module module high

`IndisputableMonolith.Papers.GCIC.Thermodynamics`

The module establishes thermodynamic bounds in the GCIC framework of Recognition Science. It proves log(phi) > 0.48 along with positivity and interval estimates for stiffness, phase barriers, and mean-field critical temperatures. Researchers deriving discrete thermal models from the J-cost function cite these results. The arguments chain interval arithmetic on logarithms with the convexity properties from DiscretenessForcing.

claimThe module asserts the inequality $log phi > 0.48$ together with positivity and interval bounds on stiffness, phase barrier, and mean-field critical temperature in the GCIC thermodynamic model.

background

This module sits in the Papers.GCIC section. It imports Constants, where the fundamental time quantum is tau_0 = 1 tick, and Cost, which defines the J-cost function. DiscretenessForcing shows that J(x) = 1/2 (x + x^{-1}) - 1 has a unique minimum at x = 1; in log coordinates this becomes J(e^t) = cosh(t) - 1, a convex bowl centered at t = 0. Numerics.Interval.Log supplies rigorous interval bounds for the natural logarithm via Taylor series error estimates, which are applied directly to the golden ratio phi.

proof idea

The module first secures log(phi) > 0.48 from the interval logarithm bounds. It then uses this inequality to prove positivity of stiffness and phase barrier. Interval bounds on the phase barrier and mean-field critical temperature follow by the same numerical chaining. The structure is a sequence of direct applications of upstream convexity and interval results.

why it matters in Recognition Science

The module supplies the thermodynamic layer that closes the GCIC analysis, linking J-uniqueness (T5) and the phi fixed point (T6) from the forcing chain to concrete thermal observables. It constrains the phi-ladder in thermal settings and supports the eight-tick octave (T7) and D = 3. No downstream theorems appear, so it functions as a terminal numerical block for the paper.

scope and limits

Does not derive exact analytic expressions for critical temperatures.
Does not incorporate quantum corrections beyond mean-field.
Does not compute response functions such as specific heat.
Limits all results to RS-native units with c = 1.

depends on (4)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import
IndisputableMonolith.Cost module_import
IndisputableMonolith.Foundation.DiscretenessForcing module_import
IndisputableMonolith.Numerics.Interval.Log module_import

declarations in this module (16)

lemma log_phi_gt_048 L21
lemma log_phi_lt_0483 L26
def gcic_stiffness L33
theorem gcic_stiffness_pos L35
theorem gcic_stiffness_bounds L41
def phase_barrier L51
theorem phase_barrier_pos L54
theorem phase_barrier_lower L64
theorem phase_barrier_upper L78
def mf_critical_temperature L109
theorem mf_critical_temperature_pos L111
theorem mf_critical_temperature_bounds L117
theorem mf_temp_eq_six_kappa L126
theorem noncompact_uniform_convexity L132
theorem jlog_above_half_square L136
theorem gcic_thermodynamics_cert L141