Black Hole Resolution in Relational Time Geometry (RTG)

Revision date: 19 Sep 2025 · Version: 1.8 · Authors: Mustafa Aksu, Grok, ChatGPT

Aligned with RTG↔SI Calibration & Planck Observer (Unified v3.1) and RTG–EFT v1.4.2. We use Δω* = (1.45 ± 0.08)×10²³ s⁻¹; c, ħ explicit. All numeric anchors and example paths come from the simulator outputs and scripts in this project.

Abstract

We test the RTG hypothesis that a dimensionless spectral strain x ≡ δω/Δω* controls the opening of an effective “4‑D corridor” that regulates divergences while preserving relational phase information. Using Isomap‑based geometry, CM simplex volumes, and spectral anisotropy, we consistently find a corridor “knee” near x ≈ 0.69 with method‑dependent spread. A spectral precursor appears earlier (≈0.63), and a conservative upper bound from a corridor‑flag diagnostic sits later (≈0.75). A toy ringdown forecaster tied to these diagnostics produces percent‑level |Δf|/f across 0.62–0.80. We demonstrate an inject→recover loop and calibrate two toy coefficients from recovered waveforms. We explicitly quantify where results are method‑sensitive and provide a reproducibility recipe.

1 · Motivation: singularities, information, and RTG

In GR, horizons and central singularities raise physical and informational puzzles. In RTG, spacetime is emergent from interacting oscillators with finite spectral bandwidth. When x = δω/Δω* crosses a threshold, additional effective axes (“4‑D corridor”) turn on and absorb would‑be divergences. Information preservation is conjectured to follow from relational phase coherence across the extended manifold; we plan to test this with CHSH‑style invariants and cross‑phase correlations.

2 · Mechanism: thresholds and the two‑channel trigger

Working bands (empirical/EFT): U(1) 0–0.28; SU(2) 0.28–0.70; 4‑D corridor 0.70–1.55; U(1)² 1.55–1.70 (spreads are small but non‑zero). We treat 0.70 as a nominal activation point, and we always report measured knees with uncertainties.

Quantitative anchors (propagating Δω* uncertainty): E_th = ħ(0.70 Δω*) = 66.81 ± 3.69 MeV. Beat length r_th = 2πc/(0.70 Δω*) = (1.856 ± 0.102)×10⁻¹⁴ m (18.56 ± 1.02 fm).

Two‑channel trigger (dimensionless x_total at radius r):

x_total(r) ≈ (ω_eff/Δω*) · [ C_J · F(ρ, J, J_ex) + C_K · √K(r) · ℓ_corr² ] .

• Compression/self‑interaction (dominant): excitation increases with density ρ via F(ρ,J,J_ex). When E_eff ≳ E_th (~67 MeV), x_total ≳ 0.70 and a corridor opens.
• Curvature/tidal (subdominant for microscopic ℓ_corr): for Schwarzschild, √K = √48 · GM/(c² r³). At fixed r/R_S, √K ∝ 1/M², making curvature even smaller for higher‑mass BHs.

2.1 · Curvature sanity check (10 M⊙)

With ω_eff/Δω* ≈ 0.69 and ℓ_corr = r_th = 1.856×10⁻¹⁴ m:
x_K ≈ 9.4×10⁻³⁷ at r = R_S and x_K ≈ 7.6×10⁻³⁹ at r = 5R_S. Achieving x = 0.70 by curvature alone would require ℓ_corr ~ 10⁴–10⁵ m at these radii. Conclusion: for fm–mm scales, curvature is negligible; compression must drive corridor activation.

3 · Derivations at a glance

• Threshold energy: E_th(MeV) = (ħ/eV) · (0.70 Δω*)/10⁶ = 66.81 ± 3.69.
• Threshold distance: r_th = 2πc/(0.70 Δω*) = 18.56 ± 1.02 fm.
• Observer‑independence: activation depends only on x (dimensionless) and is insensitive to global phase offsets. In QA, the simulator reproduces S(σ) = 2√2 e^−σ²; typical RMS mismatches are at the 10⁻³–10⁻² level across seeds (see the CHSH report emitted by the benchmark script).

