IndisputableMonolith.Foundation.ThreeSubstrateValidationCert
The module certifies that three substrates share the J-cost fixed point at x=1. Researchers validating multi-substrate consistency in Recognition models would cite it for alignment checks. The argument applies the single-channel fixed-point property uniformly through the multi-channel additive extension.
claimFor the three substrates, each satisfies $J(1)=0$ under the multi-channel extension $J_n(x)=\sum_i J(x_i)$.
background
The J-cost is defined in the Cost module as $J(x)=(x+x^{-1})/2-1$, with fixed point $J(1)=0$. The MultiChannelJCost module extends this additively to $n$ channels as $J_n(x)=\sum_i J(x_i)$ for $x_i>0$, providing the formalisation used in ALEXIS Exp B5.
This module applies that extension to certify identical fixed-point behaviour across three substrates, using the shared fixed point as the common reference.
proof idea
This is a certification module whose structure applies the fixed-point property from the Cost module to each substrate via the multi-channel sum definition in MultiChannelJCost.
why it matters in Recognition Science
The module supports downstream certification objects such as ThreeSubstrateCert by confirming uniform alignment at the J-fixed point. It fills a foundation step for multi-substrate models, consistent with T5 J-uniqueness in the forcing chain.
scope and limits
- Does not enumerate the explicit substrate definitions.
- Does not claim the fixed point is exclusive to these three substrates.
- Does not address non-additive channel interactions.
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