No Unpaid Discounts: A Ledger-Conservation Law for Recognition Barriers, with Machine-Checked Consequences for Fusion Screening and Sub-Tick Gravity

Recognition Science treats an interaction barrier as a path of recognition events that must be posted to a double-entry ledger before the interaction completes. Coherence in the environment can lower the effective barrier. The question this paper answers is by how much, and at whose expense. We state a single conservation law, no unpaid discounts: a discount on a recognition barrier must equal recognition cost already posted into that channel by the environment. Nothing else can lower a barrier, and pre-payment can at most cancel it. We then machine-check the law's consequences in the two places the framework was actually exposed to it, and the two outcomes have opposite signs. On the fusion side, the screening factor S that scales the Gamow exponent had been a modeled seam, S = 1/(1 + C_φ + C_σ) with free coherence parameters. The conservation law replaces the seam with a derived object: S = 1 − f where f is the fraction of the bare recognition path pre-posted by the environment, the rate enhancement is exactly e^Λ for pre-payment Λ, never more, and the modeled form is recovered as a theorem under the exact reparameterization f = x/(1 + x). Salpeter's weak-screening factor is the classical instance, with the Debye channel energy in the pre-payment slot. On the gravity side, the information-limited gravity (ILG) weight kernel, extrapolated below the recognition tick, had appeared to offer a weight reduction approaching 1 − φ⁻⁵ ≈ 9% under optical-band coherent driving. The conservation law closes this: no channel pre-pays a source mass's recognition events, so every admissible sub-tick extension of the kernel satisfies w ≥ 1, the saturated kernel (ratio floored at the tick) is the pointwise-least admissible extension, and the 9% figure is retracted as an artifact of extrapolating a numerically regularized formula across the tick. Both developments are formalized in Lean 4 in the IndisputableMonolith library, as the modules PrepaidScreening (fusion) and SubTickSaturation (gravity): 27 theorems, zero sorry, with transitive axiom closure {propext, Classical.choice, Quot.sound} for every one. We tag exactly what is theorem, what is derived but unformalized, and what remains open, chiefly the magnitude of driven pre-payment in a φ-coherent plasma, which the law converts from a free parameter into a priced unknown.