IndisputableMonolith.Masses.StepValueEnumeration
The module enumerates the natural invariants consisting of Q₃ cell counts and their linear combinations at D=3. These integers receive direct combinatorial meaning and supply the concrete values required by the alpha derivation and sector-dependent torsion constraints. The module consists of definitions together with short lemmas that verify summation, uniqueness, and partitioning properties of the listed pairs.
claimAt dimension 3 the natural invariants are the cell counts $V=8$, $E=12$, $F=6$, $C=1$ together with the combinations $V+F=14$, $2V+1=17$, $V+F-C=13$, $V+E=20$, $E+F=18$, $V+E+F=26$, $F+C=7$, $E+C=13$, $E-C=11$ and the relation $E-A=11$.
background
Recognition Science works at D=3 with the cubic ledger Q₃ whose cell counts are fixed by the geometry. The module records these counts and the integer combinations that arise as wallpaper-group orders or Euler characteristics. Upstream AlphaDerivation derives 4π from vertex deficits of Q₃. SectorDependentTorsion proves that the set {13,11,6,8} satisfies the cyclic-chain and 2D^D+1=55 partition constraints.
proof idea
The module proceeds by direct enumeration of the cell counts followed by algebraic verification of each listed summation and uniqueness property. Sibling lemmas check middle-pair and endpoint-pair relations that partition the relevant totals and confirm distinctness modulo the edge-pair filter.
why it matters in Recognition Science
The enumerated invariants close the combinatorial side of the D=3 forcing chain and supply the integer inputs required by the alpha derivation from the cubic ledger and by the sector-dependent torsion constraints. They therefore stand immediately before any mass-formula application that uses the phi-ladder.
scope and limits
- Does not derive the cell counts from a more primitive axiom set.
- Does not connect the invariants to the phi-ladder or mass formulas.
- Does not address higher-dimensional generalizations of the counts.
- Does not treat the relation to the eight-tick octave or J-uniqueness.
depends on (2)
declarations in this module (14)
-
def
natural_invariants_D3 -
def
middle_pairs_summing_to_17 -
theorem
middle_pairs_are_11_6 -
theorem
middle_pair_unique_nonzero -
def
endpoint_pairs_summing_to_21 -
theorem
endpoint_pairs_not_unique -
theorem
current_chain_partition -
theorem
current_chain_middle -
theorem
current_chain_endpoints -
theorem
current_chain_distinct -
theorem
chain_unique_given_edge_pair -
theorem
current_chain_unique_modulo_edge_pair_filter -
theorem
euler_identity_Q3 -
theorem
thirteen_natural_interpretations