Square-Root Boundary-Temperature Attention

(1) Exact integration point: replace the logit computation in one standard Transformer self-attention block, or the edge-message computation in a graph neural network, when inputs contain partially observed regions or explicit boundary markers. Normalize positions x_i to an interval [a,b]. Split each token into ordinary features h_i and measurement features z_i; do not put z_i into the geometric gate. (2) Pseudocode: compute beta_i = beta0 * (((x_i-a)*(b-x_i))/(b-a)^2)^0.25; compute Q, K, and V from h; compute u_i = linear_mu(z_i); set logits[i,j] = beta_i*beta_j*dot(Q[i],K[j])/sqrt(head_dim) + u_i + u_j + mask[i,j]; apply row-wise softmax and multiply by V. Initialize beta0 so that the average absolute logit is unchanged relative to the baseline. (3) Taken from the paper: the separation between geometry-controlled beta and outcome-controlled mu, and the endpoint exponent one-half. Estimated during training: beta0, the potential projection, and any optional density-to-potential network. (4) First cheap experiment: create synthetic sequences with two boundary blocks containing binary measurements and an interior target depending on both long-range context and measurement outcomes. Compare ordinary attention, learned positional gating, square-root gating without u, and square-root gating with u. Sweep beta exponent alpha in beta_i proportional to [d_left*d_right]^(alpha/2), where d_left and d_right are distances to the two boundaries. The prediction is a validation-loss minimum near alpha = 0.5, geometric attention weights scaling approximately as distance^0.5 close to each boundary, and outcome-flip sensitivity appearing mainly when the u channel is enabled. Also measure gradient norms by boundary distance: the proposed gate predicts systematic suppression near the boundaries relative to ungated attention, with the suppression exponent approaching one-half before softmax saturation.