IndisputableMonolith.Foundation.T7CycleRealization
Module realizing the T7 eight-tick cycle as a closed walk on the D-cube in D-bit pattern space. It ties Gray-code Hamiltonian cycles to image-type statements (circle, not higher sphere) and defect predicates used on the T7/T8 dimension route. Anyone citing cycle realization before D=3 forcing would land here. Content is definitional structure plus short lemmas on adjacency, edge-distinctness, and topological image.
claimA closed walk on the $D$-cube is a phase-indexed path through the space of $D$-bit patterns (maps $\mathrm{Fin}\, D\to\{\mathrm{true},\mathrm{false}\}$). The module packages: Hamiltonian and edge-distinct closed walks; the image of such a walk as a topological circle; the claim that the image is not a higher-dimensional sphere; and recognized/realized defect predicates supporting T7 cycle realization.
background
Recognition Science forces an eight-tick octave at T7 (period $2^3$) and spatial dimension $D=3$ at T8. The substrate side of that route needs an explicit combinatorial model of the tick cycle: states are patterns on the $D$-cube, adjacency means Hamming distance one, and a full period is a closed walk of length $2^D$.
Upstream, GrayCycle upgrades pattern counting to an adjacent Hamiltonian cycle on the hypercube: a length-$2^d$ closed walk visiting every pattern once. SubstrateAxioms records predicate-level structural inputs for the T7/T8 dimension route in the same style as the Alexander-duality layer: named commitments without claiming Mathlib already supplies the full topology stack.
This module sits between those inputs and dimension forcing. It names closed walks on the cube, Hamiltonian and edge-distinct variants, circle versus sphere-of-dimension image predicates, and recognized versus realized defects, so later modules can treat the T7 cycle as realizing a circle in pattern space.
proof idea
Not a single theorem: a small foundation layer. Definitions introduce closed walks on the $D$-cube, Hamiltonian and edge-distinct properties, circle and sphere-image predicates, and recognized/realized defects. Short lemmas then connect the pieces: edge-distinctness for dimension at least two; the image of a closed walk is a circle; no higher sphere arises from such a closed walk; and the packaged T7 statement that the cycle realizes a circle. The combinatorial backbone is imported from the Gray-cycle Hamiltonian construction; substrate axioms supply the predicate interface the dimension route expects.
why it matters in Recognition Science
T7 in the forcing chain is the eight-tick octave. This module is the combinatorial realization step: the tick cycle as a closed walk whose image is a circle, not a higher sphere, which is the topological hook dimension forcing needs.
It is imported by DimensionForcing (the module that proves spatial $D=3$ is forced, including the linking/topological argument), by UnifiedForcingChain (T0–T8 as inevitabilities from the cost foundation), and by the T6–T8 spine audit that tags what is theorem versus forced-conditional. Without a named cycle-realization layer, the T7→T8 bridge would be informal. Landmarks: T7 period $2^3$, T8 $D=3$.
scope and limits
- Does not by itself prove spatial dimension $D=3$; that lives in DimensionForcing.
- Does not derive the eight-tick period from the cost law; it realizes the cycle combinatorially.
- Does not supply full Mathlib algebraic topology; image-as-circle is predicate-level.
- Does not construct the Gray Hamiltonian cycle; that is imported from Patterns.GrayCycle.
- Does not address mass ladders, $\alpha$, or constants outside the T7/T8 route.
used by (3)
depends on (2)
declarations in this module (16)
-
structure
ClosedWalkOnCube -
def
Hamiltonian -
def
EdgeDistinct -
def
ImageIsCircle -
def
ImageIsSpherePofDim -
inductive
RecognizedDefect -
def
Circle -
def
RealizedDefect -
theorem
edge_distinct_of_dim_ge_two -
theorem
closed_walk_image_is_circle -
theorem
no_higher_sphere_from_closed_walk -
theorem
t7_cycle_realizes_circle -
def
grayCycle3ClosedWalk -
theorem
grayCycle3ClosedWalk_hamiltonian -
theorem
grayCycle3_realizes_circle -
theorem
grayCycle3_no_higher_sphere