cardinalitySpectrumCert
The cardinalitySpectrumCert definition assembles a concrete record witness for the CardinalitySpectrumCert structure, confirming that spectrum cardinalities decompose into products and powers of spatial dimension 3 and configuration dimension 5. Cross-domain analysts in Recognition Science cite it to verify the structured spectrum arises canonically from RS primitives rather than arbitrarily. The construction is a direct record builder that assigns reflexivity to base dimensions and references prior decidable equalities for derived relations.
claimLet $D_s=3$ denote spatial dimension and $D_c=5$ configuration dimension. The certificate asserts $gap_{45}=D_s^2 D_c$, $eightTick=2^{D_s}$, $cubeFaces=2 D_s$, $360=eightTick gap_{45}$, $70=binom{8}{4}$, $125=D_c^3$, $3125=D_c^5$, together with the spectrum being finite, pairwise distinct, and bounded above by 3125.
background
The CrossDomain.CardinalitySpectrum module collects witnesses that RS cardinalities form a structured spectrum generated from cube-generators {2,3}, configuration dimension 5, and gap45. The structure CardinalitySpectrumCert bundles the required equalities: Dspatial_is_3, Dconfig_is_5, gap_as_D, eightTick_as_D, cubeFaces_as_D, full_turn, choose_central, D_cubed, D_fifth, plus spectrum length, pairwise ordering, and boundedness. Upstream results supply the decidable identities eightTick_eq : eightTick = 2^Dspatial, gap45_eq : gap45 = Dspatial^2 Dconfig, cubeFaces_eq : cubeFaces = twoFace Dspatial, and oneTwentyFive_is_Dcubed : 125 = Dconfig^3, each proved by decide.
proof idea
The definition builds the CardinalitySpectrumCert record by direct field assignment. Reflexivity discharges Dspatial_is_3 and Dconfig_is_5. The remaining fields are populated by the upstream decidable theorems gap45_eq, eightTick_eq, cubeFaces_eq, oneTwentyFive_is_Dcubed, threeOne25_is_D_fifth, rsSpectrum_length, rsSpectrum_pairwise_lt, and rsSpectrum_bounded.
why it matters in Recognition Science
This supplies the canonical witness for the module's structural claim that RS produces a non-random numerical spectrum decomposable into its primitives. It aligns with the eight-tick octave (T7) and D=3 spatial dimensions (T8) from the forcing chain. With zero downstream uses it functions as a terminal certificate closing the collection of exemplar witnesses rather than an intermediate lemma.
scope and limits
- Does not prove the listed spectrum exhausts every RS cardinality.
- Does not derive the spectrum from the J-cost functional equation.
- Does not address continuous or real-valued extensions.
- Does not connect to the mass ladder or fine-structure constant bounds.
formal statement (Lean)
133def cardinalitySpectrumCert : CardinalitySpectrumCert where
134 Dspatial_is_3 := rfl
proof body
Definition body.
135 Dconfig_is_5 := rfl
136 gap_as_D := gap45_eq
137 eightTick_as_D := eightTick_eq
138 cubeFaces_as_D := cubeFaces_eq
139 full_turn := threeSixty_is_tick_gap
140 choose_central := seventy_is_choose_8_4
141 D_cubed := oneTwentyFive_is_Dcubed
142 D_fifth := threeOne25_is_D_fifth
143 spectrum_length := rsSpectrum_length
144 spectrum_pairwise := rsSpectrum_pairwise_lt
145 spectrum_bounded := rsSpectrum_bounded
146
147end IndisputableMonolith.CrossDomain.CardinalitySpectrum