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Entropy in Recursive Survival Geometry

Shannon entropy is the scalar readout of the normalised survival measure over generated histories. Recursive geometry generates histories, survival filtering weights them, normalisation turns them into represented measure, and entropy reports how much unresolved survival-compatible history space remains live. This page follows the merged main entropy note and its survival-closure diagnostics.

Definitional update note Reading order 03 RSG_entropy.pdf Open updated TeX source Open full text PDF

Placement of the update

This update belongs to the entropy document, not to the main RSG paper. It refines the measurement step already implied by the entropy note: generated histories become survival weights, survival weights become a normalised representation measure, and Shannon entropy is then applied to that measure.

generated histories
  -> survival weights
  -> normalised representation measure
  -> Shannon scalar

No new empirical claim is added. The locked analogue-test claim remains in the main recursive survival weighting formalism.

Survival measure over histories

A generated recursive history is a sequence of structured states carrying topological support, phase or transport data, measure data, and survival weight.

γ_i = (σ_{i,0}, σ_{i,1}, σ_{i,2}, ...)
σ_n = (X_n, φ_n, μ_n, S_n)
φ_n = (Θ_n, Π_n)

dΘ/dt = Π
dΠ/dt = -Ω²Θ

Exposure-weighted survival loss combines the local loss coefficient Γ with the action exposure W.

J(φ) = Θ² + ℓ²Π²
W(φ) = Θ² / J(φ)
λ_i(t) = Γ_i(σ_i(t)) W_i(φ_i(t))

A_i(t) = ∫_0^t λ_i(τ) dτ
S_i(t) = exp(-A_i(t))
p_i(t) = S_i(t) / Σ_j S_j(t)
       = exp(-A_i(t)) / Z(t)
Z(t) = Σ_j exp(-A_j(t))

The represented weight p_i is not the raw probability that the history was generated. It is the probability with which that history remains represented after the survival filter has acted.

Shannon readout

For a represented history i, Shannon information content is -log_b p_i. The base only fixes units: base 2 gives bits, base e gives nats. Because the survival law uses exp(-A_i), the entropy note uses natural logarithms.

I_b(i) = -log_b p_i
H_b(p) = Σ_i p_i I_b(i)
       = -Σ_i p_i log_b p_i

H_surv(t) = -Σ_i p_i(t) ln p_i(t)

p_i = exp(-A_i) / Z
H_surv(t) = Σ_i p_i(t) A_i(t) + ln Z(t)
          = _p + ln Z

N_eff(t) = exp(H_surv(t))

Shannon entropy is therefore not imported as primitive disorder. It is the scalar readout of the normalised survival distribution.

K_info = s ∘ S_Sh ∘ n ∘ d
K_info* ~ s · S_Sh · n · d

Entropy flow under survival filtering

The represented measure changes by comparing each history’s current effective loss with the represented average loss.

ṗ_i = p_i(<λ>_p - λ_i)
<λ>_p = Σ_j p_j λ_j

A history gains represented weight exactly when its effective loss is below the represented average.

λ_i < <λ>_p  =>  ṗ_i > 0

H_surv = _p + ln Z
dH_surv/dt = -Cov_p(A, λ)

If high accumulated-loss histories also keep suffering high current loss, the covariance is positive and represented entropy falls. The measure is concentrating into a narrower survival basin. If every history has the same effective loss, the represented weights and entropy do not change.

Generated, represented, and exported entropy

The merged entropy note keeps three quantities separate. H_gen measures the multiplicity of generated candidate histories before selection. H_surv measures represented multiplicity after survival filtering. S_env denotes entropy exported to unresolved environmental, loss, or discarded degrees of freedom.

H_gen,    H_surv,    S_env

A local concentration process can lower visible or represented entropy while increasing exported entropy.

H_surv decreases while S_env increases

Local concentration rule

For a region R of projected history space, represented density grows when the region’s average effective loss is lower than the global represented average.

ρ_R(t) = Σ_{i: γ_i(t) ∈ R} p_i(t)

dρ_R/dt = ρ_R(<λ>_p - <λ>_R)
<λ>_R = (1/ρ_R) Σ_{i: γ_i(t) ∈ R} p_i λ_i

dρ_R/dt > 0  ⇔  <ΓW>_R < <ΓW>_p

Open and closed survival regimes

For a finite active family of N histories, maximum entropy is ln N. Openness and closure indices can then be derived after the active history family or coarse-graining has been fixed.

H_max = ln N
O_surv = H_surv / H_max
C_surv = 1 - H_surv / H_max

O_surv near 1 means many histories remain unresolved. C_surv near 1 means representation has concentrated into a narrow basin. Entropy is not identical with closure; it is the scalar diagnostic from which this closure index can be derived.

Class and coarse-grained entropy

The same Shannon readout applies after topological or observational coarse-graining.

p_C(t) = Σ_{i: class(X_i)=C} p_i(t)
H_class(t) = -Σ_C p_C(t) ln p_C(t)

p_M(t) = Σ_{i: γ_i ∈ M} p_i(t)
H_coarse(t) = -Σ_M p_M(t) ln p_M(t)
S_phys = k_B H_coarse

Continuous history spaces

For infinite or continuous history families, a base history measure must be fixed before entropy is meaningful.

dP(γ) = (exp(-A[γ]) / Z) dμ(γ)
Z = ∫ exp(-A[γ]) dμ(γ)
0 < Z < ∞

p(γ) = dP/dμ = exp(-A[γ]) / Z
H_surv^μ = -∫ p(γ) ln p(γ) dμ(γ)

Entropy is the Shannon readout of the represented survival measure, relative to a specified history family, base measure, and coarse-graining.

Light-like and matter-like regimes

A light-like regime is associated with lossless, norm-preserving, non-closing transport. In that limit, survival entropy is not primarily reduced by dissipation, and the transport remains represented without settling into recurrent rest-frame-like closure.

Γ -> 0
||σ_{n+1}||_Φ = ||σ_n||_Φ
r not in Q

Matter-like concentration belongs to the complementary regime where survival loss varies across histories. Lower-loss histories gain representation relative to higher-loss histories after normalisation.

Γ_i W_i != Γ_j W_j
p_i(t) = exp(-A_i(t)) / Σ_j exp(-A_j(t))

Geometric and entanglement bridge

The geometric bridge is deliberately cautious. In a continuum or geometric limit, field equations may correspond to stationarity of an entropy functional under fixed boundary, volume, closure, or support constraints. This is a bridge principle, not a derivation of Einstein's equation.

δ(H_surv + η A_{∂X}) |_{V, class, μ} = 0

Compact insertion text

In RSG, Shannon entropy supplies the scalar readout of the normalised survival measure. Recursive dynamics generate histories; the geometry-dissipation functional A_i = ∫Γ_i W_i dt filters them; normalisation gives p_i = exp(-A_i)/Z; and entropy is:

H_surv = -Σ_i p_i ln p_i
       = _p + ln Z

High H_surv means many survival-compatible histories remain live. Low H_surv means the survival filter has concentrated representation into a narrower basin. The entropy-flow identity records whether filtering is actively resolving the represented measure:

dH_surv/dt = -Cov_p(A, ΓW)

This is a diagnostic update to the entropy note, not an additional empirical claim in the main RSG formalism.