t8_triple_route_bridge_holds
plain-language theorem explainer
Given that spatial dimension is forced (T8), the three independent routes (linking, eight-tick, gap-sync) each constrain D, and their conjunction uniquely yields D = 3. Anyone citing the complete T0–T8 inevitability chain needs this bridge certificate. The proof wires the linking field from the T8 hypothesis, quotes the eight-tick and spinor lemmas, and rules out D ≥ 4 by showing 16 cannot divide the sync period 360.
Claim. Assume spatial dimension is forced: nontrivial linking implies $D=3$, the eight-tick identity $2^D=8$ implies $D=3$, and there is a unique RS-compatible dimension. Then the triple-route bridge holds: the linking route, the eight-tick route, the gap-sync upper bound $D\le 3$, and the spinor route are all available, and any dimension satisfying the three RS-compatibility conditions equals $3$.
background
The Unified Forcing Chain module shows that T0–T8 are inevitabilities from the Recognition Composition Law plus normalization and calibration. T8 is the last geometric step: spatial dimension is not a free parameter. The structure T8_Dimension_Forced packages three facts: nontrivial ledger linking forces $D=3$, the eight-tick identity $2^D=8$ forces $D=3$, and a unique RS-compatible dimension exists.
The triple-route bridge is the certificate that surfaces those routes side by side. Linking is the Alexander-duality constraint (nontrivial circle linking in $S^D$ only for $D=3$). The eight-tick route is the pure arithmetic $2^D=8$. Gap-sync uses the forced synchronization period $360$: if $D\ge 4$ then $16\mid 2^D$, hence $16\mid 360$, which is false. A spinor/eight-tick characterization is recorded as a fourth column.
In RS units the fundamental tick is $\tau_0=1$ and one octave is eight ticks; the constant $D:=3$ is the value this step freezes for all later derivations (Clifford, spinors, $\alpha$).
proof idea
Structure construction, not a deep new argument. The linking field is copied from the hypothesis: h8.linking_forces_D3. The eight-tick and spinor fields are the existing DimensionForcing lemmas eight_tick_forces_D3 and spinor_eight_tick_forces_D3.
The gap-sync upper bound is proved by contradiction: rewrite the sync period as $360$; if $D\ge 4$ then $2^4\mid 2^D$, so $16\mid 360$, absurd by decide. The uniqueness clause unpacks the three RS-compatibility hypotheses, applies linking and eight-tick each to get $D=3$, re-runs the same $16\nmid 360$ argument for $D\le 3$, and returns the linking conclusion. All routes agree on $D=3$.
why it matters
This is the T8 bridge in the complete inevitability chain. Downstream, complete_forcing_chain assembles T-1 through T8 into one unconditional certificate; without the triple-route packaging, the chain cannot close the dimension step.
Framework landmark: primer T8 ($D=3$ spatial dimensions), fed by T7 (eight-tick octave, period $2^3$). The three routes are independent cross-checks, not redundant restatements: linking is topological, eight-tick is arithmetic, gap-sync is a divisibility obstruction. Once $D=3$ is forced, the next module block opens the Clifford/spinor bridge: $\mathrm{Cl}_3\cong M_2(\mathbb{C})$, $\mathrm{Spin}(3)\cong\mathrm{SU}(2)$, spinor dimension $2$, and Bott period 8 tying back to the eight-tick.
No open scaffold remains here; the claim status is fully proved.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.