structureFormationFromBITCert
The declaration constructs an explicit inhabitant of the StructureFormationFromBITCert structure, which encodes the φ-ladder ratios for cosmological peak wavenumbers. Cosmologists modeling structure formation from the BIT kernel would cite it to confirm scale-invariant peak ratios of φ and φ². It is a direct record construction that assembles the four required properties from sibling definitions of positivity, adjacent ratios, and invariance.
claimThere exists a certificate satisfying: for all base wavenumbers $k_0 > 0$ and integers $n$, the peak wavenumber $k_*(k_0, n) > 0$, the adjacent ratio $k_*(k_0, n+1)/k_*(k_0, n) = φ$, the third-to-first ratio $k_*(k_0, 3)/k_*(k_0, 1) = φ^2$, and the ratios $k_*(k_0, n+m)/k_*(k_0, n)$ are independent of the choice of base scale.
background
The module treats the matter power spectrum as inheriting the φ-ladder structure of the BIT kernel, so that characteristic wavenumbers satisfy $k_n = k_0 φ^n$. It isolates the first three CMB acoustic-peak ratios as $k_2/k_1 = φ$ and $k_3/k_2 = φ$, hence $k_3/k_1 = φ^2$, together with the statement that these ratios are positive constants of φ independent of the base scale $k_0$ (parameter-free in the ratio sector). The master certificate StructureFormationFromBITCert collects exactly these four properties: positivity of peaks for every positive base, equality of adjacent ratios to φ, the explicit φ² relation for the third-to-first peak, and scale invariance across different bases.
proof idea
The definition is a direct record construction. It supplies the four fields of the certificate by direct assignment: positivity from the sibling k_peak_pos, adjacent ratio from k_peak_adjacent_ratio, the φ² relation from peak_3_1_ratio, and scale invariance from peak_ratios_scale_invariant.
why it matters in Recognition Science
This declaration closes Track F4 by exhibiting an explicit inhabitant of the master certificate, establishing the φ-rational ratio structure as a theorem with zero sorries. It supports the module's hypothesis of numerical match to Planck/DESI data and connects to the Recognition Science framework via the φ self-similar fixed point. The module falsifier is any of the first three CMB acoustic peaks observed at a wavenumber ratio more than 5% off the predicted φ or φ² values.
scope and limits
- Does not derive the explicit functional form of the matter power spectrum P(k).
- Does not fix a numerical value for the base wavenumber k0.
- Does not address peak amplitudes or overall normalization.
- Does not incorporate baryonic acoustic oscillations or dark-energy effects.
formal statement (Lean)
102def structureFormationFromBITCert : StructureFormationFromBITCert where
103 k_pos := k_peak_pos
proof body
Definition body.
104 adjacent_ratio := k_peak_adjacent_ratio
105 peak_3_1_eq_phi_sq := peak_3_1_ratio
106 scale_invariant := peak_ratios_scale_invariant
107
108end
109
110end StructureFormationFromBIT
111end Cosmology
112end IndisputableMonolith