Consolidation Is a Measurement-Frame Effect
Ongoing research. This is an experimental result from active work, not a settled conclusion. The numbers are what we measured and the method is described so you can judge it, but the programme is still running and later experiments may revise what it means. More about this programme.
EXPLORATORY. Not a preregistered study. No new experiments, no training, no GPU, no cost. Reads only the committedresults/bundles, the captured activation tensors inanalysis/representations.ptand the final output head incheckpoint.pt.
Program v2 tier 0, item D6. Decisive computation: . Output: analysis/consolidation_fate.py. Reproduce with analysis/consolidation_fate.jsonpython analysis/consolidation_fate.py in about two minutes on CPU.
The question
The confirmed phenomenon is that rank-r residual energy rises about 50% around the accuracy transition and then falls back. The rise has been measured five ways. The fall, the thing this project has been calling consolidation, has never been decomposed. Three fates were posed in the backlog, each meaning something different:
- Decay, the expanded directions lose their energy. The expansion was transient activity.
- Rotation into the read subspace, the expanded directions become the ones the head reads. The expansion is where the solution gets built.
- Persistence as unused variance, the energy stays and the head keeps ignoring it.
The answer is none of the three. It is a fourth thing the backlog did not anticipate, and it changes what the word "consolidation" can be used for.
Method
For each of the 41 eligible task-only delayed-copy runs (all 41 qualified), take the final_norm capture at the step where residual energy peaks and freeze the tail subspace at the peak: the orthogonal complement of the top-r principal subspace, which is exactly the subspace whose energy share the confirmed rise measures. Then follow that fixed frame to the last capture.
Every quantity here has a nonzero chance value, a random (W - r)-dimensional frame already captures (W - r)/W = 0.8333 of any fixed direction, so two controls run alongside:
- a random-subspace calibration, which came back at
0.8334[0.8284, 0.8384], confirming the analytic chance level empirically; - a random-time-matched control, in the spirit of this program's standing rule: the same measurements on a tail frozen at a randomly chosen other capture step.
Only task-only runs are analysed; a penalised run drives residual energy through its own loss, which would manufacture the structure being looked for.
Result 1: the fall is a frame rotation, not a contraction
Between the peak capture and step 500, measured inside the frozen peak-tail frame:
| Quantity | Peak → final | Runs favourable |
|---|---|---|
| Energy inside the frozen peak tail | x2.1776 [2.0181, 2.3371] | 41 / 41 |
| Energy in its complement (the peak's retained subspace) | x1.5096 [1.4567, 1.5625] | 41 / 41 |
| Total representation energy | x1.8297 [1.7328, 1.9266] | 41 / 41 |
The expanded directions do not decay. They gain energy after the peak, and they gain it faster than the subspace that was being retained. Consequently the frozen tail's share of total energy rises from 0.4808 at the peak to 0.5662 at the end, in 40 of 41 runs.
Meanwhile, measured in each capture's own moving basis, the way every previous analysis in this project measured it, residual share falls from 0.4808 to 0.4233, in 41 of 41 runs.
Both statements are true of the same runs. The reconciliation is that the top-r subspace is not a fixed set of directions: it is redefined at every capture as whichever r directions currently hold the most energy. After the peak, that redefinition captures more of the still-growing energy than it did before.
The post-peak fall in residual energy is the retained subspace rotating, not the expansion unwinding. Nothing contracts.
Result 2: the rise is real, and larger than reported
The same frozen-frame test applied to the rise gives the opposite verdict, which is what makes the first result credible rather than an artifact of the method:
| Pre-transition capture → peak | Share change |
|---|---|
| In each capture's own moving basis | +0.0998 [+0.0847, +0.1149] |
| In a frame frozen at the pre-transition capture | +0.2783 [+0.2489, +0.3077] |
Energy genuinely moves into those directions across the transition, by 2.8x more than the moving-basis measurement shows, because the moving basis chases the energy and so understates it. The confirmed rise is not weakened by this analysis; it is strengthened.
Result 3: the head under-reads the tail, but not because of the peak
Measure (chance = 0.8333) | Value | Runs above chance |
|---|---|---|
| Peak tail's alignment with the output head | 0.6093 [0.5935, 0.6251] | 0 / 41 |
| Peak tail's overlap with the final retained subspace | 0.4164 [0.3824, 0.4504] | 0 / 41 |
| Random subspace, same measure (calibration) | 0.8334 [0.8284, 0.8384] | 20 / 41 |
The output head systematically under-reads the expanded tail, and the final retained subspace sits mostly inside what was already retained at the peak. So fate 2, rotation into the read subspace, is excluded: the expanded directions do not become the ones the head reads.
But this is not a property of the peak. Against the random-time-matched control, paired within run:
| Paired difference (peak frame − random-time frame) | Mean | 95% interval | Material? |
|---|---|---|---|
| Head alignment | +0.0060 | [-0.0431, +0.0552] | no |
| Overlap with final retained subspace | +0.0759 | [-0.0230, +0.1748] | no |
A tail frozen at an arbitrary capture behaves the same way. "The head reads the top of the spectrum and ignores the tail" is a standing fact about these runs, not something the transition creates.
Verdict against the kill test
The kill test was "no consistent post-peak structure across the 41 archived task-only runs." Structure is consistent: 41/41 on all three energy ratios, 40/41 on the frozen-share rise, 0/41 above chance on both subspace measures. D6 is not killed. But the structure found is not specific to the peak, and it reinterprets rather than explains the fall.
What this costs the program
- A2 (consolidation onset as a stopping criterion) is undermined. Its signal is the post-peak fall, and that fall is a property of the measurement frame rather than an event in the representation. A2 should either be re-specified against a frame-fixed quantity or dropped. As written, it would be detecting the top-
rsubspace catching up with the energy. - Any use of the word "consolidation" now needs a frame. The term entered this project from the moving-basis curve. In a fixed frame nothing consolidates.
- The confirmed rise is unaffected and understated. It should be re-quoted in fixed-frame terms in any future write-up.
Limits
- Captures are every 50 steps, so the peak is located on a coarse grid: peak-minus-transition offsets span
-10to+110, against the+14.1the 10-stepmetrics.csvgrid gives. The frozen-frame conclusions do not depend on peak placement, a random-time frame shows the same pattern, but the peak itself is imprecise here. D0 (mid-run checkpoints) would fix this. - Only the final output head is archived. Head alignment is therefore measured with the end-of-training head at every frozen time. That is the right question for "did these directions become read?", but it cannot say what the head was reading at the peak. Also fixed by D0.
- One architecture (
tiny-gru), one task (synthetic-delayed-copy-v1), one optimizer (AdamW), one learning rate. D8 and E3 still scope all of it.