# Morphic Topology Math Catalog
## Mathematical Equations from Internet Scan

**Date:** 2026-04-26T19:30:00
**Purpose:** Catalog of mathematical equations relevant to morphic topology system from internet sources

---

## Neural Coding Equations

### Rate Coding
**Spike-Count Rate (Temporal Average):**
```
r = N_spikes / T
```
Where:
- r = firing rate
- N_spikes = number of spikes in time window
- T = duration of time window (typically 100ms or 500ms)

**Reference:** Neural coding - Wikipedia

### Temporal Coding
**Binary Spike Representation:**
```
spike_train(t) = Σ_i δ(t - t_i)
```
Where:
- δ(t - t_i) = Dirac delta function at spike time t_i
- t_i = time of i-th spike

**Temporal Code Example:**
- Sequence 000111000111 ≠ 001100110011 (same mean rate, different temporal pattern)

**Reference:** Neural coding - Wikipedia

### Population Coding
**Population Vector Coding:**
```
v = Σ_i r_i v_i
```
Where:
- v = population vector (direction of motion)
- r_i = firing rate of neuron i
- v_i = preferred direction vector of neuron i

**Maximum Likelihood Reconstruction:** [BEAUTIFUL_PROVISIONAL - Standard statistical method; applicability to morphic topology system requires verification evidence]
```
P(s|r) ∝ Π_i P(r_i|s)
```
Where:
- s = stimulus
- r = population response vector
- r_i = response of neuron i

**Reference:** Neural coding - Wikipedia

---

## Synaptic Plasticity Equations

### Spike-Timing Dependent Plasticity (STDP)
**Weight Change Equation:**
```
Δw_j = Σ_f=1^N Σ_n=1^N W(t_i^n - t_j^f)
```
Where:
- Δw_j = weight change of synapse j
- t_j^f = presynaptic spike arrival times at synapse j
- t_i^n = postsynaptic firing times
- W(x) = STDP function (learning window)

**STDP Function (Exponential):**
```
W(x) = A_+ exp(-x/τ_+) for x > 0
W(x) = -A_- exp(x/τ_-) for x < 0
```
Where:
- A_+ = potentiation amplitude
- A_- = depression amplitude
- τ_+ = potentiation time constant (~10ms)
- τ_- = depression time constant (~10ms)

**Reference:** Scholarpedia - Spike-timing dependent plasticity

### Hebbian Learning
**Basic Hebbian Rule:**
```
Δw_ij = η x_i x_j
```
Where:
- Δw_ij = weight change from neuron j to neuron i
- η = learning rate
- x_i = activation of neuron i
- x_j = activation of neuron j

**Average Over Training Patterns:**
```
w_ij = (1/p) Σ_k x_i^k x_j^k
```
Where:
- p = number of training patterns
- x_i^k = k-th input for neuron i
- x_j^k = k-th input for neuron j

**Reference:** Hebbian theory - Wikipedia

---

## Signal Processing Equations

### Fourier Transform
**Forward Fourier Transform:**
```
f̂(ξ) = ∫_{-∞}^∞ f(x) e^{-i2πξx} dx
```
Where:
- f̂(ξ) = Fourier transform of f(x)
- f(x) = original function
- ξ = frequency variable

**Inverse Fourier Transform:**
```
f(x) = ∫_{-∞}^∞ f̂(ξ) e^{i2πξx} dξ
```

**Reference:** Fourier transform - Wikipedia

### Convolution Theorem
**Convolution in Time Domain:**
```
h(x) = (f * g)(x) = ∫_{-∞}^∞ f(y) g(x - y) dy
```

**Multiplication in Frequency Domain:**
```
ĥ(ξ) = f̂(ξ) ĝ(ξ)
```
Where:
- h = convolution of f and g
- ĥ = Fourier transform of h
- f̂ = Fourier transform of f
- ĝ = Fourier transform of g

**Reference:** Fourier transform - Wikipedia

### Cross-Correlation Theorem
**Cross-Correlation:**
```
h(x) = (f ⋆ g)(x) = ∫_{-∞}^∞ f(y)̄ g(x + y) dy
```

**Cross-Correlation in Frequency Domain:**
```
ĥ(ξ) = f̂(ξ)̄ ĝ(ξ)
```

**Autocorrelation:**
```
h(x) = (f ⋆ f)(x) = ∫_{-∞}^∞ f(y)̄ f(x + y) dy
ĥ(ξ) = |f̂(ξ)|²
```

**Reference:** Fourier transform - Wikipedia

---

## Information Theory Equations

### Shannon Entropy
**Entropy Definition:**
```
H(X) = -Σ_x p(x) log_b p(x)
```
Where:
- H(X) = entropy of random variable X
- p(x) = probability mass function
- b = base of logarithm (2 for bits, e for nats, 10 for bans)

**Expected Value Form:**
```
H(X) = E[I(X)] = E[-log p(X)]
```
Where:
- I(X) = information content of X
- E = expected value operator

**Conditional Entropy:**
```
H(X|Y) = -Σ_{x,y} p_{X,Y}(x,y) log(p_{X,Y}(x,y)/p_Y(y))
```
Where:
- p_{X,Y}(x,y) = joint probability P[X=x, Y=y]
- p_Y(y) = marginal probability P[Y=y]

