Skip to content
ASTRAL GATELaboratory

Public research · PMR

PMR Adaptive Angular-Convergence Gate

Insufficient angular-mode support identified. Adaptive convergence gate prospectively validated. Sparse ω=10,12 later accepted under the same frozen gate. Not an HH RSET reproduction. Not a gateway result.

ADAPTIVE GATE PROSPECTIVE PASSSPARSE 10/12 PASSω=12 CONDITIONAL ON ℓCNG DISCOVERY · STRONG EVIDENCEDECAYING TAIL FALSEABEL NOT VALIDATEDHH RSET NOT CLAIMED

ELL_SUPPORT_CAUSE_SUPPORTED = TRUE · L_ceiling = 180 · confirmation_ok = TRUE · B1_HH_REPRODUCTION_CLAIMED = FALSE

Validation checkpoint VAL-B1-PMR-ANGULAR-GATE. Live work: CNG discovery study.

The finding

Point-split / mode-sum / radial-Green high-ω probes (PMR) are a numerical-instrument check on whether a frequency point is eligible to enter later frequency integration.

Earlier sparse high-frequency probes using approximately fixed ℓ_max ≈ 35 appeared pathological for ω ≳ 5: regularized residuals grew, F/F_sing deteriorated, and |F_reg|/|F| became O(1) and eventually >1.

A precision/rtol ladder showed that this behavior was reproducible and was not explained by ordinary float64/ODE tolerance noise.

A controlled ℓ-support isolation then changed ONLY ℓ_max at fixed r = 6M, ξ = 5, ω = 8, Hartle–Hawking state, same Levi–Ori F_sing, same equations / BCs / precision. Cancellation was restored at ℓ_max = 75.

A dedicated ω=8 angular staircase then closed a late-settling plateau between L = 75, 90, and 105. The ω=8 issue was the same insufficient-ℓ-support problem, not a separate systematic.

A reusable adaptive angular-support gate was implemented so an under-resolved frequency point cannot enter later frequency integration merely because the radial/mode calculation completed.

The gate was tested prospectively on previously unused ω = 7, starting deliberately under-supported at L = 30. It selected L* = 90 without a hand-chosen final ℓ_max. Confirmation at L = 105 left F_reg unchanged within the frozen criterion.

A certified sparse extension then accepted ω=10 and ω=12 under the same frozen gate with predeclared L_ceiling=180. Both points are ANGULAR_ACCEPTED with confirmation. This is not a dense spectrum.

At those certified points, |F_reg^tt(10)|/|F_reg^tt(8)|=2.18019 and |F_reg^tt(12)|/|F_reg^tt(8)|=4.02997. A decaying high-ω tail is not demonstrated. Analytic expectation: after F_sing, F_reg is an oscillating remainder for the generalized integral, not a quantity that must decay.

Chronology — failures retained

  1. 01

    Fixed low-ℓ high-ω probe

    FAIL · residuals grew; |F_reg|/|F| became O(1)

    FAIL

  2. 02

    Precision / rtol ladder

    Failure reproducible · ordinary float64/ODE noise disfavored

    HOLD

  3. 03

    Controlled ℓ-support isolation

    ONLY ℓ_max changed · cause supported

    PASS

  4. 04

    Matched ω/ℓ re-probe

    Cancellation restored at L=75 · ω=8 not yet plateaued vs L=60

    HOLD

  5. 05

    Dedicated ω=8 angular staircase

    ANGULAR_PLATEAU_STATUS = PASS

    PASS

  6. 06

    Adaptive angular-convergence gate

    Implemented · silent accept forbidden

    PASS

  7. 07

    Prospective unused-ω validation

    ω=7 · L*=90 · confirmation_ok TRUE

    PASS

  8. 08

    Certified sparse extension ω=10,12

    SPARSE_EXTENSION_PASS · L_ceiling=180 · both ANGULAR_ACCEPTED

    PASS

  9. 09

    Analytic large-ω expectation audit

    GROWTH_EXPECTED_PRE_GI · ASYMPTOTIC_STRUCTURE_MIXED

    HOLD

  10. 10

    Dense CNG discovery scan

    51/51 · ε₁ STRONG EVIDENCE · NOT CONFIRMED

    REVIEW

  11. 11

    Old surrogate discriminator

    INVALIDATED_DO_NOT_USE_FOR_CNG_DECISION

    FAIL

  12. 12

    Offline re-score

    UNCLASSIFIED_CRITERIA_GAP · D FAIL retained

    HOLD

  13. 13

    Independent confirmation scan

    NARROW PASS · 2026-09-21 · this pipeline’s window only

    PASS

  14. 14

    Finite-support self-cancellation

    IMPLEMENTED BY 2026-10-03 · infinite support not proved

    HOLD

  15. 15

    Abel production as an infinite-support integral

    NOT VALIDATED

    LOCKED

High-ω failure explained · ℓ-support isolation

Only ℓ_max changed. Cancellation returned.

