t7_to_canonical_shift_bridge_holds
plain-language theorem explainer
Given that the eight-tick cycle is forced (T7), the canonical cyclic shift on eight-mode signals is fully certified: defining equation, period-8 law, eighth-root eigenvalues (including i and -i at modes 2 and 6), and the universal property that any operator obeying the advance-by-one rule equals that shift. Downstream complete-chain assembly cites this bridge. The proof is mostly definitional plus the existing complex-structure period and eigenvalue lemmas.
Claim. Assume the eight-tick is forced: $8 = 2^3$ and the minimal ledger-compatible cycle at $D=3$ is that eight-tick. Then the canonical cyclic shift bridge holds: the spectral advance-by-one on eight-mode signals equals $f\mapsto f((k+1)\bmod 8)$, agrees with the complex-structure shift, has period $8$, has eigenvalues $\zeta^k$ (eighth roots of unity) with $\zeta^2 = i$ and $\zeta^6 = -i$, and is universal among operators satisfying the same defining equation.
background
The Unified Forcing Chain module aims to show T0–T8 as inevitabilities from the Recognition Composition Law plus normalization and calibration. T7 asserts 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.
The bridge structure packages the link from that forced eight-tick to the concrete cyclic shift on Signal8 (eight-component complex signals). The shift advances the mode index by one tick modulo eight. Its spectrum is built from a primitive eighth root $\zeta$, with eigenvalue at mode $k$ equal to $\zeta^k$. Upstream complex-structure results already record period eight, $\zeta^2 = i$, and $\zeta^6 = -i$.
In RS-native units the fundamental time quantum is one tick ($\tau_0 = 1$); one octave is eight ticks. This bridge is the operator-level reading of that octave once T7 is in hand.
proof idea
Construct the bridge structure field by field from the T7 hypothesis.
Defining equation, agreement with the complex-structure shift, and the mode-eigenvalue equation are pure definitional equalities (rfl). Period eight is imported as ComplexStructureForcing.shift_period_8. The eighth-root law unfolds the eigenvalue definition, uses $\zeta^8 = 1$, and a short power-arithmetic calculation: $(\zeta^{k})^8 = (\zeta^8)^k = 1$. Modes 2 and 6 are the existing theorems eigenvalue_2_is_I and eigenvalue_6_is_neg_I. Universality (and the nextIdx variant) is funext plus rewriting by the assumed defining equation, then rfl.
No new analysis: the work is packaging known eight-tick operator facts under the T7 hypothesis.
why it matters
In the forcing chain, T7 forces the eight-tick octave from $D=3$. This theorem turns that discrete period into a certified canonical shift operator with full spectral data, including the appearance of $i$ and $-i$ as eigenvalues. That is the natural complex structure on the octave.
The sole recorded consumer is complete_forcing_chain, which threads T-1 through T8 bridges into the unconditional complete inevitability bundle. Without this certificate the chain would have an eight-tick cardinality fact but no operator bridge to the spectral/cyclic machinery used later in the foundation.
Framework landmarks: T7 (eight-tick octave, period $2^3$) and the link toward T8 ($D=3$). The bridge is proved, not scaffolding; it closes the T7-to-operator step inside the stronger "complete inevitability" claim of the module.
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