module module high

`IndisputableMonolith.Information.PhysicsComplexityStructure`

The module PhysicsComplexityStructure defines the J-cost function and proves its non-negativity as theorem IC-005.1 within the Recognition Science information domain. Researchers deriving complexity bounds from RS would cite it for foundational cost properties. The lemmas follow from algebraic identities imported from the Cost module and the computation-limits structure.

claim$J(x) = (x + x^{-1})/2 - 1$ satisfies $J(x) > 0$ for $x > 0$, $x ≠ 1$, with equality only at the fixed point $x = 1$.

background

The module sits in the Information domain and imports Constants (τ₀ = 1 tick), Cost (source of the J-cost definition), and ComputationLimitsStructure. The latter states that computation limits in RS emerge from three sources: Bremermann's limit, Landauer's bound, and quantum computation limits. Sibling declarations establish J-cost non-negativity, uniqueness of minimum, squared form, symmetry, and derivative signs.

proof idea

This is a definition and lemma module. Non-negativity (jcost_nonneg) and related properties are obtained by direct algebraic reduction using the imported Cost definition of J together with the quadratic representation.

why it matters in Recognition Science

The module supplies IC-005.1 (J-cost non-negativity) as a building block for physics-complexity claims. It connects to the forcing chain at T5 (J-uniqueness) and supports downstream information-theoretic results, though the current graph shows no direct used_by edges.

scope and limits

Does not derive numerical values of constants such as α or G.
Does not prove the full set of computation limits; only imports the structure.
Does not address spatial dimension forcing (T8) or the eight-tick octave.
Does not treat mass formulas or the phi-ladder.

depends on (3)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import
IndisputableMonolith.Cost module_import
IndisputableMonolith.Information.ComputationLimitsStructure module_import

declarations in this module (25)

structure of L15
theorem jcost_nonneg L46
theorem jcost_unique_minimum L52
theorem jcost_squared_form L59
theorem jcost_pos_away_from_one L65
theorem jcost_symmetric L76
def jcost_deriv L83
theorem jcost_deriv_zero_at_one L87
theorem jcost_deriv_pos_of_gt_one L92
theorem jcost_deriv_neg_of_lt_one L108
structure LedgerConfig L120
def totalJCost L125
theorem total_jcost_nonneg L129
theorem balanced_config_zero_cost L138
lemma sum_nonneg_zero_iff L144
theorem verification_equivalence L153
def physics_complexity_from_ledger L171
theorem physics_complexity_structure L174
theorem physics_complexity_implies_limits L178
theorem phi_hierarchy_exponential L184
theorem phi_rung_complexity_unbounded L189
theorem jcost_gradient_descent_converges L195
theorem jcost_complexity_gap L211
def rs_complexity_classes L219
def ic005_certificate L230