T8_To_CanonicalGap45_Bridge
plain-language theorem explainer
Certificate bundling the arithmetic identities that link forced spatial dimension D=3 to the canonical gap-45 parameter. Gap-45 is the 9th triangular number (fence-post closure 9=8+1 on an eight-tick cycle), and the sync period is lcm(8,45)=360=2³·3²·5. Anyone assembling the complete T0–T8 forcing chain cites this Prop record. It is a pure interface structure; uniqueness is by subsingleton, and the inhabitant is filled by a separate holds theorem.
Claim. Given that spatial dimension $D=3$ is forced, the following hold at once: the dimension-forcing gap parameter equals $45$; the 9th triangular number equals $45$; the fence-post closure number of a closed 8-tick cycle equals $9$; the cumulative phase over that cycle equals $45$; the synchronization period equals $\mathrm{lcm}(8,45)=360=2^3\cdot 3^2\cdot 5$; $2^3$ divides the sync period; $45=3^2\cdot 5$; and equivalently the legacy form $45=9\cdot 5$.
background
In the Unified Forcing Chain, T0–T8 are claimed as inevitabilities from the Recognition Composition Law plus normalization and calibration. T8 asserts that spatial dimension is not free: $D=3$ is the unique value compatible with nontrivial linking (ledger conservation), the eight-tick period $2^D=8$, and gap-45 synchronization.
Gap-45 is the arithmetic hinge of that last clause. The 8-tick octave is the fundamental closed evolution period once $D=3$. The fence-post count on a closed 8-cycle is $9=8+1$, and the 9th triangular number $T(9)=9\cdot 10/2=45$ is identified with the cumulative phase (and with the dimension-forcing gap parameter). The joint sync period is then $\mathrm{lcm}(8,45)=360$, whose prime factorization $2^3\cdot 3^2\cdot 5$ surfaces the $2^3$ factor as the dimensional signature.
The structure takes a T8 witness as a dummy hypothesis and packages the named numerical identities (gap equals 45, triangular identity, closure number, phase, sync period and factorizations, including the legacy $9\times 5$ form) as a single Prop certificate rather than scattering them as free lemmas.
proof idea
No proof body: this is a Prop-valued structure definition, not a theorem. Each field is a named equality or divisibility statement about the gap-45 and sync-period constants. Propositional uniqueness for a fixed T8 instance is immediate (subsingleton by rfl on the Prop record). The actual inhabitant is constructed downstream by t8_to_canonical_gap45_bridge_holds, which fills every field by rfl or by the Gap45 physical-motivation lemmas (triangular_9_is_45, closure_number_eq_9, gap_45_from_phase, and the corresponding factorization facts).
why it matters
Closes the T8 end of the forcing chain by naming the canonical T(9)=45 reading of gap-45, rather than leaving only the algebraically equivalent legacy form $9\times 5$. Downstream, CompleteForcingChain requires this bridge among the T0–T8 layers, and t8_to_canonical_gap45_bridge_holds is the theorem that supplies a concrete certificate from any T8 witness.
Framework landmarks: T7 (eight-tick octave, period $2^3$) and T8 ($D=3$ from linking plus gap-45 sync) in the primer chain. The $2^3$ factor inside $360$ is the explicit dimensional footprint; the triangular/fence-post story is the physical motivation that makes gap-45 canonical rather than an ad hoc integer. Without this bundle, the complete inevitability claim would still have a gap between "D=3 is forced" and the concrete sync arithmetic used by later constant and measurement layers.
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