IndisputableMonolith.Foundation.DAlembert.FourthGate
The FourthGate module defines the d'Alembert functional equation for cost functions and classifies solutions that force the J-cost form. Researchers proving inevitability of the composition law would cite its classification and forcing results. The module assembles definitions, counterexamples ruling out weaker combiners, and curvature-based classification into a single gate.
claimA function $H$ satisfies the d'Alembert equation when $H(xy) + H(x/y) = 2H(x)H(y) + 2H(x) + 2H(y)$ for all $x,y > 0$.
background
The module operates inside the Foundation.DAlembert hierarchy and imports Cost, FunctionalEquation (lemmas for T5 cost uniqueness), Counterexamples, and CurvatureGate. Counterexamples records that existence of some combiner $P$ with $F(xy) + F(x/y) = P(F(x),F(y))$ does not force d'Alembert structure on the log-lift. CurvatureGate requires constant nonzero curvature on the metric $ds^2 = G''(t) dt^2$ where $G(t) = F(e^t)$.
proof idea
This is a definition module collecting SatisfiesDAlembert, HasDAlembert, dAlembert_classification, dAlembert_forces_Jcost and related statements. It applies imported lemmas from CurvatureGate and Counterexamples to separate the d'Alembert case from quadratic counterexamples.
why it matters in Recognition Science
The module supplies the fourth gate to TriangulatedProof, which combines four gates into a unified inevitability theorem. It also feeds InevitabilityEquivalence by supplying concrete d'Alembert conditions that bridge abstract claims to CPM definitions. It completes the step showing d'Alembert forces Jcost inside the T5 uniqueness chain.
scope and limits
- Does not prove interaction or entanglement gates.
- Does not derive full T5 uniqueness without the other three gates.
- Does not compute numerical values of constants such as alpha.
- Does not address higher-dimensional extensions beyond D=3.
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