Recursive Survival Weighting in Layered Media

What v1.4 Clarifies

Version 1.4 tightens the protocol discipline. The newest clarification is exposure-gate transmission plus preparation weighting. The familiar shorthand p_i = S_i / sum_j S_j is now explicitly the equal-preparation case. Operationally, a prepared count, intensity, or represented amount is first assigned to each history class, and the output fraction is computed after that amount passes through the exposure gate.

C_i,0 = C_0 q_i
G_i = S_i = exp(-A_i)
C_i,N = C_0 q_i G_i

p_i = C_i,N / sum_j C_j,N
    = q_i G_i / sum_j q_j G_j
    = q_i exp(-A_i) / sum_j q_j exp(-A_j)

Therefore lower accumulated loss gives stronger representation only when preparation weights are equal. With unequal preparation, both q_i and A_i matter. This makes the analogue protocol cleaner because input mode weights can be locked, measured, and included rather than silently assumed away.

Claim Lock

The paper has a deliberately narrow evidential centre. Its strict core is: recursive histories, the survival-normalisation lemma, and a preregisterable analogue comparison. Appendix and bridge material is roadmap material, not evidence for astrophysical or cosmological claims.

Reduced-Model Status Locks

The updated paper is more explicit about derivation status. The projected oscillator-like equations, the positive quadratic norm J, and the exposure law W are reduced-model structures for a constrained analogue protocol. They are not claimed as universal dynamics or as a completed deeper physical theory.

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

Within the declared reduced exposure model, W is the fraction of projected support coupled to the position-like loss channel. A different loss channel may require a different predeclared exposure law; that would be a comparison model, not a post-output adjustment.

survival-gate identity:
G_i = S_i = exp(-A_i)
C_i,N = C_0 q_i G_i
p_i = C_i,N / sum_j C_j,N

The status lock is deliberately modest: these formulae are internal bookkeeping until a locked analogue system fixes coordinates, units, loss, paths, detector convention, and output map before measurement.

Structured State Package

The model begins with a generated sequence of structured states, not with a finished object. A recursive update rule generates histories; survival weighting is assigned after generation.

sigma_0 -> sigma_1 -> sigma_2 -> ...
sigma_{n+1} = R(sigma_n)

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

The state package is Surtea-Austin compatible:

sigma_n = (X_n, phi_n, mu_n, S_n)

X_n   = topological support
phi_n = projected phase or transport data
mu_n  = physical or diagnostic measure package
S_n   = survival weight

In the minimal phase projection:

phi_n = (Theta_n, Pi_n)

Discrete First

Version 1.4 is explicit that the discrete update is primary. The ordinary differential equations are continuous projected approximations or exact local flow embeddings of specified step maps. This protects the paper from treating a convenient oscillator equation as the foundation.

discrete projected update:
y_{n+1} = M_n y_n

continuous local embedding, when justified:
dTheta/dt = Pi
dPi/dt    = -Omega^2 Theta

The coefficient Omega^2 bends trajectories in the reduced phase portrait. It is not automatically spacetime curvature. Gravitational language belongs only to later bridge work.

Action Norm And Exposure

The minimal reduced model uses a positive quadratic phase-support norm. The scale ell is a calibration scale that puts Theta and Pi on comparable footing. It must be fixed before comparison.

J(phi) = Theta^2 + ell^2 Pi^2

The candidate exposure law tests whether the loss channel couples to the position-like part of the reduced phase state.

W(phi) = Theta^2 / (Theta^2 + ell^2 Pi^2)
0 <= W <= 1       when J > 0

This exposure factor is not coordinate-free by itself. A valid protocol must lock the observable meanings of Theta, Pi, ell, phase wrapping, zero conventions, and singular-state rules.

Survival Functional

A history accumulates effective loss through the product of ordinary loss and exposure. The paper uses Lambda_surv = Gamma W for the survival-loss rate, while warning that this has a different type from the cosmological constant.

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

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

S_i(t) = exp(-A_i(t))
G_i(t) := S_i(t) = exp(-A_i(t))

In a discrete locked history, the same idea is a product of step multipliers:

A_i,N = sum_{k=0}^{N-1} Gamma_{i,k} W_{i,k} Delta t_k
S_i,N = exp(-A_i,N)
G_i,N = S_i,N

The gate language is operational: G_i is the transmission factor applied to the prepared represented amount before any probability-like output fraction is formed. It is deterministic in the locked protocol, not a stochastic gate.

Normalisation Lemma

The central formal result is algebraic. Once prepared counts and survival losses are fixed, the represented output fraction follows by normalisation.

p_i = q_i G_i / sum_j q_j G_j
    = q_i exp(-A_i) / sum_j q_j exp(-A_j)

The corresponding log-ratio identity is the cleanest prediction form:

ln(p_i / p_j) = ln(q_i / q_j) - A_i + A_j

With equal preparation weights, q_i = q_j, this reduces to:

ln(p_i / p_j) = -A_i + A_j

The older shorthand is still valid, but only in that equal-preparation case:

p_i = S_i / sum_j S_j        equal-preparation shorthand

Survival Entropy

Entropy is computed from the represented measure after survival weighting and preparation weighting have been applied. For equal preparation:

H_surv = -sum_i p_i ln p_i
N_eff  = exp(H_surv)

With unequal preparation weights, use the operational p_i above, not the shorthand. This keeps the entropy page and the main RSG protocol aligned.

