t8_dimension_four_route_equivalence
plain-language theorem explainer
Given T8 (spatial dimension forced), the four independent routes to D=3 (linking, eight-tick, gap-sync, spinor) agree pairwise and all pin the physical dimension at 3. Cited by anyone assembling the complete T0–T8 inevitability chain or the canonical D=3 bridge. Proof fills the certificate fields from DimensionForcing lemmas, plus a short divisibility contradiction ruling out D≥4 against the gap-sync period 360.
Claim. Assume T8: nontrivial linking forces $D=3$, the eight-tick condition forces $D=3$, and there is a unique RS-compatible dimension. Then the four-route equivalence holds: the physical dimension equals $3$; each of linking, eight-tick, gap-sync, and spinor forces $D=3$; any dimension forced by one route is forced by every other; and RS-compatibility alone forces $D=3$.
background
In the Unified Forcing Chain, T0–T8 are derived as inevitabilities from the Recognition Composition Law plus normalization and calibration. T8 is the dimension step: spatial dimension is not a free parameter. The T8 package asserts that nontrivial linking (ledger conservation via Alexander duality), the eight-tick identity $2^D=8$, and uniqueness of an RS-compatible dimension all force $D=3$.
The four-route certificate strengthens that package. It records that the canonical physical dimension is exactly 3, and that the four named routes (linking, eight-tick, gap-sync with period 360, and spinor/eight-tick) are pairwise equivalent: a dimension forced by any one is forced by the others. Gap-sync uses the arithmetic constraint that $2^D$ divides the sync period 360; spinor ties the Clifford/spinor structure to the same eight-tick count.
Upstream DimensionForcing supplies the individual route lemmas (linking requires $D=3$, eight-tick forces $D=3$, spinor eight-tick forces $D=3$, dimension uniqueness). Constants fix $D:=3$ and the tick/octave conventions ($2^3=8$).
proof idea
Structure construction under the T8 hypothesis. Physical dimension equals 3 by reflexivity on the canonical constant. Linking, eight-tick, spinor, and uniqueness fields are one-line applications of the corresponding DimensionForcing lemmas (linking_requires_D3, eight_tick_forces_D3, spinor_eight_tick_forces_D3, dimension_unique).
Gap-sync is proved by contradiction: rewrite the sync period as 360; if $D\ge 4$ then $2^4=16$ divides $2^D$, hence 16 divides 360, which decide refutes. Pairwise route equalities reduce to the same forcing lemmas: any $D$ satisfying one hypothesis is sent to 3, so the routes agree.
why it matters
Closes the multi-route side of T8 inside the complete inevitability chain. Downstream complete_forcing_chain threads T−1 through T8 as an unconditional package; this certificate is the T8 four-route witness that chain consumes, converting the existential unique-compatible-dimension claim into a canonical $D=3$ with explicit route agreement.
Framework landmark: primer T8 ($D=3$ spatial dimensions) and T7 (eight-tick octave $2^3$). The bridge comment immediately below records the intended upgrade from $\exists! D$ to the iff $D=3$ plus period $2^3=8$ and the four named routes. Without pairwise agreement, a referee could treat linking, gap-sync, and spinor as independent postulates rather than forced equivalents of one dimension law.
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