# n233 — Does the derived nonlinearity pin parameters? Better: it is already pinned — Mercury becomes a pure test

**Date:** 2026-07-12 · **Question (the human's):** does n232 freeze any parameters? **Answer, in three steps, each computed:**

## 1. λ_g factors out exactly

b₃/a = +0.1229 at λ_g = 0.5, 1, 2, 4 — identical to four digits. The constrained minimizer is invariant under overall rescaling of the cost, so the derived nonlinearity ratio is **scale-free**: it cannot pin, and is not pinned by, the overall ledger scale. (Registered so nobody spends λ_g on it.)

## 2. What it does depend on: the cost mix u″/λ — which the theory has already pinned

Sweeping the mix: b₃/a = 0.128 → 0.123 → 0.104 → 0.079 as u″/λ goes 0.02 → 0.5. The dependence is real and monotone. But u″_g/λ_g is not free — **it is fixed by the marginality condition C1** (the sandpile: u″_g = R_g(0) − λ_g·G_g(0)), which the theory needs independently for zero-at-rest (n211). So:

**b₃/a is fully determined by the already-pinned structure. It consumes nothing; it is consumed.** The second-order gravity is a *parameter-free output* of the pinned two-ledger vacuum.

## 3. The consequence: over-determination, the honest kind

The equal-cone family had exactly one residual freedom, reserved for Gate D leg 2 (the Coulomb/Newton normalizations). The perihelion — once the metric dictionary (b₃ → the g₀₀ Φ² coefficient) and the orbit machinery exist — does **not** get to spend that freedom: its second-order source is already fixed. So Mercury enters the theory as a **pure test with zero dials**: the pinned structure predicts a definite β-class number, and the sky either agrees or the theory dies at that node. This legitimately rebuilds, per-sector and with derived links, the "falsification-amplified" structure that n226 suspended: the pinning ledger now reads — isotropy → weights; clusters → R = 1; sandpile → u″_g/λ_g; photon masslessness → u″_e/λ_e; GW170817 → λ_e(λ_g); Newton/Coulomb → the last freedom (leg 2, pending); **Mercury → nothing left to pin: pure verdict.**

Walls unchanged and named: the b₃ → β_PPN dictionary, the matter cone (Gate F), units. N = 64, one seed; the mix-dependence curve wants a seed ensemble.

**Class: §1–2 [A-model, computed]; §3 [structural consequence, B]. Short answer to the question: no — it freezes nothing, because everything it depends on was already frozen; and that is exactly what makes Mercury dangerous again, in the only way that counts.**