module module high

`IndisputableMonolith.Information.PhysicsComplexityStructure`

This module introduces the J-cost function and establishes its core algebraic properties, starting with non-negativity per IC-005.1, to structure complexity measures in Recognition Science information physics. Researchers deriving computational bounds from RS primitives would cite it when linking cost to Landauer's and Bremermann limits. The module is organized as a sequence of definitions followed by short algebraic lemmas that reduce directly to the J functional equation.

claimThe J-cost satisfies $J(x) = (x + x^{-1})/2 - 1$ with $J(x) = J(1/x)$ and $J(x) > 0$ for all $x > 0$, $x ≠ 1$.

background

Recognition Science derives computational limits from the single functional equation whose solution yields the J-cost. The upstream Constants module fixes the RS time quantum as τ₀ = 1 tick. The ComputationLimitsStructure module states that RS imposes three sources of limits on computation: Bremermann's limit, Landauer's bound, and quantum constraints, all expressed through cost quantities. This module supplies the J-cost as the explicit non-negative measure that quantifies deviation from the fixed point x = 1.

proof idea

This is a definition module whose lemmas are one-line wrappers or direct algebraic reductions. Non-negativity follows from rewriting J(x) as cosh(log x) - 1 and invoking the known minimum of cosh at zero. Symmetry and derivative signs are obtained by elementary differentiation and substitution x → 1/x.

why it matters in Recognition Science

The module supplies the non-negative cost measure required by the IC-002 computation-limits framework and the T5 J-uniqueness step of the forcing chain. It feeds downstream results that convert J-cost into explicit bounds on information processing rates. The IC-005.1 non-negativity theorem closes the gap between the abstract functional equation and concrete physical complexity estimates.

scope and limits

Does not compute numerical values for physical constants.
Does not treat specific quantum algorithms or hardware.
Does not include empirical data or experimental comparisons.
Does not extend beyond the algebraic properties of J.

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