IndisputableMonolith.Foundation.DAlembert.FullUnconditional
This module establishes that reciprocal symmetry of the cost functional F forces the same symmetry on P. Researchers deriving the uniqueness of the d'Alembert equation for RCL consistency in Recognition Science cite it. The argument assembles the unconditional inevitability results from the imported modules to obtain the symmetry implication with no extra hypotheses on P.
claimIf $F(x)=F(x^{-1})$ for the cost functional $F:\mathbb{R}_+\to\mathbb{R}$, then $P(x)=P(x^{-1})$.
background
The module sits inside the Foundation.DAlembert hierarchy and imports the core cost-functional machinery together with the two d'Alembert sub-modules. IndisputableMonolith.Foundation.DAlembert.Inevitability shows that the d'Alembert equation is the unique form compatible with multiplicative consistency of any cost functional F. IndisputableMonolith.Foundation.DAlembert.Unconditional strengthens this to the claim that, once F is fixed by symmetry, normalization, calibration and smoothness, P is computed from the equation rather than assumed. The supplied doc-comment isolates the concrete symmetry transfer that follows.
proof idea
The module is an assembly of imported theorems rather than a single tactic script. It re-exports the unconditional RCL inevitability and the inevitability core, then applies the symmetry transfer lemma (P_symmetric_of_F_symmetric) that follows directly from the d'Alembert relation once F is known to be reciprocal-symmetric.
why it matters in Recognition Science
The module supplies the final symmetry step that completes the unconditional d'Alembert foundation. It feeds the T5 J-uniqueness and RCL inevitability chain by guaranteeing that the computed P inherits the symmetry already imposed on F. The doc-comment records the precise implication used downstream in the Recognition Science forcing sequence.
scope and limits
- Does not assume any a-priori properties of P.
- Does not treat cases in which F itself lacks reciprocal symmetry.
- Does not supply numerical values or physical units.
- Does not extend the result to non-multiplicative consistency conditions.
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