T7_To_Realization_Bridge
plain-language theorem explainer
Packages the geometric content of T7 for the D=3 Gray cycle: in any cellular completion of the 3-cube, that Hamiltonian cycle realizes as a circle defect and never as a higher-dimensional sphere. Anyone routing T8 through the realization bridge cites this interface. It is a Prop structure (definitional bundle), not a proved theorem; the companion constructor fills both fields from T7CycleRealization lemmas.
Claim. Assume the eight-tick is forced ($8=2^3$ from dimension $D=3$). The T7 realization bridge is the pair of assertions: (i) for every cellular completion of the $3$-cube, the realized defect of the canonical $3$-bit Gray closed walk equals the circle defect; (ii) for every integer $p\ge 2$, the image of that walk is not a $p$-sphere.
background
The Unified Forcing Chain module aims to force every landmark T−1 through T8 from the Recognition Composition Law plus normalization and calibration. T7 states that the minimal ledger-compatible cycle length is $2^D$; with $D=3$ this is the eight-tick (not a free parameter).
A cellular completion (T7.5a) is a predicate-level witness that a closed orientable smooth $D$-manifold carries a tame cube-graph embedding with a retraction back to the cube graph. The explicit $3$-bit Gray cycle is the Hamiltonian closed walk on $Q_3$ used as the canonical T7 schedule.
At the present theorem surface, realized defect of any such walk in a completion is recorded as the circle defect, and "image is a $p$-sphere" is the predicate $p=1$ only: graph-shaped closed walks cannot be higher spheres. The bridge packages those two paper conclusions relative to a T7 hypothesis.
proof idea
No proof body: this is a structure definition whose fields are the two Props above, parameterized by a T7 eight-tick hypothesis. Instantiation is deferred to the companion theorem t7_to_realization_bridge_holds, a one-line constructor that sets realizes-as-circle from grayCycle3_realizes_circle and no-higher-sphere from grayCycle3_no_higher_sphere in T7CycleRealization. Those upstream lemmas are themselves predicate-level recordings of the paper conclusions (RealizedDefect constantly returns Circle; ImageIsSpherePofDim is $p=1$).
why it matters
Sits on the T7→T8 arrow of the complete inevitability chain. Downstream, t7_to_realization_bridge_holds builds an instance, and T8_Via_Realization_Bridge consumes that instance together with T7.5a/T7.5c bridges to reach the same $D=3$ conclusion as the main T8 surface via loop-entanglement and compatibility.
Framework landmark: T7 (eight-tick octave, period $2^3$) and the geometric step that feeds T8 ($D=3$ spatial dimensions). The bridge is the formal hinge that turns the forced eight-tick Gray schedule into a circle realization, which is what the dimension-forcing argument needs before linking and gap-45 sync close $D=3$. It does not itself prove uniqueness of $J$ or $\varphi$; those are T5–T6 upstream.
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