onlyD3_satisfies_RSCounting_Gap45_Absolute
plain-language theorem explainer
Only spatial dimension three satisfies the joint RS-counting and Gap-45 synchronization predicate. Dimensional-rigidity arguments in Recognition Science cite this necessity step. The proof unpacks the conjunction and applies the arithmetic characterization that lcm(2^D,45)=360 forces D=3; the cover witness is not needed for the equality.
Claim. For every natural number $D$, if there exists a complete $D$-dimensional cover whose combinatorial period equals $2^D$ and $\mathrm{lcm}(2^D,45)=360$, then $D=3$.
background
The Verification.Dimension module isolates a lightweight arithmetic path from RS counting plus 45-gap synchronization to spatial dimension three, aimed at PrimeClosure. It depends only on lcm facts and the RecogSpec layer, not on the full forcing chain machinery.
The predicate RSCounting_Gap45_Absolute packages two independent witnesses: existence of a CompleteCover of the D-hypercube with period exactly $2^D$, and the synchronization identity $\mathrm{lcm}(2^D,45)=360$. Coverage ensures $2^D$ is the actual combinatorial period of the cover, not an ad-hoc placeholder; the lcm identity ties rung-45 timing to that period (target 360).
Upstream, RecogSpec supplies lcm_pow2_45_eq_iff, the pure arithmetic equivalence between the lcm identity and $D=3$. Framework constants elsewhere fix $D:=3$ via T8 (spatial dimensions) and T9 (linking), which this verification restates in counting language.
proof idea
Term-mode, two steps. Destructure the absolute predicate into the cover existence hcov and the synchronization equality hsync. Apply the forward direction of RecogSpec.lcm_pow2_45_eq_iff to hsync, then simpa to conclude $D=3$. The cover hypothesis is unpacked but unused in the equality; it is present so both conjuncts are structurally witnessed.
why it matters
This is the necessity half of dimensional rigidity under RS counting. Downstream, rs_counting_gap45_absolute_iff_dim3 upgrades it to the biconditional RSCounting_Gap45_Absolute $D \leftrightarrow D=3$ by adding the sufficiency construction when $D=3$.
In the Recognition framework this verifies the T8 landmark (spatial dimension three) in a counting-and-synchronization language: the eight-tick octave is the $D=3$ case of period $2^D$, and Gap-45 synchronization at 360 selects that same $D$. The module keeps the path arithmetic and import-light so PrimeClosure can cite a self-contained necessity lemma rather than the full UnifiedForcingChain.
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