module module high

`IndisputableMonolith.Gravity.EightTickResonance`

The EightTickResonance module defines the interpolation cost of a frequency ratio together with resonant weights and bounds for use in gravity calculations. Acoustic-phase researchers cite these objects when modeling synchronization to the ledger clock under the eight-tick octave. The module contains only definitions and short lemmas that establish non-negativity, the zero-at-integer property, and the half-integer maximum.

claimFor a frequency ratio $r$, the interpolation cost is the distance to the nearest integer: $c(r) = min({r},1-{r})$, where ${r}$ denotes the fractional part. The resonant weight satisfies $w_{resonant}(r) = 1 - 2c(r)$ and reaches its maximum of 1 precisely when $r$ is integer.

background

The module imports the RS time quantum from Constants, whose doc-comment states: 'The fundamental RS time quantum (RS-native). τ₀ = 1 tick.' It therefore works in units where the ledger clock advances in discrete ticks. The supplied doc-comment defines the interpolation cost as the measure of desynchronization: zero when the ratio is integer (synchronized) and 1/2 when the ratio is half-integer (maximally desynchronized). The module therefore supplies the concrete functions needed to quantify resonance inside the eight-tick period.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

The module supplies the cost and weight functions imported by AcousticPhaseLevitation. It therefore supplies the concrete arithmetic that realizes the eight-tick octave (T7) inside the gravity domain. No open scaffolding is closed here.

scope and limits

Does not derive the eight-tick period from the forcing chain.
Does not relate the cost function to the J-uniqueness or phi fixed point.
Does not compute explicit resonant frequencies or mass rungs.
Does not address spatial dimension D=3 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