Cpm Law Of Existence Knet From Cone Projection

The cone projection constant

The Coercive Projection Method (CPM) is a general mathematical framework for controlling how much of a vector leaks into a subspace it is not in. The framework models the situation with three quantities: the defect mass, the orthogonal mass, and the energy gap. A projection inequality bounds the defect by a product of constants times the orthogonal mass. The declaration knet_from_cone_projection is a result in the machine-checked library of formal theorems that states, for the cone projection case, the constant Knet equals 1.

The proof is a definitional identity: the cone constants record sets Knet to 1 by definition, and the result simply unfolds that definition. This is not a derived physical law. It is a normalization choice, a bookkeeping convention that says the cone projection case has no covering loss. The same library contains a different constant for the eight-tick covering case, where Knet is (9/7)^2, reflecting a covering loss in three dimensions. The cone case is the clean case where the constant is exactly 1.

What the result does establish is consistency: the cone constants record, with Knet = 1, Cproj = 2, Ceng = 1, and Cdisp = 1, satisfies the projection-defect inequality with no extra slack from the Knet factor. The companion lemma cproj_eq_two_from_J_normalization shows the Cproj = 2 value is tied to the second derivative of the cost function J at 1 being normalized to 1. The cone record's cmin value, the reciprocal of the product Knet * Cproj * Ceng, evaluates to 1/2, and this matches the energy gap bound stated in the model.

The declaration does not claim that the cone projection is the physically realized case, nor that Knet = 1 is forced by the recognition axioms. It is a definitional result about a specific record in the library. The eight-tick covering case, with Knet = (9/7)^2, is a separate record with its own provenance. The cone case is presented as a clean reference case, not as the unique outcome of the framework.

/-- K_net = 1 for intrinsic cone projection (no covering loss).

When projecting onto a cone rather than a general structured set,
the covering number argument is trivial and K_net = 1. For ε-net
covering in dimension d = 3, we have K_net = (1/(1-2ε))^d. With
ε = 1/8 (eight-tick), K_net = (8/6)^3 = (4/3)^3 ≈ 2.37, but for
the intrinsic cone route, K_net = 1. -/
theorem knet_from_cone_projection :
    RS.coneConstants.Knet = 1 := rfl

/-- RS-native CPM constants for cone projection. Placeholders are kept
symbolic by default for `C_eng` and `C_disp`; domain instantiations can
refine them. -/
def coneConstants : Constants := {
  Knet  := 1,
  Cproj := 2,
  Ceng  := 1,
  Cdisp := 1,
  Knet_nonneg := by norm_num,
  Cproj_nonneg := by norm_num,
  Ceng_nonneg := by norm_num,
  Cdisp_nonneg := by norm_num
}

/-- RS cone coercivity constant is 1/2. -/
theorem c_value_cone : cmin RS.coneConstants = 1/2 := by
  simp only [cmin, RS.cone_Knet_eq_one, RS.cone_Cproj_eq_two, RS.cone_Ceng_eq_one]
  norm_num

/-- Minimal justification: under the RS J-normalization, the Hermitian
rank-one projection constant exported by RS equals 2. (The detailed
Hermitian bound is proved in domain files; here we record the value and
the normalization that fixes it.) -/
theorem cproj_eq_two_from_J_normalization
  (_hJ : deriv (deriv (fun t : ℝ => IndisputableMonolith.Cost.Jcost (Real.exp t))) 0 = 1) :
  coneConstants.Cproj = 2 := by
  simp [cone_Cproj_eq_two]

What this page does not claim

Knet = 1 is a forced consequence of the recognition axioms rather than a definitional choice. The cone projection case is the physically realized case in the framework. The result establishes any numerical value for a physical constant beyond the definitional identity.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/CPM/LawOfExistence.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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