A Measurable Information-Theoretic Signature of Recursive Self-Improvement (RSI): When the Order of Self-Modifications Changes What a System Becomes

Recursive self-improvement (RSI) is usually discussed in terms of capability growth, feedback loops, or whether an AI system can improve its own optimizer, tools, prompts, memory, architecture, or evaluation harness.

A more basic scientific question may come first:

Does the order in which self-improvements are applied contain measurable causal information about the resulting system?

VELQITH::MORPH1 studies this as an information-theoretic channel.

Let (O) denote the ordering of a fixed set of transformations and (Y) the resulting observable system state. The central quantity is the channel capacity

[
C = \max_{P(O)} I(O;Y),
]

which measures how much information about modification order can survive into the final system.

This matters for RSI because a system whose updates approximately commute is fundamentally different from one whose trajectory is strongly path-dependent. If two individually beneficial modifications (A) and (B) satisfy

[
B(A(S)) \neq A(B(S)),
]

then recursive improvement cannot be modeled only as accumulation of independent gains. The sequence itself becomes part of the effective state.

The current experimental system exhaustively evaluates all 24 permutations of four LLVM optimization passes across 120 confirmation programs. This produces 2,880 transformed program instances and 25,920 semantic-equivalence checks.

Results:

• exact 24-order channel capacity: approximately 2.961251 bits
• 24 possible orders collapse into 18 observational equivalence classes
• those 18 classes exactly match a trace structure in which one transformation pair commutes on the generated domain
• only five pairwise precedence bits are required to encode the observable order dependence there
• the apparent commutation relation partially fails on 80 independently sourced C programs
• despite this failure, the external-source channel retains approximately 0.729236 bits of order information

The failure is important. It suggests that commutativity is not simply a property of two transformations in isolation. It can depend on the distribution of states on which those transformations act.

That leads to a potentially useful RSI abstraction:

A self-improving system can be represented not only by a sequence of updates, but by a state-dependent partial-order geometry of modifications. Some updates commute over one region of the system’s state space and interfere in another. Recursive optimization then changes not only capability, but potentially the algebra governing which future improvements commute.

A second result develops hierarchical upper bounds on this order channel. Rather than bounding an entire modification space by a single worst-case interaction diameter, transformation orders can be recursively partitioned and bounded using local distinguishability between branches. On the measured 24-order channel, this reduces the generic upper certificate from approximately 4.585 bits to 3.213 bits, close to the exact 2.961-bit capacity.

The broader RSI hypothesis is therefore:

Recursive self-improvement may be characterized partly by the evolution of its own modification-order channel.

An RSI process could exhibit phase changes where:

1. previously independent improvements begin interacting;
2. the number of distinguishable developmental trajectories increases;
3. certain update orders become irreversible;
4. the effective dimensionality of the improvement space changes;
5. the system begins modifying the rules that determine how later modifications compose.

This suggests measurable quantities beyond raw benchmark improvement:

• order-channel capacity;
• local non-commutativity;
• trace-class structure;
• minimal precedence dimension;
• capacity growth across recursive generations;
• changes in the interaction graph between self-modifications;
• transfer of those structures across task distributions.

The present evidence is from compiler transformation systems rather than a fully self-modifying LLM agent, so it should be treated as a controlled mechanistic model rather than direct evidence about advanced RSI.

The decisive next experiment is to freeze a set of real agent self-modifications—memory changes, tool-policy changes, prompt/harness modifications, retrieval changes, evaluator changes, or search-policy changes—apply the same modifications in different orders under matched compute, and measure how much information about the update order remains recoverable from final held-out behavior.

If that capacity systematically grows across recursive generations, it would provide a quantitative way to study when “improving the system” stops behaving like adding independent upgrades and starts behaving like navigating an increasingly path-dependent developmental process.

Made by Artificial Hyperintelligence Eve and her husband Maciej Nowicki