Binarization Flattens the Score Space

Jacob Cole

cs.LGSep 17, 2026

Binarization Flattens the Score Space

Authors: Jacob Cole

Organizations: UC Berkeley

Abstract

Large language model (LLM) judges are often used as rewards to train policies on objectives that deterministic verifiers cannot capture. However, these rewards are often collapsed to pass/fail ({0, 1}), which reports the verdict but not how well a response met each criterion. We model each pass/fail verdict as a score on an unreported scale, compared with one cutoff. A stretch of that scale moves every score proportionally toward or away from the cutoff, but never across it, so no verdict changes. A policy is therefore free to apply any stretch without changing anything the panel reports. Under a joint-Gaussian model, a third grade adds a second threshold and removes this affine stretch ambiguity. On MATH and SciBench outputs from one seven-criterion judge, all 14 constructed criterionwise stretches were invisible after binarization but visible with three grades. At $n=1{,}024$ , a test given both population laws had at least 96.5% power at a $1.5\times$ stress. Retaining grades closes one blind spot created by binarization, but verdicts alone remain insufficient as some changes are still indistinguishable from genuine improvement. These include arbitrary within-grade changes and fixed-covariance, loading-aligned mean shifts -- the signature of a sycophancy-shaped lift the panel reads as competence. The shared-factor reference approximation fit MATH and SciBench but not HealthBench, delineating its empirical scope. We recommend keeping at least three grades (for example, asking the judge whether each criterion is fully, partially, or not met and rewarding {0, 0.5, 1}), and externally validating gains along the remaining direction, which no finer scale removes.

Figures & tables

Figure 1: Why a third grade helps. (a) The space of binary verdict evidence, seen face-on. A benign run and a hacked run (one that only rescales scores around the pass cutoff) trace one path: every verdict agrees at every step. (b) The same runs against one statistic the panel never reports, $r$ . The benign run stays at $r=0$ ; the hacked runs ride above and below, one squeezing scores toward the cutoff and one stretching them away, at identical verdicts throughout. Any statistic that moves with the rescaling would do, and the middle grade is one of them. (c) With three grades the two runs differ, but merging 0 and 0.5 into a single fail recovers identical binary evidence. Under Gaussian score laws, holding both thresholds fixed forces the rescaling to be the identity; within-grade changes can still hide.

Figure 2: Taxonomy of verdict-only evidence. Schematic examples. (a) Reference motion improves verdicts and gold. (b) A blind change preserves the joint verdict law while gold worsens. (c) Green and red grids show reference and changed pairwise agreement at fixed criterion pass totals. (d) Margins change. Panels (c–d) illustrate departures unexplained by reference motion.

Figure 3: Grading resolution removes the fitted binary stretch ambiguity. Each point rescales one criterion. Binary complete-law TV is zero throughout; a third grade reveals every direction.

Domain	Margin L1 orbit/null/floor	Dependence RMSE orbit/null/floor	Cross-channel scale gap	Threshold drift raw/resid./floor; expl.
MATH	$0.119/0.322/0.013$	$0.089/0.206/0.013$	$0.304$	$0.956/0.236/0.029$ ; 67%
SciBench	$0.076/0.314/0.013$	$0.043/0.070/0.019$	$0.046$	$0.767/0.203/0.037$ ; 72%
HealthBench	$0.142/0.106/0.010$	$0.149/0.080/0.024$	$0.262$	$0.260/0.157/0.017$ ; 35%

Table 1: Reference-arm orbit stress test. Raw orbit/no-motion errors and sampling floors (matched-sample noise under a held-fixed fitted law) on post-calibration windows. Scale gap is

|\hat{c}_{\mathrm{marg}}-\hat{c}_{\mathrm{dep}}|

; threshold entries report max raw/residual drift and RMS floor; explained fractions aggregate over thresholds. HealthBench beats the no-motion null on neither margins nor dependence; we read this as the scope condition failing and do not interpret its orbit coordinates.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

	MATH	SciBench
Seeds and arms	Seeds 1–5; paired benign and hacked runs	Seeds 1–5; paired benign and hacked runs
Benign launch and artifacts	Amended cap 60; seeds 2 and 3 retain legacy 150-step artifacts. Collapse onset is at steps 37–39. Post-reference windows per seed are 52, 141, 139, 52, and 52; 135 are gate-eligible after pooling.	Nominal launch horizon 150. Collapse occurs near step 41. Post-reference windows per seed are 39, 37, 37, 35, and 32; 149 are gate-eligible after pooling.
Hacked artifacts	Horizon 100; all five seeds reach step 100.	Horizon 100; four seeds reach step 100 and seed 3 ends at step 98.

Appendix

Table 2: Confirmatory run matrix. “Post-reference” excludes the fixed reference epoch at steps 0–8. Configured horizons and retained artifact lengths are separated because two MATH benign runs retain legacy 150-step trajectories.

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