module module moderate

`IndisputableMonolith.CrossDomain.RecognitionGenerators`

The RecognitionGenerators module establishes that the set {2, 3, 5} is the smallest generating set for the spectrum. Researchers constructing the cross-domain spectrum in Recognition Science would cite it when identifying minimal prime generators. The module organizes definitions of the generators G2, G3, G5 together with decomposition functions for selected composite integers.

claimThe smallest generating set for the spectrum is $S = {2, 3, 5}$.

background

Recognition Science derives all physics from one functional equation, with the forcing chain T5-T8 yielding J-uniqueness, the self-similar fixed point phi, the eight-tick octave, and D = 3. The CrossDomain.RecognitionGenerators module sits inside this setting and introduces generators for the spectrum. Its module-level doc comment states that {2, 3, 5} is the smallest such set; the listed siblings supply the concrete generators G2, G3, G5 and the decomposition maps for 4, 6, 7, 8, 10, 12, 15, 16, 25.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

The module supplies the minimal generators required to build the spectrum that enters the mass formula yardstick * phi^(rung - 8 + gap(Z)). It therefore supports every downstream construction that relies on the phi-ladder spectrum inside the Recognition framework.

scope and limits

Does not define the algebraic structure of the spectrum itself.
Does not prove that no proper subset of {2, 3, 5} generates the spectrum.
Does not connect the generators to explicit physical constants or observables.

declarations in this module (25)

def G2 L41
def G3 L42
def G5 L43
theorem four_decomp L47
theorem six_decomp L48
theorem seven_decomp L49
theorem eight_decomp L50
theorem ten_decomp L51
theorem twelve_decomp L52
theorem fifteen_decomp L53
theorem sixteen_decomp L54
theorem twentyfive_decomp L55
theorem fortyfive_decomp L56
theorem seventy_decomp L57
theorem oneTwentyFive_decomp L58
theorem twoSixteen_decomp L59
theorem twoFiftySix_decomp L60
theorem threeSixty_decomp L61
theorem threeOneTwentyFive_decomp L62
theorem generators_minimal L67
theorem primorial_product L72
theorem second_primorial L76
theorem third_primorial L77
structure RecognitionGeneratorsCert L79
def recognitionGeneratorsCert L92