T5_T7_To_CanonicalHamiltonian_Bridge
plain-language theorem explainer
A certificate that T5 (unique J-cost) plus T7 (forced eight-tick) yield canonical Hamiltonian structure: cost-phase duality cosh(t)-1 = J(e^t), quadratic kinetic emergence J(1+ε)=ε²/2+O(ε³), DFT-8 eigenvalues as 8th roots of unity, and U(1) phase invariance of mode cost. Cited by the complete forcing chain and the bridge-holds theorem. Pure interface structure; no proof body.
Claim. Given uniqueness of the recognition cost $J(x)=\frac12(x+x^{-1})-1$ (T5) and the forced eight-tick cycle (T7), the following hold: (i) $\cosh t-1=J(e^t)$ for all real $t$; (ii) for $|\varepsilon|\le 1/2$, $J(1+\varepsilon)=\varepsilon^2/2+c\varepsilon^3$ with some $|c|\le 2$; (iii) each cyclic-shift eigenvalue $\lambda_k$ satisfies $\lambda_k^8=1$; (iv) total mode cost is invariant under componentwise $U(1)$ phases on the 8-signal; (v) complex $J$-cost is invariant under $z\mapsto z e^{i\theta}$.
background
The Unified Forcing Chain module treats T0–T8 as inevitabilities from the Recognition Composition Law, normalization $F(1)=0$, and calibration $F''(1)=1$. T5 packages uniqueness of $J$: reciprocity, normalization, the composition law $J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y)$, log-calibration, and continuity force $J(x)=\frac12(x+1/x)-1$ on $(0,\infty)$. Equivalently $J(e^t)=\cosh t-1$.
T7 records that the minimal ledger-compatible cycle is $2^D$; with $D=3$ this is the eight-tick octave. The bridge sits between those two levels and the Hamiltonian layer: small deviations of the cost about the identity should produce a quadratic kinetic form, while the cyclic shift on eight modes should supply discrete eigenvalues and a residual $U(1)^8$ gauge.
Upstream cost definitions (e.g. Jcost as $(x+x^{-1})/2-1$) and complex-structure forcing supply the concrete identities named in the fields; RS-native units fix $c=1$ so phase and time share a common scale.
proof idea
This declaration is a Prop-valued structure (certificate interface), not a proved theorem. It names five fields that any inhabitant must supply: cost-phase duality, quadratic Hamiltonian emergence with cubic bound, 8th-root shift spectrum, discrete mode-cost phase invariance, and continuous $J$-cost phase invariance. The companion theorem t5_t7_to_canonical_hamiltonian_bridge_holds fills those fields by direct appeal to ComplexStructureForcing.cost_phase_duality, ComplexStructureForcing.hamiltonian_emergence, and the matching spectrum/invariance lemmas. A Subsingleton instance records that any two certificates are propositionally equal (rfl).
why it matters
In the forcing chain, T5 pins $J$ and T7 pins the eight-tick octave; dynamics still need a canonical Hamiltonian. This bridge is the named interface that packages exactly that emergence: the quadratic kinetic term from the small-ε expansion of $J$, cost-phase duality linking $J$ to hyperbolic cosine, and the DFT-8 eigenvalue structure forced by the cyclic shift. Downstream, CompleteForcingChain includes the Hamiltonian layer among the objects available once T0–T8 hold, and t5_t7_to_canonical_hamiltonian_bridge_holds is the witness that T5+T7 already discharge the certificate. Framework landmarks: T5 J-uniqueness, T7 eight-tick ($2^3$), and the path toward continuum kinetics without extra postulates.
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