Earth-Directed Transport-Survival Kernel

• The paper outlines the kernel as a donor-class dependent quantitative model assessing hard panspermia viability by integrating survival penalties from launch to post-delivery establishment.
• The kernel systematically combines shock survival, escape probability, transit dynamics, atmospheric entry, and impact factors into a unified metric to rank donor channels.
• Applications indicate that early Mars remains the only nonterrestrial donor competitive with terrestrial origins, while extrasolar sources are suppressed by transit and shielding constraints.

The Earth-directed transport-survival kernel is a donor-class dependent quantitative construct for evaluating hard panspermia to Earth as a transport-and-establishment problem rather than a theory of abiogenesis. In the formulation developed in "Natural Panspermia to Earth: Quantitative Limits on Donor Classes, Transport, Survival, and Establishment" (Turyshev, 5 Apr 2026), the kernel encodes the cumulative penalties imposed by launch, escape, transit, Earth interception, atmospheric entry, terminal loading, and post-delivery establishment. Its purpose is Earth-specific: to determine whether any nonterrestrial donor remains competitive with terrestrial origin once the full chain of transport and survival constraints is imposed. Within that framework, indigenous terrestrial origin remains the default inference, early Mars is the only quantitatively serious external hard-panspermia alternative, and extrasolar or intergalactic hard panspermia is not competitive for Earth’s actual origin history (Turyshev, 5 Apr 2026).

1. Formal definition and scope

For donor class $i$, the formulation defines $N_{\rm ej,bio}^{(i)}$ as the total number of biologically loaded ejecta launched, and introduces the carrier-state vector

$\Pi \equiv (R,d,p,t_{\rm fl},v_\infty),$

where $R$ is carrier radius, $d$ is burial depth of microbes within the carrier, $p$ is peak launch pressure on that microenvironment, $t_{\rm fl}$ is flight time from donor to Earth, and $v_\infty$ is asymptotic encounter speed at Earth before gravitational focusing (Turyshev, 5 Apr 2026).

The expected number of successful Earth-seeding events is written as

$$N_{\rm seed}^{(i)} = N_{\rm ej,bio}^{(i)}\;\mathcal{K}_\oplus^{(i)}.$$

The normalized Earth-directed transport-survival kernel is

$$\mathcal{K}_\oplus^{(i)} = \frac{1}{N_{\rm ej,bio}^{(i)}} \int d\Pi\;\Gamma_i(\Pi)\; S_{\rm sh}(p)\; P_{\rm esc}^{(i)}\; P_{\rm dyn}^{(i)}(t_{\rm fl})\; S_{\rm rad}(R,d,t_{\rm fl})\; P_\oplus(v_\infty)\; S_{\rm ent}(R,d,v_{\rm imp})\; S_{\rm imp}(R,d,v_{\rm imp}),$$

with $N_{\rm ej,bio}^{(i)}$0 the differential production rate of carriers with state $N_{\rm ej,bio}^{(i)}$1 (Turyshev, 5 Apr 2026).

Each factor has a specific operational meaning. $N_{\rm ej,bio}^{(i)}$2 encodes shock survival in the low-shock spall tail; $N_{\rm ej,bio}^{(i)}$3 gives escape from the donor system; $N_{\rm ej,bio}^{(i)}$4 gives dynamical transfer onto Earth-crossing trajectories; $N_{\rm ej,bio}^{(i)}$5 captures radiation survival during transit; $N_{\rm ej,bio}^{(i)}$6 accounts for Earth interception with gravitational focusing; and $N_{\rm ej,bio}^{(i)}$7 and $N_{\rm ej,bio}^{(i)}$8 describe survival through atmospheric entry and impact, respectively (Turyshev, 5 Apr 2026).

The Earth-interception term scales with the focused geometric cross section,

$N_{\rm ej,bio}^{(i)}$9

This places low-$\Pi \equiv (R,d,p,t_{\rm fl},v_\infty),$0 encounters at a premium. A plausible implication is that donor classes are not ranked by transit feasibility alone; they are ranked by the conjunction of low encounter speed, short enough flight time, and sufficient shielding.

2. Protected-depth envelope and survival geometry

A central element of the kernel formalism is the minimum protected-depth envelope $\Pi \equiv (R,d,p,t_{\rm fl},v_\infty),$1, which combines atmospheric-entry sterilization and cumulative radiation shielding requirements (Turyshev, 5 Apr 2026).

