IndisputableMonolith.Foundation.DAlembert.FourthGate
This module defines the d'Alembert functional equation structure for cost functions under curvature conditions in Recognition Science. Researchers proving inevitability of the J-cost from the composition law would cite it as the fourth gate. The module supplies predicates for the structure, verifies the hyperbolic cosine case, derives the implied ODE, classifies solutions, and summarizes the gate.
claimA function $H: (0,∞) → ℝ$ satisfies the d'Alembert functional equation when $H(xy) + H(x/y) = 2 H(x) H(y)$ for all $x,y > 0$. The module shows the J-cost acquires this structure once the curvature gate holds.
background
Recognition Science starts from a single functional equation for the cost and derives all physics via the forcing chain. This module belongs to the d'Alembert series in the Foundation layer. It imports the Cost module for the J-cost definition and the FunctionalEquation module for T5 uniqueness helpers. The CurvatureGate requires constant nonzero curvature via the log-coordinate metric $ds^2 = G''(t) dt^2$, while Counterexamples shows that an arbitrary combiner P does not force the d'Alembert form.
proof idea
This is a definition module with supporting lemmas. It introduces the predicates SatisfiesDAlembert and HasDAlembert, proves the hyperbolic cosine satisfies the equation, shows the J-cost inherits the structure, derives the associated ODE, classifies solutions, and ends with a gate summary.
why it matters in Recognition Science
The module supplies the fourth gate for the triangulated inevitability theorem. It feeds TriangulatedProof, which unifies the four gates, and InevitabilityEquivalence, which bridges abstract and concrete claims. By forcing the d'Alembert form from curvature, it advances the step toward T5 uniqueness, the phi fixed point, and the recognition composition law.
scope and limits
- Does not combine gates into the full inevitability theorem.
- Does not treat the interaction or entanglement gates.
- Does not derive phi, D=3, or the eight-tick octave.
- Does not perform unit calibration or compute alpha.
- Does not extend to discrete, quantum, or non-real domains.
used by (2)
depends on (4)
declarations in this module (12)
-
def
SatisfiesDAlembert -
def
HasDAlembert -
theorem
cosh_satisfies_dAlembert -
theorem
Jcost_has_dAlembert_structure -
theorem
dalembert_deriv_ode -
theorem
dAlembert_classification -
theorem
dAlembert_with_unit_calibration -
theorem
dAlembert_forces_Gcosh -
theorem
Hquad_not_dAlembert -
theorem
Fquad_not_dAlembert_structure -
theorem
fourth_gate_summary -
theorem
dAlembert_forces_Jcost