The proposed composition
The original program combined a JEPA-style latent objective, a continuous coordinate-addressable world representation, and a recurrent query controller. It predicted that the controller could retrieve less information as the world grew while maintaining accuracy on exact synthetic tasks.
The formal core proved a narrower statement. A reasoner cannot recover an arbitrary uninspected cell from a trace that does not contain it. The Lean file also makes the compute ledger monotone: reading more cells cannot be recorded as cheaper. These are interface and accounting theorems. They do not prove that training converges or that a waveform is efficient.
The rank-cost conflict
Take a global basis with K coefficients over n independent cells. Exact representation of all possible cell assignments requires rank at least n; otherwise two worlds collide in the representation while differing at a queried cell. Once K >= n, evaluating one point in a global sum touches at least n coefficients. The exact representation has recovered the same linear per-query cost it was meant to avoid.
The escape route is locality. A compactly supported multiresolution basis can evaluate a point in constant or logarithmic support. At the exact-world limit, that representation approaches an indexed lookup. A waveform can still help on worlds with compressible multiscale structure, but random independent cells contain no such structure.
Why the JEPA objective disappears on the first benchmark
For independent random cells, the mutual information between context and a held-out target is zero. A latent prediction objective cannot learn a relationship that the data distribution does not contain. The constant predictor is optimal for that term, and anti-collapse machinery can prevent representational collapse without creating predictive signal.
The first benchmark therefore disabled two proposed mechanisms at once. The global field lost its cost advantage under exactness, and the JEPA term had no predictive information to use. A result from that benchmark could test the harness for leaks, but it could not fairly test the intended structured-world hypothesis.
The component that survived
Adaptive querying survived at the theorem level. On a two-hop pointer task, an adaptive reader can follow the first pointer and then the second for two reads. Any fixed schedule that omits a possible destination fails on some world. The asymptotic separation comes from conditional choice, not from the waveform representation.
Subsequent experiments confirmed that an adaptive policy can be learned in distribution, while also showing that its current implementation fails to generalize across identifiers and scale. That split should shape the repaired program: preserve hard-read adaptivity, replace the global basis with local or indexed structure, and use JEPA only on data where I(context; target) > 0.
Evidence ledger
Formal: interface and monotone-ledger properties in
formal/WaveformJEPA.lean.Argument: the rank-cost and zero-mutual-information analyses.
Executed elsewhere: the adaptive query controller experiments reported in the companion negative result.
Not established: training convergence or an efficient waveform representation for exact random worlds.
