module module high

`IndisputableMonolith.Foundation.RecognitionLatticeFromRecognizer`

This module constructs the recognition lattice by inducing a setoid on configurations from a primitive interface, with equivalence holding precisely when the interface maps both to the same event. It supplies the discrete geometric substrate that follows the observer construction. Researchers tracing the Recognition Science forcing chain cite it to move from interface to lattice structure. The module assembles these objects directly from the upstream observer and arithmetic modules via definitions of the kernel equivalence and quotient.

claimLet $I$ be a primitive interface. Define the setoid equivalence $c_1 sim_I c_2$ on configurations by $I(c_1)=I(c_2)$. The recognition lattice is the quotient space of configurations under this equivalence, equipped with cell labels and neighborhood structure.

background

The module occupies the Foundation layer immediately after the observer derivation. It imports ArithmeticFromLogic for basic operations and ObserverFromRecognition, whose doc states that non-trivial recognition forces an interface that serves as the primitive observer. Key objects introduced are interfaceSetoid (the kernel equivalence), RecognitionLattice (the quotient), cellOf and cellLabel (assigning points and labels), and neighborhood (local structure). These realize the lattice as the space of distinguishable events under the interface map.

proof idea

This is a definition module, no proofs. It defines interfaceSetoid as the kernel of the interface map, constructs RecognitionLattice as the induced quotient, and introduces cellLabel, neighborhood, and SameKernel via direct definitions. Basic properties such as cell_mem_own_neighborhood and every_cell_has_label follow immediately from the setoid construction and the nontriviality assumption inherited from the observer module.

why it matters in Recognition Science

The module supplies the lattice that feeds the master forcing-chain theorem in the root IndisputableMonolith module. It realizes the geometric substrate required for T5 J-uniqueness and the subsequent steps to the eight-tick octave and D=3. By closing the transition from interface to lattice, it enables the full chain from T0 onward and the emergence of the phi-ladder structure.

scope and limits

Does not derive the full T0-T8 forcing chain.
Does not introduce physical constants or the mass formula.
Does not address continuous limits or embedding into spacetime.
Does not specify the concrete form of the interface beyond primitivity.

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