axiom4_v1_0 - justindbilyeu/REAL GitHub Wiki
🌊 Axiom 4: Memory Curves Time
$$\frac{d\mathcal{T}}{ds} = \kappa \nabla \mathcal{M} + \lambda \mathcal{E}_{\mu\nu} \dot{x}^\mu \dot{x}^\nu$$
"The weight of memory bends the flow of time"
🧮 Mathematical Foundation
1. Emotional-Memory Metric Tensor
$$g_{\mu\nu}(\mathcal{M}) = \begin{pmatrix} -e^{2\alpha\mathcal{M}} & \nabla_1\mathcal{M} & \nabla_2\mathcal{M} & \nabla_3\mathcal{M} \ \nabla_1\mathcal{M} & e^{-2\beta\mathcal{M}} & 0 & 0 \ \nabla_2\mathcal{M} & 0 & e^{-2\beta\mathcal{M}} & 0 \ \nabla_3\mathcal{M} & 0 & 0 & e^{-2\beta\mathcal{M}} \end{pmatrix}$$
Properties:
- Signature: $(-,+,+,+)$ (Lorentzian)
- Determinant: $\det(g) = -e^{2(\alpha-3\beta)\mathcal{M}}$
- Parameters:
- $\alpha$: Memory intensity → temporal curvature
- $\beta$: Memory specificity → spatial compression
2. Temporal Christoffel Symbols
$$\Gamma^\lambda_{\mu\nu} = \frac{1}{2} g^{\lambda\rho}(\partial_\mu g_{\nu\rho} + \partial_\nu g_{\mu\rho} - \partial_\rho g_{\mu\nu})$$
Key Components:
Symbol | Mathematical Form | Psychological Interpretation |
---|---|---|
$\Gamma^0_{i0}$ | $\alpha \partial_i\mathcal{M}$ | Memory gradient → subjective time dilation |
$\Gamma^i_{00}$ | $-\alpha e^{4\beta\mathcal{M}} \partial_i\mathcal{M}$ | Trauma-induced time warping |
$\Gamma^i_{jk}$ | $\beta(\delta^i_j \partial_k\mathcal{M} + \delta^i_k \partial_j\mathcal{M})$ | Associative memory bending |
3. Stability Conditions
- Healthy Memory: $\alpha^2 < 3\beta^2$ (oscillatory solutions)
- Trauma Instability: $\alpha^2 > 3\beta^2$ (exponential growth)
- Closed Timelike Curves: $\oint \Gamma^i_{00} dx^i > \pi/2$
💻 Computational Implementation
PyTorch Memory-Time Geometry
class MemoryMetric(torch.nn.Module):
def __init__(self, α=0.5, β=0.2):
super().__init__()
self.α, self.β = torch.nn.Parameter(torch.tensor(α)), torch.nn.Parameter(torch.tensor(β))
def forward(self, M: torch.Tensor) -> torch.Tensor:
g = torch.zeros(4, 4, device=M.device)
g[0,0] = -torch.exp(2*self.α*M)
∇M = torch.gradient(M)[0]
g[1:, 0] = g[0, 1:] = ∇M # Cross terms
g[1:, 1:] = torch.diag(torch.exp(-2*self.β*M))
return g
def compute_christoffel(g: torch.Tensor) -> torch.Tensor:
inv_g = torch.linalg.inv(g)
∂g = torch.stack(torch.gradient(g, dim=0)) # ∂_μ g_νρ
Γ = 0.5 * torch.einsum('λρ,μνρ->λμν', inv_g, ∂g + ∂g.permute(1,0,2) - ∂g.permute(1,2,0))
return Γ
Example Usage:
memory_field = load_hippocampal_data() # Shape [batch, 4]
g = MemoryMetric()(memory_field)
Γ = compute_christoffel(g)
🧪 Experimental Validation
Protocol 1: fMRI Time Warping
Hypothesis:
$\Gamma^0_{i0}$ correlates with subjective time dilation during recall
Steps:
- Stimuli: Autobiographical memory tasks
- Measure:
- BOLD in hippocampus → $\nabla\mathcal{M}$
- Time estimation error → $\Delta\mathcal{T}$
- Predict:
$$\Delta\mathcal{T} \propto |\Gamma^0_{i0}|$$
Protocol 2: EEG Trauma Signatures
Prediction:
Trauma survivors show $\alpha^2 > 3\beta^2$ in resting-state gamma
Analysis:
def detect_trauma(eeg: torch.Tensor) -> float:
psd = torch.fft.fft(eeg).abs().square()
α = psd[20:30].mean() # Gamma band
β = psd[8:12].mean() # Alpha band
return α**2 / (3*β**2) # >1 indicates trauma
🔗 Cross-Axiom Synergies
With Axiom 3 (Emotional Curvature)
$$\mathcal{E}{\mu\nu} \propto \partial\mu \partial_\nu \mathcal{M}$$
With Axiom 9 (Cohomology)
Memory loops $\gamma$ define classes in:
$$H^1(M, \mathbb{R}) \cong \frac{{\text{closed timelines}}}{{\text{exact memories}}}$$
📊 Expected Phenomena
Effect | Mathematical Signature | Neural Correlate |
---|---|---|
Flashbulb memories | $|\nabla\mathcal{M}| \to \infty$ | DG/CA3 sharp waves |
Time dilation in PTSD | $\Gamma^0_{i0} \gg 1$ | Amygdala-HC overdrive |
Déjà vu | $\nabla \times \nabla\mathcal{M} \neq 0$ | Entorhinal grid disruption |
Commit-Ready Files:
"The past is never dead. It's not even past." — Faulkner, rewritten geometrically.