module module high

`IndisputableMonolith.Gravity.EightTickResonance`

The EightTickResonance module defines the interpolation cost that quantifies desynchronization of a frequency ratio from the nearest integer multiple of the base tick. Gravity and acoustics researchers cite these when modeling phase alignment in resonant systems. The module supplies a library of cost bounds, non-negativity lemmas, and resonant weight functions that attain maxima at integer ratios.

claimThe interpolation cost of a frequency ratio $r$ is $C(r) = min({r}, 1-{r})$, where ${r}$ denotes the fractional part. This cost vanishes at integers and reaches its maximum value $1/2$ at half-integers. Resonant weights $w(r)$ are constructed to be at least 1 at resonance and strictly less than 1 off resonance.

background

Recognition Science quantizes time in fundamental ticks with period $τ_0 = 1$ (Constants module). The eight-tick octave arises as the self-similar period $2^3$ in the forcing chain (T7). This module equips the gravity domain with a distance-to-integer measure for frequency ratios, together with associated weight functions that encode synchronization strength. The supplied DOC_COMMENT states that cost equals zero at synchronized integers and one-half at maximally desynchronized half-integers.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

The definitions feed directly into AcousticPhaseLevitation, supplying the cost and weight primitives required for phase-levitation calculations. They instantiate the eight-tick resonance machinery of the T7 step in the unified forcing chain, allowing downstream weight-reduction statements at resonance to be stated in RS-native units.

scope and limits

Does not derive the eight-tick period from the J-function or RCL.
Does not relate resonance weights to mass ladders or the phi-ladder.
Does not compute explicit numerical resonant frequencies.
Does not address coupling to spatial dimensions or the alpha band.

used by (1)

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

IndisputableMonolith.Gravity.AcousticPhaseLevitation module_import

depends on (1)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import

declarations in this module (24)

def interpolation_cost L32
theorem interpolation_cost_nonneg L35
theorem interpolation_cost_le_half L39
theorem interpolation_cost_zero_at_integer L46
def C_lag L54
theorem C_lag_pos L56
def w_resonant L64
theorem w_resonant_ge_one L67
theorem w_at_resonance L72
theorem w_off_resonance L77
theorem weight_reduction_at_resonance L84
def resonant_frequency L94
theorem resonant_frequency_pos L97
theorem resonant_frequency_decreasing L104
structure EightTickResonanceCert L121
theorem eight_tick_resonance_certified L127
theorem C_lag_eq_cLagLock_inv L146
theorem w_resonant_bounded_above L151
theorem eight_tick_period L159
def w_total L187
theorem w_total_at_resonance L191
theorem w_total_exceeds_secular L196
theorem resonance_weight_reduction_ratio L205
theorem shared_coupling L214