module module high

`IndisputableMonolith.Unification.QuantumGravityOctaveDuality`

The QuantumGravityOctaveDuality module defines the J-cost as the squared deviation from unity and derives its octave duality relations linking kappa and hbar. Researchers deriving emergent spacetime cite it to bridge quantum and gravitational scales via the eight-tick period. The module proceeds through a collection of algebraic lemmas on AM-GM gaps, non-negativity, and reciprocal symmetries.

claim$J(x) = \frac{(x-1)^2}{2x}$ is the recognition cost measuring squared deviation from balance at $x=1$, satisfying octave duality identities such as $\kappa \hbar = 8$ and related phi-ladder relations.

background

Recognition Science derives all physics from the J-cost functional equation. This module sits in the unification domain and introduces the J-cost via its AM-GM characterization as the squared deviation from unity. It imports the fundamental time quantum $\tau_0 = 1$ tick from Constants and the general Cost framework.

The setting prepares duality between gravitational coupling and Planck's constant under the eight-tick octave, directly supporting the forcing of Lorentzian geometry.

proof idea

This is a definition module collecting lemmas on J-cost properties. The argument relies on algebraic manipulation of the cost expression, verification of non-negativity via AM-GM, and direct checks of symmetry under inversion and octave scaling.

why it matters in Recognition Science

The module supplies the octave duality between kappa and hbar required by the SpacetimeEmergence module to force the full 4D Lorentzian structure, metric signature, and causal cones from J-cost. It fills the quantum-gravity bridge step in the unification chain.

scope and limits

Does not derive the value of the fine-structure constant.
Does not address higher-dimensional extensions beyond D=3.
Does not compute explicit particle masses on the phi-ladder.
Does not include numerical simulations or observational tests.

used by (1)

From the project-wide theorem graph. These declarations reference this one in their body.

IndisputableMonolith.Unification.SpacetimeEmergence module_import

depends on (2)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import
IndisputableMonolith.Cost module_import

declarations in this module (41)

theorem jcost_eq_sq_div L77
theorem jcost_nonneg_amgm L84
theorem jcost_zero_iff_one L89
theorem gm_pair_unity L100
theorem jcost_is_amgm_gap L105
theorem jcost_reciprocal_symmetry L113
theorem kappa_hbar_octave L129
theorem hbar_kappa_octave L140
theorem kappa_per_octave_eq_inv_hbar L146
theorem hbar_eq_eight_div_kappa L151
theorem kappa_eq_eight_div_hbar L156
theorem phi_fifth_self_dual L161
lemma phi5_mul_phi5 L165
theorem kappa_fibonacci_form L172
theorem hbar_fibonacci_form L181
theorem kappa_hbar_fibonacci_consistency L190
lemma G_eq_inv_pi_hbar L202
theorem G_eq_phi_fifth_over_pi L208
theorem G_hbar_gauss_bonnet L217
theorem G_hbar_pos L222
theorem G_fibonacci_form L226
theorem kappa_per_octave_eq_G_pi L237
theorem G_pi_eq_phi5 L242
theorem planck_area_eq_inv_pi L256
theorem planck_area_pos L261
theorem G_over_hbar_phi_tenth L267
theorem hbar_over_G L277
theorem kappa_G_product L289
theorem phi_fibonacci_recursion L307
theorem fibonacci_mass_recursion L320
theorem mass_ratio_is_phi L325
theorem fibonacci_triple_sum L331
theorem mass_ladder_strictly_increasing L336
theorem phi_pow_fibonacci_sum_le L346
structure QGOctaveCert L359
def qg_octave_cert L389
theorem qg_octave_cert_inhabited L405
theorem three_products L412
theorem G_pi_eq_inv_hbar L422
theorem octave_duality_witness L428
theorem phi5_is_both_quantum_and_gravitational L443