SPECTER LABS

K Equations From CATWD 2.0

Explains the P and K equations used in this addendum, grounded in Robert Chis-Ciure and Michael Levin, Cognition all the way down 2.0: neuroscience beyond neurons in the diverse intelligence era (Synthese, 2025), https://doi.org/10.1007/s11229-025-05319-6.

The paper’s core move is to treat intelligence as search efficiency in a shared problem space, not as a vague label for “complex behavior”. The addendum here implements that claim as a small reference calculus and a set of runnable toy domains.

The Two Equations

The paper defines search efficiency with:

K = log10(tau_blind / tau_agent)

The article describes this as “the decimal logarithm of the ratio” between the cost of a blind walk and the cost of an agentic policy. In plain terms:

  • tau_blind is the expected cumulative cost of an unbiased or maximal-entropy search policy over the same admissible moves.
  • tau_agent is the expected cumulative cost of the agent policy over that same search problem.
  • K is measured in orders of magnitude.

That last point matters. K = 1 means the agent is about 10x more efficient than blind search. K = 2 means about 100x. K = 0 means no gain. Negative K would mean the supposed agent is worse than the blind baseline.

The paper’s second important equation is not a scalar but a structured object:

P = <S, O, C, E, H>

The paper calls this a “scale-agnostic quintuple”. It is the thing that defines the search problem whose efficiency is being measured.

What Each Part Means

  • S: the state space. What configurations count as reachable states?
  • O: the operator alphabet. What elementary moves are allowed?
  • C: constraints. Which state-operator combinations are disallowed or impossible?
  • E: the evaluation functional. What makes a state or trajectory good, bad, or goal-satisfying?
  • H: the predictive horizon or budget. How far is the comparison allowed to search?

The addendum also tracks:

  • w: the operator cost function, because different operators can have different costs even inside the same O.
  • H_unit and w_unit: because the paper’s insistence on commensurate units is not bookkeeping trivia; it is what keeps the ratio meaningful.

What The Metric Actually Measures

The paper says K records how much an agent “prunes the futile branches” of a problem space relative to a maximal-entropy walk. That is the right intuition.

K is not:

  • raw success rate by itself
  • raw speed by itself
  • a generic intelligence score detached from a modeling choice

K is:

  • a comparison between two policies
  • inside one explicitly modeled problem space
  • under one explicit cost currency
  • under one explicit horizon

So the real object being measured is not “the organism” or “the algorithm” in the abstract. It is the efficiency gain of one policy over another once you commit to a particular search model.

Why The Shared Problem Space Matters

This is the most important modeling discipline in the whole paper and in this repository.

If tau_agent and tau_blind are computed in different spaces, with different operators, different constraints, or different cost units, then the ratio is no longer interpretable. The paper explicitly warns that choices about O, C, and w shift the result.

That is why this addendum keeps agent and blind policies inside one ProblemSpace and, when available, records shared operator semantics in PolicySpec. The policy may change; the problem definition may not.

How This Addendum Maps The Paper To Code

The main reference surface is core.py.

The core datatypes line up with the paper like this:

  • ProblemSpace: the implementation of P = <S, O, C, E, H> plus units and optional S_init / S_goal descriptors for reporting.
  • PolicySpec: an explicit record of what the policy means operationally.
  • PairedTrial: one paired observation of agent cost and blind cost under the same setup.
  • paper_k_from_paired_trials(...): compute K from trial data.
  • paper_k_from_expectations(...): compute K when expected costs are already known.

The demos in demos/ instantiate that calculus with small spaces where the modeling choices are easy to inspect:

  • sorting: adjacent swaps
  • grid: four-neighbor motion
  • bitstring repair: single-bit flips
  • synthesis: bounded RPN program construction
  • chemotaxis, Hanoi, GRN, and compositional variants
  • two paper-facing examples for amoeboid chemotaxis and planarian regeneration

How To Read A Reported K Value

When you see a result from this addendum, ask these questions in order:

  1. What exactly is S?
  2. What exactly counts as one operator in O?
  3. What is the cost function w?
  4. What is the horizon H, and what happens when a trial fails within it?
  5. Is the blind baseline truly operating over the same admissible graph?

If those answers are crisp, then K is meaningful. If they are fuzzy, K is mostly a placeholder for hidden assumptions.

What The Metric Gives You

When modeled carefully, K gives a compact answer to a hard question:

How much better than blind search is this policy, measured in the natural cost currency of the task?

That is why the paper uses it as a bridge concept across very different substrates. The value is not that all systems reduce to one scalar. The value is that the scalar forces the modeler to expose the search space, the move set, the constraints, the cost measure, and the comparison policy.

That is also the point of this addendum. It is not a benchmark leaderboard. It is a reference implementation for making the paper’s commitments explicit, runnable, and inspectable.