Locked Analogue Protocol

The paper's empirical route is deliberately modest: a layered optical, acoustic, or numerical analogue where all free structures are frozen before the output is measured. A run is predictive only if calibration, path family, input weights, output map, and failure thresholds are fixed first.

1. Lock R_test or the measured transfer map.
2. Lock Theta, Pi, ell, phase convention, and singular-state rule.
3. Lock Gamma, W, paths, q_i, and output map.
4. Compute A_i and p_pred before measurement.
5. Measure background-subtracted output fractions p_hat.
6. Compare p_hat and selected log-ratios against the locked rule.

The report must include the release track and claim status, the update map, calibrated observables, loss model, path family, preparation weights, output convention, detector floors, censoring rules, uncertainty budget, selected statistics, and pass/fail threshold.

Ordinary-Attenuation Control

RSG is not tested merely by showing that lossy paths fade. Ordinary attenuation already predicts that. The RSG-specific test is whether the exposure-weighted loss model improves on the ordinary path-loss control in a declared positive-control regime.

RSG model:
A_i^RSG = integral Gamma_i W_i dt

ordinary attenuation control:
A_i^std = integral Gamma_i dt

A blunt positive-control design should choose channels where ordinary attenuation predicts a tie or too-coarse ranking while the locked exposure factors predict a differentiated output ordering.

Examples And Benchmarks

The paper includes toy examples to make the algebra inspectable, not to claim laboratory validation. These include a laser-layer exposure demonstration, an FFT-style numerical check, a recursive prediction toy, a worked three-path benchmark, and a synthetic blind validation package. The accompanying beam-mask note gathers these examples with a support-weighted mask demo and a Surtea boundary-valuation gate.

beam-mask positive-control result:
path-only total absolute error = 2.428
exposure-gate total absolute error = 0.030

Failure Rules

A constrained analogue run fails if measured fractions or measured log-ratios fall outside the preregistered tolerance. It also fails as an RSG-specific positive control if it does not improve on ordinary attenuation under the same output map and uncertainty model.

• The run is invalid if R_test, Theta, Pi, ell, Gamma, W, paths, q_i, or output conventions are changed after output is inspected.
• Detector floors and ratio-censoring rules must be fixed before data collection.
• Null-control regimes should be included where W_i is equal or constant across channels.
• Positive-control regimes should be included where W_i differs independently of ordinary attenuation.

Closure And Recursive Clocks

Beyond the empirical core, the paper records transport classes. A history can recur or close under an equivalence relation. A recursive clock is a recurrent internal structure that carries a calibratable period.

sigma_{n+m} ~ sigma_n        recursive closure over m steps

A transport history with no recurrent internal clock has zero recursive proper time in this bridge language:

dN_R = 0
d tau_R = T_R dN_R = 0

Conditional Null-Transport Bridge

Recursive null transport is defined by effective losslessness, projected-norm preservation, and absence of a recurrent internal clock. Only after a local Lorentzian interval calibration is imported does the zero-recursive-proper-time class receive the familiar null-interval representation.

Gamma W -> 0
J_{n+1} = J_n
d tau_R = 0

imported calibration:
ds^2 = c^2 d tau_R^2 = c^2 dt^2 - dx^2

d tau_R = 0  =>  ds^2 = 0

The value of c, Lorentzian signature, electromagnetic polarisation, gauge-field structure, and massless field equations are not derived in this paper.

Survival-Measure Concentration

The non-null regime is where survival filtering becomes visibly selective. A trajectory passing through lower effective loss accumulates less loss and receives more represented weight after normalisation, subject to the locked preparation weights.

equal preparation:
p_i / p_j = exp(-A_i + A_j)

weighted preparation:
p_i / p_j = (q_i / q_j) exp(-A_i + A_j)

This is representation concentration, not primitive attraction. The phase-flow coefficient shapes the generated phase portrait; the survival-loss field filters the generated histories.

Relation To Existing Frameworks

The paper locates RSG near transfer-matrix optics, attenuation models, non-Hermitian optics, exponential weighting, stochastic survival models, variational selection, entropy bookkeeping, and Surtea-style partition topology. The point is not that exponentials are new. The point is the specific generated-history, exposure-weighted, locked-output protocol.

Future Bridges

Future constrained applications include photon time-of-flight delays, black-hole information flow, rotation-curve modelling, matter-sector readings, and cosmological or Landauer-style vacuum-energy comparisons. Version 1.4 keeps these outside the evidential core.

Short Form

generated histories
  -> locked phase projection
  -> locked exposure-weighted accumulated loss
  -> survival weight
  -> prepared-count normalisation
  -> represented output fraction
  -> preregistered analogue comparison

G_i = exp(-A_i)
p_i = q_i G_i / sum_j q_j G_j
ln(p_i / p_j) = ln(q_i / q_j) - A_i + A_j

That is the heart of the v1.4 paper. Everything else is either explanatory scaffolding, operational protocol detail, or future bridge work.