V_reciprocal_symm
plain-language theorem explainer
The inflaton potential on the recognition manifold obeys V(φ) = V(1/(1+φ) - 1) for φ > -1. Cosmologists working with recognition-derived inflation models cite this result to confirm that the slow-roll trajectory inherits the universal J-cost symmetry. The proof is a short tactic sequence that unfolds the definition of V, rewrites the transformed argument via ring, and invokes the Jcost_symm lemma.
Claim. For all real numbers φ > -1, let V(φ) := J(1 + φ) where J denotes the J-cost function. Then V(φ) = V(1/(1 + φ) - 1).
background
The module constructs the inflaton field φ_inf as the deviation of the Z-coordinate from the consciousness-gap reference rung. The potential is defined by V(φ_inf) = Jcost(1 + φ_inf), so that all algebraic identities of the J-cost transfer directly to cosmology. The module's setting extends prior work on gap-45 identities by supplying an explicit scalar field whose slow-roll parameters vanish at the reference point in canonical units. The central upstream lemma states that Jcost(x) = Jcost(x^{-1}) for x > 0, which encodes the reciprocal symmetry of the recognition composition law.
proof idea
The tactic sequence unfolds V to expose Jcost(1 + phi_inf). It obtains positivity of the argument by linarith. A ring equality rewrites the transformed argument to the reciprocal of the original argument. The proof then applies the Jcost_symm lemma by exact.
why it matters
This theorem supplies the reciprocal_symm field inside the inflatonPotentialCert certificate that aggregates all structural properties of the potential. It imports the J-cost reciprocal symmetry (originating in T5 J-uniqueness of the forcing chain) into the cosmological sector, ensuring the slow-roll trajectory remains canonical and universal. The result therefore grounds the earlier n_s ≈ 0.9556 and r ≈ 0.017 predictions in an explicit scalar field on the recognition manifold.
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