Atmospheric entry imposes a nearly constant floor,

$\Pi \equiv (R,d,p,t_{\rm fl},v_\infty),$2

Radiation imposes a time-dependent depth requirement,

$\Pi \equiv (R,d,p,t_{\rm fl},v_\infty),$3

The conservative envelope is then

$\Pi \equiv (R,d,p,t_{\rm fl},v_\infty),$4

Physically, carriers must be buried at least to depth $\Pi \equiv (R,d,p,t_{\rm fl},v_\infty),$5 to survive both entry heating and accumulated radiation dose. The hierarchy implied by these scales is sharp. For $\Pi \equiv (R,d,p,t_{\rm fl},v_\infty),$6–$\Pi \equiv (R,d,p,t_{\rm fl},v_\infty),$7 yr flights, the required depth is compatible with cm–dm carriers; for $\Pi \equiv (R,d,p,t_{\rm fl},v_\infty),$8–$\Pi \equiv (R,d,p,t_{\rm fl},v_\infty),$9 yr flights, the requirement rises to $R$0–$R$1 m, forcing viability into meter-scale shielding regimes; for $R$2 yr, the shielding requirement becomes $R$3 m (Turyshev, 5 Apr 2026).

This depth envelope is not an auxiliary detail but one of the main selection rules in the model. It couples transit duration directly to admissible carrier size and therefore constrains which parts of the ejecta population can remain biologically relevant after transport.

3. Survival-weighted buried-volume fraction

To quantify what fraction of the biologically loaded low-shock ejecta remains protected, the framework introduces the survival-weighted buried-volume fraction $R$4 (Turyshev, 5 Apr 2026).

Assuming a phenomenological size spectrum for low-shock fragments,

$R$5

and uniform volumetric spacing of biological cargo within each fragment, the protected-volume fraction in a fragment of radius $R$6 is

$R$7

Here $R$8 is the Heaviside step function. Only the central sphere of radius $R$9 contributes to protected volume.

The population-averaged buried-volume fraction is

$d$0

The factor $d$1 weights each radius by fragment volume, and hence by biological cargo (Turyshev, 5 Apr 2026).

This formalism makes explicit why long-duration transport suppresses hard panspermia even before capture is considered. As $d$2 rises with $d$3, the surviving core of each fragment contracts, and fragments below the threshold $d$4 contribute zero protected volume. The paper states that $d$5 collapses for small $d$6, and that only meter-to-multi-meter spall fragments remain viable on Myr flights (Turyshev, 5 Apr 2026). This suggests that any donor channel dominated by small fragments is strongly penalized even if ejection and orbital transfer are otherwise favorable.

4. Step-function approximation and effective transport factors

For practical estimates, the kernel is simplified by introducing a step-function approximation for shock survival,

$d$7

with $d$8–$d$9 GPa, and by similarly approximating entry and impact survival through the requirement $p$0 (Turyshev, 5 Apr 2026).

Under these approximations, the time-dependent radiation-plus-entry penalty can be factored through $p$1. For the martian channel, the resulting effective transport factor is

$p$2

where

$p$3

with $p$4 on $p$5 yr, $p$6 on $p$7 yr, and $p$8–$p$9 the fraction of ejecta in the fast tail (Turyshev, 5 Apr 2026).

The compact donor-class expression is

$t_{\rm fl}$0

This representation exposes the architecture of the ranking problem. The kernel is not a single survival probability but a product of coupled filters. Extrasolar classes are suppressed not by one catastrophic bottleneck alone, but by the simultaneous action of low-$t_{\rm fl}$1 capture penalties, long-$t_{\rm fl}$2 shielding penalties, and timing penalties relative to Earth’s origin history (Turyshev, 5 Apr 2026).

5. Donor-class hierarchy and the martian exception

The paper gives an explicit hierarchy of transport factors $t_{\rm fl}$3 for major donor classes (Turyshev, 5 Apr 2026).

Donor class	$t_{\rm fl}$4	Status
Early Mars, fast tail	$t_{\rm fl}$5 with $t_{\rm fl}$6	Viable
Martian meteorite (Myr regime)	$t_{\rm fl}$7–$t_{\rm fl}$8 with $t_{\rm fl}$9	Suppressed
Birth-cluster siblings	$v_\infty$0	Strongly suppressed
Galactic-field donors	$v_\infty$1	Negligible
Intergalactic donors	$v_\infty$2	Excluded

For early Mars, the conservative parameters are specific. Mars has escape speed $v_\infty$3 km/s, so the shock tail with $v_\infty$4 GPa survives; the fast-transfer fraction is $v_\infty$5–$v_\infty$6 over $v_\infty$7–$v_\infty$8 yr; and the corresponding depth requirement is $v_\infty$9–$$N_{\rm seed}^{(i)} = N_{\rm ej,bio}^{(i)}\;\mathcal{K}_\oplus^{(i)}.$$0 m, compatible with cm–dm carriers (Turyshev, 5 Apr 2026). By contrast, the typical Myr-tail regime has $$N_{\rm seed}^{(i)} = N_{\rm ej,bio}^{(i)}\;\mathcal{K}_\oplus^{(i)}.$$1–$$N_{\rm seed}^{(i)} = N_{\rm ej,bio}^{(i)}\;\mathcal{K}_\oplus^{(i)}.$$2 yr, requiring $$N_{\rm seed}^{(i)} = N_{\rm ej,bio}^{(i)}\;\mathcal{K}_\oplus^{(i)}.$$3–$$N_{\rm seed}^{(i)} = N_{\rm ej,bio}^{(i)}\;\mathcal{K}_\oplus^{(i)}.$$4 m and yielding $$N_{\rm seed}^{(i)} = N_{\rm ej,bio}^{(i)}\;\mathcal{K}_\oplus^{(i)}.$$5.

