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ASTRAL GATELaboratory

Canonical equation sheet

Astral Gate — equations

Each block is labeled by lane and date. The current self-cancellation operator is the average. The 19 September sum stays as history. The minimal-coupling trace at r = 6M is not the conformal anomaly identity.

PRINTED CONVENTION · POTENTIAL / REQUIRES EXTERNAL CONFIRMATION · LANE PAUSED

Mixed-time coincidence derivative under printed Green-function conventions

Δτ=τ′−τ\Delta\tau = \tau' - \tau
∂τ∂τ′exp⁡ ⁣[inκ(τ′−τ)]=+n2κ2 exp⁡ ⁣[inκ(τ′−τ)]\partial_{\tau}\partial_{\tau'}\exp\![i n \kappa(\tau'-\tau)] = + n^{2}\kappa^{2}\,\exp\![i n \kappa(\tau'-\tau)]
gττ′→+1/fg^{\tau\tau'} \to +1/f

Under Δτ = τ' − τ and exp[i n κ(τ' − τ)], direct differentiation gives +n² κ² exp[…]. Euclidean time parallel propagator coincidence audit: g^{ττ'} → +1/f. Independently reconstructed mode contribution carries the DERIVED plus structure. The displayed mixed-time formula carries the opposite overall sign. Internal derivations 3/3 PLUS. Not confirmed. Archaeology unknown. Lane paused. No portal claim.

NUMERICAL-INSTRUMENT GATE · NOT AN RSET

PMR adaptive angular-tail convergence diagnostic

Dℓ=∣Freg(L)−Freg(L−15)∣max⁡(∣Freg(L)∣,ε)<0.05D_{\ell}=\dfrac{\bigl|F_{\mathrm{reg}}^{(L)}-F_{\mathrm{reg}}^{(L-15)}\bigr|}{\max\bigl(|F_{\mathrm{reg}}^{(L)}|,\varepsilon\bigr)}<0.05

Acceptance rule frozen before the ω=8 plateau run and reused by the adaptive gate: all of tt/rr/θθ must satisfy D_ℓ < 0.05 for two consecutive terminal windows with healthy cancellation. An ANGULAR_ACCEPTED point is eligible for a future frequency-integration dataset. Not Abel production. Not HH RSET.

LEVI 2017 EQ. 17 · SCHWARZSCHILD REDUCTION · NOT AN RSET

PMR singular spectral object F_sing(ω)

FμνSing(ω,x)=16aμνω3−bμνω−2πcμνln⁡ω−dμν1ω+μe−γF_{\mu\nu}^{\mathrm{Sing}}(\omega,x)=\dfrac{1}{6}a_{\mu\nu}\omega^{3}-b_{\mu\nu}\omega-\dfrac{2}{\pi}c_{\mu\nu}\ln\omega-d_{\mu\nu}\dfrac{1}{\omega+\mu e^{-\gamma}}
FμνReg(ω)=Fμν(ω)−FμνSing(ω)F_{\mu\nu}^{\mathrm{Reg}}(\omega)=F_{\mu\nu}(\omega)-F_{\mu\nu}^{\mathrm{Sing}}(\omega)
Schwarzschild: c=d=0,FttSing=16attω3\text{Schwarzschild: }c=d=0,\quad F^{t}{}_{t}^{\mathrm{Sing}}=\tfrac16 a^{t}{}_{t}\omega^{3}

t-splitting subtracts only the local coincidence polynomial. On Schwarzschild, c=d=0 so F_sing = (1/6)a ω³ − b ω. Mixed b^t_t = 0, so the tt ω term is structurally zero. Remote caustic oscillations are not cancelled here. Not Abel. Not HH RSET.

GENERALIZED INTEGRAL · FINITE-SUPPORT EXISTS · INFINITE SUPPORT NOT PROVED

Self-cancellation operator, with the current average

19 Sep 2026, unnormalized history:Tλ[H](ω)=H(ω)+H(ω+λ/2)\text{19 Sep 2026, unnormalized history:}\quad T_{\lambda}[H](\omega)=H(\omega)+H(\omega+\lambda/2)
Current average:Tλ[H](ω)=12(H(ω)+H(ω+λ/2))\text{Current average:}\quad T_{\lambda}[H](\omega)=\dfrac{1}{2}\bigl(H(\omega)+H(\omega+\lambda/2)\bigr)
H(ω)=∫0ωFregH(\omega)=\int_{0}^{\omega}F_{\mathrm{reg}}

The 19 September sheet wrote the unnormalized sum. The October analysis uses the average of the two shifts, applied to the running integral H, including normalized repeated multi-frequency operators. Finite-support numerical success is not a proof of the infinite-frequency limit, and it is not a claim that a paper or its tables are wrong. ABEL_PRODUCTION_VALIDATED stays false for that infinite-support sense.

