tc_rung_pos
The theorem shows that critical temperature on the phi-ladder remains strictly positive for every integer rung. Materials modelers in Recognition Science cite it to guarantee that any phi-coherent pairing yields a finite transition temperature above zero. The proof is a one-line term wrapper that unfolds the rung definition and invokes positivity of phi to any integer power.
claimFor every integer $n$, the critical temperature on the phi-ladder satisfies $0 < T_c(n)$ where $T_c(n) := phi^n$ in the normalized units of the Recognition Science framework.
background
The Engineering.RoomTempSuperconductivityStructure module derives room-temperature superconductivity conditions from the phi-ladder energy structure. T_c_rung(n) is defined as phi^n, absorbing the coherence quantum E_coh and Boltzmann factor into the units so that each rung directly scales the transition temperature. The module documentation states that E_coh exceeds thermal energy at 300 K, allowing coherent pairing to overcome fluctuations at ambient conditions.
proof idea
The proof is a one-line term wrapper. It unfolds T_c_rung to expose phi^n and applies the lemma zpow_pos together with the established positivity of phi.
why it matters in Recognition Science
This fills the EN-002.4 position in the room-temperature superconductivity hierarchy. It is invoked by coherent_material_has_positive_tc to confirm positive T_c for any coherence coupling and by room_temperature_superconductivity_from_ledger to establish the ambient superconductivity claim. Within the Recognition framework it anchors the phi-ladder scaling that follows from the forcing chain and the eight-tick octave, confirming that transition temperatures stay positive across all rungs.
scope and limits
- Does not compute numerical T_c values for concrete materials.
- Does not specify the structural conditions that produce phi-coherence.
- Does not derive the numerical value of phi or the rung spacing.
- Does not address pressure or magnetic-field dependence of the transition.
Lean usage
exact tc_rung_pos nformal statement (Lean)
82theorem tc_rung_pos (n : ℤ) : 0 < T_c_rung n := by
proof body
Term-mode proof.
83 unfold T_c_rung
84 apply zpow_pos phi_pos
85
86/-- **THEOREM EN-002.5**: Higher rungs give higher critical temperatures.
87 T_c(n+1) = φ · T_c(n) > T_c(n) since φ > 1. -/