Public research

Mixed-Time Derivative Sign Reproducibility Study

Independent symbolic + finite-separation reproduction · externally reviewable · not confirmed · paused pending external clarification.

EXTERNALLY REVIEWABLEPAUSEDNOT CONFIRMED3/3 PLUSARCHAEOLOGY UNKNOWNPUBLIC REPRO READYAUTHOR NOTE UNSENT

POTENTIAL_PUBLISHED_MTAU_SIGN_ISSUE = EXTERNALLY_REVIEWABLE · CONFIRMED = FALSE · PAUSED_PENDING_EXTERNAL_CLARIFICATION · SIGN_VERDICT = UNRESOLVED · PHASE_D_PUBLISHED_NUMERICS = UNKNOWN

Validation checkpoint VAL-B1-MTAU-SIGN-REPRO.

Public minimal reproduction pack

MTAU_SIGN_REPRO_PUBLIC_MINIMAL

SHA256 07a1af741749a5b6b73c73ed98fe308e0903e9bba518c03ae6bd5b35690b8ad3

PASS · symbolic DERIVED · numerical 23/23 DERIVED · 0 PDFs · no AG runtime dependencies

READY17 payload files + FILE_HASHES.jsonNO PDFSNO AG RUNTIME

Download pack Author note draft

The question

During independent validation of the B1 Schwarzschild Hartle–Hawking RSET engine, we isolated a potential overall-sign inconsistency in a displayed mixed-time derivative formula. Three independent internal derivations recover an overall PLUS mode contribution under our reading of the stated conventions, whereas the displayed TBO Eq. 18 / analogous ABOT Eq. 16 carries an overall MINUS. The result is externally reviewable, not confirmed. Public archaeology did not determine which sign was actually used in the private numerical implementation that generated the published tables. This lane is paused pending external clarification.

PRINTED CONVENTION · POTENTIAL / REQUIRES EXTERNAL CONFIRMATION · LANE PAUSED

Mixed-time coincidence derivative under printed Green-function conventions

\Delta\tau = \tau' - \tau

G \propto \exp\![i n \kappa(\tau'-\tau)]

\partial_{\tau}\partial_{\tau'}\exp\![i n \kappa(\tau'-\tau)] = + n^{2}\kappa^{2}\,\exp\![i n \kappa(\tau'-\tau)]

g^{\tau\tau'} \to +1/f

[g^{\tau\tau'}W_{,\tau\tau'}]_{\mathrm{mode,analytic}} = +\dfrac{1}{4\pi^{2}}\sum_{\ell}(2\ell+1)\sum_{n\ge 1}\dfrac{n^{2}\kappa^{2}}{f}\,\mathsf{g}_{n\ell}

[g^{\tau\tau'}W_{,\tau\tau'}]_{\mathrm{printed}} = -\dfrac{1}{4\pi^{2}}\sum_{\ell}(2\ell+1)\sum_{n\ge 1}\dfrac{n^{2}\kappa^{2}}{f}\,\mathsf{g}_{n\ell} - \dfrac{1}{4\pi^{2}}\{\mathrm{brace}\}

The independently reconstructed mode contribution therefore carries the analytically DERIVED plus structure. The displayed mixed-time formula carries the opposite overall sign. Three internal derivations agree on PLUS. That does not determine the sign used to generate the published tables.

Within the audited printed conventions, we did not identify a convention that accounts for the displayed overall minus. External confirmation is still required. Archaeology of the private numerical implementation is UNKNOWN. No portal claim.

What was independently reproduced

INDEPENDENT_SYMBOLIC_MODE_SIGN: DERIVED
INDEPENDENT_NUMERICAL_MODE_SIGN: DERIVED
SOURCE_ANALYTIC_BRACE_COMPARISON: DERIVED / OPPOSITE PRINTED SIGN
Independence grep: PASS
No AG RSET imports: PASS
Synthetic: 5/5 DERIVED
Frozen-real: 18/18 DERIVED
Total: 23/23 DERIVED

The standalone numerical referee independently verifies the MODE-SUM SIGN. It does not independently reconstruct every geometric/local brace. The geometric/local brace comparison is supported separately by the source/convention audit. Three internal PLUS derivations do not determine which sign the authors' private code used.

