Retunable objective memory: study 02 findings

# Retunable objective memory: a passed mechanism, not an application breakthrough

The frozen 36-record panel passes its capability and computational gates for
both Hermite budgets, both Chebyshev budgets, and the 64-value natural cubic
table. Linear tables fail at both budgets; the 32-value natural cubic table
passes only 45 of 108 record/merge-version checks. No budget or domain was
changed after development. Study 01 remains closed with a failed admission gate.

## What the retained object can do

An 802-byte object can approximately evaluate a previously selected family of
single-pole filter losses and terminal states at a later-selected time constant,
including a different incoming state. Chronological summaries can be combined
without the raw observations. It cannot recover arbitrary history, accept new
targets retrospectively, or train an arbitrary recurrent network. The encoder
knows the model family, parameter interval and squared-loss definition. The
individual query parameters are drawn after all objects are serialized.

All 10,080 serialized block/final objects, 36 archive hashes, source hashes,
864 final versions and regenerated raw references were audited by `analyze_02.py`.
Reference recomputation agrees exactly in this environment. There are six
independent input seeds, not 864 independent trials. Every query is synthetic.

| Representation | Bytes | Passing versions / 108 | Worst loss error | Worst terminal error | Median warm speedup |
|---|---:|---:|---:|---:|---:|
| Linear 32 | 802 | 0 | 3.64e-4 | 1.26e-3 | 468x |
| Linear 64 | 1,570 | 0 | 8.80e-5 | 3.16e-4 | 455x |
| Natural cubic 32 | 802 | 45 | 9.70e-5 | 2.99e-4 | 622x |
| Natural cubic 64 | 1,570 | 108 | 2.26e-5 | 6.42e-5 | 618x |
| Cardinal cubic Hermite 32 | 802 | 108 | 7.88e-6 | 5.28e-5 | 634x |
| Cardinal cubic Hermite 64 | 1,570 | 108 | 4.17e-7 | 2.96e-6 | 627x |
| Chebyshev 32 | 802 | 108 | 4.60e-11 | 1.76e-9 | 218x |
| Chebyshev 64 | 1,570 | 108 | 2.27e-14 | 5.48e-14 | 131x |

Errors follow the frozen normalizations. Speedups are paired medians for a
259-query workload at N=65,536, against the same compiled raw-data filter-bank
reference. They are not speedups over the best possible competing summary.
The gate uses the ratio of median query times; both that quantity and paired
ratios are in `SUMMARY_02.json`. No measured quadratic became materially
indefinite. Chebyshev-32 maximum objective regret is 1.27e-15; Hermite-32 maximum
regret is 1.80e-8, against the declared raw optimizer, not a proved global minimum.

The speed/accuracy tradeoff is informative. The local Hermite evaluator is
faster in this Python/SciPy implementation and passes at the smaller budget;
global Chebyshev interpolation is dramatically more accurate. Its analytic
response functions favor spectral approximation. This is not evidence for
universal spline superiority, and optimized polynomial evaluation may change
the timing comparison.

## Costs and limitations

The 802 bytes include serialized method/domain/time-step/length metadata and
96 float64 values. They exclude interpreter overhead and the reusable numerical
library. Evaluator numerical arrays occupy 2,480 bytes for Hermite-32 and
1,040 bytes for Chebyshev-32. These are array-byte counts, not total allocator
or process memory. Whole-record encoding breaks even after a median 0.0745
259-query batches for Hermite-32 and 0.1264 for Chebyshev-32 at the larger N;
this does not make encoding free for a single later question. Block encoding
and both merge costs are separately retained in each complete record.

The panel process peaked at 146,898,944 bytes (140.1 MiB). The evaluation harness
retains raw data, all blocks and references. It is not a demonstration of a
fully streaming device implementation. A raw two-channel float64 record at
N=65,536 is 1,048,576 bytes, but dividing that by 802 is only a task-specific
memory reduction: the object deliberately discards almost all other questions
about those samples. It is not a waveform compression ratio at equal fidelity.

The confirmation records are stored losslessly in 36 TAR.GZ archives. Their
archival compression is not included in the 802/1,570-byte model accounting.
All content is recoverable and all hashes were checked. The old development
files and original runner remain unchanged.

## Interpretation and next boundary

This establishes a compact, retunable loss/response surface for a scalar
stable dynamical model, including chronological composition. Approximate
sufficient statistics, interpolation and affine/quadratic scan algebra all
have substantial prior art. In particular, PASS-GLM already combines polynomial
approximation with streaming/distributed sufficient statistics without extra
aggregation approximation. A novelty claim therefore cannot rest on those
ingredients alone. See the primary paper:
https://www.cs.princeton.edu/~rpa/pubs/huggins2017pass.pdf

An audit found whole/sequential/balanced response functions agreeing to at most
3.04e-14 across all arms on a separate 1,025-point grid. This motivates the
explicit algebra audit in study 03: fixed-node interpolation may preserve the
merge operation exactly at the nodes, unlike arbitrary projection. That is a
post-exposure mathematical explanation, not an independent performance trial.
It clarifies, rather than changes, the cautious generic recompression statement
in `PROTOCOL_02.md`.

The admission decision is to investigate the scope and prior art of this
retunable-memory mechanism before choosing a real workload. A cheap objective
surrogate is not yet evidence of better calibration, useful learning, safe
control, general NN compression, or a practical/SOTA breakthrough.