module module moderate

`IndisputableMonolith.Information.PhysicsComplexityStructure`

This module develops the J-cost analysis for physical complexity and information bounds in Recognition Science. It establishes non-negativity of J-cost (IC-005.1) along with symmetry, derivative signs, and unique minimum at unity. Information physicists deriving RS-native computation limits cite these results. The module consists of definitions plus short algebraic lemmas imported from Cost and Constants.

claim$J(x) = \frac{x + x^{-1}}{2} - 1 \geq 0$ with unique minimum $J(1)=0$, symmetry $J(x)=J(x^{-1})$, and derivative sign changes across $x=1$.

background

The module sits in the Information domain and imports Constants (defining the RS time quantum $\tau_0 = 1$ tick), the Cost module, and ComputationLimitsStructure. The upstream ComputationLimitsStructure states that computation limits emerge from three sources in RS and addresses Bremermann's limit and Landauer's bound. Local definitions include LedgerConfig, totalJCost, and the family of jcost lemmas that encode the J-cost function and its first derivative.

proof idea

This is a structure module with multiple short theorems rather than a single proof. jcost_nonneg follows from algebraic rearrangement of the J expression. jcost_unique_minimum, jcost_symmetric, and the derivative lemmas (jcost_deriv, jcost_deriv_zero_at_one, jcost_deriv_pos_of_gt_one, jcost_deriv_neg_of_lt_one) are one-line wrappers or direct applications of calculus rules on the imported Cost definitions.

why it matters in Recognition Science

The module supplies the IC-005.1 non-negativity result and supporting J-cost properties that underpin the Recognition Composition Law and the phi-ladder mass formula. It extends ComputationLimitsStructure and prepares material for downstream physics-complexity arguments in the forcing chain (T5 J-uniqueness through T8). No direct used_by edges are recorded, indicating it functions as an intermediate lemma collection.

scope and limits

Does not derive numerical values for alpha or G.
Does not treat multi-particle or field-theoretic extensions.
Does not prove the full set of computation limits stated in the imported module.
Does not address time-dependent or non-equilibrium cases.

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