Public research · PMR

PMR Asymptotic Expectation Audit

Analytic / code-mapping lane, now closed. ASYMPTOTIC_STRUCTURE_MIXED remains the correct classification. Global confirm is FALSE. Not Abel. Not an HH RSET reproduction.

GROWTH EXPECTED PRE-GIASYMPTOTIC STRUCTURE MIXEDGLOBAL CONFIRM FALSESPARSE 10/12 PASSDECAYING TAIL FALSEABEL NOT VALIDATED

PMR_ASYMPTOTIC_EXPECTATION_AUDIT = CLOSED · VERDICT = GROWTH_EXPECTED_PRE_GI · STRUCTURE = MIXED · GLOBAL CONFIRM = FALSE · FREQUENCY_SUPPORT_CONVERGED = FALSE

Validation checkpoint VAL-B1-PMR-ASYMPTOTIC-AUDIT.

The audit

Closed 19 September 2026 analytic audit. After local subtraction, F_reg is not expected to decay before a generalized integral. Sparse ω=10 and 12 passed the angular gate, and |F_reg^tt| grew. Global mixed-structure confirm is FALSE. This is not the current frontier. The current self-cancellation operator is the average of the two shifts; the 19 September sheet wrote the unnormalized sum.

Analytic expectation was frozen from Levi–Ori t-splitting before the ω=8,10,12 ratios were read as evidence. Those ratios are not a derivation of an envelope power.

Predeclared outcomes

GROWTH_EXPECTED_PRE_GI

SELECTED

The analytic PMR prescription permits — and in the published t-splitting RSET construction exhibits — qualitative growth / non-decay of F_reg(ω) before generalized integration.

DECAY_EXPECTED_POST_SUBTRACTION

NOT SELECTED

Would mean the implemented regularized spectrum should already decay after F_sing, so certified growth would identify another unresolved implementation/subtraction problem. Not selected.

ASYMPTOTIC_EXPECTATION_UNRESOLVED

NOT SELECTED

Would mean available derivation/literature is insufficient to determine the expected qualitative behavior. Not selected for the qualitative pre-GI question. Selected only for the unclaimed envelope-power / regime-identification question.

Referee verdict

Growth of the regularized frequency remainder is expected before generalized integration

Levi–Ori t-splitting subtracts only the local coincidence singularity of the Christensen / DeWitt–Schwinger counterterm. That subtraction removes a polynomial in ω (ω³ and ω for the Schwarzschild RSET). It does not cancel remote null-geodesic / caustic singularities of the two-point function.

Those remote singularities Fourier-transform into undamped oscillations of F_reg(ω). For ⟨φ²⟩ the published envelope is ∼ω^{1/2}. For the RSET, Levi (2017) shows at the same diagnostic radius r=6M that F_rr^Reg is dominated by oscillations with amplitude proportional to ω^{5/2}. The ordinary integral ∫ F_reg dω therefore need not converge. The remainder is defined to be taken as a generalized integral (Abel / self-cancellation).

Therefore a decaying high-ω tail of |F_reg(ω)| is not the analytic prediction of the implemented PMR prescription prior to generalized integration. Observing growth of |F_reg^tt| between certified angular-gated points at ω=8, 10, and 12 does not, by itself, establish a missing F_sing term.

DECAY_EXPECTED_POST_SUBTRACTION
NOT SELECTED
Rejected as the literature prediction. F_sing removes the local polynomial, not the remote-caustic oscillations. Levi 2017, r=6M, F_rr^Reg: amplitude ∝ ω^{5/2}.
ASYMPTOTIC_EXPECTATION_UNRESOLVED
NOT SELECTED
The qualitative pre-GI behavior — non-decaying oscillations, ordinary integral need not converge — is resolved by the published t-splitting RSET construction. The precise envelope power for the project's F_reg^tt at these three points is not identified.

DECAYING_HIGH_OMEGA_TAIL_DEMONSTRATED = FALSEASYMPTOTIC_REGIME_IDENTIFIED = FALSEFREQUENCY_SUPPORT_CONVERGED = FALSEASYMPTOTIC_STRUCTURE_MIXED_GLOBAL_CONFIRM = FALSEABEL_PRODUCTION_VALIDATED = FALSEB1_HH_REPRODUCTION_CLAIMED = FALSE

LITERATURE DEFINITION · PUBLIC DIAGNOSTIC MAPPING

What raw spectral object F(ω) is being computed

The public PMR diagnostic computes, at a fixed radial location, the angular-summed coincidence of the Christensen-type mode bilinear at frequency ω, before the local singular piece is subtracted. In Levi's t-splitting notation this is F_μν(ω, x).