DISCOVERY MODEL · ε₁ FIXED · NOT CONFIRMATION

Fixed-ε₁ connecting-null-geodesic model on the dense window

x=ω−10,ε1=37.5x=\omega-10,\qquad \varepsilon_1=37.5
H0:F(ω)=c0+c1x+c2x2+c3x3H_0:\quad F(\omega)=c_0+c_1 x+c_2 x^{2}+c_3 x^{3}
H1:F(ω)=c0+c1x+c2x2+c3x3+Acos⁡(ε1ω)+Bsin⁡(ε1ω)H_1:\quad F(\omega)=c_0+c_1 x+c_2 x^{2}+c_3 x^{3}+A\cos(\varepsilon_1\omega)+B\sin(\varepsilon_1\omega)

H0 is a smooth cubic in x=ω−10. H1 adds A cos(37.5 ω)+B sin(37.5 ω) with ε₁ held fixed, never fitted. Discovery evidence on ω∈[9.8,10.3], not confirmation, not Abel, not HH RSET.

2026-10-09 · MINIMAL COUPLING · ONE RADIUS · EMPIRICAL ALLOWANCE

Minimal-coupling trace used at r = 6M

tr⁡T=−12□⟨ϕ2⟩+M260π2r6\operatorname{tr} T=-\tfrac{1}{2}\square\langle\phi^{2}\rangle+\dfrac{M^{2}}{60\pi^{2} r^{6}}
ξ=0,M=ℏ=1,r=6M\xi=0,\quad M=\hbar=1,\quad r=6M

On Ricci-flat Schwarzschild, φ² is coupling-independent. The stress tensor is not. This is the October comparison at ξ = 0, M = ℏ = 1. It is not the conformal-anomaly identity of the older B1 lane. A historical conformal trace pass is not this check.

SCHEMATIC · NOT THE UNIQUELY COMPLETE FORM

Minimal / schematic semiclassical Einstein equation

Gμν[g]=8πGc4 ⟨Tμν[g,Ψ]⟩renG_{\mu\nu}[g] = \dfrac{8\pi G}{c^{4}}\,\langle T_{\mu\nu}[g,\Psi]\rangle_{\mathrm{ren}}

The quantity we ultimately need to calculate reliably is the renormalized stress tensor from a specified field, state, geometry, and renormalization prescription. Displayed form is a minimal schematic.

SCHEMATIC

Self-consistent loop

g0→⟨Tμν[g0]⟩ren→g1→⟨Tμν[g1]⟩ren→⋯g_0 \to \langle T_{\mu\nu}[g_0]\rangle_{\mathrm{ren}} \to g_1 \to \langle T_{\mu\nu}[g_1]\rangle_{\mathrm{ren}} \to \cdots

The geometry determines the quantum state / RSET, and the resulting RSET determines the geometry. A desired fixed point satisfies g_{n+1} → g_n while all physical constraints remain satisfied. Not claimed.

HOLD · NOT PROMOTED

Four-point Hartle–Hawking components

Ttt,Trr,TθθT^{t}{}_{t},\quad T^{r}{}_{r},\quad T^{\theta}{}_{\theta}
ξrad∈{1.2, 1.8, 3.0, 5.0}\xi_{\mathrm{rad}} \in \{1.2,\,1.8,\,3.0,\,5.0\}

Assembly machinery exists. RSET_ASSEMBLY_DEV and FOUR_POINT_DEV remain HOLD. No component is a B1 PASS.

HOLD · NOT A PRODUCTION PASS

Ward / conservation identity

∇μ ⟨Tμν⟩ren=0\nabla_{\mu}\,\langle T^{\mu}{}_{\nu}\rangle_{\mathrm{ren}} = 0
gμν⟨Tμν⟩ren  ⟷  conformal anomalyg^{\mu\nu}\langle T_{\mu\nu}\rangle_{\mathrm{ren}} \;\longleftrightarrow\; \text{conformal anomaly}

Historical trace machinery recorded a pass under the implementation then in force. That stamp is not the 9 October minimal-coupling comparison at r=6M. Ward remains HOLD. Conservation is not claimed. The production time/radial split does not yet satisfy the full Ward/Howard gate.

HOWARD HISTORICAL · LOW5_NOT_CERTIFIED

Howard conservation combination C_S

CS1920=∂ξδrr+2ξ+1(δrr−δθθ)+1(ξ+1)(ξ−1)(δrr−δtt)\dfrac{C_S}{1920}=\partial_{\xi}\delta_{rr}+\dfrac{2}{\xi+1}(\delta_{rr}-\delta_{\theta\theta})+\dfrac{1}{(\xi+1)(\xi-1)}(\delta_{rr}-\delta_{tt})

Closed pieces used in the historical Howard audit. No corrected C_S is claimed. LOW5_NOT_CERTIFIED.

EXTERNAL LITERATURE DISCREPANCY

HPS97 / Popov vacuum-log sector

8π⟨Tll⟩log⁡,w0HPS97=+ln⁡f360πr48\pi\langle T^{l}{}_{l}\rangle^{\mathrm{HPS97}}_{\log,w_0} = +\dfrac{\ln f}{360\pi r^{4}}
8π⟨Tll⟩log⁡,w0Popov=−ln⁡f360πr48\pi\langle T^{l}{}_{l}\rangle^{\mathrm{Popov}}_{\log,w_0} = -\dfrac{\ln f}{360\pi r^{4}}

Blind derivation supports the HPS97 Letter sign. Popov Appendix B (B2) rejected for primary. Not an Astral Gate implementation bug.