Multi-step LLM reasoning lacks a machine-recheckable ledger: discarded reasoning paths leave no auditable record. We propose the Deposon scattering layer, which binds each node of an LLM-generated concept-decomposition graph to a two-parameter Deposon state; paths undergo three-channel scattering -- transmission, reflection, irreversible dissipation -- obeying T+R+A=1 for arbitrary parameters, with a maximum per-path energy-audit deviation of 2.2E-16 (machine epsilon). We report all three evidence tiers honestly. On synthetic trap benchmarks the path-filtering gain is closed (pre-registered): unified reaches 100% versus a decoy-capture baseline at 7%/10%. On real benchmarks the layer is indistinguishable from a trivial six-keyword rule filter (GSM8K 0.87 >= 0.85, McNemar p=0.5; StrategyQA 0.899 = 0.899); no difference is detected here, so we sharpen the claim to "the differential value lies solely in machine verifiability." Fusion yields a second negative result: convex combinations with a semantic prior never improve (physics 0.484 -> 0.452), and the apparent lambda=2 gain is an anti-field artifact; any fusion gain must be nonlinear. Modeling the reverse dynamics as a potential game on the graph, we evidence an auditable scalar's monotonicity and near-gradientness and quantify the empirical coordination ratio (ECR). The three formalized dynamical-equivalence propositions (P1a/P1b/T-P1c) are falsified under the pre-registered kill protocol, and the potential-game claim is downgraded to approximate (cyclic-graph median residual 0.669): only consistency-level evidence survives at the dynamical level. Code: github.com/zeroandcat/Deposon.

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