T7_T8_To_CanonicalSchrodinger_Bridge
plain-language theorem explainer
From the forced eight-tick cycle and D=3, every recognition tick is discrete Schrödinger evolution: the cyclic shift acts as exp(-i E_k τ₀/ℏ) on each DFT-8 eigenmode, with E_k = ℏ π k/(4 τ₀). The structure packages seven canonical steps (eigenmodes, phase factor, Hermitian non-negative spectrum, linearity, unitarity) plus inhabitation of the master Schrödinger certificate. Anyone citing the T7–T8 link into quantum dynamics uses this interface. It is a Prop-structure certificate, not a proved theorem; inhabitation is discharged downstream.
Claim. Given that the eight-tick octave is forced ($8=2^3$ from $D=3$) and that spatial dimension $D=3$ is forced, the following hold as a bridge certificate: (1) each DFT-8 mode $k$ is an eigenmode of the cyclic shift with factor $\omega_8^k$; (2) $\omega_8^k=\exp(-i E_k\tau_0/\hbar)$ with quarter-turn energies $E_k$; (3) the shift acts as that phase on every scaled mode; (4)–(5) the $E_k$ are real and non-negative; (6) the shift is $\mathbb{C}$-linear; (7) it preserves pointwise mode norms; and the master Schrödinger-equation certificate is inhabited.
background
The Unified Forcing Chain module derives T0–T8 as inevitabilities from the Recognition Composition Law plus normalization and calibration. T7 states that the minimal ledger-compatible cycle is $2^D$, hence eight-tick once $D=3$. T8 states that $D=3$ is the unique dimension with nontrivial linking, eight-tick sync, and gap-45 synchronization.
On the spectral side, the eight-tick ledger is analyzed via the length-8 DFT: modes dft8_mode k and root of unity $\omega_8$. The cyclic shift is the discrete time-one map of one recognition tick of duration $\tau_0$. Native action is $\hbar=\varphi^{-5}$ (RS units), so the phase $\exp(-i E\tau_0/\hbar)$ is the discrete Schrödinger factor. Quarter-turn energies $E_k=\hbar\pi k/(4\tau_0)$ are the eigenvalues attached to those modes.
The structure does not re-prove T7 or T8; it consumes them as hypotheses and records the exact interface that turns the forced octave into canonical discrete Schrödinger data, including inhabitation of the master certificate SchrodingerEquationCert.
proof idea
This declaration is a Prop-valued structure (a certificate interface), not a theorem with a tactic proof. Its body is the bundle of eight fields listed in the signature: eigenmode evolution under cyclic shift, identification of $\omega_8^k$ with the Schrödinger phase factor, the induced discrete flow on scaled modes, reality and non-negativity of the quarter-turn spectrum, $\mathbb{C}$-linearity of the shift, pointwise norm preservation on modes, and Nonempty of the master Schrödinger certificate.
Inhabitation is supplied by the sibling theorem t7_t8_to_canonical_schrodinger_bridge_holds, which fills each field from SchrodingerDerivation lemmas (eigenmode_evolution_exact, omega8_pow_eq_evolution_factor, discrete_schrodinger_eigenmode, and the remaining spectrum/linearity/unitarity facts). A Subsingleton instance records propositional uniqueness of any two such certificates.
why it matters
In the forcing chain, T7 (eight-tick) and T8 ($D=3$) close the discrete geometric layer. This bridge is the handoff from that geometry into quantum dynamics: one recognition tick equals unitary Schrödinger evolution on the DFT-8 eigenbasis, with the standard phase $\exp(-iH\tau_0/\hbar)$ and Hermitian non-negative energies.
Downstream, CompleteForcingChain includes this bridge among the layers that must be available once T0–T8 are forced (quarter-turn, Hamiltonian, projective, measurement). The companion theorem t7_t8_to_canonical_schrodinger_bridge_holds is the inhabitation witness. Framework landmarks: T7 eight-tick octave ($2^3$), T8 spatial dimension three, and RS-native $\hbar=\varphi^{-5}$ entering the phase factor. Without this certificate, the chain would stop at discrete geometry and would not reach canonical Schrödinger form inside the monolith.
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