module module high

`IndisputableMonolith.Thermodynamics.ErrorCorrection`

This module defines recognition defects as states with positive J-cost and introduces error-correction objects such as syndromes, protocols, and phi-code distance built on the eight-tick cycle. Researchers modeling fault tolerance in J-minimizing dynamics would cite it when extending the T=0 foundation. The module supplies definitions and supporting lemmas that rest directly on free-energy monotonicity.

claimA recognition defect satisfies $J(x)>0$ with energy equal to the J-cost itself. Error syndromes are detected via $jcost_syndrome$, correction proceeds by eight-tick cycles, and code distance is measured by the phi-ladder quantity $phi_code_distance$.

background

The module belongs to the Thermodynamics domain and imports FreeEnergyMonotone, whose doc-comment states that Recognition Free Energy is non-increasing under RS dynamics and constitutes the Recognition Science version of the Second Law. It therefore inherits the T=0 setting in which reality is defined by absolute minimization of the universal cost $J(x)=½(x+1/x)-1$ introduced in the parent Thermodynamics module. The supplied DOC_COMMENT identifies a recognition defect as any state with $J>0$ and equates its energy to the J-cost.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

The module feeds the parent Thermodynamics module, whose doc-comment describes the development of statistical mechanics from the T=0 J-minimization foundation. It supplies the defect and correction machinery required to discuss fault tolerance inside the eight-tick octave (T7) and the three-dimensional spatial structure (T8) of the Recognition framework.

scope and limits

Does not derive the Second Law from first principles.
Does not compute a numerical fault-tolerance threshold.
Does not prove existence or completeness of any concrete code.
Does not treat continuous-time or relativistic extensions.

used by (1)

From the project-wide theorem graph. These declarations reference this one in their body.

IndisputableMonolith.Thermodynamics module_import

depends on (1)

Lean names referenced from this declaration's body.

IndisputableMonolith.Thermodynamics.FreeEnergyMonotone module_import

declarations in this module (17)

structure RecognitionDefect L38
def defect_energy L45
theorem defect_energy_pos L49
structure ErrorSyndrome L56
def jcost_syndrome L63
structure CorrectionProtocol L70
def eight_tick_cycle L80
def is_complete_correction L84
structure FaultToleranceThreshold L91
def is_fault_tolerant L98
def code_distance L110
def phi_code_distance L115
theorem phi_code_distance_pos L119
structure LogicalOperator L129
def id_logical_op L136
def is_protected_observable L144
theorem conservation_is_protected L149