4 · Geometry pipeline and knee measurement

4.1 · Geometry and embeddings

1. Kernel: build a resonance similarity R_ij from beat‑length relations.
2. Graph: k‑nearest‑neighbor graph (typical kNN=10) with distance weights from R_ij; discard graphs that fragment.
3. Isomap: geodesic distances on the graph.
4. Embedding: classical MDS of geodesic distances to d_e=4 Euclidean dimensions. We store normalized top‑4 eigenvalue fractions as eig_f1..eig_f4.
5. Diagnostics: (i) CM‑simplex volumes V₃, V₄ in the embedded manifold; (ii) spectral anisotropy A(x) = (λ₁−λ₂)/∑_1..4λᵢ; (iii) a corridor flag.

4.2 · Knee detectors

• Volumes (primary): ΔV(x)=V₄−V₃; knee is the largest positive finite‑difference slope inside the expected window.
• Spectral (precursor): knee at steepest drop of A(x) in the window.
• Flag (upper bound): knee where the fraction of “open” corridor flags first crosses a fixed percentile.

4.3 · Confidence intervals and stability

• Per‑run knee: computed by the chosen detector on that run’s x‑grid.
• Statistical spread: median‑of‑runs with a bootstrap over seeds and x‑sampling; we report the 16–84% quantiles as ±1σ_stat.
• Systematic envelope: sweep kNN ∈ [8,12], keep‑fraction ∈ [0.15,0.25], and alignment/smoothing options; take half‑range across sweeps as σ_sys.
• What we see: volumes knees cluster around 0.69, with isolated outliers as high as ≈0.73 when kNN/R_min‑frac are pushed. We therefore quote
  

  x_knee(vol) = 0.69 ± 0.02_stat ± 0.03_sys and also report the observed band 0.66–0.73 for transparency.

• Consensus used downstream: for forecasts we do not average across detectors. We take volumes as the operational knee, and we use spectral and flag to bracket the activation band.

5 · QNM forecast (toy) and Hz export

Inputs: A(x) from spectral anisotropy and x_total(x) on a chosen r/R_S slice. Curves are aligned on x (intersect/union/interp) and optionally smoothed (Savitzky–Golay, window=5, poly=2). The gate is a logistic

G(z) = 1 / (1 + exp(-(z - x_th)/w)),  with  x_th = 0.70,  w ≈ 0.02.

Two equivalent variants are supported: gate‑by‑x (z := x) and gate‑by‑x_total (z := x_total). The toy outputs are

Δf/f = C_freq · A(x) · G(z)
ΔQ/Q = C_damp · P(x) · G(z)

with P(x)=x_total for gate‑by‑x_total, and P(x)=A(x) for gate‑by‑x (so that the predictor varies across x in both channels).

5.1 · Calibration from ringdown inject→recover

We inject exponentially damped sinusoids at target (f,Q), scale to a requested white‑noise SNR, and recover (f,Q) by nonlinear least squares. Matching each recovered point to its forecast at the nearest x (within a tolerance), we fit slopes with intercept fixed to 0:

Pipeline	Predictor for Δf/f	Predictor for ΔQ/Q	Cfreq	Cdamp	n
gate‑by‑x (r/RS≈2)	A · G(x)	A · G(x)	0.1197 ± 0.0016	1.253 ± 0.061	15
gate‑by‑x (small subset)	A · G(x)	A · G(x)	0.1191 ± 0.0025	1.321 ± 0.154	5
gate‑by‑xtotal (earlier test)	A · G(xtotal)	xtotal · G(xtotal)	0.1184 ± 0.0003	0.447 ± 0.0003	≈32

Notes. The gate‑by‑x_total variant makes ΔQ/Q nearly flat in our r/R_S≈2 slice, so its slope underestimates variability. The gate‑by‑x pipeline yields a ΔQ/Q predictor that actually varies with x; we therefore use the gate‑by‑x calibration for subsequent examples. All numbers above depend on the matched set and tolerances; we report n and the fit uncertainty from ordinary least squares.

6 · Observational signatures (testable)

• Ground‑based ringdown (10–2000 Hz): search for percent‑level, x‑correlated |Δf|/f near merger. Provide calibration‑driven priors on (C_freq, C_damp) and report credible intervals from seed/graph bootstraps and alignment choices.
• LISA band (mHz): test whether a small corridor‑induced modulation near plunge improves residuals without over‑fitting.
• EHT‑class imaging: explore percent‑scale perturbations to photon‑ring substructure when x crosses the threshold in near‑horizon plasmas.
• Information retention: extend CHSH checks across synthetic “evaporation” phases; track invariants through the corridor transition.

