module module high

`IndisputableMonolith.Unification.QuantumGravityOctaveDuality`

The QuantumGravityOctaveDuality module develops the octave duality layer inside the J-cost framework. It links the recognition cost to the eight-tick period and establishes self-dual relations for phi^5 together with kappa-hbar reciprocity. Researchers deriving emergent spacetime from recognition cost cite these identities. The module proceeds by direct algebraic identities and fixed-point arguments applied to the AM-GM form of J-cost.

claimThe module centers on the J-cost $J(x) = (x-1)^2/(2x)$ together with the octave relations $J(x) = J(1/x)$, $J(x) = 0$ iff $x=1$, and the duality identities $hbar = 8/kappa$, $kappa = 8/hbar$, and the self-dual fixed point at $phi^5$.

background

The module sits inside the Unification domain and imports the base time quantum tau_0 = 1 tick from Constants together with the J-cost definition from Cost. J-cost is introduced as the squared deviation from balance, J(x) = (x-1)^2/(2x), which is the AM-GM gap measuring departure from x=1. The setting uses the Recognition Composition Law and the self-similar fixed point phi to generate the eight-tick octave period that relates the quantum constant hbar = phi^{-5} to its gravitational dual.

proof idea

This is a definition module, no proofs. The structure consists of direct algebraic lemmas: non-negativity and zero condition follow from the AM-GM inequality applied to the squared term; reciprocal symmetry is immediate from substitution x to 1/x; the octave identities kappa_hbar_octave and phi_fifth_self_dual are obtained by substituting the fixed-point equation into the J-cost formula and counting the eight-tick period.

why it matters in Recognition Science

The module supplies the octave duality required by the downstream SpacetimeEmergence module, which forces the complete 4D Lorentzian structure (signature (-,+,+,+), light cones, arrow of time) from J-cost. It fills the T7 eight-tick octave step of the UnifiedForcingChain and supplies the concrete constants hbar = phi^{-5} and the self-dual phi^5 that later fix D=3 and the alpha band.

scope and limits

Does not derive the spacetime metric tensor or causal structure.
Does not prove the emergence of three spatial dimensions.
Does not address the fine-structure constant or its numerical bounds.
Does not include numerical matching to experimental particle masses.

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