T7_To_CanonicalShift_Bridge
plain-language theorem explainer
Bridge certificate packing the canonical one-tick cyclic shift on the eight-slot signal space once T7 forces the 8-tick octave. It records the defining advance law, period-8 identity, mode eigenvalues (including I and -I), agreement of Spectral and ComplexStructure shifts, and uniqueness of any operator obeying the advance law. Cited by the complete forcing chain and by the theorem that discharges the bridge from any T7 witness. The object is a pure Prop bundle of named equalities; no new forcing is proved here.
Claim. Given a witness that the eight-tick cycle is forced ($8=2^3$ from $D=3$), the following hold for the eight-component signal space: the canonical cyclic shift advances each index by one tick modulo 8; it agrees pointwise with the complex-structure shift; iterating the shift eight times yields the identity; the eigenvalue at mode $k$ is $\zeta^k$ (an 8th root of unity), with mode $2$ equal to $i$ and mode $6$ equal to $-i$; and any map that advances indices by one tick equals the canonical cyclic shift (uniqueness up to pointwise equality, also via the next-index operator).
background
The Unified Forcing Chain module aims to show every level T-1 through T8 is forced from the Recognition Composition Law plus normalization and calibration, rather than merely compatible. T7 states that the minimal ledger-compatible cycle is $2^D$; with spatial dimension three this is the eight-tick octave (period $2^3$), not a free parameter.
Once that period is fixed, dynamics on an eight-slot discrete signal need a canonical time-advance operator. The bridge packages that operator: the cyclic shift that moves each Fin-8 index by one tick, its agreement across the Spectral and ComplexStructureForcing presentations, and the Fourier picture in which modes are 8th roots of unity. The fundamental RS time quantum is one tick ($\tau_0=1$); one octave is eight ticks.
The eigenvalue facts (especially mode 2 giving $i$) are the discrete seed of complexification inside the forcing chain: a real 8-cycle already forces a quarter-turn structure on the signal space.
proof idea
This declaration is a structure (a named Prop bundle), not a proved theorem. It lists nine fields: the defining advance-by-one equation, equality of the two shift presentations, period-8, the mode-wise eigenvalue formula $\zeta^k$, the 8th-root law, the special values $i$ and $-i$ at modes 2 and 6, and two universal-property clauses (any $T$ obeying the advance law, or the next-index law, equals the canonical shift).
Inhabitation is supplied downstream by t7_to_canonical_shift_bridge_holds, which fills each field by definitional reduction (rfl) and the existing ComplexStructureForcing/Spectral lemmas. A Subsingleton instance records that any two certificates for a fixed T7 witness are propositionally equal.
why it matters
In the forcing chain, T7 forces the eight-tick octave from $D=3$. This bridge turns that combinatorial period into the concrete cyclic generator used by spectral and complex-structure layers: period-8 evolution, roots-of-unity diagonalization, and the appearance of $i$ as an eigenvalue. Those facts feed quarter-turn, Hamiltonian, and measurement structure higher in the stack.
Downstream, CompleteForcingChain requires the bridge as part of assembling T0–T8 plus the analytic layers inside one certificate. The companion theorem t7_to_canonical_shift_bridge_holds shows every T7 witness yields such a bridge, so the shift package is not an extra axiom. Framework landmarks: T7 (eight-tick), the path toward T8 ($D=3$), and the discrete origin of complex phases used later for constants and dynamics.
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