module module high

`IndisputableMonolith.Foundation.RecognitionLatticeFromRecognizer`

The module constructs the recognition lattice as the quotient setoid induced by a primitive interface, with configurations identified exactly when the interface outputs the same event. Researchers tracing the recognition-to-geometry steps in the forcing chain cite it after the observer construction. It assembles the lattice via imported results on interfaces and arithmetic, then proves basic lattice properties and cell-labeling lemmas.

claimLet $I$ be a primitive interface. Define the setoid $\sim_I$ on configurations by $c_1 \sim_I c_2$ if and only if $I(c_1)=I(c_2)$. The recognition lattice is the quotient set equipped with the induced lattice operations and neighborhood structure.

background

This module follows ObserverFromRecognition, whose doc states that non-trivial recognition forces an interface and that an interface is the primitive observer. It also imports ArithmeticFromLogic. The central object is the interfaceSetoid, whose equivalence classes are the kernels of the interface map. RecognitionLattice then turns the quotient into a lattice whose cells carry labels and neighborhoods.

proof idea

This is a definition module. It introduces interfaceSetoid, RecognitionLattice, cellOf, neighborhood, and the supporting lemmas cellLabel_cellOf, every_cell_has_label, and nontrivial_recognition_forces_lattice by direct construction from the imported interface and arithmetic primitives.

why it matters in Recognition Science

The module supplies the lattice geometry required by the root IndisputableMonolith module for the master forcing-chain theorem. It completes the transition from the primitive observer (interface) to the discrete lattice that later yields the eight-tick octave and D=3. The downstream doc describes the umbrella exposing the T-1 through T8 surfaces.

scope and limits

Does not derive the full T0-T8 forcing chain.
Does not supply physical interpretations of cells or labels.
Does not address embeddings into continuous space.
Does not prove uniqueness of the lattice up to isomorphism.

used by (1)

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

IndisputableMonolith module_import

depends on (2)

Lean names referenced from this declaration's body.

IndisputableMonolith.Foundation.ArithmeticFromLogic module_import
IndisputableMonolith.Foundation.ObserverFromRecognition module_import

declarations in this module (22)

def interfaceSetoid L48
def RecognitionLattice L54
def cellOf L58
theorem cell_eq_iff_kernel L64
def cellLabel L72
theorem cellLabel_cellOf L78
def neighborhood L82
theorem cell_mem_own_neighborhood L87
theorem every_cell_has_label L92
theorem nontrivial_recognition_forces_lattice L101
def pointLattice L109
def SameKernel L116
def latticeEquivOfSameKernel L121
theorem latticeEquivOfSameKernel_cell L143
def iteratedCarrier L150
def logicNatToLattice L155
theorem logicNatToLattice_zero L159
theorem logicNatToLattice_step L163
theorem logicNat_interprets_into_lattice L170
structure RecognitionLatticeCert L180
def recognitionLatticeCert L198
theorem recognitionLatticeCert_inhabited L205