Fixed point: r = 6M, ξ = 5, ω = 8, Hartle–Hawking, Levi–Ori F_sing. ℓ-support isolation changed ONLY ℓ_max. Same r, ξ, ω, state, F_sing, equations, BCs, and precision.

ℓ_max

25

|F_reg|/|F|_tt

8.698

F/F_sing_tt

0.103

cancellation

FAIL

ℓ_max

35

|F_reg|/|F|_tt

3.775

F/F_sing_tt

0.209

cancellation

FAIL

ℓ_max

50

|F_reg|/|F|_tt

1.011

F/F_sing_tt

0.497

cancellation

MARGINAL

ℓ_max

75

|F_reg|/|F|_tt

0.0124

F/F_sing_tt

1.013

cancellation

HEALTHY

Cancellation healthy in tt, rr and θθ at ℓ_max = 75. ELL_SUPPORT_CAUSE_SUPPORTED = TRUE. This is not proof that the full PMR RSET is correct.

ω=8 angular plateau

Late settling, same cause

All tt / rr / θθ components must satisfy D_ℓ < 0.05 for TWO consecutive terminal windows while cancellation remains healthy. Acceptance rule frozen BEFORE the run.

Dℓ=∣Freg(L)−Freg(L−15)∣max⁡(∣Freg(L)∣,ε)<0.05D_{\ell}=\dfrac{\bigl|F_{\mathrm{reg}}^{(L)}-F_{\mathrm{reg}}^{(L-15)}\bigr|}{\max\bigl(|F_{\mathrm{reg}}^{(L)}|,\varepsilon\bigr)}<0.05

window

L=75 vs 60

D_tt

0.510

D_rr

0.136

D_θθ

1.152

rule

FAIL

window

L=90 vs 75

D_tt

1.98×10⁻¹²

D_rr

2.86×10⁻¹²

D_θθ

6.81×10⁻¹²

rule

PASS

window

L=105 vs 90

D_tt

0

D_rr

0

D_θθ

0

rule

PASS

ANGULAR_PLATEAU_STATUS = PASS. The ω=8 issue was late angular settling of the same insufficient-ℓ-support problem, not evidence for a separate new systematic.

Adaptive angular-convergence gate

Under-resolved points cannot silently enter frequency integration

Prevent an under-resolved frequency point from entering later frequency integration merely because the radial/mode calculation completed.

  1. 01Begin at a deliberately finite angular cutoff L.
  2. 02Extend in ΔL = 15 increments.
  3. 03Evaluate angular-tail convergence D_ℓ for tt, rr, θθ.
  4. 04Require healthy cancellation.
  5. 05Require D_ℓ < 0.05 in ALL THREE components.
  6. 06Require TWO CONSECUTIVE passing terminal windows.
  7. 07Only then issue ANGULAR_ACCEPTED.
  8. 08If a predefined angular ceiling is reached without satisfying the gate: ANGULAR_UNRESOLVED. Never silently accept the point.

Prospective validation · unused ω=7

The gate chose L* = 90 without a hand-picked cutoff

ω = 7, ξ = 5, r = 6M, HH, Levi–Ori F_sing, rtol = 1e-8. Deliberate under-supported start at L = 30.

L

30

D_tt

0.191

D_rr

—

D_θθ

—

cancel_ok

FALSE

gate

FAIL

L

45

D_tt

0.913

D_rr

—

D_θθ

—

cancel_ok

FALSE

gate

FAIL

L

60

D_tt

38.25

D_rr

—

D_θθ

—

cancel_ok

TRUE

gate

ANGULAR FAIL

L

75

D_tt

5.20×10⁻⁶

D_rr

4.87×10⁻⁶

D_θθ

1.55×10⁻⁵

cancel_ok

TRUE

gate

PASS

L

90

D_tt

0

D_rr

0

D_θθ

0

cancel_ok

TRUE

gate

PASS · L*

ADAPTIVE_GATE_PROSPECTIVE_PASS. ANGULAR_GATE_VALIDATED_PROSPECTIVELY = TRUE. confirmation_ok = TRUE at L = 105. Important because the gate was not merely fitted retrospectively to the already-inspected ω = {4, 5, 6, 8} points.

