THE CLOSURE CONDITION
E(F(X, a)) = g(E(X), a)
If two complete states share a memory code, applying the same permitted action must lead to the same next code.
01 A FUTURE-SUFFICIENT REPRESENTATION
AN EXPERIMENT ABOUT FUTURE-SUFFICIENT MEMORY 2026 / 10
What must a memory keep?
Two systems can look identical right now—and still behave differently after the same interaction. Explore the residue that makes the difference.
32DFULL DYNAMICS
4DPRESENT ANSWERS
+1DFUTURE-RELEVANT RESIDUE
27DDISPENSABLE FOR THIS TASK
01 / INTERACTIVE INSTRUMENT
The residue experiment.
Keep four core numbers—or keep the fifth, initially invisible bridge. Then apply actions and watch their consequences.
LIVE IN YOUR BROWSER
Same now. Different later?
The full system and a compact model begin with matching answers. Both receive exactly the same action.
FULL SYSTEM 32 DIMENSIONS
Current answer+0.0000
COMPACT MODEL 4 DIMENSIONS
Predicted answer+0.0000
Response trajectory
Actual answer versus compact prediction
Full Code
IDENTICAL PRESENT0.000000Difference between twins before any action
→
POSSIBLY DIFFERENT FUTURE0.000000After identical actions with opposite hidden bridge
What you are operating: a self-contained, deterministic 32-variable illustration of the same nonlinear mechanism class as the repository's frozen experiment. Its coefficients are chosen for the web demo, so its live numbers are not the eight-seed Python receipt. The results below come from the measured code.
02 / THE MATHEMATICAL TEST
Compression is not enough.
A code must survive the next interaction, not merely answer the current question.
THE RESIDUE LEDGER
X = core + Rbridge + Rnuisance
Keeping every residue reconstructs the past. Keeping only future-relevant residues may be enough to preserve the computation.
02 SIHTI-INSPIRED BOOKKEEPING
What the hidden fifth number does: it changes no current answer. But in the bridge world it alters the next core state. The 27 other coordinates continue to evolve without feeding back into the permitted answers.
03 / RECORDED RESULTS
The measured boundary.
These results belong to the preregistered synthetic Python experiment, not the browser illustration above.
01 CLOSED WORLD
≈ 0 ERRORFour numbers correctly update every measured future response when the bridge cannot feed back.
02 BRIDGE WORLD, 4 STORED
0.274 NRMSEAll initial answers match, but future predictions fail after the causally relevant residue is discarded.
03 BRIDGE WORLD, 5 STORED
≈ 0 ERRORKeeping one additional coordinate restores machine-precision prediction under the allowed actions.
8 / 8 fixed gates passed8 random seeds32 / 32 full Jacobian rank11 repository tests
The closure is engineered: the experimenter knows the hidden basis and the update law. The full simulator calls that same compact-law implementation. These checks establish the intended mechanism and a paired-state failure case; they do not establish a learned visual encoder, a biological memory or a cheaper AI model.
04 / LEARNED MEMORY, WITHOUT THE ORACLE
Can the system learn what survives?
This time the encoder does not know the hidden 32-dimensional basis. It sees only twelve mixed sensor channels and three frames of history before predicting six action-dependent futures.
Held-out normalized error · shorter bars are better
Average across two fixed apparatus seeds · 400 held-out episodes per seed
View data ↗
WHAT CHANGED AFTER THE ORIGINAL PROOF
No privileged encoder.
No exact answer.
Each candidate must infer a compact, self-updating state from observations and supervised consequences. After the initial window, no new sensor data are supplied during the six-step prediction.
20 bytesFive stored float32 values per memory, excluding shared model weights, the observation buffer, and training cost.
381Shared trainable parameters in the five-coordinate predictive model. The conventional GRU baseline uses 501.
The inconvenient detail matters: five coordinates also outperform four in the closed world, where no bridge is causally required. So this does not establish that the trained encoder independently discovered the true fifth residue. All predictions retain significant error. The PCA and GRU comparisons use one fixed training recipe, not exhaustive architecture searches.
The original browser instrument above still illustrates the hand-designed v1 mechanism. This section reports a separate v2 PyTorch learning experiment, trained on two synthetic apparatus seeds. Neither is yet an image model.
05 / THE CLASSICAL CHECK
Does learning beat the textbook answer?
Realization theory computes a minimal memory in closed form: the rank of the map from pasts to futures. Same data, same split, same metric — no gradient descent.
Six-step held-out error · shorter bars are better
Mean of apparatus seeds 41 and 97 · 400 held-out episodes per seed
View data ↗
WHAT THE HANKEL RANK SAYS
Pure ERA loses.
But it sees the bridge.
Value of the fifth coordinate (six-step error, 5 minus 4). The closed form spends it only where the bridge is causal.
−0.143ERA, bridge world. Closed world: −0.002 — essentially nothing.
−0.043Learned, bridge world. Closed world: −0.014 — the learned fifth coordinate helps where no bridge exists.
The honest ledger: under the frozen protocol, pure ERA loses by ~0.25 — its shift law is fitted from two-action words and extrapolated to six. A post-hoc, strictly linear refinement on the full horizon (same 381 parameters) beats the learned model in the closed world (0.156 vs 0.179) and trails it in the bridge world (0.180 vs 0.157). So the learned memory's surplus over textbook system identification is confined to the nonlinear bridge case.
ERA is verified to recover an exact linear switched system to below 10⁻⁶. The refinement rows were added after the frozen result to locate the gap and are labelled as such.