defect_bound_constant_value
The defect bound constant in the coercive projection for ILG gravity equals exactly 162/49. Researchers modeling information-limited gravity would cite this to fix the coercivity bound that guarantees a unique energy minimizer. The proof is a direct term-mode reduction that unfolds the definitions of the net constant K_net and projection constant C_proj then applies numerical normalization.
claimIn the coercive projection law for information-limited gravity, the defect bound constant equals $162/49$.
background
The module formalizes the Coercive Projection Law of Gravity from two source papers. It shows the ILG energy functional admits a unique minimizer whose coercivity constant is 49/162, with the net constant K_net = (9/7)^2 arising from the eight-tick epsilon = 1/8. The local setting equates the ILG modified Poisson equation nabla^2 Phi = 4 pi G a^2 (w rho_b delta_b) to the standard Poisson equation with effective pressure source p = w rho_b delta_b, so ILG modifies only the source term.
proof idea
The term proof unfolds defect_bound_constant, K_net and C_proj then applies norm_num to obtain the exact rational 162/49.
why it matters in Recognition Science
This supplies the explicit numerical value required to close the coercivity argument for the unique ILG energy minimizer and to support the pressure-equivalence section that follows. The value traces directly to the eight-tick octave (T7) in the Unified Forcing Chain and the phi-ladder discretization. No open scaffolding remains.
scope and limits
- Does not derive the constant from first principles without the eight-tick discretization.
- Does not apply outside the ILG energy functional.
- Does not supply error bounds or higher-order corrections.
- Does not address stability under cosmological evolution.
formal statement (Lean)
66theorem defect_bound_constant_value :
67 defect_bound_constant = 162 / 49 := by
proof body
Term-mode proof.
68 unfold defect_bound_constant K_net C_proj; norm_num
69
70/-! ## Pressure Equivalence -/
71
72/-- The ILG modified Poisson equation is EQUIVALENT to the standard Poisson
73 equation with an effective pressure source:
74
75 ILG: nabla^2 Phi = 4*pi*G * a^2 * (w * rho_b * delta_b)
76 Standard: nabla^2 Phi = 4*pi*G * a^2 * p
77
78 where p = w * rho_b * delta_b is the "effective pressure."
79
80 This equivalence means ILG is NOT a modification of GR's field equations
81 but rather a modification of the SOURCE SIDE only. -/