Points are split in the Killing-time direction: x = (t, r, θ, φ), x′ = (t+ε, r, θ, φ). Each mode picks up a factor e^{−i ω ε}. The coincidence limit of the point-split stress tensor is rewritten as a cosine transform of F_μν(ω).

F_{\mu\nu}(\omega,x)\equiv\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\Re\bigl\{f_{\omega\ell m,\mu}(x)\,f_{\omega\ell m,\nu}^{*}(x)+g_{\omega\ell m,\mu}(x)\,g_{\omega\ell m,\nu}^{*}(x)\bigr\}

\bigl\langle\tilde T_{\mu\nu}(x)\bigr\rangle_{\mathrm{ren}}=\lim_{\varepsilon\to 0}\Biggl[\hbar\int_{0}^{\infty}F_{\mu\nu}(\omega,x)\cos(\omega\varepsilon)\,d\omega-\tilde L_{\mu\nu}(x,\varepsilon)\Biggr]

f are the in (or equivalent) modes; g are the second set required on an eternal black hole (up modes). Hartle–Hawking changes the occupation of those modes. It does not change the local Hadamard / DeWitt–Schwinger singularity, so it does not change F_sing.
The project's public diagnostic reports tt, rr, and θθ components of this object after angular summation at r=6M, ξ=5, HH. This audit does not invent an unpublished mode-normalization convention.
F(ω) as displayed here is not yet F_reg, not an RSET component, and not an Abel integral.

LEVI 2017 EQ. 17 · SCHWARZSCHILD REDUCTION

Exact implemented F_sing(ω), component by component

The local counterterm in the time split has the coincidence expansion L̃_μν(ε)/ℏ = a_μν ε^{−4} + b_μν ε^{−2} + c_μν ε^{−1} + d_μν ln(μ ε) + e_μν + O(ε). Distributional cosine transforms convert those powers of ε into polynomials / logs in ω. Levi's singular spectral object is therefore

F_{\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}}

F_{\mu\nu}^{\mathrm{Reg}}(\omega,x)=F_{\mu\nu}(\omega,x)-F_{\mu\nu}^{\mathrm{Sing}}(\omega,x)

Distributional cosine identities

\int_{0}^{\infty}\omega^{3}\cos(\omega\varepsilon)\,d\omega=6\,\varepsilon^{-4}

\int_{0}^{\infty}\omega\cos(\omega\varepsilon)\,d\omega=-\varepsilon^{-2}

\int_{0}^{\infty}\ln\omega\,\cos(\omega\varepsilon)\,d\omega=-\dfrac{\pi}{2}\,\varepsilon^{-1}

\int_{0}^{\infty}\dfrac{\cos(\omega\varepsilon)}{\omega+\mu e^{-\gamma}}\,d\omega=-\ln(\mu\varepsilon)+O(\varepsilon)

These identities are used in the generalized / Abel sense. They are how a local ε-expansion becomes a spectral F_sing, not ordinary L¹ Fourier transforms.

a_μ^ν(r): 1/(2π² f²) · diag{−3, 1, 1, 1}
b_μ^ν(r): M/(12π² r³ f²) · diag{0, −2, 1−3M/r, 1−3M/r}
c_μ^ν(r): 0
d_μ^ν(r): 0
e_μ^ν(r): finite local tensor (Levi 2017)

On Schwarzschild, F_sing(ω) = (1/6) a ω³ − b ω. For the mixed tt component the ω term drops, leaving a pure cubic: F^t_t Sing(ω) = (1/6) a^t_t ω³ with a^t_t < 0.

tt (mixed t t)

(1/6) a^t_t ω³

a^t_t = −3 / (2π² f²)

b^t_t = 0

rr (mixed r r)

(1/6) a^r_r ω³ − b^r_r ω

a^r_r = +1 / (2π² f²)

b^r_r = −2M / (12π² r³ f²)

θθ (mixed θ θ)

(1/6) a^θ_θ ω³ − b^θ_θ ω

a^θ_θ = +1 / (2π² f²)

b^θ_θ = M(1−3M/r) / (12π² r³ f²)

Expected large-ω expansion after subtraction

Write F(ω) = F_local-poly(ω) + F_remote(ω) + F_smooth(ω).

F_sing is constructed to cancel F_local-poly, i.e. the Fourier image of the DeWitt–Schwinger / Christensen coincidence singularity at ε=0. After subtraction, F_reg(ω) = F_remote(ω) + F_smooth(ω) + (any mismatch of local polynomials).

