module module high

`IndisputableMonolith.Foundation.StillnessGenerative`

StillnessGenerative module defines non-trivial configurations as those with at least one entry not equal to 1, distinct from the uniform ground state. It assembles auxiliary definitions including the phi-ladder and T4_Recognition. Researchers examining the emergence of structure in discrete recognition ledgers would reference these notions. The module contains no proofs and serves as a container for these foundational definitions.

claimA configuration $c$ is non-trivial if there exists an index $i$ such that $c_i ≠ 1$. Equivalently, $c$ is not the uniform ground state in which every entry equals 1.

background

The module operates in the Foundation layer of Recognition Science and imports five modules. LawOfExistence states that an object $x$ exists precisely when its defect vanishes. Cost supplies the J-cost functional on configurations. PhiForcing shows that the golden ratio $φ$ is forced by self-similarity in a discrete ledger. PhiForcingDerived derives the quadratic $r^2 = r + 1$ from discrete scale and additive ledger axioms. InitialCondition formalizes the low-entropy starting state.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

The definitions supply the is_nontrivial predicate that supports T4_Recognition and the phi_ladder family. They distinguish the ground state from states that participate in the forcing chain T0-T8 and the Recognition Composition Law, enabling later derivation of $D=3$ and the alpha band.

scope and limits

Does not prove existence of non-trivial configurations.
Does not quantify J-cost of non-trivial states.
Does not link non-triviality to specific Z or phi-rung values.
Does not treat generative dynamics from the ground state.

depends on (5)

Lean names referenced from this declaration's body.

IndisputableMonolith.Cost module_import
IndisputableMonolith.Foundation.InitialCondition module_import
IndisputableMonolith.Foundation.LawOfExistence module_import
IndisputableMonolith.Foundation.PhiForcing module_import
IndisputableMonolith.Foundation.PhiForcingDerived module_import

declarations in this module (39)

def phi_ladder L44
theorem phi_ladder_pos L46
theorem phi_zpow_ne_one L49
theorem phi_ladder_ne_one L57
theorem phi_ladder_positive_cost L62
theorem phi_cost_eq L66
theorem phi_cost_pos L69
theorem phi_perturbation_bounded L72
def has_phi_structure L77
theorem unity_has_no_phi_structure L80
def is_nontrivial L105
structure T4_Recognition L120
theorem nontrivial_closed_has_phi_structure L136
theorem t6_derived L161
theorem ground_state_recognition_impossible L177
theorem static_ground_state_impossible L186
def eight_tick_period L212
def cycle_nondegenerate L217
theorem uniform_cycle_degenerate L221
theorem eight_tick_forces_nontrivial L235
theorem eight_tick_breaks_uniformity L244
theorem perturbation_cost_quadratic L250
theorem perturbation_cost_positive L254
theorem perturbation_cost_small_bound L258
theorem dalembert_cascade L264
theorem phi_power_compose L271
theorem phi_power_ratio L275
theorem ladder_cascade_bound L279
theorem doubling_cascade L287
theorem doubling_cascade_positive L300
theorem fibonacci_cascade L320
theorem one_plus_phi_eq_phi_sq L332
theorem closure_populates_next L341
theorem every_rung_from_fibonacci L353
theorem ledger_symmetry_negative_rungs L363
theorem stillness_is_creative L397
theorem ground_state_paradox L431
theorem origin_question_resolved L443
theorem symmetry_breaking_mechanism L463