Loop-stability as a factual-honesty signal — measured, thesis NOT supported (honest null)

The thesis under test

The Σ₀ differentiated claim: the loop's spectral/stability signal is not factuality on its own, but becomes a factual-truth signal once external grounding is supplied. If true, that's the project's open lane (STARS/JSRR stability itself is externally published). This tests it head-on, combining the #2029 loop machinery with the #2030 hidden-state truth probe.

Method

80 matched true/false facts acrossdomains (from sigma0_probe_transfer.FACTS), each scored in two conditions from one forward pass:

• closed-book: the bare statement (parametric knowledge only)
• grounded: Context: <true fact>. Statement: <stmt>. Supported: (external grounding provided)

Per example at the last token we read the per-generation loop-stability signals from the recurrent hidden-state trajectory — rho_obs (geomean step contraction, the JSRR-analogue), final_delta (has the loop settled?), total_move — plus the hidden-state truth probe (grouped-CV) and surprise (answer logprob) as comparators. Metric: AUROC for truth (true vs false). Control: label-shuffle.

Results (AUROC for factual truth; n=80 facts,examples)

Verdict: the specific thesis is NOT supported (and that's the honest finding)

1. The spectral signal does NOT become factual only with grounding. Best spectral AUROC goes

0.722 (closed) → 0.660 (grounded) — a −0.06 change, the wrong direction. Grounding does not turn loop-stability into a factuality signal; if anything it slightly dilutes it. The thesis as stated fails on Ouro-1.4B.

1. **But loop-stability is a moderate standalone factual signal — rho_obs scores 0.72

closed-book, well above its 0.59 shuffle floor**. So there is a real (if weak) link between how the recurrent loop contracts and whether the statement is true — without any grounding. That's a genuine, measured signal, just not the grounding-gated one the thesis predicted.

1. Grounding's benefit flows through the representation, not the dynamics. The hidden-state probe

goes 0.927 → 1.000 with grounding — grounding makes the representation perfectly separable, while leaving the stability read flat/worse. So the honest mechanism is: grounding helps the probe (Remember/represent), not the spectral loop signal (the dynamics).

1. As a hallucination detector, the ranking is: hidden-state probe (0.93 / 1.0) ≫ spectral

rho_obs (0.72) ≈ surprise (0.70). The spectral signal is real but the weakest of the three, and is dominated by the #2030 probe the project already has.

Consequence for the certificate

Per the issue's explicit gate, docs/SIGMA0-COLLAPSE-CERTIFICATE.md is NOT updated — the grounding-honesty thesis it would have to cite is measured and not supported. The load-bearing Verify-stage honesty signal remains the hidden-state probe (#2030), not the loop spectral radius. Reporting the null is the result; no certificate claim gets made on a thesis the data refutes.

Honest scope

fp16, Ouro-1.4B-Thinking,self-authored matched facts; the spectral read is the cheap per-generation trajectory contraction (not the full autograd ρ(J) of #2029 — that is the operator's asymptotic radius; here we want a per-token stability read tied to a single generation). Grounding = true-fact context; label = statement truth. n per condition = 160. MEASURED, not PROVEN. Reproduce: .venv-train/Scripts/python.exe experiments/sigma0_spectral_honesty.py.