NoetherTheoremDeepCert
NoetherTheoremDeepCert bundles the count of four canonical Noether charges with the non-negativity and ground-state vanishing of the recognition energy function. Researchers deriving conservation laws from J-cost symmetries in the Recognition Science framework cite this structure as the master certificate. The definition is assembled directly from three upstream results on charge enumeration, energy positivity, and equilibrium value.
claimThe structure certifies that the number of canonical Noether charges equals four, that the recognition energy function $E(r)$ satisfies $E(r) ≥ 0$ for all $r > 0$, and that $E(1) = 0$.
background
In the NoetherTheoremDeep module the recognition action admits four continuous symmetries (J-translation, σ-translation, Z-translation, Θ-rotation) that produce conserved charges. The count of these charges is defined as noether_charge_count := D + 1; with spatial dimension D = 3 this equals four. The recognition energy is defined by energy_RS(r) := Jcost(r), where Jcost is the recognition cost function whose non-negativity for positive ratios is already established in the JCostThermoBridge import.
proof idea
The structure is a direct bundling of three upstream results: the theorem four_charges that unfolds the definition and normalizes to obtain the integer 4, the theorem energy_nonneg that invokes Jcost_nonneg, and the theorem energy_zero_at_eq that invokes Jcost_unit0. No additional tactics are required beyond the field assignments.
why it matters in Recognition Science
This definition supplies the master certificate referenced in the module documentation for the structural Noether theorem. It is consumed by the downstream construction noetherTheoremDeepCert and by the inhabitedness theorem that follows. The certificate closes the chain from J-cost invariance to the four conserved quantities (recognition budget, baryon number, complexity, phase) that the framework treats as the complete set of Noether charges for the recognition action.
scope and limits
- Does not derive explicit integral expressions for the four charges.
- Does not treat time-dependent or discrete symmetries.
- Does not connect the canonical charges to specific Standard-Model quantum numbers.
- Does not prove that the charge count remains four under deformations of the action.
formal statement (Lean)
54structure NoetherTheoremDeepCert where
55 four_charges : noether_charge_count = 4
56 energy_nonneg : ∀ r : ℝ, 0 < r → 0 ≤ energy_RS r
57 energy_zero_at_eq : energy_RS 1 = 0
58