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:

  1. Stimuli: Autobiographical memory tasks
  2. Measure:
    • BOLD in hippocampus → $\nabla\mathcal{M}$
    • Time estimation error → $\Delta\mathcal{T}$
  3. 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.