Skip to main content
Paper 4 of the TFPT 4.5 seriesConditional Closure
TFPT 4.52026-04-27385 KBSHA-256 27ab3e60…f60d
Paper 4Conditional Closure

Admissibility, Strong CP, and Nonperturbative QFT Closure

Selector vs. dynamics on the TFPT branch

The analytic closure layer of TFPT. The selector P_adm is treated as a physical admissible-sector construction, while the dynamics is carried by Z_rel, admissible Schwinger distributions, Osterwalder–Schrader reconstruction, the local Minkowski net, stable massive scattering, and the exact admissible RG flow.

Inputs
  • ›Paper 1 supplies P_prim.
  • ›Paper 2 supplies the carrier and discrete determinant data.
  • ›Paper 3 is not logically required except for cross-references.
Contribution
  • ›Conditional nonperturbative closure: P_adm = P_prim · P_sing · P_Θ, with θ_eff = 0 and arg det M_u = arg det M_d = 0.
  • ›Reflection positivity, OS reconstruction, local Minkowski net, stable massive scattering, exact admissible RG flow.
Not claimed here
  • ›No α detail calculation.
  • ›No CMB, no E8.
  • ›No full empirical tables.
Falsification surface
  • ›Fails if selector and dynamics are conflated, if positivity/gap hypotheses are hidden, or if strong-CP closure uses an inadmissible phase convention.
Download Paper 4View PDF
Highlights
θ_eff0Theorem-level null, not a tuned parameter
OS✓Osterwalder–Schrader reconstruction inside the admissible sector
SelectorP_admThree projectors: primitive, singlet, theta

Key formulas

  • Admissibility selector
    Padm=Pprim Psing PΘP_{\mathrm{adm}} = P_{\mathrm{prim}} \, P_{\mathrm{sing}} \, P_\Theta
    Composition of three admissibility projectors.
  • Strong-CP null
    θeff=0\theta_{\mathrm{eff}} = 0
    Theorem-level null on the admissible branch.

Selector vs. Dynamics

The central distinction: P_adm selects the physical sector, while dynamics is carried by Z_rel, then S^T_n, then OS reconstruction, then the local net, the flow Γ_k, and the renormalized observable layer. This separation is the main defence against overclaiming.

Padmselects the physical sectorP_{\mathrm{adm}} \quad \text{selects the physical sector}
Zrel⇒{SnT}⇒(Hadm,Aadm)⇒Γk⇒GTFPTrenZ_{\mathrm{rel}}\Rightarrow\{S_n^T\}\Rightarrow(\mathcal{H}_{\mathrm{adm}},\mathfrak{A}_{\mathrm{adm}})\Rightarrow \Gamma_k \Rightarrow \mathfrak{G}^{\mathrm{ren}}_{\mathrm{TFPT}}

Full Admissibility Complex

After carrier and determinant data are fixed, the full selector is composed of three admissibility projectors.

Padm=Pprim Psing PΘP_{\mathrm{adm}} = P_{\mathrm{prim}} \, P_{\mathrm{sing}} \, P_\Theta

Strong CP Closure

The strong-CP sector is stated as an admissibility result. The argument connects hadronic singlet selection, determinant structure, γ₅-Hermiticity, and the sheet involution without importing phenomenological tuning.

θeff=0\theta_{\mathrm{eff}} = 0
arg⁡det⁡Mu=arg⁡det⁡Md=0\arg\det M_u = \arg\det M_d = 0

Exact Admissible RG Flow

The exact admissible flow is the analytic continuation of the same sector, with the admissible projection included in the flow definition.

∂kΓk=12STr⁡ ⁣[(Γk(2)+Rk)−1∂kRk]adm\partial_k \Gamma_k = \tfrac{1}{2}\operatorname{STr}\!\left[(\Gamma_k^{(2)} + R_k)^{-1}\partial_k R_k\right]_{\mathrm{adm}}

Key formulas at a glance

  • Admissibility selector
    Padm=Pprim Psing PΘP_{\mathrm{adm}} = P_{\mathrm{prim}} \, P_{\mathrm{sing}} \, P_\Theta

    Composition of three admissibility projectors.

  • Strong-CP null
    θeff=0\theta_{\mathrm{eff}} = 0

    Theorem-level null on the admissible branch.