F_smooth is the state-dependent, locally analytic remainder. It is not required to dominate the large-ω diagnostic. F_remote comes from singularities of the two-point function at nonzero time separations ε_n along null geodesics that connect x to a distinct point with the same (r,θ,φ) after n revolutions (caustics / “remote singular points”).

For ⟨φ²⟩, Levi & Ori (2015) show that a remote singularity ∼ |ε−ε_s|^{−3/2} produces oscillations in F_reg whose amplitude grows as ω^{1/2}. The two extra derivatives in the Christensen stress-tensor operator raise the local coincidence from ∼ε^{−2} to ∼ε^{−4} (Fourier ω → ω³) and raise the remote singularity to ∼|ε−ε_s|^{−7/2} (Fourier envelope ω^{5/2}).

Levi (2017), at r=6M — the same radius as this project's PMR diagnostic — states that F_rr^Reg is dominated by oscillations with an amplitude proportional to ω^{5/2}. The integral function H_rr(ω)=∫_0^ω F_rr^Reg also oscillates with that envelope. The ordinary integral does not converge. The generalized / self-cancellation integral is the definition of the ω-integral in the method.

F_{\mu\nu}(\omega)=\underbrace{\tfrac16 a_{\mu\nu}\omega^{3}-b_{\mu\nu}\omega}_{\mathrm{local\ coincidence}} + F_{\mu\nu}^{\mathrm{remote}}(\omega)+F_{\mu\nu}^{\mathrm{smooth}}(\omega)

F_{\mu\nu}^{\mathrm{Reg}}(\omega)=F_{\mu\nu}^{\mathrm{remote}}(\omega)+F_{\mu\nu}^{\mathrm{smooth}}(\omega)+\Delta_{\mathrm{poly}}(\omega)

F_{\mu\nu}^{\mathrm{remote}}(\omega)\sim\sum_{n}\Re\bigl(A_{n}\,\omega^{p}\,e^{i\omega\varepsilon_{n}}\bigr),\qquad p=\tfrac12\ (\langle\phi^{2}\rangle),\quad p=\tfrac52\ (\mathrm{RSET\ at\ }r=6M\mathrm{\ published\ for\ }rr)

Polynomial growth of F itself

YES, ∼ω³, cancelled by F_sing

Decay of |F_reg| after F_sing, before GI

NO — not the literature prediction

Oscillation of F_reg at large ω

YES — remote null geodesics / caustics

Growth of the oscillation envelope

YES — published ω^{5/2} for F_rr^Reg at r=6M

Ordinary ∫_0^∞ F_reg dω converges

NOT REQUIRED

Generalized / Abel / self-cancellation integral

YES — this is the production ω-integral

Which residual terms vanish only under the generalized-integral prescription

Levi explicitly allows F_reg to contain “blind spots”: spectral pieces B(ω) whose cosine transform vanishes for ε≠0 but whose ordinary integral ∫_0^∞ B(ω) dω diverges or fails to exist. Oscillatory remainders with amplitude ∼ω^p, 0<p, are of this type.

The generalized integral is lim_{α→0+} ∫_0^∞ e^{−α ω} F_reg(ω) dω, equivalently Abel summation. The operator written here on 19 September is the unnormalized sum T_λ[H](ω)=H(ω)+H(ω+λ/2). Dated correction, 9 October 2026: the analysis now in use averages those two shifts, T_λ[H]=(H(ω)+H(ω+λ/2))/2, and applies normalized repeated multi-frequency operators to the running integral H. A finite-support readout is not a rigorous infinite-support result. This correction does not say that a paper or its numerical tables are wrong.

Terms that are supposed to vanish, or to settle, only under that prescription:

Remote-caustic oscillations ∼ ω^p e^{i ω ε_n}
Not cancelled by F_sing. Ordinary integral diverges. Generalized / self-cancellation integral exists for p below the operator threshold.
Polynomial blind spots B_n(ω) that Fourier-vanish for ε≠0
Not part of F_sing. Defined to be handled by the generalized integral, not by raising ℓ_max and not by ε→0 of a truncated Ω integral in the wrong order.
Finite local tensor e_μν
Not a large-ω remainder. Subtracted after the ω-integral. Irrelevant to whether |F_reg| decays.
State-dependent smooth remainder
This is the physical integrand. It is what one hopes survives GI. It is not yet isolated at three sparse certified frequencies.

ABEL_LIMIT_ORDER remains Ω → ∞ FIRST. Do not claim that raising polynomial degree in ε repairs missing ω support. Do not run Abel in this update.

