Survival weighting

The Four-Step Pipeline

generated histories
  -> accumulated loss A_i
  -> exposure-gate transmissions G_i
  -> prepared-count normalisation
  -> represented weights p_i

The visualisation's "What survives?" panel is a live picture of this pipeline: make paths, move them, score cost, fade weak histories, and rank the survivors.

Action Norm And Exposure

A projected phase state combines a position-like coordinate and a motion-like coordinate. The scale \ell must be fixed by the model or experiment; it is not a decoration.

J_n = J(\phi_n) = \Theta_n^2 + \ell^2 \Pi_n^2

The exposure factor asks how much of that projected action lies in the loss-exposed channel. In the simplest form used throughout the companion:

W_n = \Theta_n^2 / J_n
0 <= W_n <= 1, when J_n > 0

Effective Survival Loss

\Gamma is a non-negative local loss coefficient. The effective survival-loss rate is not just \Gamma; it is the product of local loss and exposure.

\Lambda_surv(\sigma,\phi) = \Gamma(\sigma) W(\phi)

Along a generated history, this rate accumulates into an action-like loss:

A_i(t) = integral_0^t \Gamma(\sigma_i(\tau)) W(\phi_i(\tau)) d\tau

The PDF repeatedly stresses the operational point: in an analogue test, \Gamma, W, the path family, preparation weights, and the detector convention must be fixed before output is measured.

Coordinate Convention Lock

The exposure law is not invariant under arbitrary phase-coordinate changes. It becomes meaningful only after the observable meanings of \Theta, \Pi, \ell, the zero of phase, phase wrapping, and singular-state rules are declared.

W = Theta^2 / (Theta^2 + ell^2 Pi^2)

Theta' = a Theta
Pi'    = b Pi
ell'   = (a / b) ell
fixed a,b > 0 -> W unchanged

A translation of the zero of Theta, a nonlinear reparametrisation, a rotation mixing Theta and Pi, or a post-output change in ell changes the model unless it was declared as the model before the run.

Survival Weight

Survival decreases exponentially with accumulated loss. This is why the model resembles a Gibbs weighting while still being framed as survival filtering rather than thermal equilibrium.

dS_i/dt = -\Gamma_i W_i S_i
S_i(t) = exp(-A_i(t))
G_i(t) := S_i(t) = exp(-A_i(t))

discrete-first form:
A_{i,n+1} = A_{i,n} + Gamma_{i,n} W_{i,n} Delta t
S_{i,n+1} = S_{i,n} exp(-Gamma_{i,n} W_{i,n} Delta t)
G_i(n) = S_i(n)

The updated PDF names this multiplier the exposure-gate transmission G_i. The gate language does not add randomness; it says the prepared represented amount is multiplied by G_i before normalisation.

Representation By Normalisation

Raw survival weights are not yet represented probabilities. In v1.4 they become represented weights only after the exposure-gate transmission and the prepared counts or input intensities have also been included. The unweighted form is the equal-preparation shorthand.

equal preparation:
Z(t) = sum_j exp(-A_j(t))
p_i(t) = S_i(t) / sum_j S_j(t)
p_i(t) = exp(-A_i(t)) / Z(t)

This is the formal heart of the visual bars labelled "Most represented histories". Each bar is a share of the surviving measure, not an independent brightness score.

with preparation weights q_i:
p_i = q_i G_i / Z_q
    = q_i exp(-A_i) / Z_q
Z_q = sum_j q_j G_j

Equal preparation weights are the special case. If q_i is not equal across histories, it must be fixed before the comparison and carried into the prediction.

Survival-Ratio Identity

Once the gate transmissions and prepared amounts are normalised, the log ratio between two represented histories is determined by their accumulated-loss difference only in the equal-preparation case.

ln(p_i / p_j) = -A_i + A_j

with unequal preparation:
ln(p_i / p_j) = ln(q_i / q_j) - A_i + A_j

Inside the formalism this is an algebraic identity. It becomes empirical only when A_i and A_j are independently specified and compared with measured output ratios.

History Measure And Normalisability

For a finite family, the normalised weights are enough. Infinite, branching, or continuous history spaces need a base measure and a finite normaliser before entropy or representation weights are meaningful.

dP(gamma) = q(gamma) exp(-A[gamma]) dmu(gamma) / Z_q
Z_q = integral q(gamma) exp(-A[gamma]) dmu(gamma)
0 < Z_q < infinity

Formal Lemma Versus Empirical Claim