Manuscript XI-A-ST@1 · XI-A-ST.tex · section · c3f0929d

The quadratic normalization

The signed action and its Euler derivatives › The quadratic normalization · lines 2075–2183

XI-A-ST@1#eq:Hfinite, XI-A-ST@1#app:comparisons

Set

u_{0}^{μν} = η^{μν}, u_{1}^{μν} = - γ^{μν}, u_{2}^{μν} = γ^{μ}_{ρ} γ^{ρ ν} .

For $n = 1, 2, 3$ , put

Γ_{n}_{μν}^{ρ} = \frac{1}{2} u_{n - 1}^{ρ σ} (\partial_{μ} γ_{ν σ} + \partial_{ν} γ_{μ σ} - \partial_{σ} γ_{μν}) .

Define

R_{n, μν} = r_{n} = C_{n, μν} = \partial_{ρ} Γ_{n}_{μν}^{ρ} - \partial_{ν} Γ_{n}_{μ ρ}^{ρ} + a + b = n a, b \geq 1 \sum (Γ_{a}_{ρ λ}^{ρ} Γ_{b}_{μν}^{λ} - Γ_{a}_{ν λ}^{ρ} Γ_{b}_{μ ρ}^{λ}), i + j = n i \geq 0, j \geq 1 \sum u_{i}^{μν} R_{j, μν}, r_{0} = 0, R_{n, μν} - \frac{1}{2} η_{μν} r_{n} - \frac{1}{2} γ_{μν} r_{n - 1} .

With $b_{0} = 1$ and $b_{1}, b_{2}$ from (eq:bs),

H_{n}^{μν} = i + j + ℓ + s = n i, j, ℓ \geq 0, s \geq 1 \sum b_{i} u_{j}^{μα} u_{ℓ}^{ν β} C_{s, α β}, 1 \leq n \leq 3.

Only indices through two occur on $u$ and $b$ in the required sums.

These formulas are the coefficient expansion of the displayed Christoffel, curvature, inverse-metric, and density expressions. They fix $H_{1}, H_{2}, H_{3}$ without leaving an unknown classical vertex in (eq:eexp). They do not construct a quantum product involving those coefficients.

comparisons and non-inputs

The following are role comparisons, not premise transfers.

{@{}p{.28}p{.66}@{}} Record & Role and non-identification
Record & Role and non-identification

I-B; X-A-ST & Prescribed regular host, signature ledger, destructive clock, and the existing refusal to infer Einstein dynamics from records.

X-L-ST & Storage and supported-release discipline only. Its action and free operator are not this action.

VIII-A, VIII-B, VIII-C & Conical certificates, persistent packet, and finite composition comparisons. No Einstein–scalar action is imported CheungVIII.

Oai282-A & Finite Ward, coverage, invisibility, and identification boundaries. No verification of OpenAI Family 282.

Oai283-A & A finite supplied-instance comparator, not the Einstein jet, tree, or field-acquisition map.

En08-A; En09-A & Scheduler/geometry separation and finite clock readout. Neither supplies this action or a free ghost sector.

III; VI; VII-A & Different equation spaces, signatures, or metric prescriptions. They do not replace XI-A's second-order classical sheet.

VII-C; VII-H; IX-A & Separate extension, background, and admission questions. No theorem is applied or contradicted by the present finite table.

Displayed formulas in this passage

unlabeled · line 2092
$R_{n, μν} = r_{n} = C_{n, μν} = \partial_{ρ} Γ_{n}_{μν}^{ρ} - \partial_{ν} Γ_{n}_{μ ρ}^{ρ} + a + b = n a, b \geq 1 \sum (Γ_{a}_{ρ λ}^{ρ} Γ_{b}_{μν}^{λ} - Γ_{a}_{ν λ}^{ρ} Γ_{b}_{μ ρ}^{λ}), i + j = n i \geq 0, j \geq 1 \sum u_{i}^{μν} R_{j, μν}, r_{0} = 0, R_{n, μν} - \frac{1}{2} η_{μν} r_{n} - \frac{1}{2} γ_{μν} r_{n - 1} .$
XI-A-ST@1#eq:Hfinite · line 2113
$H_{n}^{μν} = i + j + ℓ + s = n i, j, ℓ \geq 0, s \geq 1 \sum b_{i} u_{j}^{μα} u_{ℓ}^{ν β} C_{s, α β}, 1 \leq n \leq 3.$
unlabeled · line 2078
$u_{0}^{μν} = η^{μν}, u_{1}^{μν} = - γ^{μν}, u_{2}^{μν} = γ^{μ}_{ρ} γ^{ρ ν} .$
unlabeled · line 2084
$Γ_{n}_{μν}^{ρ} = \frac{1}{2} u_{n - 1}^{ρ σ} (\partial_{μ} γ_{ν σ} + \partial_{ν} γ_{μ σ} - \partial_{σ} γ_{μν}) .$

The quadratic normalization

Displayed formulas in this passage

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