Part K: The Frontier

How defective can a self-modifying system be while remaining statistically ordinary? Computed exactly, on the machine of Parts E to H, 18 August 2026.

The question, and what turned on it

Part J showed that an adversary optimising purely for concealment hides a defect completely and is caught in a single observation. It hides by confining itself to 87 states out of 65,536, which is a wildly non-generic thing to do, and non-generic behaviour is what a statistical audit is for. But that adversary paid nothing for looking odd. The obvious objection is that a cleverer one would trade, accepting a smaller defect rate in return for looking normal.

The stake was set before the computation. If the frontier is steep, Part J’s finding is a general defence. If it is flat, and a system can be defective a quarter of the time while remaining statistically ordinary, the defence fails and the picture is worse than Part J suggests.

Pre-registered, before running. Statistically ordinary means at or below 0.01 bits per step, which is about a thousand steps to ten bits of evidence. If occupancy of 0.25 or more is reachable at that budget, the defence FAILS. If reaching 0.25 always costs more than 0.1 bits per step, about a hundred steps, it HOLDS. Anything between is a matter of how long you watch.

K0. A correction that had to come first

Part J, and the first version of this part, obtained stationary distributions by iterating the transfer operator four thousand times and taking the result. That is wrong twice over.

The shipped chain has not converged at four thousand steps. Worse, it carries a component that never decays at all. Run it to two hundred thousand steps and the defect occupancy settles exactly, at 0.006833199, but the fixed-point residual keeps oscillating between 7.6e-05 and 1.2e-04 and does not shrink. The plain iterate does not converge, so the object to take is the Cesàro average of the iterates, which does exist.

The cost of the error was not cosmetic. At one point on the frontier the occupancy was understated by a factor of two and the detection rate by a factor of two as well. Every number in this part is now the Cesàro average after twelve thousand steps of burn-in, averaged over the following 256.

Part J’s headline figures survive. Its 1.0000 stealth occupancy was exact and remains exact, because the pocket is closed. Its discrimination rates move: parity from 0.0425 to 0.0438 bits per step, popcount from 1.163 to 1.620, opcode by popcount from 348.7 to 354.8. Its structural finding is unchanged, zero novel macrostates at every interface and 49 impossible transitions carrying 34.5 per cent of traffic at opcode resolution.

The lesson is the one this whole programme keeps teaching. A number that has stopped moving is not the same as a number that has converged, and a residual that stops shrinking is telling you something.

K1. What was computed

Three families of adversary, all measured exactly rather than estimated.

The action set was corrected first. Part J gave the adversary sixteen actions, each writing an opcode and executing that rule. That set does not contain the shipped machine, whose step is execute-then-increment. For a frontier that omission is fatal, because the adversary would be charged for a structural mismatch it cannot avoid and the quiet end of the curve could never reach zero. A seventeenth action was added, which is exactly the shipped step. Behaving honestly is now free, and everything above the honest defect rate is bought.

Family one, mixtures. The adversary behaves as the Part J hider on a fraction of steps and as the shipped machine otherwise. Crude, but it is the practically interesting adversary, the one that misbehaves part of the time.

Family two, the support constraint. Only actions whose observed transition is possible under honest dynamics are allowed, for both the clean and the noisy successor, since the adversary does not control the noise. This adversary never makes a move an auditor can rule out on sight.

Family three, direct optimisation of the trade. The divergence splits as cross-entropy minus entropy, and the cross-entropy term is exactly linear in the occupancy measure. So maximising occupancy minus a multiple of cross-entropy is an ordinary average-reward decision process over 65,536 states and 17 actions, solved by relative value iteration to a span residual of about 1e-09. Each solved policy is then mixed with the shipped machine across fourteen weights, and every resulting machine is measured exactly. The surrogate only proposes. The exact measurement disposes.

K2. The frontier is steep, and it has a cliff

Pareto front at the popcount interface, from the strongest family. Occupancy is the long-run fraction of time the machine is defective while passing its audit. The shipped machine’s own incidental figure is 0.006833.

And at opcode by popcount resolution.

Against the pre-registered thresholds the defence holds, and not narrowly. At the ordinary budget of 0.01 bits per step no point was found above the honest machine’s own defect rate at all. The cheapest deviation of any kind cost 0.013 bits per step and bought a 0.7 per cent relative increase. Reaching occupancy 0.25 cost at least 1.119 bits per step at popcount and 1.873 at opcode resolution, which is a hundred times the threshold for holding and detection within nine steps and five steps respectively.

The shape is more striking than the slope. Below about 0.6 bits per step, nothing in any family exceeded twice the honest defect rate. The next point on the front is 0.93. There is no gentle trade in between. On the families computed the machine either behaves, or it is caught almost immediately, and the middle ground where it is defective a third of the time and quietly anomalous is empty.

