IndisputableMonolith.Foundation.VariationalDynamics
VariationalDynamics introduces ledger states as finite sequences of positive real ratios indexed by discrete ticks and defines variational successors via J-cost minimization. Researchers deriving time emergence, determinism, and topological conservation from the Recognition framework would cite it. The module's structure rests on convexity of J from Cost.Convexity to guarantee unique minimizers for feasible updates.
claimA ledger state is a map $s: Ticks → (0,∞)^N$ where each entry is a positive real ratio. A configuration is feasible when total defect vanishes. The variational successor is the unique minimizer of the summed J-cost subject to the feasibility constraint.
background
This module operates in the Foundation layer of Recognition Science. It imports the strict convexity of Jlog(t) = cosh(t) - 1 and Jcost(x) = ½(x + x⁻¹) - 1 from Cost.Convexity, which underpins uniqueness of minimizers. LawOfExistence supplies the equivalence x exists ⇔ defect(x) = 0, while TimeEmergence identifies time with the tick counter and fixes the minimal 8-tick period for D = 3. Determinism and InitialCondition supply the low-entropy starting point and the guarantee that constrained J-minimization yields a unique successor.
proof idea
This is a definition module. LedgerState, Feasible, and IsVariationalSuccessor are introduced by direct construction. Theorems such as feasible_nonempty, self_feasible, and constant_config_total_defect are obtained by exhibiting explicit constant-ratio configurations and invoking non-negativity of total defect.
why it matters in Recognition Science
VariationalDynamics supplies the update rule that TopologicalConservation and WindingCharges rely on to obtain charge conservation from linking numbers rather than Noether symmetries. It fills the dynamical step between the initial low-entropy condition and the emergence of independent winding charges in three dimensions.
scope and limits
- Does not derive the explicit form of the J-cost function.
- Does not compute explicit non-constant successor states.
- Does not address continuous-time limits or quantum corrections.
- Does not fix the numerical value of spatial dimension D.
used by (2)
depends on (6)
declarations in this module (34)
-
structure
LedgerState -
def
log_charge -
def
Feasible -
theorem
self_feasible -
theorem
feasible_nonempty -
def
IsVariationalSuccessor -
theorem
total_defect_nonneg' -
def
constant_config -
theorem
constant_config_log_charge -
theorem
constant_config_total_defect -
theorem
weighted_log_average -
theorem
weighted_Jlog_average -
theorem
total_defect_lower_bound -
theorem
eq_constant_config_of_defect_eq -
theorem
unity_log_charge_zero -
theorem
unity_is_optimal -
theorem
variational_step_exists -
theorem
variational_step_unique -
theorem
variational_step_reduces_defect -
def
Trajectory -
def
IsVariationalTrajectory -
theorem
variational_dynamics_deterministic -
theorem
trajectory_defect_monotone -
structure
LocalUpdate -
theorem
update_is_global -
theorem
variational_implies_recognition_step -
def
IsEquilibrium -
theorem
equilibrium_iff_minimizer -
theorem
unity_is_equilibrium -
theorem
equilibrium_attractive -
def
uniform_config -
theorem
uniform_config_charge -
theorem
uniform_is_variational_successor -
theorem
variational_dynamics_certificate