Recursive survival weighting

Claim Lock First

This matters because the same notation can be used in three different ways: as an internal definition, as a conditional bridge, or as a measured claim. The live chapters keep those layers apart so the visualisation stays useful without pretending to be validation data.

strict empirical target:
locked analogue output fractions and log-ratios
match precomputed accumulated-loss differences
within preregistered uncertainty

Three-Layer Scope

The v1.4 paper is deliberately layered. It gives a strict formal core, one locked analogue-test route, and then a separate bridge roadmap. Reading the page this way prevents the future physical applications from being mistaken for results already established by the survival-normalisation lemma.

Structured Recursive State

The basic object is not a bare particle and not a smooth spacetime point. It is a structured recursive state carrying four pieces: topological support, projected phase or transport data, measure data, and survival weight.

\sigma_n = (X_n, \phi_n, \mu_n, S_n)

Discrete Update Before Continuous Approximation

The paper starts from a recursive update. Continuous equations are allowed only as projected approximations or local-flow embeddings of a specified step. If the continuous flow is chosen after the desired output is known, it has no predictive content.

\sigma_{n+1} = R(\sigma_n)

\pi_\Phi : \Sigma -> \Phi
\pi_\Phi(\sigma_n) = \phi_n

The visualisation uses this distinction constantly. The canvas shows smooth curves, waves, and fields because they are readable, but the conceptual engine is a stepwise generation-filter-normalise loop.

y_{n+1} = F_{Delta t,n}(y_n)
y_{n+1} = M_n y_n + O(||y_n||^2)
M_n = exp(Delta t A_n) + O(Delta t^2)

In other words, a local ODE is allowed when it faithfully summarises locked step data. It is not allowed to become a smooth curve chosen after the output has been seen.

Generated Histories

A history is a generated sequence of structured states. RSG does not begin by declaring one history real and all others irrelevant. It generates a family, assigns accumulated loss, and then asks which histories remain represented.

\gamma_i = (\sigma_{i,0}, \sigma_{i,1}, \sigma_{i,2}, ...)

Projected Phase Portrait

The common reduced phase picture uses a position-like coordinate and a motion-like coordinate. These are model coordinates. In a test, they must be operationally defined before output comparison.

\phi_n = (\Theta_n, \Pi_n)

In a continuous projected approximation, a locally constant positive phase-flow cell can be represented by oscillator-like equations:

d\Theta/dt = \Pi
d\Pi/dt = -\Omega^2 \Theta

The coefficient \Omega^2 creates curvature-like bending in this reduced phase portrait. The PDF is careful here: this is not automatically a spacetime curvature invariant.

Omega^2 > 0  -> oscillator-like phase rotation
Omega^2 = 0  -> free linear phase motion
Omega^2 < 0  -> hyperbolic projected cell

For a locally constant, lossless cell, E = 1/2(Pi^2 + Omega^2 Theta^2) is conserved. With variable Omega^2, the same expression is a local diagnostic unless an extended invariant is supplied.

Transport Classes

RSG v1.4 distinguishes transport behaviours by recurrence, closure, projected norm preservation, and effective loss. The live visualisation turns those classes into readable modes.

Gamma W -> 0
||sigma_{n+1}||_Phi = ||sigma_n||_Phi
no recurrent internal clock -> d tau_R = 0

The null-interval result is conditional. Under an imported local Lorentzian interval calibration, d tau_R = 0 is represented by ds^2 = 0. The metric signature, the value of c, and electromagnetic field structure are not derived by the survival-normalisation lemma.

What The Paper Does Not Claim

RSG v1.4 does not report a completed lab validation. It also does not derive Maxwell's equations, gauge structure, photon polarisation, matter formation, a universal astrophysical loss law, or the cosmological constant. Those topics remain bridge programmes until they supply their own fixed variables, constants, maps, and falsifiers.

Lambda_surv = Gamma W
Lambda_surv is not Lambda_CC
without a dimensional map, conservation law, and equation of state

How To Use This Companion

1. Read this chapter to learn the objects: state, update, history, projection, and transport class.
2. Read Survival weighting for the loss, survival, and normalisation machinery.
3. Read Analogue tests before treating anything as an empirical claim.
4. Use Visual guide to map these terms onto the live canvas.