Cross‑Ratios as Universal Atoms of Meaning: A Projective‑Invariant Framework for Likelihood, Information, and Decision Theory

In the paper's words

Dimensionless descriptors such as physical constants, likelihood ratios, and mutual information are central to inference across physics, statistics, and machine learning, yet they are vulnerable to changes of units, model parametrizations, or data distributions. This work bridges projective geometry and quantitative inference by proposing the cross‑ratio—the unique invariant of four points under projective transformations—as a universal “meaning atom.” We formalize meaning units as cross‑ratios over distinction spaces, prove that likelihood ratios in additive Gaussian channels and Poisson channels are exact cross‑ratios, and show that mutual information in Gaussian feature alignment equals the expected log‑cross‑ratio. A concrete illustration computes the fine‑structure constant α as a cross‑ratio (α ≈ 0.007297357 ≈ 1/137.036) and a simple four‑point cross‑ratio (CR = 4/3).

The abstract, verbatim; every line below is quoted from the text.

What it claims

that any epistemic agent detecting a stable cross‑ratio can obtain invariant likelihood‑ratio and information‑theoretic descriptors—would be falsified if an agent extracts a cross‑ratio that varies significantly under admissible projective transformations of the data. Concretely, in a controlled additive Gaussian experiment, if the computed cross‑ratio deviates from the directly evaluated likelihood ratio by more than 10^-6 across a range of signal‑to‑noise ratios, the invariance hypothesis fails.

Numbers it reports

1. Using reported low‑false‑positive improvements of the Pairwise Likelihood‑Ratio Membership Inference Attack (22.94 %, 27.29 %, 27.65 %), we derive an average gain of 25.96 %.

   from Abstract

2. Computation of the average low‑false‑positive improvement of PL‑MIA over baselines (22.94 %, 27.29 %, 27.65 %) yielding 25.96 % (Section 4).

   from 1. Introduction

3. Assuming a baseline true‑positive rate (TPR) of 50 % at 0 % false‑positive rate on CIFAR‑10, the reported 22.94 % relative improvement for PL‑MIA [7] projects an absolute TPR of $$ \text{TPR}_{\text{proj}} = 50\% \times (1 + 0.2294) = 61.47\%, $$ i.e. an absolute gain of 11.47 % .

   from (e) Projected Performance Bounds (Projection)

4. Numerical illustrations of the fine‑structure constant and a simple geometric cross‑ratio, together with the average 25.96 % gain of PL‑MIA, provide concrete evidence of the framework’s predictive power.

   from 8. Conclusion

5. Empirical gains of PL‑MIA over baseline attacks are reported as 22.94 %, 27.29 %, and 27.65 % low‑FPR improvements on CIFAR‑10, CIFAR‑100, and CINIC‑10 respectively (Section 5.2 of [7]).

   from 2. Background and Related Work

6. Reported low‑FPR improvements: 22.94 %, 27.29 %, 27.65 % (Section 5.2 of [7]). $$ \text{Sum}=22.94+27.29+27.65=77.88,\qquad \text{Average}=\frac{77.88}{3}=25.96\%. $$ All calculations involve only elementary arithmetic and the definitions above.

   from 3.4 Numerical Illustration

How it is done

1. Using reported low‑false‑positive improvements of the Pairwise Likelihood‑Ratio Membership Inference Attack (22.94 %, 27.29 %, 27.65 %), we derive an average gain of 25.96 %.

   from Abstract

2. Proof that likelihood ratios in additive Gaussian channels [1] and Poisson channels [2] are exact cross‑ratios (Section 3).

   from 1. Introduction

3. For points $A,B,C,D$ on a line, $$ \operatorname{CR}(A,B,C,D)=\frac{(A-C)(B-D)}{(A-D)(B-C)}. $$ Its invariance under any projective transformation $T\in\operatorname{PGL}(2,\mathbb{F})$ follows from the theorem of the QNFO synthesis [12].

   from 2. Background and Related Work

4. Concretely, in a controlled additive Gaussian experiment, if the computed cross‑ratio deviates from the directly evaluated likelihood ratio by more than $10^{-6}$ across a range of signal‑to‑noise ratios, the invariance hypothesis fails.

   from 6.2 Failure Modes and Falsifiability

What would falsify it

1. The central claim—that any epistemic agent detecting a stable cross‑ratio can obtain invariant likelihood‑ratio and information‑theoretic descriptors—would be falsified if an agent extracts a cross‑ratio that varies significantly under admissible projective transformations of the data.

   from 6.2 Failure Modes and Falsifiability

2. The result predicts that any epistemic agent that can extract a stable cross‑ratio will obtain invariant likelihood‑ratio and information‑theoretic descriptors, enabling robust inference without domain‑specific recalibration.

   from Abstract

3. Limitations, falsifiability criteria, and a concrete experimental pathway (Sections 5–6).

   from 1. Introduction

Where the claim stops

1. Assumption Sensitivity A1: Reference hypothesis chosen as pure‑noise case Essential for exact cancellation; alternative references yield a scaled ratio.

   from 6.1 Limitations

2. A3: Mutual information derivation assumes exact Gaussianity and known covariances Estimation errors in $\Sigma$ affect the equality $I=\mathbb{E}[\log\operatorname{CR}]$.

   from 6.1 Limitations

3. A4: Numerical illustrations ignore measurement uncertainties Propagating uncertainties would slightly shift the computed $\alpha$ value.

   from 6.1 Limitations

Where to read

1. Abstract187 words · 1 min
2. 1. Introduction304 words · 1 min
3. 2. Background and Related Work396 words · 2 min
4. 3. Methods269 words · 1 min
5. 4. Analysis239 words · 1 min
6. 5. Results87 words · 1 min
7. 6. Discussion197 words · 1 min
8. 7. Practical Implications and Engineering Pathways315 words · 1 min
9. 8. Conclusion121 words · 1 min
10. Correction Notice122 words · 1 min
11. Provenance and AI Involvement232 words · 1 min
12. Appendix A. Divergence report15 words · 1 min