module module high

`IndisputableMonolith.Information.PhysicsComplexityStructure`

Module PhysicsComplexityStructure proves J-cost non-negativity as IC-005.1 together with symmetry, derivative signs, and positivity away from unity. Information theorists working on RS-derived computation bounds cite these results. Proofs reduce each property to the closed form of J and elementary inequalities or calculus.

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

background

The module sits in the Information domain and imports Constants (where the RS time quantum satisfies τ₀ = 1 tick), Cost, and ComputationLimitsStructure. The latter frames fundamental limits of computation as emerging from three RS sources that include Bremermann's limit and Landauer's bound. J-cost is the cost measure built from the J-uniqueness relation in the forcing chain.

proof idea

The module collects separate lemmas on J-cost. Non-negativity is obtained by algebraic reduction to the AM-GM inequality on x and x^{-1}. Derivative lemmas follow by direct differentiation of the closed form. Symmetry and sign lemmas use substitution and case analysis on x > 1 versus x < 1.

why it matters in Recognition Science

Delivers the non-negativity result IC-005.1 that anchors J-cost usage in information structures. It supplies the base case for totalJCost and LedgerConfig definitions inside the same module and supports downstream work on physics complexity bounds.

scope and limits

Does not derive numerical values inside the alpha band.
Does not extend J-cost to multi-particle or field configurations.
Does not address the phi-ladder mass formula or gap(Z) corrections.
Does not close any open questions on the RCL identity.

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