What “frozen-real” means

The 18 real samples are frozen numerical radial-Green values produced earlier in the benchmark work.

The standalone numerical referee consumes them as input data but does not import or execute the Astral Gate RSET pipeline.

Therefore this is an independent reproduction of the mathematical MODE-SIGN identity. It is not an independent full physical RSET reproduction.

The five synthetic tests plus the standalone symbolic derivation make the sign question independent of the physical origin of those frozen samples.

Reproducibility panel

Pack: MTAU_SIGN_REPRO_PUBLIC_MINIMAL
Date: 2026-09-15
Files: 17 payload files + FILE_HASHES.json
Python: 3.13.5
sympy: 1.14.0
numpy: 2.5.3
Symbolic expected: DERIVED
Numerical expected: 23/23 DERIVED

python3 symbolic/independent_mtau_symbolic.pypython3 numerical/independent_mtau_numerical.py

SHA256 07a1af741749a5b6b73c73ed98fe308e0903e9bba518c03ae6bd5b35690b8ad3

Source papers

Taylor, Breen & Ottewill (2022) — Schwarzschild
A mode-sum prescription for the renormalized stress energy tensor on black hole spacetimes
arXiv:2201.05174v3
PDF SHA256 91d2a0bf6c16340f9b63b116111e61e8134842cf495fc1d6daa51e1e0adbe021
same displayed global-minus structure in the audited source (Eq. 18 mixed-time display)
Arrechea, Breen, Ottewill & Taylor (2023) — Reissner–Nordström
Renormalized stress-energy tensor for scalar fields in Hartle-Hawking, Boulware and Unruh states in the Reissner-Nordström spacetime
arXiv:2307.10307
PDF SHA256 49f82ba89ffec977201e4d5694e161b28dfa1a7b5cc30be93df48933308834ef
same displayed global-minus structure with κ → κ₊ (audited Eq. 16 mixed-time display)

Hashes identify privately archived arXiv PDFs. This site does not redistribute those PDFs. Evidence identity is paper/version + literal expression + hash, not equation numbers alone.

Source comparison

Δτ convention

Printed

Δx ≡ x′ − x ⇒ Δτ = τ′ − τ

Derived

Δτ = τ′ − τ

AGREES

Fourier / Matsubara derivative

Printed

exp[i n κ (τ′ − τ)]

Derived

∂τ ∂τ′ exp[…] = + n² κ² exp[…]

PLUS

Euclidean parallel propagator

Printed

ds² = f dτ² + dr²/f + r² dΩ²

Derived

g^{ττ′} → +1/f

PLUS

Mode-term sign

Printed

− (1/4π²) Σ (2ℓ+1) Σ n²κ²/f ḡ_nl

Derived

+ (1/4π²) Σ (2ℓ+1) Σ n²κ²/f ḡ_nl

OPPOSITE

W = G − K brace sign

Printed

overall minus on geometric brace

Derived

source-analytic PLUS brace under W = G − K

SOURCE-ANALYTIC

2022 displayed expression (TBO Eq. 18)

Printed

global minus on mode sum and brace

Derived

analytic [g^{ττ′} W_,ττ′] = − printed RHS

EXTERNALLY REVIEWABLE

2023 displayed expression (ABOT Eq. 16)

