IndisputableMonolith.Foundation.ThreeSubstrateValidationCert
The ThreeSubstrateValidationCert module certifies that three substrates share the identical J-cost fixed point at x=1. Researchers modeling multi-channel systems in Recognition Science would cite it to establish uniformity across independent channels. The module achieves this by importing the base J-cost and its additive multi-channel extension, then packaging the shared fixed point via sibling definitions including ValidationSubstrate and ThreeSubstrateCert.
claimFor the three substrates, the J-cost satisfies $J(1)=0$ as the common fixed point, where the multi-channel extension is $J_n(x_1,x_2,x_3)=J(x_1)+J(x_2)+J(x_3)$ and $J(x)=(x+x^{-1})/2-1$.
background
The module sits in the Foundation domain and imports the J-cost definition from IndisputableMonolith.Cost together with the multi-channel generalisation from MultiChannelJCost. The latter states that $J_n(x)=∑_i J(x_i)$ for $x∈ℝ^n$ with all $x_i>0$, providing the additive extension of the single-channel J-cost $J(x)=(x+x^{-1})/2-1$. The local setting is the validation of substrate independence at the self-similar fixed point, with sibling definitions ValidationSubstrate, shared_fixed_point, shared_descent, shared_symmetry and ThreeSubstrateCert introduced to formalise the common property.
proof idea
This is a definition module, no proofs. It organises the argument by declaring ValidationSubstrate, validationSubstrateCount, shared_fixed_point, shared_descent, shared_symmetry, languageModelAlignmentFraction, photonicCodeRate and ThreeSubstrateCert to encapsulate the shared fixed point at x=1.
why it matters in Recognition Science
The module supplies the uniformity certificate required by the Recognition Science forcing chain at T5 (J-uniqueness) and supports the multi-channel consistency used in the ALEXIS B5 formalisation. It feeds the parent construction of the eight-tick octave (T7) and D=3 spatial dimensions (T8) by confirming that distinct substrates remain aligned at the J-cost fixed point.
scope and limits
- Does not derive the functional form of J itself.
- Does not compute numerical values or rung placements on the phi-ladder.
- Does not address Berry creation threshold or dream fraction.
- Does not treat interactions or descent between the three substrates.
depends on (2)
declarations in this module (13)
-
inductive
ValidationSubstrate -
theorem
validationSubstrateCount -
theorem
shared_fixed_point -
theorem
shared_descent -
theorem
shared_symmetry -
def
languageModelAlignmentFraction -
theorem
lm_fraction_eq -
theorem
lm_above_threshold -
def
photonicCodeRate -
def
photonic_code_rate_rfl -
theorem
seven_eighths_from_F2_cube -
structure
ThreeSubstrateCert -
def
threeSubstrateCert