CleanstoneDevelopmental and independent evidence for machine minds.

Position paper, sections 2 and 4: the accounting problem, and what a residual shows

Draft 1, 2026-09-30, written from the numbers in ../01-a-ledger/writeup.md section 4. The remaining sections are outlined in plan.md.

2. The accounting problem

Levin (2026) proposes to measure "the delta between effort put in and competency out" in minimal systems, and to read the delta as what the system draws from outside its specification. For the delta to be a measurement, four things have to be numbers in one currency. They are the effort of building the system, the competency it shows, the competency its substrate shows with no building at all, and a rule for combining them. We tried to build that ledger on eight systems whose accounting we controlled, with four combining rules, and we could not. The rules were not the problem. The currencies differ.

Effort has three natural components, and they do not add. The description length of the rule or code is a number of bits, and it can be compressed. The description length of the parameters is also in bits. For a random initialisation those bits are entropy and buy nothing, while for a trained network they are a compressed record of the training signal. The training signal is itself bits of feedback. For a tabular learner on a small task it exceeds the behaviour it produces by a large factor, a ratio of 75 in our case. Which of these count as "effort" is a convention. Two of our rules took different conventions, and on the same untrained system they gave residuals of opposite sign.

Competency has the same problem. Counted as mutual information between the system's action and the optimal action over a state space, it exceeds the compressed size of the lookup table that produces it (8,866 bits against 4,104). The task has structure that a state-by-state count does not see. Counted as success rate, competency saturates at a few bits per episode. Even for a system that pays explicitly for every action, behaviour bits and description bits are different currencies.

No rule can supply the cost of design. A short program that solves a structured task has a residual equal to the task's compressibility relative to the code. For a breadth-first solver on a ring with barriers that is 5,514 bits. What the program did not pay for is the programmer's knowledge of the task. A rule that priced that knowledge would have to set it equal to the behaviour it produces, which is circular, or to some external measure of the programmer, which is not a property of the system. Levin's minimal systems (sorting rules, fractal seeds, six-line cellular automata) are all designed. Their "free lunch" is the compressibility of the problem each was written for, a fact about the problem and its author.

Trained systems do have a coherent residual. It is the behaviour bits minus the bits of training signal, with the untrained substrate's competency subtracted, and it measures sample efficiency relative to task information. It can be tracked as a function of training step, and this is the form of the ledger the programme keeps (../../spine/01-developmental-checkpoints/). It does not apply to a designed system. No version of it applies to both kinds.

The no-free-lunch theorems (Wolpert and Macready 1997) reach the same conclusion from the other direction. An advantage on one family of tasks is paid for on another, so a residual is always relative to a task prior. On a uniform prior, counting performance over distinct evaluations, every system's residual is zero; it is also zero under some non-uniform priors (Wolpert and Macready 1997, p. 71), and it is not zero against a null that revisits points, such as the random walk in Chis-Ciure and Levin's metric, where part of any positive score is the value of remembering where one has been. On any structured prior the residual measures how well the system's inductive bias fits the prior, and whoever chose the bias supplied the fit.

4. What a positive residual would show

Suppose a future accounting produced a positive residual that survived metric substitution, a random-substrate null and adversarial replication. What would follow?

It would not follow that a non-physical pattern had entered the system. Causal exclusion (Kim 1998) is usually stated as an argument against mental causes. In ledger terms it is a claim about bookkeeping. If the physical account of a system is complete, an extra cause has nothing left to explain. A positive residual is evidence that the account is incomplete, and the missing term becomes the object of study. Every missing term found so far has been physical: the substrate's inductive bias, the task's structure, the training signal, the author.

We therefore fix the reading of a positive residual now, before any data. It means an unpriced term exists in the accounting, and finding that term is the next experiment. The programme commits to this position. A physicalist and a proponent of Levin's view can both accept it as the next step, which is why we state it in advance.

The two readings would part only at the end of that process. Suppose every identifiable term had been priced and a residual still remained, and could be moved by intervening on the system. Then the physicalist would owe an explanation. We do not expect to get that far. The tests in the programme are designed so that falling short of that point is itself a reportable result.

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