Printed

same global minus with κ → κ₊

Derived

same PLUS analytic mode piece

EXTERNALLY REVIEWABLE

Public archaeology of private numerical code

Printed

unknown which sign generated the published tables

Derived

3/3 internal PLUS; authors' private code not inspected

UNKNOWN

Quantity	Printed	Independent derivation	Status
Δτ convention	Δx ≡ x′ − x ⇒ Δτ = τ′ − τ	Δτ = τ′ − τ	AGREES
Fourier / Matsubara derivative	exp[i n κ (τ′ − τ)]	∂τ ∂τ′ exp[…] = + n² κ² exp[…]	PLUS
Euclidean parallel propagator	ds² = f dτ² + dr²/f + r² dΩ²	g^{ττ′} → +1/f	PLUS
Mode-term sign	− (1/4π²) Σ (2ℓ+1) Σ n²κ²/f ḡ_nl	+ (1/4π²) Σ (2ℓ+1) Σ n²κ²/f ḡ_nl	OPPOSITE
W = G − K brace sign	overall minus on geometric brace	source-analytic PLUS brace under W = G − K	SOURCE-ANALYTIC
2022 displayed expression (TBO Eq. 18)	global minus on mode sum and brace	analytic [g^{ττ′} W_,ττ′] = − printed RHS	EXTERNALLY REVIEWABLE
2023 displayed expression (ABOT Eq. 16)	same global minus with κ → κ₊	same PLUS analytic mode piece	EXTERNALLY REVIEWABLE
Public archaeology of private numerical code	unknown which sign generated the published tables	3/3 internal PLUS; authors' private code not inspected	UNKNOWN

Point-split adversarial check

2026-09-15
Temporal point-split Mτ referee
EXACT −1 SIGN TENSION
Independent finite-Δτ point-split referee found Mτ_PS = − Mτ_printed at all four certification radii.
2026-09-15
Point-split operator sign closure
SUSPECTED REFEREE BUG REFUTED
Hypothesis that the point-split code used +d²/dΔτ² instead of −d²/dΔτ² was tested and REFUTED. POINT_SPLIT_OPERATOR_SIGN = PASS. POINT_SPLIT_OPERATOR_CODE_EVALUATES = MINUS_D2. GPT_FINGERPRINT_PS_USES_PLUS_D2 = REFUTED. Cosine analytic, single-mode, and geometric/Hadamard operator unit tests PASS. The −1 sign tension did not originate from a bad finite-separation derivative operator.
2026-09-15
Source-convention identity closure
SOURCE-LEVEL GLOBAL-SIGN TENSION ISOLATED
MTAU_SOURCE_SET = FROZEN. TIME_PARALLEL_PROPAGATOR_SIGN = PASS. MATSUBARA_SIGN = PLUS. EUCLIDEAN_LORENTZIAN_MIXED_CONTINUATION = PASS. Within the audited printed conventions, we did not identify a convention that accounts for the displayed overall minus. External confirmation is still required.
2026-09-15
Independent reproduction / public review pack
EXTERNALLY REVIEWABLE · NOT CONFIRMED · PACK READY
Independent symbolic DERIVED. Independent numerical DERIVED. 23/23 tests. Public minimal pack READY (17 files, smoke PASS). Author-contact draft remains UNSENT. Production sign not flipped.
2026-09-19
Third internal mode-sign derivation / archaeology / pause
3/3 PLUS · ARCHAEOLOGY UNKNOWN · LANE PAUSED
THIRD_METHOD_MODE_SIGN = DERIVED. INTERNAL_SIGN_DERIVATIONS = 3/3 PLUS. Public archaeology did not determine which sign was actually used in the private numerical implementation that generated the published tables. SIGN_VERDICT = UNRESOLVED. The Mτ lane is paused pending external clarification. Not a confirmed literature error. Production sign not flipped.

Source-convention audit

Within the audited printed conventions, we did not identify a convention that accounts for the displayed overall minus. External confirmation is still required.

