module module high

`IndisputableMonolith.Foundation.PhiEmergence`

The PhiEmergence module defines self-similarity for ratios and establishes basic properties of the golden ratio phi within Recognition Science. Model builders working on the forcing chain cite it to anchor phi as the unique positive fixed point. The module supplies definitions and elementary lemmas that rest on the imported Inequalities and Constants modules.

claimA ratio $r$ is self-similar when $r^2 = r + 1$. The unique positive solution is the golden ratio $phi = (1 + sqrt(5))/2$, with its conjugate, positivity, and closure properties under multiplication and division on the associated ladder.

background

The module occupies the Foundation layer and imports the RS time quantum tau_0 = 1 tick from Constants together with fundamental inequalities. Its central definition states that a ratio r is self-similar precisely when r squared equals r plus one. Sibling declarations then prove phi is positive and greater than one, confirm it satisfies the self-similarity equation, establish the conjugate is negative, and construct the phi-ladder with ratio and closure lemmas.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

The module supplies the self-similarity step that feeds Experiments.Protocols, where it enables the EEG phi-frequency predictions. It directly implements the T6 forcing step that selects phi as the self-similar fixed point of the J-cost function. The phi-ladder results are used downstream to organize mass rungs and the eight-tick octave.

scope and limits

Does not derive the full forcing chain T0-T8.
Does not address spatial dimensions or coupling constants.
Does not contain numerical evaluations or experimental protocols.
Does not treat the J-cost functional equation itself.

used by (1)

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

IndisputableMonolith.Experiments.Protocols module_import

depends on (2)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import
IndisputableMonolith.Foundation.Inequalities module_import

declarations in this module (23)

def IsSelfSimilar L23
theorem phi_pos L26
theorem phi_gt_one L29
theorem phi_is_self_similar L34
theorem phi_conjugate_self_similar L39
theorem phi_conjugate_neg L48
theorem phi_unique_positive L58
def PhiLadder L83
theorem phi_pow_pos L86
theorem phi_ladder_ratio L90
theorem phi_ladder_mul_closed L97
theorem phi_ladder_div_closed L104
theorem J_at_phi L114
def J_at_phi_pow L118
theorem J_at_phi_approx L122
def H_ThresholdFromPhi L141
theorem phi_inv_eq L146
theorem phi_inv_pos L149
theorem phi_inv_lt_one L154
theorem phi_series_sum L161
def IsStablePosition L198
theorem phi_ladder_stable L202
def H_StableIffPhiLadder L211