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Document 7 of the TFPT 5.0 setOpen research gates
TFPT 5.02026-06-08577 KBSHA-256 6aa91814…4930
Paper 7Open research gates

Research Contracts for the Two Open Gates

(U_wall) the parabolic flavor wall-selection · (G_metric) the full QG measure

After the compiler closure the entire residual is Rest = (U_wall) ⊕ (G_metric) ⊕ (F_frontier). This note turns the two genuine research gates into contracts: a numbered chain of lemmas, the single theorem that closes each gate, and — for every step — whether it is machine-certifiable today. F_frontier is not a gate. Priority: (U_wall) first (finite, algebraic, falsifiable), then (G_metric) (deep analytic programme).

Inputs
  • ›The closed compiler and the two named residual gates from the introduction's status card.
Contribution
  • ›Contract 1 (U_wall) — now complete: the quark ratios are closed (Readout Rigidity) and the absolute amplitude U_point reduces to one overall scale v_geo (ratios + Grand Mass Volume) — the same dimensionful anchor as gravity's 1/G. The two [A] anchors collapse to one.
  • ›Contract 2 (G_metric): IR tier closed under RP + gap (Decoupling Theorem, Δ_eff = 1.648 > 0); the ambient measure G6 is holographically reduced to a finite seam-boundary measure that is the rigorously-constructed (E₈)₁ lattice net (c = 8 = 5 + 3, conformal embedding (D₅)₁×(A₃)₁, coset c = 0) — so G6 is imported into existing RCFT/conformal-net rigor, not built anew.
  • ›The quark ratio c_u/c_d = 55/117 is closed (Readout Rigidity); the '11' is the Pascal sum 16 − g_car.
Not claimed here
  • ›U_point is not a free transcendental input but the single overall scale v_geo (shared with 1/G); the strict claim is only that one dimensionful anchor remains.
  • ›The ambient boundary projective measure (G6) is reduced but not closed; it blocks certification as a strict physical TOE, but its absence does not affect the bounded IR claim.
Falsification surface
  • ›Each contract names its closing theorem and certifiability; fails if a lemma certified [F] does not in fact machine-check, or if the closing theorem is asserted before its chain completes.
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Highlights
Gates2(U_wall) flavor + (G_metric) quantum gravity
U_point→ v_geoGate 1 closed: the single overall scale (= 1/G anchor)
c_u/c_d55/117Closed by Readout Rigidity
G6→ (E₈)₁ netReduced to the rigorously-constructed E₈ level-1 lattice net (c = 8 = 5 + 3)
v_geo1 scaleDimensional-analysis floor: one scale + π; shared by flavor & gravity

Key formulas

  • Gate 1 closed
    Upoint→vgeo=the 1/G anchorU_{\mathrm{point}} \to v_{\mathrm{geo}} = \text{the } 1/G \text{ anchor}
    Ratios + Grand Mass Volume ⇒ one overall scale. [I]/[A]
  • Quark ratio closed
    cucd=5⋅119⋅13=55117\frac{c_u}{c_d} = \frac{5\cdot 11}{9\cdot 13} = \frac{55}{117}
    Readout Rigidity on the discrete stratum. [I]
  • Gate 2 reduction
    2∥V∥=314π2<Δ=6log⁡32⇒Δeff=1.6482\|V\| = \tfrac{31}{4\pi^2} < \Delta = 6\log\tfrac32 \Rightarrow \Delta_{\mathrm{eff}} = 1.648
    IR closed (decoupling); G6 reduced to a seam-boundary measure. [I]/[P]

Contract 1 — (U_wall), the flavor wall

The goal is to select the one D₄-symmetric realisation on the family curve. The selectors det R = 8 and Spec(Q₊) = {1,2,3} are read off the bundle; the quark ratio is closed by Readout Rigidity, leaving only the absolute amplitude scale.

det⁡R=8=n⋅a,Spec⁡(Q+)={1,2,3}=3α+1\det R = 8 = n\cdot a, \qquad \operatorname{Spec}(Q_+) = \{1,2,3\} = 3\alpha + 1
cucd=gcar ∥Pl(K)∥1Nfam2 ΔQ=5⋅119⋅13=55117\frac{c_u}{c_d} = \frac{g_{\mathrm{car}}\,\|\mathrm{Pl}(K)\|_1}{N_{\mathrm{fam}}^2\,\Delta_Q} = \frac{5\cdot 11}{9\cdot 13} = \frac{55}{117}

Theorem U — the four-way split

The remaining flavor bridge splits into four pieces: unitarity (polystable ⇒ unitary, finite linear algebra), the H2 readoff, the Λ² readout rigidity, and U_point (the full amplitude normalisation) — which now reduces to the single overall scale v_geo (the same anchor as 1/G). Gate 1 is complete.

(Uwall)=Uunitary+UH2+UΛ2+Upoint(U_{\mathrm{wall}}) = U_{\mathrm{unitary}} + U_{\mathrm{H2}} + U_{\Lambda^2} + U_{\mathrm{point}}
∥Pl(K)∥1=∑k=02(4k)=11=16−gcar\|\mathrm{Pl}(K)\|_1 = \textstyle\sum_{k=0}^{2}\binom{4}{k} = 11 = 16 - g_{\mathrm{car}}

Contract 2 — (G_metric), the QG measure

The goal is the reflection-positive projective-limit measure over the diffeomorphism-quotiented metric sector. G2 (Seeley–DeWitt R + R²) and G5 (gap dominance, Decoupling Theorem) are certified; the projective limit G6 is now holographically reduced — because the seam is a finite causal boundary, the bulk measure is reconstructed from a finite seam-boundary (Calderón) measure, so G6 is a boundary projective limit (conditional on RP + tightness), not a diffuse bulk problem.

a2=−R3,a4∣R2=R272a_2 = -\tfrac{R}{3}, \qquad a_4\big|_{R^2} = \tfrac{R^2}{72}
2∥Vmetric∥=0.785<Δ=6log⁡32=2.433,Δeff=1.648>02\|V_{\mathrm{metric}}\| = 0.785 < \Delta = 6\log\tfrac{3}{2} = 2.433, \qquad \Delta_{\mathrm{eff}} = 1.648 > 0

Certifiability and order

(U_wall) is finite, algebraic and falsifiable today; (G_metric) is a deep analytic programme. The recommended order freezes the frontier status in between.

(Uwall)→freeze frontier status→(Gmetric)(U_{\mathrm{wall}}) \rightarrow \text{freeze frontier status} \rightarrow (G_{\mathrm{metric}})

Key formulas at a glance

  • Gate 1 closed
    Upoint→vgeo=the 1/G anchorU_{\mathrm{point}} \to v_{\mathrm{geo}} = \text{the } 1/G \text{ anchor}

    Ratios + Grand Mass Volume ⇒ one overall scale. [I]/[A]

  • Quark ratio closed
    cucd=5⋅119⋅13=55117\frac{c_u}{c_d} = \frac{5\cdot 11}{9\cdot 13} = \frac{55}{117}

    Readout Rigidity on the discrete stratum. [I]

  • Gate 2 reduction
    2∥V∥=314π2<Δ=6log⁡32⇒Δeff=1.6482\|V\| = \tfrac{31}{4\pi^2} < \Delta = 6\log\tfrac32 \Rightarrow \Delta_{\mathrm{eff}} = 1.648

    IR closed (decoupling); G6 reduced to a seam-boundary measure. [I]/[P]