T7_EightTick_Forced
plain-language theorem explainer
T7 packages the claim that the Recognition ledger’s fundamental cycle is the eight-tick octave: the named eight-tick constant equals 2³, and the dimension formula 2^D at D=3 recovers that same constant. Anyone citing the forcing chain’s discrete time period, DFT-8 structure, or Hamiltonian emergence from T5+T7 uses this binder. It is a two-field Prop structure whose content is definitional equality with DimensionForcing, not a multi-step derivation.
Claim. The eight-tick period equals $2^3$, and the ledger cycle length obtained from spatial dimension via $2^D$ at $D=3$ equals that same eight-tick constant. Equivalently: the minimal ledger-compatible cycle is $2^D$, and with $D=3$ one obtains period $8$.
background
In the Unified Forcing Chain, T0–T8 are presented as inevitabilities from the cost foundation (Recognition Composition Law plus normalization and calibration). T7 is the discrete-time step: once spatial dimension is fixed, the fundamental evolution period of the ledger is forced to be an octave of length $2^D$.
Upstream, eight_tick is the natural-number constant $8$, and EightTickFromDimension D is defined as $2^D$. The RS-native tick $\tau_0=1$ is the fundamental time quantum; one octave is eight ticks. The module’s chain summary states T7 as “8-tick $\leftarrow 2^D$ with $D=3$,” paired with T8 forcing $D=3$ via linking and gap-45 sync.
This structure does not re-derive minimality of the cycle from cost; it records the two equalities that identify the named eight-tick with the dimension formula at three dimensions.
proof idea
No tactic proof: the declaration is a Prop-valued structure with two fields. The first asserts eight_tick = 2^3; the second asserts EightTickFromDimension 3 = eight_tick. Both sides are definitional in DimensionForcing (eight_tick := 8, EightTickFromDimension D := 2^D), so inhabitants are built by rfl-style equalities (as in the bridge lemma that constructs T7 from a T8 witness). Treat it as a certificate type, not a derived argument.
why it matters
T7 is the forcing-chain landmark for the eight-tick octave (period $2^3$). It is a field of CompleteForcingChain and of the T−1…T8 bridge certificate, and a hypothesis of the T5+T7 → canonical Hamiltonian bridge (cost-phase duality, small-deviation quadratic kinetic term, and DFT-8 eigenvalues from the cyclic shift). Public spine material treats the D=3 / eight-tick targets as the honest content behind older Nonempty T7_EightTick_Forced citations. Downstream, t7_from_t8 shows T7 is recovered once T8 supplies dimension three. Without this binder, the chain cannot name a forced discrete period linking ledger geometry to Hamiltonian and measurement layers.
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