module module high

`IndisputableMonolith.Unification.QuantumGravityOctaveDuality`

The QuantumGravityOctaveDuality module defines J-cost as the squared deviation from balance and assembles octave duality identities linking constants such as kappa and hbar. Researchers deriving spacetime from recognition cost cite these algebraic relations. The module consists of targeted lemmas on J-cost symmetry and phi-self-duality, each obtained by direct manipulation of the cost formula.

claim$J(x) = \frac{(x-1)^2}{2x}$ is the recognition cost measuring squared deviation from unity; the module derives octave relations including $\kappa \hbar = 8$ and $\phi^5$ self-duality in RS-native units.

background

The module sits inside the Recognition Science unification layer and imports the time quantum $\tau_0 = 1$ tick together with the Cost library. J-cost is introduced exactly as the AM-GM gap $J(x) = \frac{(x-1)^2}{2x}$, which quantifies departure from the fixed point $x=1$. Sibling lemmas establish non-negativity, zero only at unity, reciprocal symmetry, and octave scaling of the constants.

proof idea

This is a definition module, no proofs. Individual lemmas apply algebraic rearrangement of the J-cost expression or the AM-GM inequality; no tactic scripts or external theorems beyond the imported Cost identities are required.

why it matters in Recognition Science

The module supplies the J-cost and octave-duality machinery required by the downstream SpacetimeEmergence module, whose central theorem forces the full 4D Lorentzian structure (metric signature, light-cone, arrow of time) from recognition cost. It therefore occupies the unification step that connects the eight-tick octave (T7) and phi-ladder constants to geometric emergence.

scope and limits

Does not derive the spacetime metric or causal structure.
Does not compute numerical values of G, alpha, or particle masses.
Does not address interactions outside the octave duality.
Does not contain experimental predictions or falsification criteria.

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