Minimal Conserved Recursive-Survival Sector v0.2

Reading Position

This page has been rebuilt from the local PDF source. The document is the formal partner to the plain-speak jigsaw summary that follows it in the reading order. Read this page as the model statement: it tells you what the minimal conserved Recursive-Survival sector is allowed to do, what it must conserve, what variables it tracks, and what would make the model fail.

The paper's central discipline is conservation. The recursive sector is not added as a free mystery density. It is an effective sector that exchanges with the ordinary represented sector while the sum remains conserved. The model is intentionally low-dimensional so it can be scanned numerically before stronger claims are made.

Claim Boundary

The PDF states its limits clearly. It does not replace general relativity, does not supply a completed microscopic derivation, does not give a full perturbation theory, and does not claim that the universe literally computes every possible history. It gives a projected, smooth, lowest-mode effective layer suitable for first tests.

Plain-Language Model

The ordinary cosmological sector contains the usual represented density. The recursive-survival sector is a temporary bookkeeping sector for unresolved survival-weighted structure. It tracks histories or structures that have not yet settled into the ordinary classical description but may still influence what becomes represented.

Three ingredients drive the minimal model. First, recursive density says how much effective density remains in the recursive sector. Second, coherence memory says whether that sector is still coherent enough to affect ordinary representation. Third, exchange transfers density from the coherent recursive sector into the ordinary represented sector. A successful run is one where this transfer produces a finite early window and then switches itself off.

Conservation Ledger

The model begins with an effective Einstein equation whose source has two pieces: ordinary stress-energy and recursive-survival stress-energy. The Bianchi identity forces the total source to be conserved, so any transfer into one sector is balanced by loss from the other.

G_{mu nu} = 8 pi G (T_{mu nu} + T^R_{mu nu})

nabla_mu (T^{mu nu} + T_R^{mu nu}) = 0

nabla_mu T^{mu nu} = Q^nu

nabla_mu T_R^{mu nu} = -Q^nu

In homogeneous FLRW, the exchange current is aligned with the cosmological four-velocity. The sign convention is chosen so that positive exchange means recursive-sector density is transferred into the ordinary represented sector.

Q^nu = Xi u^nu

Xi > 0  means  recursive-sector density -> ordinary sector

Survival Variables

Recursive Survival is expressed through accumulated loss rather than raw survival weight. The full survival machinery is then compressed for the homogeneous model into an effective density, an equation of state, and a coherence memory variable.

A = -ln S

A = integral Gamma W d_tau

s_mu = nabla_mu A

T^R_{mu nu} = T^R_{mu nu}[A, s_mu, C, boundary stability, closure diagnostics, g_{mu nu}]

E = dA/dN = Gamma W / H

The exposure-per-e-fold variable matters because the numerical model uses e-fold time, not ordinary time. The ultra-minimal exchange law absorbs a representative exposure value into the coupling.

FLRW Reduction

The background reduction assumes a spatially flat FLRW metric and treats both sectors as perfect fluids at homogeneous order. Radiation, baryons, cold dark matter, and vacuum density are not denied; they are suppressed in the first pass so the recursive exchange mechanism can be isolated.

ds^2 = -dt^2 + a(t)^2 delta_{ij} dx^i dx^j

T^{mu nu} = (rho + p) u^mu u^nu + p g^{mu nu}

T_R^{mu nu} = (rho_R + p_R) u^mu u^nu + p_R g^{mu nu}

p = w rho,  p_R = w_R rho_R

H^2 = (8 pi G / 3) (rho + rho_R)

rho -> rho_r + rho_b + rho_c + rho_Lambda

Minimal Exchange Law

The exchange law is the core modelling move. Transfer requires recursive abundance, coherence memory, and the cosmological activity scale. If any of these vanish, exchange vanishes.

rho_dot + 3H(1 + w) rho = +Xi

rho_R_dot + 3H(1 + w_R) rho_R = -Xi

rho_tot_dot + 3H[(1 + w)rho + (1 + w_R)rho_R] = 0

Xi = alpha_R C H rho_R

Xi / (H rho_R) = alpha_R C

less-compressed: Xi = alpha_0 C E H rho_R

alpha_R = alpha_0 E_eff

This is not the only possible exchange law. It is the smallest homogeneous law that keeps the survival-sector interpretation visible while remaining tractable for numerical scans.

Coherence Memory

The variable C tracks whether recursive histories remain coherently represented enough to influence the ordinary sector. It is bounded between zero and one, so it behaves as a state variable rather than an unlimited source.

Omega_R = rho_R / (rho + rho_R)

C_dot = H[-gamma_C C + eta Omega_R(1 - C)]

C' = -gamma_C C + eta Omega_R(1 - C)

if gamma_C >= 0, eta >= 0, 0 <= Omega_R <= 1, then 0 <= C <= 1 is invariant

C_*(Omega_R) = eta Omega_R / (gamma_C + eta Omega_R)

Canonical Minimal System

The final v0.2 numerical core evolves exchange, coherence, recursive fraction, and Hubble rate per e-fold. This is the model's compact centre.

