Gravity Btfremergence

Baryonic Tully-Fisher Relation Emergence

The baryonic Tully-Fisher relation (BTFR) is an empirical correlation between the total baryonic mass of a galaxy and its asymptotic rotation velocity. Recognition Science treats this correlation as a consequence of its modified gravity law, not as a separate fact. The module BTFREmergence establishes the algebraic bridge between the Recognition Science rotation curve modification and the BTFR power law, and proves the key deep-regime result.

The empirical BTFR states that baryonic mass M_b scales as rotation velocity v_f to a power β, with β approximately 3.5 to 4 and intrinsic scatter around 0.1 to 0.2 dex. The module derives this power law from the Recognition Science assumption that observed acceleration is a modified version of the baryonic acceleration, using a power-law form for the modification. The derivation assumes a flat rotation curve at large radius, where centripetal acceleration equals the observed acceleration and the baryonic acceleration is Keplerian.

The module proves a theorem, btfr_mass_velocity_relation, that under these assumptions the baryonic mass is proportional to the rotation velocity raised to the power 4/(2 − α), where α is the modification exponent. This is the general BTFR slope. The naive formula gives β ≈ 2.48 for α = 0.389, which is too low. The module's docstring explains that the correct derivation, accounting for how the modification produces flat rotation, gives β = 4 in the deep modification regime.

The central theorem is the deep-regime result. In the deep modification regime, where baryonic acceleration is much smaller than the acceleration scale a₀, the exponent is exactly 4. The theorem btfr_deep_regime proves that if M_b = v_f^4 / (G * a₀), then there exists a positive constant C = 1/(G * a₀) such that M_b = C * v_f^4. This constant is independent of the galaxy's mass. The theorem btfr_constant_universal then proves that for any two galaxies in this regime, the ratio of their masses equals the fourth power of the ratio of their velocities. This universality of the constant explains the tight scatter observed in the BTFR.

What the module does not do is prove a fully structural BTFR emergence theorem with a nontrivial constant independent of M_b. The file provides a mechanically checkable power-law form wrapper, but the docstring states that a fully structural theorem is still in progress. The observed β ≈ 3.5 is attributed to finite radii, mixed disk and halo contributions, and the transition between Newtonian and modified regimes, not to a single derived exponent.

THEOREM btfr_mass_velocity_relation · IndisputableMonolith/Gravity/BTFREmergence.lean

THEOREM btfr_deep_regime · IndisputableMonolith/Gravity/BTFREmergence.lean

THEOREM btfr_constant_universal · IndisputableMonolith/Gravity/BTFREmergence.lean

What this page does not claim

This answer does not claim that the BTFR is fully derived from Recognition Science first principles; only the algebraic power-law form is established. This answer does not claim that the observed β ≈ 3.5 is derived; the module attributes it to finite radii and mixed regimes. This answer does not claim that the acceleration scale a₀ or the exponent α are derived from the forcing chain; they are parameters in the modification law.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/BTFREmergence.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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