Implementation mapping and omitted-term audit

This laboratory website does not contain the production PMR source. The mapping is from the frozen public diagnostic (Levi–Ori F_sing, angular-summed F, F_reg = F − F_sing, r=6M, ξ=5, HH, adaptive ℓ gate) onto the published t-splitting RSET construction. It is not a line-by-line code dump.

Raw object

MAPPED

Public diagnostic F(ω) is the angular-summed Christensen mode bilinear at fixed (r,ω), i.e. Levi F_μν(ω) after ℓ-sum. Adaptive gate now requires that ℓ-sum to be ANGULAR_ACCEPTED.

F_sing leading ω³

FIRING · DISFAVORS OMISSION

At the already-certified ω=8, L=75 point, |F_reg|/|F|_tt ≈ 0.0124 and F/F_sing_tt ≈ 1.013. If the cubic term were omitted, |F_reg|/|F| would remain O(1). The leading local polynomial is being subtracted.

F_sing subleading ω (tt)

STRUCTURALLY ZERO IN MIXED tt

Levi's mixed b^t_t = 0 on Schwarzschild. An omitted ω term cannot explain tt growth.

F_sing ln ω and 1/ω

STRUCTURALLY ZERO ON SCHWARZSCHILD

c=d=0. Not available as missing counterterms for this background and field.

Remote-caustic remainder / GI step

FINITE SUPPORT LATER · INFINITE SUPPORT NOT PROVED

As written in this 19 September audit, the generalized-integral step had not been applied, so absence of decay was not an omitted F_sing term. By 3 October a normalized finite-support self-cancellation was implemented. That is not a proof of the infinite-frequency limit. ABEL_PRODUCTION_VALIDATED stays false in that sense.

Index placement / HH weights / exact Christensen combination

NOT LINE-AUDITED HERE

Cannot be certified from this sandbox. Remaining possible implementation defects are not excluded. They are not the leading explanation of qualitative growth after healthy |F_reg|/|F|≪1.

Angular-splitting F(l) subtraction

NOT THIS OBJECT

Levi–Ori 2016 angular-splitting F_sing(l) is a different regularization. Do not mix F(l) decay statements into F(ω).

A missing leading coincidence counterterm is disfavored by the certified ω=8 cancellation. The remaining growth of the small F_reg remainder is the term the generalized-integral prescription is designed to handle. This does not certify that every subleading implementation choice is correct.

Sparse extension 10/12 · frozen numbers only

Certified sparse extension 10/12

Frozen adaptive angular-convergence gate. Predeclared L_ceiling = 180. Silent accept forbidden. Status SPARSE_EXTENSION_PASS. L_ceiling = 180.

ω = 8

ANGULAR_ACCEPTED

Confirmation: YES (plateau L=90 vs 75 and 105 vs 90)

Historical certified point. Ratio denominator.

ω = 10

ANGULAR_ACCEPTED

Confirmation: YES

Sparse extension. L* not published in this freeze.

ω = 12

ANGULAR_ACCEPTED

Confirmation: YES

Sparse extension. L* not published in this freeze.

CERTIFIED_POINTWISE_DOMAIN_TO_OMEGA12 = PASS_CONDITIONAL_ON_ELL_GATE.
This is not a dense spectrum. It is not a claim that every integer ω ≤ 12 is certified. It is not frequency-support convergence. It is not Abel.
Do not automatically extend to ω>12. Additional certified frequencies are not authorized until this audit is digested.

Numerical data, used only after the analytic expectation was frozen

|F_reg^tt(10)| / |F_reg^tt(8)|: 2.18019
|F_reg^tt(12)| / |F_reg^tt(8)|: 4.02997

The ratios are greater than 1. |F_reg^tt| grew. A decaying high-ω tail is therefore not demonstrated. DECAYING_HIGH_OMEGA_TAIL_DEMONSTRATED = FALSE stands.

Qualitative growth of |F_reg| after F_sing is what the frozen analytic expectation permits. The observation is consistent with GROWTH_EXPECTED_PRE_GI. It is not a measurement of p=5/2, and it is not a demonstration that the remote-caustic regime has been reached.

ASYMPTOTIC_REGIME_IDENTIFIED = FALSE. FREQUENCY_SUPPORT_CONVERGED = FALSE. No Abel. No ω>12 extension. No ε→0.

