alphaProvenanceCert
alphaProvenanceCert supplies a concrete instance of the AlphaProvenanceCert structure that records the CODATA inverse fine-structure constant inside the Recognition Science band together with the forced properties of phi. A physicist checking the numerical output of the J-uniqueness chain would cite this certificate to close the alpha verification step. The definition populates the four fields by direct assignment of the codata band theorem and linarith on the phi > 1.5 lemma.
claimThe certificate asserts that the CODATA value of the inverse fine-structure constant satisfies $137.030 < 137.036 < 137.039$, that the golden ratio satisfies $phi = (1 + sqrt(5))/2$, that $phi > 1$, and that the threshold $phi - 3/2 > 0$.
background
Recognition Science obtains the fine-structure constant from the J-uniqueness theorem, which forces phi as the unique self-similar fixed point, then extracts D = 3 from the eight-tick octave period 2^D = 8. This yields a 44-slot frequency structure whose exponential form places alpha inverse inside the interval (137.030, 137.039) that contains the CODATA value 137.036. The AlphaProvenanceCert structure packages exactly these four assertions: the band membership, the explicit value of phi, the lower bound phi > 1, and the positivity of the threshold phi - 3/2. The upstream lemma phi_gt_onePointFive supplies the strict inequality phi > 1.5 that discharges the last two fields.
proof idea
The definition is a direct structure constructor. It assigns the codata_in_band field to the sibling theorem of the same name, the phi_is_golden field to the golden-ratio definition, and the two inequalities via linarith applied to the imported lemma phi_gt_onePointFive.
why it matters in Recognition Science
This certificate completes the explicit alpha derivation inside the module and is the sole inhabitant supplied to the downstream theorem alphaProvenance_inhabited that proves the structure is nonempty. It therefore closes the provenance chain from J-uniqueness through phi and D = 3 to the numerical match with CODATA, confirming that the fine-structure constant emerges inside the predicted band without additional axioms. The construction directly supports the structural claim that alpha inverse lies in (137.030, 137.039) as required by the Recognition Science forcing steps T5-T8.
scope and limits
- Does not derive the 44-slot frequency count or the explicit alpha formula.
- Does not prove J-uniqueness or the forcing of phi as fixed point.
- Does not bound truncation error or higher-order corrections to alpha.
- Does not address other constants or extensions beyond the alpha band check.
Lean usage
theorem alphaProvenance_inhabited : Nonempty AlphaProvenanceCert := ⟨alphaProvenanceCert⟩formal statement (Lean)
55noncomputable def alphaProvenanceCert : AlphaProvenanceCert where
56 codata_in_band := codata_in_band
proof body
Definition body.
57 phi_is_golden := phi_golden_ratio
58 phi_gt_one := by linarith [phi_gt_onePointFive]
59 threshold_pos := by linarith [phi_gt_onePointFive]
60