Open Gaps

Density of replicating terms (Conjecture 50.1). The argument that the added replication machinery can always be hidden from any given HML formula requires a careful analysis of which formulae can probe the replication channel. In particular, if \(\phi = \langle K \rangle \psi\) where \(K\) involves the replication channel, the argument fails. A precise density theorem would need to show that such formulae form a meager (first-category) set in the logical topology, so that “generically” \(\phi\) does not probe replication.

Defining the elementary terms. Proposition 48.1 requires a designated set \(E\) of elementary terms. In the chemical setting, these are atoms or monomers. In the Rho calculus, the natural candidates are the behavioral primes of Part II (Section 4): the irreducible terms from which all others are built by interaction. Confirming that this identification gives assembly indices matching those of Assembly Theory requires working through the combinatorics of the Rho calculus prime factorization.

The basin of attraction. The claim that a population entering a neighborhood of \(\mathrm{Rep}\) is “sucked in” to the replicating fixed point requires a Lyapunov-style argument: a function on the process space that is monotonically decreasing along trajectories near the fixed point. The natural candidate is the distance \(d_{\HML}\) to \(\mathrm{Rep}\), but showing it decreases requires assumptions on the interaction dynamics.

The compact cluster as the new geosphere. The identification of Smith’s fourth geosphere with the compact coherence cluster centered on the replicating fixed point requires showing that this cluster is indeed compact (in the sense of Part II) and that it persists under the perturbations of the selection dynamics.

Crossover as a variant of comm. The claim that genetic recombination is one definitional step from the comm rule requires specifying precisely what “fragment substitution” means in the Rho calculus and showing that the resulting operator has the recombination properties of a genetic crossover (fitness inheritance, schema theorem, etc.).

Gradient-coupled rules and the weight map. Definition of gradient-coupled rules modifies the weight map of Chapter 13 by introducing an external parameter \(\Phi\). Half of this gap has closed since it was written. Time-varying weight maps are not an extension of the framework but its defining feature: the update functions of Definition 13.6 vary the map at every step, and Theorem 13.1 says the process remains an exactly simulable chain because the map is part of the state. What remains open is the externally driven case. An update reading a parameter the configuration does not carry leaves the chain, and the repair is to bring \(\Phi\) inside — either as a component of the configuration or as traffic on its own channels with its own keys. Which of those is right is a modelling question we have not settled, and only the first keeps the process Markov.

Metabolic viability (Conjecture 51.1). The condition that catabolic inflow exceeds anabolic cost in expectation requires computing the path integral of a driven open system — a substantially harder problem than the closed-system path integral of Part I. The quantum-jump simulator of Chapter 13 provides a simulation method, but an analytic viability criterion requires further development.

Trophic level as causal depth (Proposition 51.1). The identification of trophic level with \(d^o_{\min}(K_\Phi, B)\) requires showing that the account-flow graph \(\mathcal{F}\) is faithfully captured by the open synchronization tree structure. In particular, it requires that every account transfer corresponds to a minimal-context transition, which is plausible but needs verification.

Non-well-founded scopes. Remark 18.4 argues that a budgeted learner can inhabit only a least fixed point, since membership must terminate. But an ecology with no bottom is exactly what one would model with the greatest fixed point, and the reflection tower is bounded below only by no-self-code, which is a fact about names rather than about ecologies. Is there a coherent notion of a scope that is non-well-founded and yet surveyable stratum by stratum?

The copy series under merger. §24.8 shows that merger sets \(\bt = 0\) and destroys factorisation. What does it do to the copy series of Definition 53.2? The expectation is that it collapses two adjacent strata into one, which would make the series a direct diagnostic for whether a composite has merged or merely composed — and would give the composition/merger distinction of \(\chComp\) an observable signature.

Affordable scope and the individuation fixed point. A learner’s region is bounded by what it can pay to survey; Remark 24.11 makes individuation a fixed point. Both are circular in the same way — the region determines what can be afforded and what can be afforded determines the region. Do they have a common solution, and is the individual the least one?

The survey is an assay. A survey visits channels and decides bisimilarity at a depth; that is an experiment, and by §21.4 an experiment has three outcomes, the third being budget-relative. So a survey has a \(\bot\) column, and a copy number is properly a triple: confirmed copies, confirmed non-copies, and channels the budget could not settle. What is the right generalisation of Definition 53.1 that carries the third column, and does the biosignature criterion survive it?