IndisputableMonolith.Foundation.VariationalDynamics
VariationalDynamics defines LedgerState as N positive real ratios indexed by discrete ticks, plus feasibility and defect notions for J-cost minimization. Modules on topological conservation and winding charges cite it to ground charge definitions in ledger updates. It is a definitions module whose properties follow directly from imported convexity and determinism results.
claimA ledger state is a map $s : [N] → (0,∞)$ of positive real ratios indexed by tick; a configuration is feasible when its total defect is minimal under the strictly convex J-cost.
background
The module sits inside the Recognition Science foundation layer and imports convexity of Jcost (½(x + x⁻¹) − 1) and Jlog from Cost.Convexity, the unique-minimizer statement from Determinism, defect(x) = 0 from LawOfExistence, tick-count time from TimeEmergence, and the low-entropy initial condition. These supply the variational principle: ledger updates minimize J-cost subject to the existence constraint. Sibling definitions include Feasible, total_defect_nonneg', and weighted_Jlog_average, all built on the imported strict convexity.
proof idea
This is a definition module, no proofs. It records the LedgerState type, the Feasible predicate, self_feasible and feasible_nonempty lemmas, and basic identities such as constant_config_total_defect by direct appeal to the convexity and determinism imports.
why it matters in Recognition Science
Supplies the ledger-state formalism required by TopologicalConservation (F-012) and WindingCharges (F-013). Those modules use it to replace Noether symmetries with topological linking in D = 3 and to define independent charges via winding numbers on the eight-tick cycle. It thereby closes the gap between the deterministic J-minimization of Determinism and the topological conservation laws.
scope and limits
- Does not derive continuous-time equations of motion.
- Does not prove global existence or uniqueness of infinite sequences.
- Does not connect ledger states to specific mass or charge spectra.
- Does not address measurement or observer-dependent updates.
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