The resulting martian estimates are

$$N_{\rm seed}^{(i)} = N_{\rm ej,bio}^{(i)}\;\mathcal{K}_\oplus^{(i)}.$$6

The paper therefore identifies early Mars as the only nonterrestrial donor whose kernel is $$N_{\rm seed}^{(i)} = N_{\rm ej,bio}^{(i)}\;\mathcal{K}_\oplus^{(i)}.$$7 (Turyshev, 5 Apr 2026).

For extrasolar donors, the picture is markedly different. Birth-cluster siblings may have capture kinematics favoring $$N_{\rm seed}^{(i)} = N_{\rm ej,bio}^{(i)}\;\mathcal{K}_\oplus^{(i)}.$$8 km/s, but still satisfy only $$N_{\rm seed}^{(i)} = N_{\rm ej,bio}^{(i)}\;\mathcal{K}_\oplus^{(i)}.$$9, while flight times $$\mathcal{K}_\oplus^{(i)} = \frac{1}{N_{\rm ej,bio}^{(i)}} \int d\Pi\;\Gamma_i(\Pi)\; S_{\rm sh}(p)\; P_{\rm esc}^{(i)}\; P_{\rm dyn}^{(i)}(t_{\rm fl})\; S_{\rm rad}(R,d,t_{\rm fl})\; P_\oplus(v_\infty)\; S_{\rm ent}(R,d,v_{\rm imp})\; S_{\rm imp}(R,d,v_{\rm imp}),$$0–$$\mathcal{K}_\oplus^{(i)} = \frac{1}{N_{\rm ej,bio}^{(i)}} \int d\Pi\;\Gamma_i(\Pi)\; S_{\rm sh}(p)\; P_{\rm esc}^{(i)}\; P_{\rm dyn}^{(i)}(t_{\rm fl})\; S_{\rm rad}(R,d,t_{\rm fl})\; P_\oplus(v_\infty)\; S_{\rm ent}(R,d,v_{\rm imp})\; S_{\rm imp}(R,d,v_{\rm imp}),$$1 yr drive shielding to multi-meter depths and further reduce $$\mathcal{K}_\oplus^{(i)} = \frac{1}{N_{\rm ej,bio}^{(i)}} \int d\Pi\;\Gamma_i(\Pi)\; S_{\rm sh}(p)\; P_{\rm esc}^{(i)}\; P_{\rm dyn}^{(i)}(t_{\rm fl})\; S_{\rm rad}(R,d,t_{\rm fl})\; P_\oplus(v_\infty)\; S_{\rm ent}(R,d,v_{\rm imp})\; S_{\rm imp}(R,d,v_{\rm imp}),$$2. Galactic-field donors have relative speeds $$\mathcal{K}_\oplus^{(i)} = \frac{1}{N_{\rm ej,bio}^{(i)}} \int d\Pi\;\Gamma_i(\Pi)\; S_{\rm sh}(p)\; P_{\rm esc}^{(i)}\; P_{\rm dyn}^{(i)}(t_{\rm fl})\; S_{\rm rad}(R,d,t_{\rm fl})\; P_\oplus(v_\infty)\; S_{\rm ent}(R,d,v_{\rm imp})\; S_{\rm imp}(R,d,v_{\rm imp}),$$3–$$\mathcal{K}_\oplus^{(i)} = \frac{1}{N_{\rm ej,bio}^{(i)}} \int d\Pi\;\Gamma_i(\Pi)\; S_{\rm sh}(p)\; P_{\rm esc}^{(i)}\; P_{\rm dyn}^{(i)}(t_{\rm fl})\; S_{\rm rad}(R,d,t_{\rm fl})\; P_\oplus(v_\infty)\; S_{\rm ent}(R,d,v_{\rm imp})\; S_{\rm imp}(R,d,v_{\rm imp}),$$4 km/s and vanishingly small capture probability. Intergalactic donors, with $$\mathcal{K}_\oplus^{(i)} = \frac{1}{N_{\rm ej,bio}^{(i)}} \int d\Pi\;\Gamma_i(\Pi)\; S_{\rm sh}(p)\; P_{\rm esc}^{(i)}\; P_{\rm dyn}^{(i)}(t_{\rm fl})\; S_{\rm rad}(R,d,t_{\rm fl})\; P_\oplus(v_\infty)\; S_{\rm ent}(R,d,v_{\rm imp})\; S_{\rm imp}(R,d,v_{\rm imp}),$$5 km/s and $$\mathcal{K}_\oplus^{(i)} = \frac{1}{N_{\rm ej,bio}^{(i)}} \int d\Pi\;\Gamma_i(\Pi)\; S_{\rm sh}(p)\; P_{\rm esc}^{(i)}\; P_{\rm dyn}^{(i)}(t_{\rm fl})\; S_{\rm rad}(R,d,t_{\rm fl})\; P_\oplus(v_\infty)\; S_{\rm ent}(R,d,v_{\rm imp})\; S_{\rm imp}(R,d,v_{\rm imp}),$$6–$$\mathcal{K}_\oplus^{(i)} = \frac{1}{N_{\rm ej,bio}^{(i)}} \int d\Pi\;\Gamma_i(\Pi)\; S_{\rm sh}(p)\; P_{\rm esc}^{(i)}\; P_{\rm dyn}^{(i)}(t_{\rm fl})\; S_{\rm rad}(R,d,t_{\rm fl})\; P_\oplus(v_\infty)\; S_{\rm ent}(R,d,v_{\rm imp})\; S_{\rm imp}(R,d,v_{\rm imp}),$$7 yr, fail in both capture and survival (Turyshev, 5 Apr 2026).

