t7_holds
plain-language theorem explainer
The eight-tick octave is forced once spatial dimension is three: the minimal ledger-compatible cycle is $2^3=8$. Anyone assembling the T0–T8 inevitability chain cites this packaging of T7. The proof is a two-field term discharging both equalities by reflexivity on the dimension-forcing definitions.
Claim. The named eight-tick period equals $2^3$, and the eight-tick constructed from spatial dimension $D=3$ coincides with that period. Equivalently, the minimal ledger-compatible cycle forced by dimension three is exactly eight ticks.
background
In the Unified Forcing Chain module, every step from the absolute floor through T8 is packaged as a forced inevitability from the Recognition Composition Law plus normalization and calibration. T7 is the octave step: the fundamental evolution period is eight ticks, not a free parameter.
Upstream constants fix the RS-native time quantum $\tau_0=1$ (one tick) and record spatial dimension as the natural number $D=3$. The combinatorial content is the $D$-hypercube: its vertex count is $2^D$. With $D=3$ one obtains eight configurations per full cycle, the eight-tick octave of the primer (period $2^3$).
The T7 record is a Prop-structure with two fields: the bare identity $8=2^3$ for the named eight-tick constant, and the statement that building the eight-tick from dimension three recovers that same constant. The bridge copy of the same structure carries the same two equalities.
proof idea
Term-mode construction of the T7 record. Both fields are definitional equalities in the dimension-forcing layer (eight_tick = 2^3 and EightTickFromDimension 3 = eight_tick), so each is closed by rfl. No intermediate lemmas are invoked; the proof is pure definitional unfolding.
why it matters
Closes landmark T7 in the forcing chain: the eight-tick octave with period $2^3$. The module diagram states T7 as "8-tick $\leftarrow 2^D$ with $D=3$", with T8 separately forcing $D=3$ from linking plus gap-45 sync. Downstream, T0_T8_holds_proven in the gravity master theorem bundles this witness with the other t*_holds results to assert the full T0–T8 package. Without T7, the chain cannot hand the octave period to later constant and mass constructions that treat eight ticks as the fundamental evolution cycle.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.