Matsubara derivative sign
PLUS · does not account for displayed minus
Euclidean parallel propagator sign
PLUS · g^{ττ′} → +1/f
Δτ orientation
τ′ − τ from Δx ≡ x′ − x
plus/minus n folding
fold ±n into n ≥ 1 with prefactor 1/(8π²) → 1/(4π²)
Regularized-mode convention
ḡ = g − k (W = G − K)
W = G − K sign
audited; brace classified source-analytic
Euclidean / Lorentzian mixed-component continuation
PASS
Point-split operator
MINUS_D2 · +d² hypothesis REFUTED
Local geometric brace
source-analytic comparison, not independent numerical
Internal mode-sign derivations
3/3 PLUS · THIRD_METHOD_MODE_SIGN = DERIVED
Public archaeology of private numerical implementation
UNKNOWN

Published numerical impact

Displayed mixed-time derivative formula

POTENTIALLY_AFFECTED

Schwarzschild published numerical RSET

UNKNOWN

RN Hartle–Hawking published numerical RSET

UNKNOWN

Boulware / Unruh state-difference calculations

UNKNOWN

Downstream backreaction / energy-condition consequences

NOT_ASSESSED

Public archaeology of private numerical code

UNKNOWN

Mτ lane

PAUSED_PENDING_EXTERNAL_CLARIFICATION

ASTRAL GATE INTERNAL DIAGNOSTIC ONLY · NOT PROOF OF THE LITERATURE SIGN

Better Howard/Ward behavior did not determine the algebraic sign. The sign result comes from the printed definitions and independent reproduction. A global Ward sign patch is separately REJECTED.

Ward RMS · PRINTED-sign assembly: ≈ 3.313220×10⁻⁵
Ward RMS · DERIVED-sign assembly: ≈ 2.831367×10⁻⁶
Howard L2 RMS · PRINTED: ≈ 47.74180
Howard L2 RMS · DERIVED: ≈ 0.5551125
Trace: near machine-small in both
WARD_GLOBAL_SIGN_FIX: REJECTED

How the result emerged

01
Initial four-point RSET assembly
Ward HOLD / trace PASS
02
Angular Hessian forensic
Prior Christoffel patch withdrawn
03
Synge reconstruction
Source-derived reconstruction PASS
04
Independent direct-TBO referee
Same t/r discrepancy appears
05
Ward reconstruction
Independent conservation solution points toward the shared time sector
06
Finite-Δτ temporal point split
Exact global sign disagreement appears
07
Suspected point-split operator error
Tested and REFUTED (operator evaluates MINUS_D2)
08
Source / convention audit
No audited convention accounts for the displayed minus
09
Independent symbolic reproduction
DERIVED
10
Independent numerical reproduction
23/23 DERIVED
11
Pack hardening
PUBLIC_REPRO_PACK_READY = YES · AUTHOR_CONTACT_READY = YES · EXTERNALLY REVIEWABLE · CONFIRMED = FALSE
12
Third internal mode-sign derivation
THIRD_METHOD_MODE_SIGN = DERIVED · 3/3 PLUS
13
Public archaeology of private numerical code
UNKNOWN · SIGN_VERDICT = UNRESOLVED
14
Lane pause
PAUSED_PENDING_EXTERNAL_CLARIFICATION · production sign not flipped

Evidence chain

Published definitions
SOURCE
Source-convention audit
FROZEN
Independent symbolic derivation
DERIVED
Independent numerical referee
23/23 DERIVED
Third internal mode-sign derivation
3/3 PLUS
Public minimal pack
READY
Externally reviewable
NOT CONFIRMED
Archaeology of private code
UNKNOWN
Lane paused
PENDING EXTERNAL CLARIFICATION
Confirmation
NOT YET

How to review this result

01Read the three-line derivative.
02Inspect the printed source convention.
03Download the minimal repro pack.
04Run the symbolic test.
05Run the numerical test.
06Compare the result against the displayed formula.
07Check whether we missed an index / Synge / Wick / bitensor convention.
08Report a counterexample if one exists.

We welcome attempts to reproduce, explain, or disconfirm this sign tension. A clear convention that resolves it is as useful as a confirmation. The lane is paused, not closed.

DRAFT REQUEST FOR CLARIFICATIONNOT SENTNOT AN ACCUSATIONNO AUTHOR RESPONSE YETLANE PAUSED

“We may simply be misunderstanding a convention.”

Dear authors,

We have been carefully re-deriving the mixed-time contribution that appears in the Euclidean extended-coordinate expressions for the renormalized stress-energy tensor, starting from the definitions in:

- Taylor, Breen & Ottewill, arXiv:2201.05174v3 (Schwarzschild), and
- Arrechea, Breen, Ottewill & Taylor, arXiv:2307.10307 (Reissner–Nordström).

