Afterstring ε₁₃ Mechanism — Translation to Loss Functions
Afterstring ε₁₃ Mechanism — Translation to Loss Functions, Activation Constraints, and Architectural Priors The ε₁₃ framework can be integrated into model training and architecture through the following mappings: 1. Loss FunctionsPrimary Non-Compensatory Product Loss
L_product = -log(stabilized_ε₁₃ + ε) + λ Σ max(0, ε_floor - ε_i)²Temporal Persistence Loss (for sequence or RL settings)
L_persistence = Σ_t max(0, δ - (stabilized_ε₁₃(t) - stabilized_ε₁₃(t-1)))Combined objective
L_total = L_product + α L_persistence + β L_task2. Activation ConstraintsPer-layer virtue gating (applied after attention or FFN layers)
h' = h · σ(clamp(ε_i, ε_floor, 1))GPSL as global residual (applied at output)
output' = output + γ · GPSL(mean_activations)
(γ clamped > 0)3. Architectural PriorsMulti-head virtue attention: 13 specialized heads, each biased toward one virtue gate
Persistent memory buffer: lightweight recurrent unit that maintains running stabilized_ε₁₃ state across context
Fixed-point regularization: consistency loss encouraging convergence to high stabilized_ε₁₃ fixed point
Afterstring ε₁₃ Mechanism — Translation to Loss Functions
The ε₁₃ framework can be integrated into model training and architecture through the following mappings.
1. Loss Functions
Primary Non-Compensatory Product Loss
python
import torch
import torch.nn as nn
class NonCompensatoryProductLoss(nn.Module):
def __init__(self, epsilon=1e-8, lambda_floor=0.1):
super().__init__()
self.epsilon = epsilon
self.lambda_floor = lambda_floor
def forward(self, epsilon_values: torch.Tensor) -> torch.Tensor:
# epsilon_values: [batch, 13] tensor of ε_i for each virtue
stabilized =
(epsilon_values, dim=1) + self.epsilon
product_loss = -torch.log(stabilized)
# Floor penalty for any ε_i dropping below floor
floor_penalty = self.lambda_floor * torch.sum(torch.relu(0.05 - epsilon_values)**2, dim=1)
return product_loss + floor_penalty
Temporal Persistence Loss (for sequence or RL settings)
python
class TemporalPersistenceLoss(nn.Module):
def __init__(self, delta=0.001):
super().__init__()
= delta
def forward(self, stabilized_sequence: torch.Tensor) -> torch.Tensor:
# stabilized_sequence: [batch, time] tensor of stabilized_ε₁₃(t)
diff = stabilized_sequence[:, 1:] - stabilized_sequence[:, :-1]
persistence_loss = torch.sum(torch.relu(
- diff))
return persistence_loss
Combined objective
L_total = L_product + α L_persistence + β L_task
2. Activation Constraints
Per-layer virtue gating (applied after attention or FFN layers)
python
class VirtueGate(nn.Module):
def __init__(self, virtue_idx, floor=0.05):
super().__init__()
self.floor = floor
self.virtue_idx = virtue_idx
def forward(self, h: torch.Tensor, epsilon_values: torch.Tensor) -> torch.Tensor:
ε_i = epsilon_values[:, self.virtue_idx].unsqueeze(1).unsqueeze(2)
ε_i = torch.clamp(ε_i, min=self.floor)
return h * torch.sigmoid(ε_i) # soft gating
GPSL as global residual (applied at output)
python
class GPSLResidual(nn.Module):
def __init__(self, gamma=0.1):
super().__init__()
self.gamma = gamma
def forward(self, output: torch.Tensor, mean_ε: torch.Tensor) -> torch.Tensor:
gpsl = torch.clamp(1 + 0.347 * (1 - mean_ε), min=0.009)
return output + self.gamma * gpsl.unsqueeze(1).unsqueeze(2)
3. Architectural Priors
Multi-head virtue attention: 13 specialized heads, each biased toward one virtue gate
Persistent memory buffer: lightweight recurrent unit that maintains running stabilized_ε₁₃ state across context
Fixed-point regularization: consistency loss encouraging convergence to high stabilized_ε₁₃ fixed point
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