module module high

`IndisputableMonolith.Gravity.NoGraviton`

In Recognition Science, gravity is large-scale ledger curvature, not a force needing a gauge boson. This module packages the no-graviton stance: gravity is emergent and not force-mediated, there is no separate graviton quantum, the RS coupling κ is fixed by φ alone (Fibonacci form), and GW polarizations equal two as in GR. Downstream UnitBridge converts κ_rs = 8φ⁵ into SI BMV phase rates. Structure is theorem packaging over ZeroParameterGravity plus positivity and polarization lemmas.

claimGravity is emergent curvature of the recognition ledger, not a force mediated by a gauge boson; there is no separate graviton quantum. The RS coupling is $\kappa_{\mathrm{rs}} = 8\varphi^5$ (from $\varphi$ alone, Fibonacci form), with $\kappa_{\mathrm{rs}} > 0$. Gravitational waves have exactly two polarizations, matching GR. The BMV coupling built from $\kappa_{\mathrm{rs}}$ is positive.

background

Recognition Science treats gravity as geometry of the ledger lattice rather than a fundamental interaction. The parent ZeroParameterGravity module states G-001: gravity is not a fundamental force; it is large-scale curvature induced by defect distributions on the ledger. Constants supply the RS time quantum τ₀ = 1 tick; AlphaDerivation supplies φ-dressing and recognition-scale content used when couplings are written in RS-native units.

This module sharpens the particle-physics reading of that claim. If gravity is ledger curvature, there is no need for a spin-2 gauge boson as mediator. The coupling that appears in the quantum channel is κ_rs, forced from φ (with the band from ZeroParameterGravity.kappa_bounds), not fitted. Sibling names mark the local vocabulary: emergent gravity, no force mediation, no separate graviton quantum, κ from φ alone, two GW polarizations, and a positive BMV coupling.

proof idea

Not a single theorem: a gravity-domain module that packages consequences of ZeroParameterGravity. Core stance theorems assert gravity is emergent and not force-mediated, hence no separate graviton quantum. Positivity lemmas derive κ > 0 and κ ≠ 0 from the emergent picture. Coupling lemmas identify κ with a pure φ expression (Fibonacci form, κ_rs = 8φ⁵ in the downstream bridge). Polarization lemmas count GW modes and match GR (exactly two). BMV coupling definitions and positivity close the quantum-channel interface. Imports are Mathlib, Constants, AlphaDerivation, and ZeroParameterGravity; no independent dynamical derivation lives here.

why it matters in Recognition Science

Places the no-graviton reading of RS gravity in Lean so later quantum-channel work can cite it cleanly. Feeds IndisputableMonolith.Gravity.NoGraviton.UnitBridge, which formalizes Theorem 4 of Gravity from Recognition IV: the dimensionless κ_rs = 8φ⁵ (band from ZeroParameterGravity.kappa_bounds) converts to the dimensionful BMV entangling phase rate via an RS-native-to-SI bridge.

Aligns with framework landmarks: gravity as ledger curvature (G-001), couplings built from φ rather than free parameters, and continuity with GR at the level of two GW polarizations. Does not replace the full zero-parameter gravity derivation; it is the particle-content and coupling interface layer between ZeroParameterGravity and the BMV unit bridge.

scope and limits

Does not derive Einstein equations or full GR dynamics from the ledger.
Does not prove experimental absence of gravitons beyond the RS emergent reading.
Does not fix exact infrared α; AlphaDerivation seed identification remains open.
Does not itself perform the SI unit bridge; that lives in NoGraviton.UnitBridge.
Does not introduce free fit parameters for κ; κ is φ-fixed in this package.

used by (1)

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

IndisputableMonolith.Gravity.NoGraviton.UnitBridge module_import

depends on (3)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import
IndisputableMonolith.Constants.AlphaDerivation module_import
IndisputableMonolith.Gravity.ZeroParameterGravity module_import

declarations in this module (28)

def gravity_is_emergent L40
theorem gravity_not_force_mediated L42
theorem no_separate_graviton_quantum L46
theorem emergent_implies_kappa_pos L49
theorem emergent_implies_kappa_ne_zero L52
theorem kappa_from_phi_alone L63
theorem kappa_fibonacci_form L68
def gw_polarization_count L81
theorem gw_polarizations_eq_two L84
theorem gw_matches_gr L88
def BMV_coupling L104
theorem BMV_coupling_pos L107
theorem BMV_coupling_bounds L110
theorem kappa_integer_phi_power L128
theorem kappa_fibonacci_structure L133
structure NoGravitonCert L139
theorem no_graviton_cert L146
def lattice_tensor_components L175
def lattice_trace_constraint L178
def lattice_gauge_constraints L181
def lattice_gw_modes L184
theorem lattice_gw_modes_eq_two L188
theorem lattice_matches_continuum L192
def fibonacci_square_conjecture L218
theorem fibonacci_square_conjecture_consistent L221
def ilg_parameter_count L226
theorem ilg_zero_params_if_conjecture L229
theorem ilg_one_param_if_not L232