Public research · 2026-10-09

Scalar calculation repaired at r = 6M

Independent trace agreement at one radius, under a combined empirical allowance. The scalar-only allowance does not cover the residual. This is not a full stress-tensor validation.

TRACE_R6_NUMERICAL_PASS_UNDER_EMPIRICAL_BUDGETSCALAR ALLOWANCE SHORTFULL RSET FALSEGATE NO

Checkpoint VAL-B1-SCALAR-TRACE-R6.

What changed

The massless scalar subtraction was repaired and compared with the stored PMR trace at one radius, r = 6M. A September confirmation of the predeclared oscillation, a narrow published-curve comparison, and a target-exposed Hadamard derivation are now on the ledger as separate, narrower results.

Evidence

As of 2026-10-09. Scalar trace status TRACE_R6_NUMERICAL_PASS_UNDER_EMPIRICAL_BUDGET. Residual 1.5474844313788875×10⁻¹¹ against a combined empirical allowance 3.861830391164996×10⁻⁹. CNG confirmation is a 21 September narrow pass on ω ∈ [10.40, 10.90]. Published-curve agreement is 12/12 at four radii and does not repair the MRX_R_INVALID chain.

Limit

The allowance is empirical and dominated by the inherited PMR estimate. It is not a statistical confidence interval or a rigorous infinite-tail bound. Nine nearby scalar nodes support the derivative at r = 6M; they are not nine trace validations. No validated throat, backreaction solution, stability result, or engineering readiness.

Next step

Supported omitted-mode remainder control for the differentiated scalar sum; independent trace comparisons around r = 5.5M and 6.5M; then independent tensor components with consistent conventions, trace, conservation, and PMR comparison.

Editorial summary · not a rerun

One radius, one comparison

Schwarzschild, Hartle–Hawking, massless scalar, M = ℏ = 1, r = 6M. The tensor comparison uses minimal coupling. These figures are the public research summary. This website did not recompute them.

Status: TRACE_R6_NUMERICAL_PASS_UNDER_EMPIRICAL_BUDGET
φ²: 1.9571596066001755×10⁻⁴
First derivative: −1.4850596580409429×10⁻⁵
Second derivative: 6.65897313952517×10⁻⁶
Box(φ²): 3.1414970956971594×10⁻⁷
Scalar-inferred trace: −1.2088044476173205×10⁻⁷
Stored PMR trace: −1.2086496991741827×10⁻⁷
Absolute residual: 1.5474844313788875×10⁻¹¹
Scalar-only allowance: 5.251546195221486×10⁻¹²
Inherited PMR allowance: 3.856578844969775×10⁻⁹
Combined allowance: 3.861830391164996×10⁻⁹

The residual is about 2.95× the scalar-only allowance. That allowance does not cover it.

The inherited PMR allowance dominates the combined budget. The combined allowance covers the residual and is about nine times smaller than the sensitivity target 3.585508742915911×10⁻⁸. Exact status TRACE_R6_NUMERICAL_PASS_UNDER_EMPIRICAL_BUDGET.

\operatorname{tr} T=-\tfrac{1}{2}\square\langle\phi^{2}\rangle+\dfrac{M^{2}}{60\pi^{2} r^{6}}

NUMERICAL_BUDGET_PROTOCOL

Empirical, not a confidence interval

Primary integral: N = 24, L = 48, 256-point quadrature, Duffy transform after the coincidence singularity. A rectangular grid showed resolution-dependent drift and is saved only as a diagnostic. It is excluded from the primary result.

E(a,b) = ½((2/3)|a″ − b″| + (5/18)|a′ − b′|), so derivative errors cannot cancel. Each scalar refinement envelope is multiplied by ten and then summed. The largest scalar contribution is the profile-derivative referee. Retrospective and target-exposed. No fitted normalization, constants, or counterterms. Multiplying a finite-cutoff change by ten is not a proof of an omitted-mode bound.

1,225 mode pairs at the nine nearby radii. The nine radii 5.92, 5.94, 5.96, 5.98, 6, 6.02, 6.04, 6.06, 6.08 support the derivative check at r = 6M. They are not nine independent trace validations. The exact static mode was checked at all nine. Local or second-host agreement is reproducibility, not external peer review.