Audit steps

1
Write down raw F(ω)
Levi F_μν(ω) · angular-summed Christensen bilinear
PASS
2
Write down F_sing(ω) componentwise
Schwarzschild: (1/6)a ω³ − b ω · c=d=0
PASS
3
Derive expected large-ω remainder
Remote-caustic oscillations; local polynomial cancelled
PASS
4
Decay vs growth vs oscillation before GI
GROWTH / NON-DECAY EXPECTED · ordinary integral not required
HOLD
5
Residuals that vanish only under GI
ω^p e^{iωε_n} blind spots · Abel / self-cancellation
HOLD
6
Literature cross-check
Levi 2017 r=6M F_rr^Reg amplitude ∝ ω^{5/2}
PASS
7
Omitted-term audit
Leading ω³ firing · GI step not yet applied · subleading not line-audited
HOLD
8
Power-law fit to ω=8,10,12
FORBIDDEN · not a derivation
FAIL
9
Numerical ratios after analytic freeze
Grew · consistent with pre-GI remainder · not a regime ID
HOLD

Literature

Levi & Ori (2015)
Pragmatic mode-sum regularization method for semiclassical black-hole spacetimes
arXiv:1503.02810
t-splitting ⟨φ²⟩. Definition of F(ω), F_sing(ω) ∼ ω, F_reg oscillations with envelope ∼ω^{1/2} from remote null geodesics, generalized / self-cancellation integral, consistency with Abel summation.
Levi (2017)
Renormalized stress-energy tensor for stationary black holes
arXiv:1611.05889
t-splitting RSET. Exact F_μν(ω), F_sing = (1/6) a ω³ − b ω − (2/π) c ln ω − d/(ω+μ e^{−γ}), Schwarzschild a,b,c,d,e tensors, Fourier identities, and the r=6M statement that F_rr^Reg is dominated by oscillations with amplitude ∝ ω^{5/2}.
Levi & Ori (2016)
Mode-sum regularization of ⟨φ²⟩ in the angular-splitting method
arXiv:1606.08451
Contrast only. Angular-splitting regularizes a different object F(l), not F(ω). Its large-ω mode envelope decays. It is not the spectral object of the present PMR frequency diagnostic.
Christensen (1976)
Vacuum expectation value of the stress tensor in an arbitrary curved background: The covariant point-separation method
Phys. Rev. D 14, 2490
Origin of the bitensor differential operator whose coincidence expansion supplies the local ε^{−4}, ε^{−2}, ln ε pieces that become the ω³, ω, ln ω terms in F_sing.

After this freeze

This analytic mapping is frozen. The confirmation window is no longer the next task.

This analytic mapping is frozen. ASYMPTOTIC_STRUCTURE_MIXED remains the correct classification. ASYMPTOTIC_STRUCTURE_MIXED_GLOBAL_CONFIRM = FALSE. The 20 September confirmatory window was later computed and received a narrow pass; it is not the current next task. Current next work is omitted-mode remainder control for the differentiated scalar sum, then traces near 5.5M and 6.5M. Do not invent a new local subtraction term merely to force pointwise decay.

Not Abel integration.
Not ε → 0.
Not automatic ω>12.
Not Howard-guided tuning.
Not a new local subtraction term.
Not numerical CNG confirmation.

CNG discovery study Angular-gate study Validation checkpoint

Claim language

After Levi–Ori F_sing subtraction, F_reg is not expected to decay before the generalized integral.

ASYMPTOTIC_STRUCTURE_MIXED remains the correct asymptotic classification.

Levi (2017) exhibits growing oscillations of F_rr^Reg at r=6M with amplitude ∝ ω^{5/2}.

Certified |F_reg^tt| growth from ω=8 to 12 is consistent with that qualitative expectation.

A decaying high-ω tail has not been demonstrated.

This is not frequency-support convergence.
This is not Abel production.
This is not an HH RSET reproduction.
The ω=8,10,12 ratios are not a derived envelope power.
ASYMPTOTIC_REGIME_IDENTIFIED = FALSE.
ASYMPTOTIC_STRUCTURE_MIXED_GLOBAL_CONFIRM = FALSE.
The production PMR source was not line-audited in this website freeze.

Do not say

The high-ω tail has been shown to decay.
Ω → ∞ frequency support has converged.
We fitted |F_reg| ∼ ω^p on {8,10,12} and that is the Levi–Ori prediction.
Certified growth proves a missing F_sing term.
Abel production is validated.
Run Abel now.
Automatically extend to ω>12.
A new local subtraction term is needed to force pointwise decay.

B1_HH_REPRODUCTION_CLAIMED = FALSE · ABEL_PRODUCTION_VALIDATED = FALSE · FREQUENCY_SUPPORT_CONVERGED = FALSE · PRODUCTION_SIGN_FLIP = NO · GATE = NO