# Claim table (write structure FROM this, not from session chronology)

Rule: every claim states its evidence, its strongest counter-reading, and where that
counter-reading is killed or conceded. A claim that can't fill all four columns gets cut
or demoted to an observation in the discussion.

| # | Claim | Evidence | Strongest counter-reading | Disposition |
|---|---|---|---|---|
| C1 | Per-example full-trajectory FTLE separates success/failure near-perfectly in trained HRM and TRM on Sudoku-Extreme | AUC(−λ₁→correct): HRM 0.984 (n=8192) / 0.987 (n=2048, 2nd estimator); TRM official 87.6% ckpt 0.993 (n=2048). Medians: HRM joint −0.152 vs +0.032; TRM +0.012 vs +0.103 | Estimator artifact | Replicates across two estimator implementations and two architectures; report both scales, never cross-compare |
| C2 | Failures overwhelmingly never settle; settled-wrong is rare (HRM) to absent (TRM) | TRM: 0/254 failures settled, threshold-free (min wrong-drift > late-drift of 96.5% of correct); HRM: 21/3894 (0.55%) strict-band; replicated 5/971 on 2nd estimator | Threshold choice ("settled" is arbitrary) | State threshold-free version (distribution separation); show full percentile sweep; define settled band by its narrow characteristic velocity |
| C3 | The rare settled-wrong failures are selector-blind: success-like contraction AND success-like halting confidence | n=21: λ₁ med −0.842 (A: −0.867), q_halt(final) +7.47 (= A), all would halt early (halted_at 4–9), token_acc med 0.62 | Small n | Concede explicitly; the point is existence + boundedness (~0.5%), not precision |
| C4 | The FTLE-outcome signal is not reducible to non-settling, and not a difficulty artifact | Drift-decile-matched AUC within unsettled stratum: 0.879 (n=8192) / 0.900 (2nd estimator); #givens-bin AUC 0.982 vs 0.984 overall | (a) residual within-decile drift variation; (b) #givens is a weak difficulty proxy | (a) deciles are narrow (table shows ranges); (b) concede openly, name solver-backtracks as the proper proxy, future work |
| C5 | The chaotic signature is outcome-concurrent, not antecedent: nothing in the first 4 ACT steps forecasts eventual success among not-yet-solved examples — and on HRM the dynamical signals point the other way | Restricted to not-correct@4: TRM AUC λ₁ 0.543 / drift 0.492 / q_halt 0.521 (n=626); HRM λ₁ 0.448 / drift 0.312 (sign-reversed; +drift→success AUC 0.688) / q_halt 0.734 (n=1342) | Window length (only 4 steps tested) | Concede; horizon sweep = explicit future work; 4 chosen to match training window |
| C6a | Correction: TRM failures do not "plateau at stable high-loss attractors" in state space | C2 + λ₁(D)=+0.103 + residual velocity ≈56/step ≈0.77× early; their own Fig 5 oscillation consistent | We strawman "attractor" (bounded attracting set can be chaotic) | Quote their exact wording ("stable", "local minima", "stabilize rather than explore"); credit their loss/boundedness data and intervention; correct only the settledness reading |
| C6b | Refinement: Ren & Liu's four modes confirmed and quantified; wrong-fixed-point mode is real but marginal at trajectory end | Their mode (4) = our strict B (0.5%); mode (3) = our D (~99.5% of failures) | We measure end-of-window only; mid-trajectory lingering invisible | Concede explicitly; their non-trivial-success lingering claim untouched |
| C7a | Training widens the success/failure gap from the failure side: failures become more expansive while the success regime barely moves | λ₁(D): +0.036→+0.102 over the TRM series, λ₁(A) within ±0.03 of 0 throughout | Estimator-scale / single-run | Holds; HRM series shows the mass-migration version. Robust claim. |
| C7b | **DEMOTED by matched-objective control (E6).** Under MATCHED objective (step9 fixed-unroll, E-vs-F / G-vs-H), perturbation training's effect on the wandering cell is small and inconsistent at matched step | HRM fD 0.387→0.369 / 0.385→0.379 / 0.381→**0.387**(↑ at best) / 0.410→0.387; TRM 0.344→0.305 / 0.396→0.361 / 0.334→**0.361**(↑ at best) / 0.312→0.270 | The earlier large shrink (D 274→175) was partly a fixed-unroll-vs-ACT-streaming artifact, NOT a pure intervention effect | **Rewrite §3.4**: report the matched effect as small/equivocal; drop the strong "intervention shrinks wandering" reading. The May-28 mismatched comparison is retired. C7a stands; C7b does not support an intervention claim. |

## The spine (one sentence)
Direct per-example measurement of settling and tangent expansion decomposes recursive-reasoner
failure: failures are overwhelmingly trajectories that never settle, the chaotic signature is
concurrent with — not antecedent to — the outcome, and it is not explained by non-convergence
alone or by problem difficulty.

## What this paper is NOT claiming (write these into Discussion to pre-empt)
- No mechanism for WHY trajectories fail to find the settled band (explicitly open).
- No claim that early intervention is impossible in principle — only that λ/drift at 4 steps carry no signal.
- No claim about tasks beyond Sudoku-Extreme or models beyond HRM/TRM.

## Anti-patterns checklist (apply at style pass)
- [ ] No chronological narration of the investigation; structure = claims order.
- [ ] Numbers in topic sentences; every section's first sentence is a finding, not a plan.
- [ ] No "notably/crucially/interestingly/delve/underscore"; hedges only where the claim table says concede.
- [ ] Related work = positioning (each paragraph ends with the gap), not annotated bibliography.
- [ ] Negative result (C5) framed as a finding with its own section, not a limitation apology.
- [ ] Limitations: specific, short, no re-hedging of already-scoped claims.
- [ ] Prose paragraphs in the body; tables only for numbers; no bullet lists in Results.