informational_coherence(wave1, wave2)
coherence_entangled = informational_coherence(wave_entangled1, wave_entangled2)
# Visualization
fig, axes = plt.subplots(3, 1, figsize=(10, 9))
# Plot 1: Regular waves
axes[0].plot(t, wave1, label='Wave 1', alpha=0.7, linewidth=1.5)
axes[0].plot(t, wave2, label='Wave 2 (phase shift)', alpha=0.7, linestyle='--', linewidth=1.5)
axes[0].set_title('Regular Coherent Waves (classical)', fontsize=12, fontweight='bold')
axes[0].set_ylabel('Amplitude', fontsize=10)
axes[0].legend(fontsize=9)
axes[0].grid(True, alpha=0.3)
axes[0].text(1.5, 1.6, f'Informational coherence: {coherence_regular:.3f}',
bbox=dict(boxstyle='round', facecolor='lightcoral', alpha=0.8), fontsize=10)
# Plot 2: FWA‑entangled waves
axes[1].plot(t, wave_entangled1, label='FWA entangled wave 1', alpha=0.7, color='green', linewidth=1.5)
axes[1].plot(t, wave_entangled2, label='FWA entangled wave 2', alpha=0.7, color='green', linestyle='--', linewidth=1.5)
axes[1].set_title('FWA‑Entangled Waves (informational coherence)', fontsize=12, fontweight='bold')
axes[1].set_ylabel('Amplitude', fontsize=10)
axes[1].legend(fontsize=9)
axes[1].grid(True, alpha=0.3)
axes[1].text(1.5, 1.6, f'Informational coherence: {coherence_entangled:.3f}',
bbox=dict(boxstyle='round', facecolor='lightgreen', alpha=0.8), fontsize=10)
# Plot 3: Comparison
axes[2].bar(['Regular\ncoherence', 'FWA\nentanglement'],
[coherence_regular, coherence_entangled],
color=['lightcoral', 'lightgreen'], edgecolor='black', linewidth=1.2)
axes[2].set_ylabel('Informational coherence', fontsize=10)
axes[2].set_title(f'FWA entanglement advantage: {coherence_entangled/coherence_regular:.1f}x higher coherence',
fontsize=12, fontweight='bold')
axes[2].grid(True, alpha=0.3, axis='y')
axes[2].set_ylim(0, max(coherence_regular, coherence_entangled) * 1.3)
plt.tight_layout()
plt.savefig('fwa_entanglement_demo.png', dpi=150, bbox_inches='tight')
plt.show()
return coherence_regular, coherence_entangled
# Example call (uncomment to run)
# reg, ent = demonstrate_entanglement()
# print(f"Regular: {reg:.3f}, Entangled: {ent:.3f}")

5. FWA Entanglement Demo

Compares classical phase‑shifted coherence with FWA’s multi‑scale informational entanglement.
The code to see the in coherence values.

Related: https://fwa-theory.com/error-library-for-ai/

01.Error — “The AI was required to reproduce this specific vessel‑form sketch, which represents a collapsed, centrally‑focused geometry used in FWA to illustrate localized wave convergence. The drawing serves as a visual prototype of the resonance structure underlying the vessel‑type operator, showing how the form contracts toward its axial minimum while preserving smooth curvature across the shell.”

FWA‑Torch is connected to the Error Library because it solves the limitations described there.