T6_To_T7_Canonical_Bridge
plain-language theorem explainer
A Prop-structure certificate bridging T6 (φ forced) to T7 (eight-tick forced): it packages φ-uniqueness, Alexander duality forcing D=3, and the canonical period law Π(D)=2^D so that the octave is 2^3=8. Anyone assembling the unified forcing chain or comparing T6→T7 routes cites this bundle. It is a named hypothesis surface plus a T7 field, not a computational proof.
Claim. Given a T6 certificate that $\varphi$ is the unique positive solution of $r^2=r+1$, a T6-to-T7 canonical bridge asserts: $\varphi$-uniqueness is available; any dimension supporting nontrivial circle linking equals $3$; the canonical period satisfies $\Pi(D)=2^D$; $\Pi(D)=8$ iff $D=3$; at $D=3$ one recovers the eight-tick value; the dimensional eight-tick law agrees with $\Pi$ at every $D$; and the T7 eight-tick forcing surface holds.
background
The module UnifiedForcingChain aims at a complete inevitability chain T-1 through T8 from the Recognition Composition Law plus normalization and calibration. In that ladder, T6 states that the only positive scaling ratio compatible with a discrete self-similar ledger is $\varphi=(1+\sqrt{5})/2$, the unique positive root of $x^2=x+1$. T7 states that the minimal ledger-compatible cycle is $2^D$, hence eight ticks once $D=3$.
Independently of T6, Alexander duality is used to name the unique dimension that supports nontrivial circle linking; that forces $D=3$. The canonical period construction is the pure dimensional power of two, $\Pi(D)=2^D$. The RS tick is the unit time quantum $\tau_0=1$, and one octave is eight ticks. The bridge exists so T7 is not an unnamed sibling: it is the explicit output of T6 plus the linking/dimension facts and the period law.
proof idea
This declaration is a structure (a Prop bundle), not a proved theorem. Its fields are documentation-carrying hypotheses: expose T6's $\varphi$-uniqueness; record linking-forces-$D=3$; fix $\Pi(D):=2^D$ by definition; state the biconditional $\Pi(D)=8\leftrightarrow D=3$; evaluate at $D=3$; equate $\Pi$ with the EightTickFromDimension law; and carry a T7_EightTick_Forced witness.
The companion theorem t6_to_t7_canonical_bridge_holds fills the bundle: copy phi_unique from the T6 hypothesis, apply DimensionForcing.linking_requires_D3, discharge definitional equalities by rfl, and assemble the T7 surface from $2^3=8$. A Subsingleton instance records that any two such certificates for fixed T6 are propositionally equal.
why it matters
In the primer forcing chain, T6 pins $\varphi$ and T7 pins the eight-tick octave as $2^D$ at $D=3$. This bridge is the explicit T6→T7 link that makes the octave a derived object rather than an inserted constant. Downstream, CompleteForcingChain includes the full T0–T8 spine and needs a coherent T7 surface; T6_To_T7_RouteEquivalence compares this direct canonical-period route with the indirect T6→T8→T7 path and requires the present certificate as its direct_route field; t6_to_t7_canonical_bridge_holds is the existence theorem that populates the bundle.
The construction isolates what is independent of $\varphi$ (Alexander duality / linking ⇒ $D=3$) from what T6 contributes (scale uniqueness), so the eight-tick falls out purely as $\Pi(3)=2^3=8$. That separation is what lets the complete chain claim every level is forced, not merely compatible.
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