t6_to_t8_dimension_bridge_holds
plain-language theorem explainer
Given T6 (φ forced as the unique positive root of x² = x + 1), the named topology/dimension interface yields T8: spatial dimension D = 3. Anyone citing the complete T0–T8 inevitability chain or the T6→T7 route equivalence needs this bridge. The proof is a term-mode assembly that wires three DimensionForcing theorems into the bridge certificate.
Claim. If $\varphi$ is forced (unique positive solution of $x^2 = x + 1$), then the T6-to-T8 dimension bridge holds: ledger-compatible nontrivial linking forces $D = 3$, the eight-tick equation $2^D = 8$ forces $D = 3$, and there is a unique RS-compatible dimension $D = 3$, so T8 is obtained.
background
The Unified Forcing Chain module shows T0–T8 as inevitabilities from the Recognition Composition Law plus normalization and calibration. T6 states that in a discrete ledger with self-similar cost, the only scaling ratio is $\varphi = (1+\sqrt{5})/2$, the unique positive root of $x^2 = x + 1$.
Spatial dimension is not fixed by $\varphi$ alone. The bridge structure records an extra topological interface: nontrivial linking of ledger loops. Upstream, linking_requires_D3 is the primary T8 theorem (Alexander duality: linking exists iff $D = 3$, Hatcher Thm 3.44). Separately, eight_tick_forces_D3 shows $2^D = 8$ forces $D = 3$, and dimension_forced packages uniqueness of the RS-compatible dimension.
A module audit note clarifies that the Lean surface encodes the circle-cohomology predicate arithmetically ($k = 1$); classical topology enters only as the interpretation of that bridge predicate, not as a hidden RS axiom.
proof idea
Term-mode structure construction, not a tactic script. From the T6 hypothesis, copy phi_unique into phi_unique_available. Fill the three dimension fields by direct reference:
linking_route:=DimensionForcing.linking_requires_D3eight_tick_route:=DimensionForcing.eight_tick_forces_D3unique_dimension:=DimensionForcing.dimension_forced
The nested T8 witness reuses the same three theorems as linking_forces_D3, eight_tick_forces_D3, and unique_dimension. No new arithmetic is proved here; the bridge is a named packaging of existing DimensionForcing results under a T6 premise.
why it matters
Closes the T6→T8 step in the complete inevitability chain (primer landmarks T6 φ forced, T7 eight-tick octave, T8 $D = 3$). Downstream, complete_forcing_chain threads this bridge into the unconditional top-level certificate. t6_to_t7_route_equivalence uses it as the via-T8 path from T6 to T7, showing the canonical-period and dimension-bridge routes are the same construction. t6_to_t8_dimension_bridge_unique_eq_triple_route identifies the bridge's uniqueness field with the three-route DimensionForcing witness.
Without this certificate, T8 would sit as an unnamed sibling rather than an explicit topological output of the φ layer. The Alexander-duality dependency is audited in-module: no topology axiom enters the forcing chain itself.
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