Current PMR domain language

PMR_INPUT_VALIDITY_OMEGA_LE_8

PASS_CONDITIONAL_ON_ELL_CONVERGENCE

Within the tested ω ≤ 8 diagnostic domain, individual PMR frequency points can now be accepted only after passing the adaptive angular-convergence gate.

CERTIFIED_POINTWISE_DOMAIN_TO_OMEGA12

PASS_CONDITIONAL_ON_ELL_GATE

Angular-gated points at ω=10 and ω=12 were accepted under the frozen adaptive gate with L_ceiling=180. This extends the certified pointwise domain to ω=12. It is not a dense spectrum and not frequency-support convergence.

PMR_HIGH_OMEGA_INPUT_VALIDITY

UNESTABLISHED_BEYOND_OMEGA12

No claim is made for ω > 12. Do not automatically extend. Old under-resolved high-ω points must not be used as production Abel input.

DECAYING_HIGH_OMEGA_TAIL_DEMONSTRATED

FALSE

|F_reg^tt(10)|/|F_reg^tt(8)|=2.18019 and |F_reg^tt(12)|/|F_reg^tt(8)|=4.02997. Growth, not decay. Analytic expectation: decay was not required before GI.

ASYMPTOTIC_STRUCTURE

MIXED

After Levi–Ori / PMR local singular subtraction, pointwise decay of F_reg is not required. Sparse growth through ω=12 is not automatically subtraction failure. Global mixed-structure confirm remains FALSE.

ASYMPTOTIC_REGIME_IDENTIFIED

FALSE

Three sparse certified frequencies cannot identify the remote-caustic envelope regime. No power-law fit is a derivation.

FREQUENCY_SUPPORT_CONVERGED

FALSE

Generalized-integral / Abel ω-support has not been run and is not claimed.

It does not mean:

  • frequency support Ω → ∞ has been demonstrated
  • a decaying high-ω tail of F_reg has been demonstrated
  • the asymptotic remote-caustic regime has been identified
  • generalized-integral convergence has been demonstrated
  • Abel ε → 0+ has been validated
  • absolute HH RSET has been reproduced

Abel / generalized-integral status

ABEL_MATH_SYNTHETIC_REFEREE
PASS_DIAGNOSTIC
ABEL_PRODUCTION_VALIDATED
FALSE
ABEL_PRIMARY_SUSPECT
SUPPORT
ABEL_SUPPORT_TEST
UNRESOLVED
ABEL_CROSSCHECK
HOLD
PMR_PHASE_C
PARTIAL
ABEL_LIMIT_ORDER
OMEGA_TO_INFINITY_FIRST

The mathematical/synthetic referee established that the Abel procedure requires the large-frequency-support limit to be controlled BEFORE ε → 0+ extrapolation.

Do not claim that increasing polynomial degree in ε repairs missing ω support.

Earlier synthetic Ω ≈ 12–40 estimates remain DESIGN INFORMATION ONLY, not authorization or evidence for a physical production result.

The discovery of insufficient ℓ support means future ω-support studies must use angular-certified frequency points. Old under-resolved high-ω points must not be used as production Abel input.

Certified pointwise growth of |F_reg| through ω=12 does not authorize Abel. Analytic expectation: F_reg need not decay before the generalized integral. Do not run Abel in this freeze.

Next authorized step

Next: remainder control, then neighboring traces

SCALAR_REMAINDER_THEN_NEIGHBOR_TRACES_THEN_COMPONENTS

repaired massless scalar · minimal-coupling trace lane · editorial record through 2026-10-09. Do not retune to the stored trace. The r=6M empirical pass is not a tail bound and not component-wise validation.

Open the scalar-trace studyVAL-B1-SCALAR-TRACE-R6
  • Omitted-mode remainder control
  • Traces around r = 5.5M and 6.5M
  • Individual tensor components
  • This is not a new research job launched by the website.
  • This is not confirmation that the infinite-frequency integral has been proved.
  • This is not an independent full Hartle–Hawking stress tensor.
  • This is not a reason to flip the production sign or to email the authors.