6. Interpretation, assumptions, and relation to soft panspermia

The kernel formalism supports a restricted conclusion about hard panspermia. Hard panspermia remains physically credible only on Solar-System scales; early Mars is the only quantitatively serious external hard-panspermia alternative for Earth’s origin history; and beyond the Solar System, hard panspermia fails the combined capture and survival filters by many orders of magnitude (Turyshev, 5 Apr 2026).

Several critical assumptions are stated explicitly. The low-shock spall ejecta tail must extend to at least cm–dm scales for fast Mars transfers or meter scales for Myr Mars transfers. Radiation models for dose accumulation are taken to be conservative. Capture cross-section formulas are assumed to apply unchanged to interstellar and intergalactic bodies. Biological priors, including origination probability and establishment probability, remain unknown; transport factors alone rank channels (Turyshev, 5 Apr 2026).

These assumptions delimit the meaning of the kernel. It is not an abiogenesis model, and it does not determine the conditional probability that a viable arrival establishes life on Earth. That role is separated into $$\mathcal{K}_\oplus^{(i)} = \frac{1}{N_{\rm ej,bio}^{(i)}} \int d\Pi\;\Gamma_i(\Pi)\; S_{\rm sh}(p)\; P_{\rm esc}^{(i)}\; P_{\rm dyn}^{(i)}(t_{\rm fl})\; S_{\rm rad}(R,d,t_{\rm fl})\; P_\oplus(v_\infty)\; S_{\rm ent}(R,d,v_{\rm imp})\; S_{\rm imp}(R,d,v_{\rm imp}),$$8, which appears in the birth-cluster estimate of only $$\mathcal{K}_\oplus^{(i)} = \frac{1}{N_{\rm ej,bio}^{(i)}} \int d\Pi\;\Gamma_i(\Pi)\; S_{\rm sh}(p)\; P_{\rm esc}^{(i)}\; P_{\rm dyn}^{(i)}(t_{\rm fl})\; S_{\rm rad}(R,d,t_{\rm fl})\; P_\oplus(v_\infty)\; S_{\rm ent}(R,d,v_{\rm imp})\; S_{\rm imp}(R,d,v_{\rm imp}),$$9 expected Earth-seeding events (Turyshev, 5 Apr 2026). A plausible implication is that transport viability and biological establishment are intentionally decoupled so that donor classes can be ranked without assuming a specific origin-of-life prior.

The formalism also distinguishes hard from soft panspermia. The paper states that soft panspermia is much more plausible as chemical enrichment of early terrestrial abiogenesis, whereas hard panspermia requires survival of biological cargo through the full transport chain (Turyshev, 5 Apr 2026). This distinction addresses a common conflation: the failure of extrasolar hard panspermia does not imply failure of exogenous delivery of organics or catalysts.

In a broader methodological sense, the term "kernel" is used differently across transport problems on arXiv. For solar energetic particles, the STAT framework solves the full time-dependent three-dimensional focused transport equation numerically and does not derive an explicit Earth-directed survival kernel (Linker et al., 2019). By contrast, the panspermia formulation is explicitly kernelized and donor-class dependent (Turyshev, 5 Apr 2026). This suggests that, in the panspermia setting, the kernel is not merely a mathematical convenience but the organizing device for a comparative hierarchy of donor classes.