Implementation

Two confirmed corrections

A blanket massless skip omitted 18 nonzero logarithmic coefficients and six nonzero rational-tail coefficients. 105 stored expressions specialize exactly to massless Schwarzschild. The notebook was reparsed: 21 log cells and 15 rational-tail cells.

The archived static-mode outer-boundary third-order coefficient had the wrong sign and the wrong denominator.

\alpha=\ell+1,\quad M=1

\gamma=-\dfrac{\alpha(3\alpha+2)}{2\alpha+1}

\gamma_{\mathrm{archived}}=+\dfrac{\alpha(3\alpha+2)}{2\alpha-1}

For ℓ = 0 the correct and archived coefficients are −5/3 and +5.

Fresh static modes use exact Legendre solutions. Exact g₀₀ = log(r/(r−2))/8 = 0.05068313851352055, against an archived 0.05068410711710673. That single mode shifts φ² by about 1.22675×10⁻⁸, most of the old/new scalar difference of about 1.22716×10⁻⁸. The archived first derivative is close to the exact derivative, so this scalar offset is not itself the earlier derivative or trace discrepancy.

Do not merge lanes

Minimal here, conformal on the older B1 lane

φ² is coupling-independent on this Ricci-flat background. The stress tensor is not. The conformal anomaly-only relation is a distinct check. Older conformal TRACE PASS stamps keep their dates. They are not relabeled as this minimal-coupling result.

The 7 October stored-tensor prediction Box⟨φ²⟩ = 3.141187598810884×10⁻⁷ is a prediction from the stored tensor. It is not this 9 October independent Box.

Scope

What this pass does not certify

No validated throat, backreaction, stability, or engineering readiness. No independent validation of every tensor component. E_VALIDATED stays false. The full multi-radius chain remains MRX_R_INVALID. Mode outputs, coefficient tables, and server logs are not published. Secrets and private server paths stay off this site.

Replaces the CNG window as the current task

What comes next

01
Omitted-mode remainder control
A supported bound on the frequency and angular remainders of the differentiated full m=6 sum. Per-mode or per-shell corrected quantities, with n and ℓ limits kept separate. A single doubled rectangle or a fitted power is not enough. Primary values stay frozen as the reference.
02
Traces around r = 5.5M and 6.5M
The same independent solver and subtraction, without retuning, compared with the stored PMR traces at those radii. The present nine-radius neighborhood is only 5.92M–6.08M and is a derivative check, not those comparisons.
03
Individual tensor components
Euclidean time and angular derivatives, local coincidence terms, and the minimally coupled physical stress tensor. Check index position, Lorentzian continuation, trace, radial conservation, and the PMR components. A scalar trace pass does not certify every component or its sign.

Checkpoint VAL-B1-SCALAR-TRACE-R6.

October 2026 research summary
Editorial handoff. Not a runnable reproduction package.
October 2026 public results (JSON)
Machine-readable summary of the same editorial handoff. Not code, inputs, or logs.

Plain-language terms

PMR: Pragmatic mode-sum regularization. Levi–Ori t-splitting: subtract a local frequency polynomial, then integrate the regularized remainder in a generalized sense.
CNG: Connecting-null geodesic. A remote caustic of the two-point function. After local subtraction it can remain as an oscillation in frequency, not as a term in F_sing.
RSET: Renormalized stress-energy tensor. The quantum source that would enter the semiclassical Einstein equation. Not yet independently reproduced here.
HH: Hartle–Hawking state. The thermal quantum state regular on both past and future horizons of an eternal black hole.
GI / self-cancellation: Generalized integral of a non-decaying oscillatory remainder. Finite-support numerical success is not a proof of the infinite-frequency limit.
Ward: Covariant conservation of the stress tensor. A stored primary-tolerance diagnostic is not full independent closure of that identity.
E: The finite local Hadamard counterterm removed after the frequency integral. A derivation candidate is not E_VALIDATED.
B1: First calibration benchmark: known Schwarzschild Hartle–Hawking physics. Partial. Not a portal test.
φ²: Renormalized field variance ⟨φ²⟩. On this Ricci-flat background the scalar variance does not depend on curvature coupling; the stress tensor does.
Mτ: Mixed Euclidean-time derivative of the Green function. The displayed-sign question is paused and unconfirmed.