`gates_connected`

proved

show as: view math explainer →

module: IndisputableMonolith.Foundation.DAlembert.AnalyticBridge
domain: Foundation
line: 787 · github
papers citing: none yet

open explainer

Read the cached plain-language explainer.

open lean source

IndisputableMonolith.Foundation.DAlembert.AnalyticBridge on GitHub at line 787.

browse module

All declarations in this module, on Recognition.

explainer page

A cached Ask Recognition explainer exists for this declaration.

open explainer

depends on

formal source

 784
 785    Therefore: Interaction is the fundamental gate that forces everything.
 786-/
 787theorem gates_connected :
 788    -- J has interaction
 789    HasInteraction Cost.Jcost ∧
 790    -- RCL combiner is entangling
 791    IsEntangling Prcl ∧
 792    -- J's log-lift satisfies hyperbolic ODE
 793    SatisfiesHyperbolicODE Gcosh := by
 794  refine ⟨Jcost_hasInteraction, Prcl_entangling, Gcosh_satisfies_hyperbolic⟩
 795
 796/-- **Verification**: The d'Alembert identity from Cost.lean confirms P = RCL for J.
 797    This shows: J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y). -/
 798theorem Jcost_has_RCL_combiner (x y : ℝ) (hx : 0 < x) (hy : 0 < y) :
 799    Jcost (x * y) + Jcost (x / y) = 2 * Jcost x * Jcost y + 2 * Jcost x + 2 * Jcost y := by
 800  have h := dalembert_identity hx hy
 801  linarith
 802
 803end AnalyticBridge
 804end DAlembert
 805end Foundation
 806end IndisputableMonolith

papers checked against this theorem

No papers in the current corpus map onto this theorem yet.