module module high

`IndisputableMonolith.Cognition.AnimalZComplexityBound`

The AnimalZComplexityBound module defines Z-complexity as Z_k = phi^k on the phi-ladder for cognitive structures. Researchers modeling animal intelligence in Recognition Science cite it to bound complexity at successive rungs. The module supplies a chain of definitions establishing positivity, strict increase, and concrete rung assignments for vertebrates through life.

claim$Z_k := phi^k$ for the Z-complexity at rung $k$, with $phi$ the self-similar fixed point.

background

Recognition Science places cognitive complexity on the same phi-ladder used for mass and length scales. The module imports the RS time quantum tau_0 = 1 tick from Constants and the cost functions from the Cost module to ground the definitions. Z-complexity at rung k is introduced directly as Z_k = phi^k, with auxiliary lemmas recording that the sequence is positive, strictly increasing, and satisfies the bond relation Z_{k+1} = phi * Z_k.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

The definitions supply the rung assignments that later cognition results invoke when bounding animal intelligence. They sit inside the phi-ladder construction that begins with the self-similar fixed point and the eight-tick octave, providing the concrete Z values needed for vertebrate, octopus, cetacean, human, and life cases.

scope and limits

Does not compute numerical Z values for any specific organism.
Does not derive complexity bounds from physical measurements.
Does not address non-cognitive systems or non-animal life.
Does not connect Z_rung to the alpha band or gravitational constants.

depends on (2)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import
IndisputableMonolith.Cost module_import

declarations in this module (27)

def z_rung L84
theorem z_rung_pos L87
theorem z_rung_zero L92
theorem z_rung_succ L96
theorem z_rung_strictly_increasing L102
def z_rung_bond L114
def z_rung_cf L119
def z_rung_vertebrate L122
def z_rung_octopus L125
def z_rung_cetacean L128
def z_rung_human L131
def z_rung_life L134
theorem cf_below_bond L139
theorem bond_below_vertebrate L143
theorem vertebrate_below_octopus L147
theorem octopus_below_cetacean L151
theorem cetacean_below_human L155
theorem human_below_life L159
theorem rung_ordering L163
theorem z_rung_monotone L177
def z_cognition_floor L184
theorem z_cognition_floor_eq L187
theorem cognition_floor_below_bond_z L190
theorem cognition_floor_below_human_z L195
structure AnimalZComplexityBoundCert L217
def animalZComplexityBoundCert L231
theorem animal_z_complexity_one_statement L246