t8_holds
plain-language theorem explainer
Spatial dimension is forced to D=3: nontrivial linking (ledger conservation), the eight-tick identity 2^D=8, and uniqueness of an RS-compatible dimension all pin D uniquely. Anyone citing the T0–T8 inevitability chain or the complete forcing package needs this certificate. The proof is a three-field structure constructor that wires in the DimensionForcing theorems (Alexander duality for linking, power-of-two for eight-tick, and the uniqueness theorem).
Claim. There is a certificate that spatial dimension equals $3$: (i) every dimension supporting nontrivial linking equals $3$; (ii) every dimension whose binary tick count equals the eight-tick equals $3$; (iii) there exists a unique RS-compatible dimension.
background
The Unified Forcing Chain module records that T0 through T8 are forced from the cost foundation (Recognition Composition Law, normalization, calibration), not merely compatible with it. In that ladder, T7 is the eight-tick octave and T8 is the claim that spatial dimension is not a free parameter: D=3.
The certificate packages three DimensionForcing facts. Linking support is the topological condition that ledger conservation needs nontrivial circle linking; by Alexander duality that holds only in D=3. The eight-tick map sends dimension D to the period 2^D, and equating that period to eight forces D=3 by elementary power-of-two arithmetic. RS-compatibility bundles linking, eight-tick sync, and gap-45 synchronization; the uniqueness theorem shows the only such dimension is 3.
Upstream, linking_requires_D3 is the primary T8 theorem ("Alexander duality — no reference to 8-tick or gap-45"), eight_tick_forces_D3 is the pure 2^D route, and dimension_forced is the unique-existence package that also recovers lcm(8,45)=360.
proof idea
Term-mode structure constructor, not a tactic script. The three fields of the T8 certificate are filled by direct assignment:
- linking forces D=3 is
DimensionForcing.linking_requires_D3(Alexander duality on circle linking); - eight-tick forces D=3 is
DimensionForcing.eight_tick_forces_D3(unfold 2^D and apply the power-of-two lemma); - unique RS-compatible dimension is
DimensionForcing.dimension_forced(exhibit D=3 and invoke uniqueness).
No new algebra is done here; the declaration is the assembly point that turns those three DimensionForcing theorems into the single T8 Prop used by the rest of the chain.
why it matters
This is the T8 landmark in the forcing chain: D=3 spatial dimensions, forced rather than assumed. It closes the dimension step that T7 (eight-tick octave, period 2^3) already anticipates, and it is the topological half of the gap-45 / eight-tick synchronization story.
Downstream it is consumed by the complete T-1-to-T8 bridge (complete_forcing_chain_t8), the physical packaging (physical_forcing_chain), the public spine certificate, the extended ultimate-inevitability statement, and the realization-bridge form of T8 that sits beside the T7.5 cellular-completion and 1-acyclicity hypotheses. Cosmology also reaches for it when phase-space density scaling is specialized to the forced dimension.
Within the stronger "Complete Inevitability Chain" claim of this module, T8 is the last geometric pin: once cost forces logic, MP, discreteness, ledger, recognition, unique J, phi, and the eight-tick, dimension cannot float. Constants (c, hbar, G, alpha) then sit on a fixed geometric substrate.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.