t6_to_t7_canonical_bridge_holds
plain-language theorem explainer
Given that φ is forced as the unique positive root of x² = x + 1, Alexander duality supplies D = 3 and the canonical period Π(D) = 2^D yields the eight-tick octave as 2³ = 8. Forcing-chain authors cite this as the direct T6→T7 certificate. The proof is a term-mode structure fill: uniqueness from T6, the linking theorem, the period–dimension iff, and definitional equalities.
Claim. If $\varphi$ is forced (the unique $r > 0$ with $r^2 = r + 1$), then a canonical bridge to the eight-tick layer holds: nontrivial circle linking forces spatial dimension $D = 3$, the canonical period is $\Pi(D) = 2^D$, and $\Pi(3) = 8$, so the eight-tick octave is obtained as $2^3$.
background
The Unified Forcing Chain module shows T0–T8 as inevitabilities from the Recognition Composition Law plus normalization and calibration. In that ladder, T6 asserts that a discrete ledger with self-similar cost admits only the scaling ratio $\varphi = (1+\sqrt{5})/2$, the unique positive solution of $x^2 = x + 1$. T7 is the eight-tick octave; T8 is spatial dimension $D = 3$.
The bridge structure packages what is needed to pass from T6 to T7 on the canonical-period route: $\varphi$-uniqueness from T6, the Alexander-duality fact that only $D = 3$ supports nontrivial circle linking, the law $\Pi(D) = 2^D$, and the equivalence of that period with the eight-tick surface at $D = 3$.
Upstream, linking_requires_D3 is the primary T8 theorem proved from Alexander duality with "no reference to 8-tick or gap-45." Independently, the period–dimension lemma states that the canonical period equals eight if and only if $D = 3$.
proof idea
Term-mode construction of the bridge certificate under hypothesis h6 : T6_Phi_Forced.
phi_unique_availableis taken directly fromh6.phi_unique.linking_forces_D3is the upstream theoremDimensionForcing.linking_requires_D3(Alexander duality).- Canonical-period definition, evaluation at $D = 3$, and identification with the eight-tick-from-dimension construction are discharged by
rfl(definitional equalities for $\Pi(D) = 2^D$). - The bidirectional period–dimension law is
PeriodDependsOnDimension.period_eq_eight_iff_D_eq_three. - The embedded T7 witness is assembled with two
rflfields: eight is $2^3$, and it arises from dimension.
No new analysis; the bridge is pure packaging of already-proved dimension and period facts with T6 uniqueness.
why it matters
This is the direct T6→T7 step in the complete inevitability chain: once φ is forced, the eight-tick falls out as $\mathrm{PeriodFromDimension},3 = 2^3 = 8$, matching the primer landmarks T6 (φ fixed point), T7 (eight-tick octave), and the independent T8 input $D = 3$ from linking.
Downstream, complete_forcing_chain threads this bridge into the unconditional T−1…T8 assembly. The sibling t6_to_t7_route_equivalence uses it as the direct_route and compares it to the indirect path T6→T8→T7, certifying that both routes present the same T7 surface "up to equivalence."
Without this certificate the chain would have a gap between self-similar scale recursion and the discrete temporal octave that later feeds constants and mass-ladder structure.
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