- Pullback of a closed form is closed.
- Matter coupled to g̃ = A² g has diffeomorphism-invariant conservation in the physical frame.
- Worldline variation of −m ∫ A ds yields a^ν = −c² h^νμ ∇_μ ln A.
- The dust limit of the Euler equation reproduces that worldline force.
- u_μ f^μ = 0, so four-velocity normalization is preserved.
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Established geometry, conjectural physics
The RTCH paper is explicit: conformal matter coupling and closed-form pullbacks are differential geometry. Identifying elliptic-curve invariants with thermodynamic fields, and asserting that A depends on 𝒥_Q, are conjectural until constrained against nested nulls. Current repository hypotheses appear in the claim registry; their recorded statuses come from the canonical repository snapshot. Browser demonstrations provide no controlled or physical support.
Structural chain
- 01
State map
Φ : M → 𝒯, x ↦ Y^A(x)
- 02
Pullback
F_Q = Φ* ω, d F_Q = 0
- 03
Local scalar
𝒥_Q = ¼ F_Q² (not Q_RTCH)
- 04
Conformal factor
A = A(φ, ℳ, 𝒥_Q) > 0
- 05
Physical metric
g̃_μν = A² g_μν
- 06
Matter action
S_m[Ψ_m, g̃]
- 07
Both regimes
particles, fluids, T_m, exchange
H₀ : β_Q = 0
- Nontrivial [ω] ∈ H²(𝒯) with thermodynamic interpretation.
- A depends on 𝒥_Q rather than another invariant of F_Q.
- Y^A built from (log N, log |Δ|, r, log R, log |Ω|, …).
- Scale-dependent H_eff from period weighting.
- Arithmetic projection as a source of large-scale structure.
Limitations (paper §53)
Candidate action
The master action is a variational realization, not a uniqueness theorem.
Nontrivial class
Existence of closed non-exact ω is an assumption on 𝒯.
Choice of 𝒥_Q
¼ F² is a chosen local invariant among several possible scalars.
A(φ, ℳ, 𝒥_Q)
The conformal coupling is additional model structure.
Two-form B
Introduces extra dynamical degrees of freedom.
No entropy production
A conservative action does not automatically describe irreversible processes.
Arithmetic ≠ thermodynamic
Y^A(E) does not by itself identify rank with mass or conductor with energy.
Open empirics
Local coupling remains untested until a map from observables to (φ, ℳ, F_Q) is specified.
What would count
A meaningful RTCH-E1 result is a stable H1/H2 signature longer than matched nulls, surviving seeds, feature subsets, and isogeny-class controls, with geodesic or scalar coordinates that predict a held-out invariant. Even then it is structure in a chosen metric representation — not a physical law. Falsification is equally useful: persistence indistinguishable from N1, or topology explained by a single marginal.