module module high

`IndisputableMonolith.Gap45.RecognitionBarrier`

RecognitionBarrier formalizes the coprimeness of 8 and 45 to establish an uncomputability barrier in Gap-45 arithmetic. Researchers studying synchronization in odd dimensions cite it to justify why D=3 yields a minimal period without subharmonic alignment. The module assembles supporting lemmas from the Gap45 and SyncMinimization imports to encode the no-alignment constraint.

claimThe module asserts that $8$ and $45$ are coprime, i.e., no common sub-period exists that aligns both the eight-tick octave and the $45$-unit recognition period.

background

The upstream Gap45 module records that 9 and 5 are coprime. SyncMinimization formalizes constraint (S) from the Dimensional Rigidity paper: among odd spatial dimensions $D$ at least 3, $D=3$ uniquely minimizes the synchronization period lcm of $2^D$ and $T(D^2)$. RecognitionBarrier applies this setting to define the barrier as the direct consequence of gcd(8,45)=1, where 8 arises as the eight-tick octave period.

proof idea

This is a definition module, no proofs. It declares the core coprime_barrier fact together with auxiliary lemmas such as beat_diff, beat_is_prime, window_insufficient, and barrier_cert that encode the arithmetic constraints on period alignment.

why it matters in Recognition Science

The module supplies the coprimeness fact used by ShimmerFactor for the subjective-time factor 360/37 and by SafetyInterlock to establish the safety interlock via gap45_coprime and gap45_lcm=360. It fills the Gap-45 uncomputability barrier step in the Recognition Science chain, linking directly to the eight-tick octave at T7.

scope and limits

Does not extend the coprimeness argument to even dimensions.
Does not compute explicit values of T(D squared) for D greater than 3.
Does not address the full forcing chain from T0 to T8.
Does not provide numerical simulations of the synchronization period.

used by (2)

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

IndisputableMonolith.Gap45.ShimmerFactor module_import
IndisputableMonolith.Superhuman.SafetyInterlock module_import

depends on (2)

Lean names referenced from this declaration's body.

IndisputableMonolith.Gap45 module_import
IndisputableMonolith.Gap45.SyncMinimization module_import

declarations in this module (14)

theorem coprime_barrier L37
theorem beat_diff L40
theorem beat_is_prime L43
theorem beat_fraction_irreducible L46
theorem window_insufficient L53
theorem minimal_resolution_window L58
theorem resolution_requires_full_period L62
theorem aliasing_ratio_lt_one L72
theorem time_dilation_factor L76
theorem phase_mismatch L85
structure BarrierCert L97
def barrier_cert L105
structure ConsciousnessHypothesis L126
def consciousness_hypothesis L135