Probability in Recursive Survival Geometry

Reading Position

Read this immediately after the main RSG protocol paper and before the entropy note. The main paper tells the reader how generated histories are filtered. This paper explains what the resulting probability-like quantity means. The entropy note then reads Shannon spread from that normalised survival measure.

The safest phrase in the paper is "normalised survival-representation weight." The symbol p_i is formally probabilistic because its values are non-negative and sum to one over the declared represented family. It is not primitive chance, matter density, metric curvature, or a quantum outcome.

Claim Lock

The paper is explicit about what it does not claim. It does not derive the Born rule, quantum measurement, matter formation, gravitational curvature, or physical density from p_i. In the strict formalism, p_i is the represented weight of a history after survival filtering. Stronger readings require additional bridge laws and independent ways to fail.

generated histories
  -> accumulated exposed loss
  -> exposure-gate transmissions
  -> normalised representation probability

This makes the paper a guardrail document. It allows RSG to use probability language while preventing that language from quietly turning into primitive randomness or unearned physics.

Not Primitive Randomness

RSG does not begin with a random choice. It begins with a declared family of recursive histories. A history is a path through structured states:

gamma_i = (sigma_i,0, sigma_i,1, sigma_i,2, ...)

The index i labels a candidate history, while the step index labels recursive depth. The generated family may be finite, countable, or described by a history measure, but the paper keeps the finite case as the safest operational case.

Preparation weights q_i come before survival filtering. They describe the initial count, intensity, or represented amount supplied to each history class. They are not survival probabilities.

{gamma_i}_{i in I}, q_i first
then A_i, G_i, S_i, p_i

Exposed Survival Loss

Each history accumulates loss through an exposed loss rate. In compact continuous notation:

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

Lambda_surv,i(tau) = Gamma(sigma_i(tau)) W(phi_i(tau))

Gamma is available attenuation, dissipation, or loss. W is exposure to that loss channel. The RSG-specific empirical question is whether Gamma W predicts output better than ordinary attenuation by Gamma alone.

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

G_i is the exposure-gate transmission for history i. It is not an additional force or a new random choice; it names the multiplicative gate through which prepared representation passes before normalisation.

Operational Count Map

The paper gives an operational route from prepared counts to represented output fractions. If C_0 is the total prepared count and q_i is the preparation weight for history class i, then:

C_i,0 = C_0 q_i

C_i,k+1 = C_i,k exp[-Gamma_i,k W_i,k Delta t_k]

G_i = exp[-sum_k Gamma_i,k W_i,k Delta t_k] = exp(-A_i)

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

Normalising the surviving represented counts gives the general finite-family probability map:

p_i = C_i,N / sum_j C_j,N

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 older shorthand p_i = S_i / sum_j S_j is valid only when the histories are equally prepared and S_i = G_i. This is one of the main practical repairs in the paper: preparation bias, gate transmission, and survival filtering must not be mixed together.

Log-Ratio Test

The cleanest analogue readout is the log-ratio, because the common normaliser cancels. From the general formula:

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

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

For equal preparation this reduces to:

log(p_i / p_j) = -A_i + A_j

The phrase "lower accumulated loss wins representation" is therefore correct only in the equal-preparation case. With unequal q_i, a lower-loss history can still lose represented share if it was prepared with much less initial weight.

History Measures

For a finite declared history family, the weighted survival normaliser is:

Z_q(t) = sum_j q_j exp(-A_j(t))

The probability map is meaningful only when the normaliser is positive and finite. For continuous or branching history families, a base history measure must be declared before probability is meaningful.

dP(gamma)
  = [q(gamma) exp(-A[gamma]) / Z_q] d mu(gamma)

Z_q = integral_H q(gamma) exp(-A[gamma]) d mu(gamma)

0 < Z_q < infinity

If the base family, preparation density, loss functional, or output map is chosen after seeing the result, the construction is no longer operational. It has become post-hoc fitting.

Entropy After Probability

Once the represented weights p_i are defined, survival entropy is a Shannon readout of that represented measure:

H_surv = -sum_i p_i log p_i

N_eff = exp(H_surv)

High survival entropy means many histories remain similarly represented. Low survival entropy means the survival filter has concentrated representation into fewer histories or basins. This is why the probability paper belongs before the entropy paper: entropy is not the primitive object here. It reads from the probability measure produced by survival normalisation.

