Phase 15 · memory accounting and redundancy
The Memory Has To Live Somewhere
AT-0119 → AT-0126. Path dependence created an accounting problem. If the outcome depends on the route, some variable differs at the end — and the only escape is that the difference is genuinely redundant, not merely unwatched.
The theorem this phase is built on
COMPLETE-STATE ACCOUNTING THEOREM (toy model). In a deterministic model with complete state X and an identity observable I = I(X), two trajectories ending at the exact same complete state X_f cannot yield different identity outcomes, because I is a function of X_f alone. Therefore any endpoint path dependence must be encoded in (a) a differing state variable, (b) a path-dependent definition of the observable, or (c) an equivalence between complete states. It follows that 'path dependence with zero hidden-memory debt' is impossible when 'same final state' means literal equality of the complete state.
Model & audit limits
κ is the whole experiment. At κ = 0 the hidden coordinate influences nothing, so the endpoints are indistinguishable and the redundancy claim is trivially true. Raise κ and the coordinate starts acting — at which point the endpoints are separable by behaviour alone and the claim fails.
Same physics, different representation?
Endpoint A
- x
- 1.0000
- y
- 1.0000
- h (hidden)
- 0.0000
- m (mediator)
- 0.0000
Endpoint B
- x
- 1.0000
- y
- 1.0000
- h (hidden)
- 1.1578
- m (mediator)
- 0.0000
Observable set: declared only
| Observable | Set | A | B | Gap | Separates? |
|---|---|---|---|---|---|
| Address x | declared | 1.0000 | 1.0000 | 0.0000 | no |
| Address y | declared | 1.0000 | 1.0000 | 0.0000 | no |
| Address invariant ½(x²+y²) | declared | 1.0000 | 1.0000 | 0.0000 | no |
Inside the declared observable set the endpoints agree. This is the equivalence claim, not evidence for it. Admit more observables to test it.
If two routes seem to end in the same place but the system remembers which route you took, that memory has to live somewhere. The only escape is if the difference is genuinely redundant — two descriptions of the same physical state — not if we simply choose not to look.
Running the accounting audit…