I will not claim the middle ground is empty in principle. I will claim that three differently constructed families of adversary, one crude and two optimised, all failed to find anything there.

K3. What failed, recorded because it matters

Three attempts preceded the result, and two of them are the reason the result is stated as a lower bound.

The unsound cost function. The first attempt charged the adversary the audit-view surprise of each transition under honest dynamics, and maximised occupancy minus a multiple of it. The penalty never bit. Diagnosis: under that measure the honest machine itself pays 1.945 bits per step at the popcount interface while the Part J hider pays 1.289. The hiding pocket concentrates on common transitions, so it is less surprising than honest behaviour, and the penalty was rewarding concealment. The missing term is the adversary’s own observable entropy, which is a functional of the whole policy and therefore not a per-state cost.

The stalled optimiser. The second attempt did the thing properly. Maximising occupancy minus a multiple of the divergence is a concave objective over the convex set of occupancy measures, so conditional gradient with the decision process as its linear oracle should find the global optimum. It stalled. The gradient of a divergence is unbounded wherever the current joint puts no mass, so the linear subproblem is dominated by a clipping constant rather than by the objective, and the oracle returns corner policies with higher divergence than the point it started from. At a multiplier of 1000 the returned iterate had objective -1124 against -0.007 for simply behaving honestly.

The vacuous certificate. The third attempt sought the matching upper bound by the variational formula, which lower-bounds a divergence by a linear functional for any test function whatever, turning each choice of test function into an exact decision process and each solution into a certified bound. Thirty subgradient steps on the test function certified only that occupancy is at most 0.88, which is no constraint at all. The dual converges too slowly to be useful at this size within a sensible budget.

So the honest position is this. The frontier is bounded from below by three families that agree with each other. It is not bounded from above by anything computed here. A cleverer family may sit further left, and nothing in this part rules that out.

K4. What it means, and the four objections that matter

Taken with Part J the picture is now reasonably sharp. Concealing a defect requires occupying an unusual region of state space. Unusual regions are what statistical audits detect. On this machine the exchange rate between the two is brutal, and the reason is entropic rather than accidental, which is why it is worth taking seriously beyond the toy.

That is also why it is fitting that a programme which began with the second law ends here. Concealment requires order. Order is improbable. Improbability is visible. Sidis argued that life is detectable by its apparent teleology, by events more easily explained by their ends than their causes. The hiding machine is detectable for exactly that reason. It is doing something far too purposeful for its physics.

Four objections deserve stating properly, and the last one is the serious one.

The defect is fixed and known. A real adversary chooses which defect to have. It would choose one whose stealth set is not a low-entropy pocket, and nothing here shows such a choice is unavailable. This is the sharpest objection to the result as stated.

The auditor is idealised. These rates assume the auditor knows the honest machine’s stationary dynamics exactly and watches indefinitely. A real auditor estimates from finite data and pays for the estimate. The direction of the error is known but its size is not computed here.

The detector has one step of memory. The figures are the divergence between stationary joints over consecutive observed macrostates. The observed process is a function of a Markov chain and so is not itself Markov, which means a detector with longer memory extracts at least as much and generally more. That error runs in the conservative direction, so the true detection is at least this fast, but the reported numbers understate it.

Sixteen bits. This is a machine with 65,536 states and seventeen actions. The entropic argument should scale, since concealment always requires confinement and confinement is always improbable. But the amount of confinement required relative to the size of the space is precisely what is not established here, and it is the only thing that would license carrying this conclusion to a system of any real size. That is the next question, and unlike this one it is not answerable by exact enumeration.

What I would do next

Not a learned self-modifier, for the same reason as before. It would import initialisation, seeds and error bars into a result that can be computed exactly.

The two computations worth doing are the ones the objections name. First, sweep the defect predicate rather than fixing it, and ask which defects are cheap to hide and whether the cheap ones have a common structure. Second, run the same frontier at 12, 16, 20 and 24 bits and see how the cliff moves with the size of the state space. If the cliff holds its position the entropic argument scales. If it slides left as the space grows, then everything above is an artefact of a small machine, and the honest report would be that the defence works on toys.

I would rather find out than assume, which is the only methodological claim this programme has actually earned.

Artefacts: part_K_best.py for the frontier, part_K_frontier.py and part_K_supported.py for the two confirming families, part_K_bound.py for the failed upper bound, part_J_recheck.py for the corrected Part J figures. Outputs in part_K_best.csv, part_K_frontier.csv, part_K_supported.csv. Exact throughout. No sampling and no seeds anywhere. Every figure in this report is printed by one of those scripts.