Attenuation Control

The paper makes the positive-control comparison explicit. Ordinary attenuation integrates the loss coefficient alone:

A_i^std = integral_{gamma_i} Gamma(sigma_i(t)) dt

RSG instead uses exposed loss:

A_i^RSG = integral_{gamma_i} Gamma(sigma_i(t)) W(phi_i(t)) dt

Both predictions should be computed before measurement:

p_i^std = q_i exp(-A_i^std) / sum_j q_j exp(-A_j^std)

p_i^RSG = q_i exp(-A_i^RSG) / sum_j q_j exp(-A_j^RSG)

If the exposure factor W does not improve prediction in the declared positive-control regime, the RSG-specific empirical content collapses back to ordinary attenuation for that test.

Null And Information Bridges

The paper is careful with bridge language. A lossless class has Gamma W -> 0, hence A_i -> 0 and S_i -> 1. That says the history does not decay under the survival filter. It does not by itself make the history a photon, a light ray, or a null geodesic.

Gamma W -> 0
A_i -> 0
S_i -> 1

d tau_R = 0      bridge clock condition
ds^2 = 0         after Lorentzian bridge is declared

Information recovery is also separate from represented probability. A high p_i history is strongly represented, but it is informative only when a support, channel, detector, or recovery map is declared.

I_rec = R({p_i, gamma_i, mu_i}_{i in I})

Worked Examples

The first example uses three equally prepared histories:

A = (0, 1, 2)
S = (1, exp(-1), exp(-2))

p approx (0.665, 0.245, 0.090)

The second example is a selective laser-layer toy model. With calibrated loss eta = 0.20 and exposed-support fractions:

W = (1.00, 0.75, 0.50, 0.25, 0.00)

I_RSG(W) = exp(-0.20 W)

p_laser^RSG approx (0.181, 0.190, 0.199, 0.210, 0.221)
p_laser^std = (0.200, 0.200, 0.200, 0.200, 0.200)

The point is not that RSG rediscovers selective absorption. The point is that a predeclared exposure factor can produce a different output vector from ordinary path attenuation.

The FFT/windowing toy example makes the same lesson in a spectral setting. Raw FFT bin weights are preparation weights; survival ranking is an additional predeclared layer tested against held-out output.

The updated PDF also makes the geometric-probability reading explicit: the generated history family is the sample space, a cluster of histories is an event-like subset, and RSG adds the extra rule that the represented measure is obtained by preparation, exposed loss, gate transmission, and normalisation.

The new beam-mask companion note makes the same count chain visible in a support-weighted setting. Fixed masks compute W_m, the gate transmission is G_m = exp(-eta W_m), and only the transmitted amounts are normalised or compared. Its Surtea support example likewise uses W_X = nu_D(bd_D X)/nu_D(cl_D X) before applying the gate.

Failure Modes

The strongest practical section is the non-circularity lock. A valid analogue test must fix the recursive update, phase-flow coefficient, calibration scale, loss coefficient, exposure map, history family, preparation weights, detector convention, output map, and tolerance before measurement.

lock before measurement:
R, Omega^2, ell, Gamma, W, H, q_i, D, O, epsilon

• Ordinary-control collapse. If p^RSG does not improve on p^std, the exposure factor has not earned empirical use in that setting.
• Normalisation failure. If Z_q is not positive and finite, the represented probability measure is undefined.
• Bridge inflation. p_i must not be identified with matter density, stress-energy, or metric curvature without a declared bridge law.

p_i != rho_matter
p_i != T_mu_nu
p_i != g_mu_nu

Fit With The Site

This page should be used as the probability companion to the main RSG paper. It explains why the visualisation's represented histories are not raw generated possibilities, but survival-filtered shares. It also explains why entropy, information recovery, null transport, matter-like transport, and vacuum measure must remain separate layers unless a bridge law has been supplied.

In one sentence: probability in RSG is the normalised share of transmitted represented amount after survival filtering, not randomness first.