IndisputableMonolith.Foundation.DAlembert.FullUnconditional
The FullUnconditional module delivers the strongest axiom-free form of RCL inevitability for cost functionals, proving that reciprocal symmetry of F forces symmetry of P with no hypotheses on P required. Researchers tracing T5 uniqueness or the forcing chain cite it to close the derivation of the d'Alembert equation directly from symmetry and normalization. The structure combines imported results from Unconditional and Inevitability to compute P from the functional equation rather than assume it.
claimIf the cost functional $F:\mathbb{R}_+\to\mathbb{R}$ is symmetric under reciprocal, then the associated $P$ is symmetric. The d'Alembert equation holds unconditionally once $F$ is fixed by symmetry, normalization, calibration and smoothness.
background
This module belongs to the Foundation.DAlembert section and re-exports the core cost machinery. It imports Cost.FunctionalEquation (lemmas for T5 cost uniqueness), FunctionalEquationAczel (Aczél closure), Inevitability (d'Alembert as the unique form required by multiplicative consistency of any cost functional), and Unconditional (the strongest RCL inevitability statement).
The local setting is the Recognition Science cost functional whose multiplicative consistency must obey the Recognition Composition Law. The Unconditional doc states: "This module proves the strongest possible form of RCL inevitability: NO ASSUMPTION ON P IS NEEDED. The key insight is that if F is determined (by symmetry, normalization, calibration, smoothness), then P is COMPUTED from the functional equation, not assumed."
proof idea
The module aggregates the unconditional inevitability theorems and the inevitability lemmas imported from its dependencies. It derives the listed sibling results (P_symmetric_of_F_symmetric, dAlembert_forces_cosh_is_theorem, consistency_forces_RCL_form_hypothesis) by applying the core functional-equation reductions and the no-assumption closure already established in Unconditional.
why it matters in Recognition Science
The module completes the unconditional half of the RCL inevitability argument and feeds the T5 J-uniqueness step in the forcing chain (T0-T8). It supplies the axiom-free closure that lets higher modules treat the d'Alembert equation as forced rather than postulated, directly supporting the eight-tick octave and D=3 derivations.
scope and limits
- Does not derive numerical values for constants such as alpha or G.
- Does not address Berry creation threshold or phi-ladder mass formulas.
- Does not treat conditional hypotheses on P that appear in earlier modules.
- Does not cover the full forcing chain beyond RCL inevitability.
depends on (5)
declarations in this module (18)
-
theorem
P_symmetric_of_F_symmetric -
theorem
P_symmetric_on_range -
theorem
P_at_zero_left -
theorem
P_at_zero_right -
theorem
P_diagonal -
def
LogConsistency -
theorem
log_consistency_of_mult_consistency -
theorem
H_dAlembert_of_G_RCL -
theorem
dAlembert_even_solution -
def
dAlembert_forces_cosh_hypothesis -
theorem
dAlembert_forces_cosh_is_theorem -
def
consistency_forces_RCL_form_hypothesis -
theorem
consistency_forces_RCL_form_is_theorem -
structure
FullUnconditionalHypotheses -
theorem
full_unconditional_inevitability -
theorem
full_inevitability_explicit -
theorem
washburn_full_unconditional -
theorem
consistency_forces_RCL_polynomial