cell_Bohmian_RungPhase_positive
plain-language theorem explainer
The Bohmian–rung-phase cell asserts that the φ-rung phase delay is strictly positive, so RS predicts a nonzero black-hole echo phase that Bohmian and dynamical-collapse substrates do not. Gravity and QG auditors comparing RS echo channels to Bohmian/DP would cite it. Proof is a one-line term wrapper on the prior positivity lemma for that delay.
Claim. The Recognition Science rung-phase delay is strictly positive: $0 < \delta_{\mathrm{rung}}$. Bohmian and dynamical-collapse (DP) substrates do not produce this $\varphi$-rung phase algebra, so positivity alone discriminates RS from those rivals in the rung-phase sector.
background
Track 6.D builds a $4\times 3$ discriminator matrix: rivals LQG, string, CDT, Bohmian against sectors LeadingLog, EchoDamping, and RungPhase. Each cell is a theorem-grade inequality. For Bohmian the design is existence rather than a numerical margin: the rival predicts no quantum-gravity signal in the sector, so any strictly positive RS prediction separates the two.
The rung-phase delay is the phase shift tied to the $\varphi$-ladder rung structure in the black-hole echo analysis (upstream in BlackHoleEchoesFromBounce). It sits in the eight-tick / octave phase algebra: phases $k\pi/4$ on the period-$2^3$ clock, with the self-similar fixed point $\varphi$ fixing the rung spacing. Bohmian and DP substrates lack this $\varphi$-rung phase algebra, which is exactly the content of the cell doc-comment.
Positivity of the delay is already a proved fact; this declaration only installs that fact as the matrix cell.
proof idea
One-line term-mode wrapper. The goal $0 < \mathrm{rungPhaseDelay}$ is discharged by applying the upstream lemma that already proves the delay is positive. No new algebra is done here; the cell is a named packaging of that inequality for the discriminator matrix.
why it matters
Fills the (Bohmian, RungPhase) entry consumed by the master certificate discriminatorMatrixFull, which bundles every cell into a single DiscriminatorMatrixCert. Together with the sibling cells this closes Track 6.D of the quantum-gravity master plan and the binding success criterion: a discriminator matrix with at least one unambiguous cell per rival, and three or more theorem-grade $\varphi$-derived discriminators with named observational channels.
The distinction is structural: RS inherits a positive rung-phase delay from the $\varphi$-ladder and eight-tick octave (forcing-chain T6/T7), while Bohmian/DP substrates produce none. Echo damping and rung phase remain quarantined as algebraic cells until a horizon-consistent physical echo mechanism exists; this cell only certifies the algebraic inequality, not a completed echo model.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.