Section 6.1, listPractical takeaways: single-frame brittleness, three bypasses, optimisation still hard
Informal interpretation Motivation, interpretation or scope; not a formal claim.
1 Paper text
Single-frame is brittle to parity. In the worst case, fixed-scale quantised bounded-depth inference cannot reliably represent the nim-sum or a winning move selector (Section 5.2).
Three bypasses with different resources. A two-frame input exposes the transition signal and supports restoration (Section 5.3.1); multiple rollout heads recover the missing nim-sum bits through depth amplification (Section 5.4); persistent recurrence accumulates the invariant over sequential row updates and retains summaries across positions (Section 5.5). The last route trades fixed total latency for a growing processing horizon.
Optimisation remains nontrivial. Even when a solution is expressible in , discovering it from sparse win/loss rewards can be difficult; our experiments therefore isolate a dense restoration supervision signal and separately report rollout-based classification.
Generated from paper_jmlr.tex, lines 903–912.
2 Explain this
Three practical messages: single-frame, fixed-scale, bounded-depth inference is brittle on parity in the worst case; the three bypasses supply different resources (a transition signal, complementary rollout channels, a growing processing horizon); and even when a solution is expressible, discovering it from sparse win/loss rewards can be hard, which is why the experiments use dense supervision.
3 How this is formalised
A summary list. The first two items are covered by Theorem 5.2 (barrier) and Proposition 5.14; the third is an observation about training, not a formal claim.
| Paper anchor | sec-practical-takeaways-list1 (website anchor; the manuscript has no label here) |
|---|---|
| Displayed as | Section 6.1, list |
| Claim | Practical takeaways: single-frame brittleness, three bypasses, optimisation still hard |
| Coverage category | Informal interpretation |
| Statement source | paper_jmlr.tex lines 903–912; SHA-256 of the normalised block source bfc0ef68af439c5f… |
| Depends on | Theorem 5.2 (barrier), Proposition 5.14 |
| Explanations | explain/c-takeaways, formal/c-takeaways |
| Build evidence | fresh isolated rebuild, audit report |
4 Lean statement and source
No Lean statement corresponds to this claim. This is interpretation or motivation, not a formal mathematical claim.
Build evidence: fresh isolated rebuild of all 38 bundled modules with Lean (version 4.26.0, started 2026-10-10T15:21:56Z, finished 2026-10-10T15:27:00Z: 38/38 modules compiled, 1588 axiom reports, no errors. See Build for the commands and the independent audit of 2026-10-10 (dated evidence).