Two Saturated Steps: A Profile That Separates Every Carrier - and the Two Data It Still Needs

*2026-11-03 — draft — not deposited* I priced Feynman's path integral through a chain of papers that closed eight factored classes for item D-361-409-45, and the eighth fell to a proved pattern: a unit-exponent monomial window of length L is blind to the period-L rescalings whose factors multiply to 1 over the window, so the survivor must factor through no monomial window of any length. This paper first prices the candidate that suggests itself next — the chord profile M_j = (f_{j+1} − f_j)/(f_{j+1} + f_j), a quotient of a difference by a sum, no monomial in sight — and excludes it before its battery can mislead anyone: with r_j = f_{j+1}/f_j the ratio profile's entry, M_j = (r_j − 1)/(r_j + 1) identically — **wherever both are defined (f_j ≠ 0 and f_j + f_{j+1} ≠ 0; outside that domain the two languages are not the same constraint, a definedness defect priced in §3)** — verified on the priced grid to 1.2×10⁻¹⁶, and the map inverts, so on that domain the chord language *is* the ratio language in another dress. Its battery is the inherited one: identical where the rectangle doubles (scale not fixed), identical under rotation and against the constant-potential carrier (absolute phase not fixed), value-blind to the scrambled carrier, completable by a single datum at 3.4×10⁻¹⁴ — plus a definedness failure of its own, since the entry is undefined wherever two neighbouring cells sum to zero. The chord closes no ninth class. The paper's candidate proper is built to break every cancellation the chain has fed on: the **paired saturated-step profile** L_j = S(f_j, f_{j+1}) + S(f_{j+1}, f_{j+2}), where the saturated step S(a, b) = (b − a)/(1 + |a| + |b|) sums two non-multiplicative steps across a three-cell window — numerators that rotate with the density, denominators that do not