**Reference:** Entropy (information theory) - Wikipedia

---

## Graph Theory Equations

### Laplacian Matrix
**Definition for Simple Graph:**
```
L = D - A
```
Where:
- L = Laplacian matrix
- D = degree matrix (diagonal matrix of vertex degrees)
- A = adjacency matrix

**Normalized Laplacian (for k-regular graph):**
```
ℒ = (1/k)L = I - (1/k)A
```
Where:
- ℒ = normalized Laplacian
- I = identity matrix
- k = degree of regular graph

**Eigenvalue Properties:**
- L is symmetric and positive-semidefinite
- λ₀ = 0 (smallest eigenvalue)
- λ₁ = algebraic connectivity (Fiedler value)
- Number of connected components = multiplicity of 0 eigenvalue

**Reference:** Laplacian matrix - Wikipedia

---

## Dynamical Systems Equations

### Attractor Definition
**Forward Invariance:**
```
if a ∈ A, then f(t,a) ∈ A for all t > 0
```
Where:
- A = attractor subset of phase space
- f(t,a) = evolution function
- a = point in phase space

**Basin of Attraction:**
```
B(A) = {b : lim_{t→∞} f(t,b) ∈ A}
```
Where:
- B(A) = basin of attraction for A
- b = point in phase space

**Reference:** Attractor - Wikipedia

---

## Quantum-Inspired Equations

### Wave Function Superposition
**Superposition State:**
```
|ψ⟩ = Σ_i a_i |φ_i⟩
```
Where:
- |ψ⟩ = quantum state
- a_i = complex amplitude
- |φ_i⟩ = basis state

**Normalization:**
```
Σ_i |a_i|² = 1
```

**Wave Function Collapse (Measurement):**
```
|ψ⟩ → |φ_k⟩ with probability |a_k|²
```

**Reference:** Wave function collapse - Wikipedia

---

## Topology Equations

### Tangent Space
**Definition:**
The tangent space T_pM at point p on manifold M is the space of all tangent vectors at p.

**Properties:**
- T_pM is a vector space of dimension n (where n = dimension of M)
- Tangent vectors act as directional derivatives
- Basis: ∂/∂x_i|_p for local coordinates x_i

**Reference:** Tangent space - Wikipedia

---

## Differential Geometry Equations

### Curvature
**Scalar Curvature:**
```
R = g^{ij}R_{ij}
```
Where:
- R = scalar curvature
- g^{ij} = inverse metric tensor
- R_{ij} = Ricci curvature tensor

**Riemann Curvature Tensor:**
```
R^i_{jkl} = ∂_kΓ^i_{jl} - ∂_lΓ^i_{jk} + Γ^i_{km}Γ^m_{jl} - Γ^i_{lm}Γ^m_{jk}
```
Where:
- R^i_{jkl} = Riemann curvature tensor
- Γ^i_{jk} = Christoffel symbols

**Reference:** Manifold Diffusion Geometry (arXiv:2411.04100)

---

## Integration with Morphic Topology System

### Morphic Scalar Superposition (Quantum-Inspired)
```
Scalar(t) = Σ_i a_i |profile_i⟩
```
Where:
- Scalar(t) = morphic scalar at time t
- a_i = amplitude for profile i
- |profile_i⟩ = computational profile (|neural⟩, |signal⟩, etc.)

### Measurement (Collapse)
```
Measure(Scalar, Niche) → |profile_k⟩
```

### Amplitude Update (Learned Superposition)
```
a_i(new) = a_i(old) + Δa_i
```
Where:
- Δa_i = amplitude update based on success/failure
- Successful route: increase amplitude
- Failed route: decrease amplitude or scar

### OEPI (Operator Escalation Percentage Index)
```
OEPI = 0.25 × uncertainty + 0.25 × impact + 0.20 × time_sensitivity + 0.15 × irreversibility + 0.15 × live_voltage_risk
```

---

## Sources

1. Neural coding - Wikipedia: https://en.wikipedia.org/wiki/Neural_coding
2. Spike-timing dependent plasticity - Scholarpedia: http://www.scholarpedia.org/article/Spike-timing_dependent_plasticity
3. Hebbian theory - Wikipedia: https://en.wikipedia.org/wiki/Hebbian_theory
4. Fourier transform - Wikipedia: https://en.wikipedia.org/wiki/Fourier_transform
5. Entropy (information theory) - Wikipedia: https://en.wikipedia.org/wiki/Entropy_(information_theory)
6. Laplacian matrix - Wikipedia: https://en.wikipedia.org/wiki/Laplacian_matrix
7. Attractor - Wikipedia: https://en.wikipedia.org/wiki/Attractor
8. Tangent space - Wikipedia: https://en.wikipedia.org/wiki/Tangent_space
9. Wave function collapse - Wikipedia: https://en.wikipedia.org/wiki/Wave_function_collapse
10. Manifold Diffusion Geometry - arXiv:2411.04100

---

## Status

**Math Scan Complete:** 2026-04-26T19:30:00
**Equations Cataloged:** 25+ equations across 8 mathematical domains
**Integration Status:** Pending integration into implementation guide