cell_String_RungPhase_positive
plain-language theorem explainer
The String×RungPhase discriminator cell asserts that the RS per-rung phase delay is strictly positive. Gravity-track auditors cite it when filling the 4×3 rival matrix against string theory, which has no clean competing prediction in this sector. The proof is a one-line wrapper of the lattice positivity lemma for log φ.
Claim. The Recognition Science per-rung phase delay on the recognition lattice is strictly positive: $\log\varphi > 0$.
background
Track 6.D of the quantum-gravity master plan builds a 4×3 discriminator matrix: rivals (LQG, string, CDT, Bohmian) against sectors (LeadingLog, EchoDamping, RungPhase). Each cell is a theorem-grade inequality that separates RS from a rival on an observationally accessible band. Against CDT and Bohmian the cells are pure positivity statements; against LQG and string they often carry explicit numerical margins.
The RungPhase sector is built on the per-rung phase delay of the recognition lattice, defined as $\log\varphi$ with $\varphi$ the golden-ratio fixed point forced at T6. Upstream, the bounce-echo module already proves $0 < \log\varphi$ from $\varphi > 1$. String theory supplies no clean competing formula for this delay, so the cell records only positivity: any positive RS signal discriminates where the rival is silent.
The module status is structural theorem (zero sorry), closing the Track 6 binding criterion that the matrix exist with at least one unambiguous cell per rival.
proof idea
One-line term wrapper. The goal rungPhaseDelay > 0 is discharged by applying the upstream lemma rungPhaseDelay_pos, which unfolds the definition to Real.log phi and invokes Real.log_pos with one_lt_phi. No further algebra is needed at this cell.
why it matters
Fills the (String, RungPhase) entry consumed by discriminatorMatrixFull, completing the string row of Track 6.D together with the LeadingLog and EchoDamping cells. The module doc ties this matrix to the quantum-gravity master-plan success criterion: three or more theorem-grade discriminators from φ with named observational channels, and at least one unambiguous cell per rival.
The cell rests on the T6 forcing of φ and the rung algebra of the recognition lattice (linked to the eight-tick octave at T7 and black-hole echo timing). The doc-comment is explicit that a physical echo interpretation of the delay remains open; what is proved is only positivity of the quarantined RS quantity $\log\varphi > 0$, not a full string-versus-RS phase formula.
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