Manuscript I-B-VP@3 · I-B-VP-v3.tex · section · 93c04eb4

Retained SAE checks

Retained SAE checks · lines 3618–3709

The named files are

\nolinkurl co d e / s a e_{c} o mm o n . p y, \nolinkurl co d e / s a e_{s} y m . p y, \nolinkurl co d e / s a e_{n} u m . p y, \nolinkurl co d e / s a e_{l} e a d . p y, \nolinkurl co d e / s a e_{f} m . p y,

with /sae_*.json, including /sae_lead.json and /sae_fm.json. The source reports 30-digit work unless stated otherwise and a double-exponential quadrature rule tested on $\int x K_{ν}^{2}$ to $4 \times 1 0^{- 29}$ .

The source numbering skips C6. No missing C6 result is invented.

SAE-S1 The exact $E_{0}$ kernel reportedly gives the shifted Wick square, current, and four diagonal stress formulas with residual exactly zero in both modes. Conservation, trace, current conservation, and diagonality were reported exact. Theorem thm:shift supplies the independent proof.

SAE-S2 For the half-holonomy line family, conservation, trace, and current conservation were reported exactly zero. The compact forms reportedly agreed at rational sample points to $8 \times 1 0^{- 174}$ , and $θ = 0$ reproduced the single-mode formulas. Sampled high precision is not an exact identity certificate; Theorem thm:line gives the analytical calculation.

SAE-S3 The Krein stress integrands were reported to satisfy pointwise conservation numerically below $1 0^{- 166}$ (the abstract also reported $1 0^{- 167}$ ). Their $ν = 0$ translation to AKO (4.13)–(4.15) was reported with symbolic residual zero. The Bessel identity in Theorem thm:diagonal proves the conservation relation.

SAE-S4 A truncated comparison of the branch series $S (β \pm 1)$ was reported with residual $1 0^{- 63}$ . It is a truncation report, separate from the exact branch identity in Theorem thm:shift.

C1 Ten Krein-integral versus $E_{0}$ closed-form cases reportedly included irrational $α, β$ , mode $0$ , mode $- 1$ , and both, with $ξ = 0, 1/6, 0.25, - 0.1$ . The maximum relative deviation was $5.9 \times 1 0^{- 30}$ . Reported off-diagonal stress was at most $2 \times 1 0^{- 36}$ , radial current at most $1 0^{- 65}$ , and differentiated conservation, trace, and current residuals at most $7 \times 1 0^{- 31}$ .

C2 An eigenfunction expansion

\int_{0}^{\infty} k d k \frac{f _{k} ( r ) f _{k} ( r ^{'} ) - J _{ν} ( k r ) J _{ν} ( k r ^{'} )}{k ^{2} + p ^{2}}, f_{k} \propto t (k) J_{ν} + J_{- ν},

was reportedly compared with the Krein term at $r \neq = r^{'}$ . Four $E_{0}$ or $λ > 0$ cases gave $1.1 \times 1 0^{- 15}$ at 20 digits, described as limited by an oscillatory tail.

C3 For matrix relations the reports give: Hermiticity residual $3 \times 1 0^{- 47}$ ; boundary-condition residual $1 0^{- 42}$ from the $I_{\pm ν}$ split and $6 \times 1 0^{- 16}$ from a direct small-radius fit; and positive $C$ in the tested nonnegative cases. For 12 random Hermitian matrices, the reported determinant-root count agreed with the number of negative matrix eigenvalues, including zero, one, and two transverse bound states. Theorem thm:spectrum proves the count without random tests.

Displayed formulas in this passage

unlabeled · line 3622
$\nolinkurl co d e / s a e_{c} o mm o n . p y, \nolinkurl co d e / s a e_{s} y m . p y, \nolinkurl co d e / s a e_{n} u m . p y, \nolinkurl co d e / s a e_{l} e a d . p y, \nolinkurl co d e / s a e_{f} m . p y,$
unlabeled · line 3688
$\int_{0}^{\infty} k d k \frac{f _{k} ( r ) f _{k} ( r ^{'} ) - J _{ν} ( k r ) J _{ν} ( k r ^{'} )}{k ^{2} + p ^{2}}, f_{k} \propto t (k) J_{ν} + J_{- ν},$

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section: Retained SAE checks