IndisputableMonolith.Foundation.DAlembert.AnalyticBridge
This module defines the log-lift of a cost function to bridge discrete cost structures with continuous d'Alembert equations in log-coordinates. Researchers deriving physics from the Recognition Composition Law would cite it when translating multiplicative costs into hyperbolic metrics. The module consists of definitions for the lift and supporting consistency results that prepare the curvature and entanglement gates.
claimThe log-lift $G(t) := F(e^t)$ of a cost function $F$, with associated $H$ and consistency properties under the curvature condition $ds^2 = G''(t) dt^2$.
background
The module sits in the Foundation.DAlembert namespace and imports Cost for the base cost function together with the three gate modules. CurvatureGate requires that the log-coordinate representation $G(t) = F(e^t)$ yields a 1D metric via $ds^2 = G''(t) dt^2$. EntanglementGate enforces nonzero cross-derivatives in the combiner $P$. NecessityGates shows that symmetry plus normalization alone fail to force the full d'Alembert/RCL structure.
proof idea
This is a definition module, no proofs. It introduces the log-lift via G_of and H_of, then records consistency lemmas such as log_consistency and the boundary and ODE results that follow from the lift.
why it matters in Recognition Science
The module supplies the analytic translation layer that the DAlembert gates rely on to connect cost functions to the Recognition Composition Law. It prepares the geometric setting in which the forcing chain (T5 J-uniqueness through T8 D=3) can be recovered from the curvature and entanglement conditions.
scope and limits
- Does not derive the full RCL from the gates.
- Does not extend the lift to higher-dimensional costs.
- Does not include numerical checks or counterexamples.
- Does not link to the phi-ladder mass formulas.
depends on (4)
declarations in this module (24)
-
def
G_of -
def
H_of -
theorem
log_consistency -
theorem
G_zero -
theorem
Q_boundary_v -
theorem
Q_boundary_u -
lemma
separable_forces_additive -
theorem
separable_forces_flat_ode -
theorem
entangling_forces_hyperbolic_ode -
theorem
flat_ode_unique -
theorem
hyperbolic_ode_unique -
theorem
even_function_deriv_zero -
theorem
separable_combiner_forces_flat -
theorem
entangling_combiner_forces_hyperbolic -
theorem
differentiation_key_lemma -
theorem
entangling_with_boundary_is_RCL -
theorem
analytic_bridge -
theorem
analytic_bridge_full -
theorem
Jcost_satisfies_dAlembert -
theorem
F_forced_to_Jcost -
theorem
P_forced_to_RCL -
theorem
full_inevitability -
theorem
gates_connected -
theorem
Jcost_has_RCL_combiner