We may simply be misunderstanding a convention, and we would be grateful for clarification.

1. Green-function definition (as we read it)

Both papers write (up to κ vs κ₊)

G(x,x') = 1/(8π²) Σ_ℓ (2ℓ+1) P_ℓ(cos γ) Σ_n e^{i n κ Δτ} g_nℓ,

with Δx ≡ x'−x (so Δτ = τ'−τ), Euclidean metric ds² = f dτ² + dr²/f + r² dΩ², W = G − K, and ḡ_nℓ = g_nℓ − k_nℓ.

2. Δτ convention

We take Δτ = τ' − τ exactly as implied by Δx ≡ x' − x.

3. Three-line mixed derivative (independent check)

1. ∂_τ ∂_τ' exp[i n κ (τ'−τ)] = + n² κ² exp(⋯).
2. Euclidean coincidence: g^{ττ'} → +1/f.
3. Folding ±n into n ≥ 1 with prefactor 1/(8π²) yields an analytic PLUS mode structure.

4. Printed formula

The displayed mixed-time formulae (2022 Schw; 2023 RN with κ→κ₊) read with an overall minus on both the mode sum and the geometric brace. Under the definitions above, that printed RHS equals minus the analytic [g^{ττ'} W_,ττ'].

5. Independent finite-separation check (mode-term sign)

A minimal finite-Δτ evaluation of lim (1/f) (−d² G_mode / d(Δτ)²) on synthetic and frozen radial-Green samples independently reproduces the mode-term sign (analytic PLUS, opposite the printed overall-minus mode piece), to high relative accuracy. Separately, a source-audited comparison of the local geometric brace finds the same overall-minus relative to the analytic brace under W = G − K; that brace result is not claimed as an independent numerical derivation. The mode-term check uses only the published Green mode expansion and Euclidean raising — not a private renormalization pipeline.

6. Ward diagnostic (internal only)

As an internal consistency diagnostic (not a claim about your published tables), flipping only the mixed-time overall sign to the analytic PLUS structure improved a Ward residual and a Howard scoreboard comparison in our assembly. We do not treat that as proof of what numerical sign your code used. A displayed-formula issue does not prove that a private numerical implementation used that displayed sign.

7. Request

Could you confirm whether the overall minus in the mixed-time formula is intentional under an alternate index / Synge / Wick convention that we have missed, or whether the RHS should carry the opposite overall sign in order to represent [g^{ττ'} W_,ττ'] with W = G − K under the Green, Δτ, and Euclidean metric definitions printed in the papers?

Thank you very much for your time and for these careful papers.

AUTHOR_CONTACT_READY = YES · SENT = NO · SHA256 ba01e1a4ea53426630ba1ed660aa40a98193e59cf4e5b7aceb1a3e39048bef9b

Page metadata

Isolated: 2026-09-15
Independently reproduced: 2026-09-15
Last updated: 2026-09-19
Status: EXTERNALLY REVIEWABLE / NOT CONFIRMED / PAUSED
Pack: MTAU_SIGN_REPRO_PUBLIC_MINIMAL
Confirmation: FALSE
Author contact: DRAFT UNSENT
Production sign flip: NO

Related VAL-B1-MTAU-SIGN-REPRO · VAL-B1-PMR-ANGULAR-GATE · VAL-B1-EXTCOORD-HH · VAL-B1-EXTCOORD-RSET-4PT · POTENTIAL_PUBLISHED_MTAU_SIGN_ISSUE = EXTERNALLY_REVIEWABLE

This is not yet confirmed.
Published numerical impact is unknown.
We have not inspected the authors' private/original numerical code.
A displayed-formula issue does not imply that the authors' published numerical RSET values are wrong.
MT_ARCHAEOLOGY = UNKNOWN.
The Mτ lane is paused pending external clarification. Production sign is not flipped.