IndisputableMonolith.Verification.DimensionCRT
Defines the synchronization period S(D)=lcm(2^D,45) used in dimensional rigidity arithmetic, and proves it is uniquely minimized at D=3 with S(3)=360. Anyone citing the D=3 forcing step or eight-tick/octave synchronization would use these lemmas. Arguments are elementary LCM identities for powers of two against the fixed factor 45.
claimThe module introduces the synchronization period $S(D)=\mathrm{lcm}(2^D,45)$ and proves $S$ is uniquely minimized at spatial dimension $D=3$ with $S(3)=360$, equivalently $\mathrm{lcm}(2^D,45)=360$ if and only if $D=3$.
background
Recognition Science forces three spatial dimensions (forcing chain T8) partly by synchronization arithmetic: discrete recognition ticks live on a $2^D$-periodic lattice (the D-fold product of binary phase), while a fixed 45-factor appears from the RecogSpec band structure imported here. The natural common period is therefore the least common multiple $S(D)=\mathrm{lcm}(2^D,45)$.
The module sits in Verification and packages the pure number-theoretic facts needed downstream: how $S$ factors, its value at $D=3$, and the uniqueness statement that equality with 360 pins $D$ to three. No continuum geometry is assumed; only LCM and divisibility of integers.
proof idea
Definition module plus short arithmetic lemmas. syncPeriod is the def $S(D)=\mathrm{lcm}(2^D,45)$. Multiplicativity and explicit evaluation give $S(3)=360$. Minimization and the iff form proceed by comparing $v_2$-valuations and the odd part 45: for $D<3$, $2^D$ fails to clear the needed 8-power relative to the target period; for $D>3$, $2^D$ strictly enlarges the LCM past 360. Standard Mathlib LCM lemmas discharge the equalities.
why it matters in Recognition Science
Supplies the arithmetic spine for dimensional rigidity: the claim that the recognition synchronization period is shortest exactly in three dimensions, matching forcing-chain T8 ($D=3$) and the T7 eight-tick octave ($2^3$). Downstream dimensional-rigidity or CRT-style uniqueness arguments cite the minimization and the lcm(...)=360 ↔ D=3 equivalence rather than re-proving elementary number theory. The module itself has no further used-by edges in the current graph; it is a leaf verification package supporting the D=3 landmark.
scope and limits
- Does not derive why the odd factor is 45; that is imported from RecogSpec.Bands.
- Does not prove continuum or geometric D=3, only the LCM minimization identity.
- Does not address temporal dimension count or non-power-of-two tick lattices.
- Does not connect S(D) to physical constants (c, ħ, G, α) inside this module.