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Geometric Sufficiency Ratio

GSR measures how much target-relevant predictive value survives a declared geometric compression under a controlled comparator protocol.

A GSR estimate without a comparator protocol is not an Ageometrics result.

Metric statusDecision-theoretic working paper metric

Version0.5

AttributionSynaptient

Known riskComparator imbalance can mimic residue

1. Concise definition

GSR is a task-relative ratio comparing the risk reduction preserved by a declared geometric representation against the risk reduction available from a declared fuller reference record.

2. Why comparator discipline matters

The central failure mode is easy to state and easy to miss: if the geometry-only path is evaluated with one learner family and the fuller record with a stronger one, the measured gap may reflect learner advantage rather than representational insufficiency.

Comparator protocol diagram showing the same held-out cases, same temporal boundary, and matched learner or capacity controls feeding baseline, geometry-only, and fuller-record risks before producing a GSR estimate with uncertainty. — Same cases, same temporal boundary, declared control of learner and capacity, then a paired estimate with uncertainty.

3. Declared objects

X: declared fuller reference record available at prediction or decision time
G(X): specified geometric representation derived from X
B: declared baseline information or baseline predictor
Y: target
ell: declared loss, with lower preferred
Pi: complete empirical comparison protocol

4. Canonical Bayes-risk definition

R*_ell(Z) = inf_f E[ell(Y, f(Z))]

GSR*_{ell,Y}(G | X,B)
=
[R*_ell(B) - R*_ell(G(X))]
/
[R*_ell(B) - R*_ell(X)]

R*_ell(B) > R*_ell(X)

5. Boundedness assumptions

R*_ell(X) <= R*_ell(G(X)) <= R*_ell(B)

When the geometry is derived from the fuller record, the fuller record can reproduce the geometry-only decision rule, and the geometry is at least as informative as the baseline, the canonical population quantity is bounded:

0 <= GSR* <= 1

Boundedness is not automatic when those assumptions fail.

6. Empirical protocol-specific estimator

GSR_hat_Pi
=
[R_hat_Pi(B) - R_hat_Pi(G)]
/
[R_hat_Pi(B) - R_hat_Pi(X)]

The three empirical risks should be evaluated on the same held-out cases, under the same temporal-availability rules, with declared comparator controls and leakage safeguards.

7. Same-learner, approximately capacity-matched, and model-envelope views

Same-learner GSR: the same declared predictive family, preprocessing policy, search budget, partitions, and stopping rule are applied when technically possible. This holds much of the observer and protocol fixed while remaining observer- and architecture-relative.
Approximately capacity-matched GSR: when input forms require different model families, parameter count, training compute, data exposure, tuning effort, regularization, and stopping criteria are matched as closely as practicable and the remaining asymmetries are reported.
Model-envelope GSR: several credible predictive families are reported to expose observer dependence. This is a robustness-oriented view, not an automatically conservative one.

8. Worked 0.75 / 0.80 / 0.95 example

Accuracy form:

(0.80 - 0.75) / (0.95 - 0.75) = 0.25

Error-risk form:

(0.25 - 0.20) / (0.25 - 0.05) = 0.25

Raw geometry-only performance can sound strong while still preserving only a small share of the recoverable improvement.

9. Empirical interval violations

Empirical estimates below 0 or above 1 can occur because of finite samples, regularization, optimization, model mismatch, or comparator imbalance. These values are diagnostic and should be preserved, not silently clipped.

10. Canonical and empirical NGR

Side-by-side panels distinguishing bounded canonical non-geometric residue from signed empirical non-geometric residue diagnostics. — Canonical NGR is bounded only under the canonical assumptions. Empirical NGR is signed and protocol-relative.

NGR* = 1 - GSR*

NGR_hat_Pi = 1 - GSR_hat_Pi

NGR is not a metaphysical non-geometric substance. It is a target-, representation-, loss-, and protocol-relative performance gap.

11. Information-theoretic companion

GSR^I_Y(G | X) = I(G(X);Y) / I(X;Y)

This companion connects Ageometrics to Information Bottleneck thinking. It is not the default empirical estimator because high-dimensional mutual information can be unstable to estimate.

12. Interventional GSR

Interventional GSR asks how much of intervention-response prediction survives geometric compression, rather than passive observation alone.

13. Temporal GSR

Temporal GSR asks how much target-relevant information remains when developmental or event history is compressed into final-state or trajectory geometry.

14. Representation-stability envelope

One geometry is rarely enough. The representation-stability envelope summarizes GSR across a family of admissible geometric representations of the same record. A wide envelope means the sufficiency claim depends too heavily on representation choice.

15. Encoding cost

Chart showing geometric sufficiency rising as encoding cost increases, with a warning about merely warehousing the fuller record inside the geometry. — A geometry that becomes sufficient only after heavy augmentation may no longer be a meaningful compression.

Encoding cost forces a harder question: did the geometry expose structure, or did it merely absorb timestamps, labels, provenance, and interventions until it became a renamed full record?

16. Minimal residue-restoring channel

A useful benchmark can ask what smallest auxiliary channel restores sufficiency above a declared threshold. That channel identifies what the geometry was missing.

17. Benchmark protocol

At minimum, an Ageometrics benchmark should declare the fuller reference record, the geometry, the baseline, the target, the loss, the temporal boundary, the learner controls, the uncertainty method, and the leakage-risk plan.

18. Statistical cautions

Comparator unfairness
Model-class confounding
Target leakage and future information
Geometry smuggling through labels or timestamps
Small-denominator instability

19. Open questions

Which proper losses are best by domain?
How should uncertainty be propagated for ratio estimators?
How should model-envelope results be aggregated?
When are cross-domain comparisons meaningful?

20. Downloads and citation

Current working paper Current PDF Illustration code Repository root

Diagram distinguishing the canonical Bayes-risk quantity from the finite-data empirical estimator, showing that only the canonical quantity is bounded under stated assumptions while the empirical estimate may violate the interval and requires diagnosis. — The population quantity and the empirical estimate should never be spoken of as if they were the same object.