N = ln a

X' = dX/dN = (1/H) dX/dt

Xi = alpha_R C H rho_R

C' = -gamma_C C + eta Omega_R(1 - C)

Omega_R' = Omega_R[3(w - w_R)(1 - Omega_R) - alpha_R C]

H'/H = -(3/2)[1 + w(1 - Omega_R) + w_R Omega_R]

w_eff(N) = w(1 - Omega_R) + w_R Omega_R

Well-Behaved Regime

The model's safe parameter regime keeps densities non-negative, keeps coherence bounded, and forces the recursive contribution to fade when the integrated depletion bracket is positive.

alpha_R >= 0, gamma_C > 0, eta >= 0, 0 <= C_0 <= 1, rho_0 >= 0, rho_R0 >= 0

rho_R(N) = rho_R(N_0) exp[- integral(3(1 + w_R) + alpha_R C) dN]

late GR recovery: Omega_R -> 0, C -> 0, alpha_R C -> 0

The equation-of-state parameter w_R is important because it controls passive redshifting. If it redshifts away faster than the late ordinary sector, it helps remove the recursive remnant. If it redshifts too slowly, the model becomes dangerous unless exchange or another decay removes it.

Transient Maturation

The model is interesting only if it produces a finite early effect and then switches off. The key product is not just coherence and not just recursive fraction, but the coherent recursive abundance C Omega_R.

coherent recursive abundance: C Omega_R

exchange diagnostic: alpha_R C

M' = s_R C Omega_R M

with explicit exposure: M' = s_R C Omega_R E M

M(N) / M(N_0) = exp[integral s_R C(N_tilde) Omega_R(N_tilde) dN_tilde]

The maturation index is deliberately a toy diagnostic, not a substitute for perturbation theory. It measures the area under the early C Omega_R bump before any stronger structure-growth claim is made.

Numerical Protocol

The first numerical run should scan the reduced system rather than fit data immediately. The primary plots are designed to show whether the recursive sector appears early, coheres, transfers, and then fades.

initial values: Omega_R(N_i), C(N_i), H(N_i)

parameters: Theta = {alpha_R, gamma_C, eta, w, w_R}

primary plots: Omega_R(N), C(N), C Omega_R, alpha_R C, w_eff(N), H/H_standard

optional plots: M(N), ln[M(N)/M(N_i)]

A viable parameter point needs bounded C, bounded Omega_R, a finite early bump, small late remnant, small late exchange, acceptable expansion history, and a maturation boost that is meaningful but not arbitrary.

Quantum-History Bridge

The canonical model remains a classical effective-fluid scaffold. The later sections ask whether the recursive sector might be read as a coarse-grained contribution from unresolved quantum spacetime histories. This is framed as an extension, not as a completed derivation.

recursive histories gamma_i as coarse-grained quantum spacetime histories

S_i = exp(-A_i)

p_i ~ exp(-A_i / hbar) / Z

rho_R as coarse-grained contribution from unresolved quantum geometry

T_R^{mu nu} ~ expectation(T_hat_unresolved^{mu nu})

T_R^{mu nu} ~ delta A_survival / delta g_{mu nu}

S_surv <-> S_grav <-> A_hor / (4 G hbar)

Failure Tests

The paper is useful because it states what would break the idea. The model fails if the recursive density remains too large at late times, coherence does not decay or saturate naturally, expansion deviates too strongly, the maturation boost is always negligible, the parameters require case-by-case tuning, or no transient C Omega_R window can be produced without a late remnant.

Minimal Pseudocode

The appendix compresses the implementation into a simple loop. This is the cleanest route for a future site visualisation or notebook.

choose alpha_R, gamma_C, eta, w, w_R
choose N_i, N_f, Omega_R_i, C_i, H_i

for N from N_i to N_f:
    Omega_R_prime = Omega_R * (3*(w - w_R)*(1 - Omega_R) - alpha_R*C)
    C_prime = -gamma_C*C + eta*Omega_R*(1 - C)
    H_prime_over_H = -1.5 * (1 + w*(1 - Omega_R) + w_R*Omega_R)
    integrate Omega_R, C, ln(H)
    record C*Omega_R, alpha_R*C, w_eff, H/H_standard

Terms To Carry Forward

Recursive density fraction

Omega_R is the fraction of total effective density still in the recursive-survival sector.

Coherence memory

C is a bounded memory variable measuring coherent recursive relevance.

Exchange scalar

Xi is the homogeneous rate at which recursive density transfers into the ordinary sector.

Coherent recursive abundance

C Omega_R is the key transient window that should form early and vanish late.

Late GR recovery

The model must approach ordinary FLRW/GR behaviour as Omega_R, C, and alpha_R C decay.

Companion decoder

The plain-speak jigsaw page translates the displayed equations symbol by symbol.

How It Fits The Library

This page is reading order 12, immediately after the three-body survival-filtering note and before its plain-speak companion. It is the first compact cosmology-facing model in the route. Use it whenever the site discusses early structural maturation, recursive density, coherence memory, or conserved exchange.