cell_CDT_EchoDamping_positive
plain-language theorem explainer
The RS black-hole echo amplitude damps by the positive factor 1/φ per bounce. Causal Dynamical Triangulations predict no such echo channel, so positivity alone discriminates RS from CDT in the EchoDamping sector. Gravity-track auditors cite this cell when assembling the 4×3 discriminator matrix. The proof is a one-line appeal to the already-proved positivity of 1/φ.
Claim. The Recognition Science per-echo amplitude damping ratio equals $1/\varphi$ and satisfies $0 < 1/\varphi$.
background
Track 6.D of the quantum-gravity master plan builds a 4×3 discriminator matrix (rivals: LQG, string, CDT, Bohmian; sectors: LeadingLog, EchoDamping, RungPhase). Each cell is a theorem-grade numerical band that separates RS from a rival on an empirically accessible channel.
In the EchoDamping sector the RS prediction is the per-echo amplitude ratio $1/\varphi$, coming from the black-hole bounce ledger. CDT has no closed physical echo mechanism, so it predicts a null signal. Any strictly positive RS ratio therefore discriminates.
Upstream, echoDampingRatio is defined as $1/\varphi$ and echoDampingRatio_pos already shows $0 < 1/\varphi$ from positivity of $\varphi$. The eight-tick phase algebra and rung ladder supply the $\varphi$-native units in which the ratio is stated.
proof idea
One-line term proof: the claim is exactly the upstream theorem that the echo damping ratio is positive. That upstream result unfolds the definition $1/\varphi$ and applies positivity of the numerator and of $\varphi$. No further algebraic work is done here.
why it matters
Fills the (CDT, EchoDamping) cell of the discriminator matrix. Downstream it is wired into discriminatorMatrixFull, which packages all twelve cells into the Track 6.D certificate. Together with Session 93's three theorem-grade discriminators this meets the binding success criterion: a matrix with at least one unambiguous cell per rival, each derived from $\varphi$ with a named observational channel.
Against CDT and Bohmian the cells are existence statements ($>0$) rather than numerical margins, because those rivals predict no quantum-gravity echo signal at all. The doc-comment notes that while the RS ratio is positive, the physical echo mechanism itself is not yet closed; the cell still stands as a structural discriminator. Framework landmarks: $\varphi$ from T6, the bounce-ledger origin of the $1/\varphi$ damping.
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