Progression

  1. Observed high-ω failure

    HISTORICAL FAIL

  2. Precision / tolerance noise disfavored

    REPRODUCIBLE

  3. ℓ-support isolated

    CAUSE SUPPORTED

  4. Cancellation restored

    L=75 HEALTHY

  5. ω=8 angular plateau demonstrated

    PASS

  6. Adaptive ℓ gate implemented

    ANGULAR_ACCEPTED RULE

  7. Prospective ω=7 validation

    PASS · L*=90

  8. Sparse extension ω=10,12

    ANGULAR_ACCEPTED

  9. Decaying high-ω tail

    NOT DEMONSTRATED

  10. Analytic asymptotic-expectation audit

    MIXED · GROWTH EXPECTED PRE-GI

  11. Dense CNG discovery scan

    STRONG EVIDENCE

  12. CNG independent confirmation

    NARROW PASS · 21 SEP

  13. Global CNG structure

    NOT ESTABLISHED

  14. Finite-support self-cancellation

    IMPLEMENTED · NOT INFINITE

  15. Infinite-support generalized integral

    NOT PROVED

  16. Independent HH benchmark reproduction

    ONLY LATER

  1. Fixed low-ℓ high-ω probe

    FAIL

  2. Precision / rtol ladder

    NOISE DISFAVORED

  3. ℓ-support isolation

    CAUSE SUPPORTED

  4. Cancellation restored

    L=75 HEALTHY

  5. ω=8 angular plateau

    PASS

  6. Adaptive ℓ gate

    IMPLEMENTED

  7. Prospective ω=7

    PASS · L*=90

  8. Sparse extension ω=10,12

    PASS

  9. Analytic large-ω expectation

    MIXED

  10. Dense CNG discovery

    STRONG EVIDENCE

  11. CNG window confirmation

    NARROW PASS

  12. Global CNG structure

    NOT ESTABLISHED

  13. Infinite-support integral

    NOT PROVED

  14. Independent HH benchmark

    ONLY LATER

Ward / trace

TRACE_DEV
PASS
WARD_DEV
HOLD
WARD_GLOBAL_SIGN_FIX
REJECTED

Latest residual localization indicates there is no single universal sign error.

Near horizon: derivative / connection terms dominate depending on decomposition.

Farther out: angular anisotropy becomes important.

Therefore no global sign patch is authorized.

Branch G / closed lanes — do not leak instrument progress into physics claims

BRANCH_G
OPEN

OPEN means only that the audited Terno theorem does not currently exclude every intended Branch-G hypothesis. It is NOT evidence that Branch G is viable.

BRANCH_G_DEAD
FALSE

Do not declare Branch G dead. Do not declare it working.

TERNO_APPLICABILITY
PARTIAL

Partial applicability is not a physics pass.

NAMED ADMISSIBLE HADAMARD RSET + SELF-CONSISTENT EINSTEIN FIXED POINT
NOT PRODUCED

No named admissible Hadamard RSET plus self-consistent Einstein fixed point has been produced.

GATE
NO

Physical gate not established.

CASIMIR ROUTE
CLOSED

Not a live production path. Do not revive as a current claim.

~10⁷² SPECIES ROUTE
WITHDRAWN

Withdrawn. Do not advertise as viable.

RIG-1 HARDWARE / PROCUREMENT
FROZEN

Hardware and procurement remain frozen.

FLAT-SPACE QEI → CURVED-THROAT EXCLUSION
PROHIBITED

Prohibited unless the required hypotheses are established.

How to read a stamp

  • ESTABLISHED / REPRODUCED NUMERICAL FINDING

    A controlled numerical isolation or gate that survived its frozen acceptance rule. Still not a physical RSET.

  • PROVISIONAL / PARTIAL

    True inside a stated diagnostic domain, with an explicit remaining condition.

  • HOLD / UNRESOLVED

    Open scientific or numerical question. Not a pass.

  • CLOSED / WITHDRAWN

    A route that is no longer live. History is retained; revival is not implied.

  • NOT CLAIMED

    The project has not earned this statement. Absence of a fail is not a pass.

Must say

  • This is numerical-method progress, not an HH RSET reproduction or gateway result.
  • An ANGULAR_ACCEPTED point is merely eligible to enter a future controlled frequency-integration dataset.
  • It does not mean frequency support Ω → ∞ has been demonstrated.
  • It does not mean a decaying high-ω tail has been demonstrated.
  • It does not mean generalized-integral convergence has been demonstrated.
  • It does not mean Abel ε → 0+ has been validated.
  • It does not mean absolute HH RSET has been reproduced.
  • The latest PMR numbers are not physical RSET measurements.
  • The ω=8,10,12 |F_reg| ratios are not a derived envelope power.

Must not say

  • Schwarzschild Hartle–Hawking RSET has been reproduced.
  • Abel / PMR production RSET has been validated.
  • The published TBO/ABOT numerical tables contain a confirmed sign error.
  • A traversable gate has been demonstrated.
  • A self-consistent semiclassical throat has been found.
  • Branch G works.
  • Old fixed-ℓ high-ω PMR data are trustworthy production Abel input.
  • The high-ω tail of F_reg has been shown to decay.
  • Frequency support has converged.