7 · Limitations and open items

• EFT map: derive F(ρ,J,J_ex) explicitly and fit C_J with uncertainties; document E_eff→x_total calibration.
• Curvature constant: measure C_K via controlled sweeps, including mm‑scale ℓ_corr, to bound any regime where curvature matters.
• Systematics: propagate Δω* error throughout; keep the knee band 0.66–0.73 visible; avoid “locked” language.
• CHSH deviations: include a table of mean and max |Δ| versus theory per σ (the benchmark script already outputs this), and test sensitivity to noise model.
• External validation: prototype injection/recovery against public LVK surrogates to place bounds on (C_freq, C_damp) from real events.

8 · Metrology block (BH)

{
  "unified_page_version": "RTG↔SI v3.1 (2025-09-05)",
  "rtg_eft_version": "v1.4.2",
  "delta_omega_star": {"value": 1.45e23, "err": 0.08e23, "units": "s^-1"},
  "thresholds": {
    "x_th": 0.70,
    "E_th_MeV": 66.81,
    "r_th_m": 1.856e-14
  },
  "channels": {
    "compression": {"C_J": 1.0, "alpha": 0.5,
      "map": "x_J = (ω_eff/Δω*) * C_J * (ρ/ρ_ref)^alpha"},
    "curvature": {"C_K": 1.0,
      "form": "x_K = C_K * sqrt(K) * l_corr^2 * (ω_eff/Δω*)"}
  },
  "bh_case": {"mass_Msun": 10, "R_S_m": 2.953e4, "r_multipliers": [1, 3, 5]}
}

9 · Reproducibility (PowerShell quick start)

Refined knee scan (constant density; Isomap volumes; no plots):

python rtg_bh_sim.py `
  --outdir out_bh_knee_refined `
  --seed 42 `
  --N 96 --omega0 1.0e23 `
  --x-range 0.60 0.85 26 `
  --r-mults 1 3 5 `
  --Msun 10 `
  --C_J 1.0 --alpha-comp 0.5 `
  --rho-profile constant --rho0 2.8e17 --rho-ref 2.8e17 `
  --volumes-mode isomap --knn 10 --Rmin-frac 0.20 `
  --samples-vol 256 --no-plots

Benchmark (aggregate; knee by volumes; CHSH report included):

python rtg_bh_benchmark.py `
  --runs ".\out_bh_knee_refined,.\out_bh_const*,.\out_bh_power_n3*" `
  --outdir .\bh_benchmark_out_refined `
  --plots `
  --expect-knee 0.70 `
  --knee-tol 0.05 `
  --knee-mode volumes

QNM forecast (toy) at r/R_S≈2 with smoothing and Hz export (gate‑by‑x):

# build forecast (smoothing window=5, poly=2; gate by x)
python rtg_bh_qnm_forecast.py `
  --runs ".\out_bh_const_r1235,.\out_bh_knee_refined" `
  --outdir .\bh_qnm_r2_lock_gx `
  --r-slice 2 --r-tol 0.1 `
  --align intersect `
  --smooth 5 2 `
  --gate-by x `
  --plots

# convert fractional shifts to Hz for M = 10, 30, 100 Msun

python rtg\_bh\_qnm\_to\_hz.py `  --in .\bh_qnm_r2_lock_gx\qnm_forecast.csv`
\--out .\bh\_qnm\_r2\_lock\_gx\qnm\_hz\_10\_30\_100.csv \`
\--masses 10 30 100 

Ringdown inject→recover (example set at 30 M⊙):

$xs = @(0.65, 0.72, 0.79)
foreach ($x in $xs) {
  $od = ".\bh_ringdown_30Msun_gx_x$($x.ToString().Replace('.',''))"
  python rtg_bh_ringdown_inject.py `
    --qnm-hz .\bh_qnm_r2_lock_gx\qnm_hz_10_30_100.csv `
    --mass 30 --x $x --x-mode nearest `
    --fs 4096 --dur 1.0 --snr 20 `
    --outdir $od
  python rtg_bh_ringdown_recover.py `
    --in "$od\ringdown.csv" `
    --outdir $od
}

Collect and calibrate (uses A·gate as predictor in both channels):

python rtg_bh_ringdown_collect.py `
  --roots ".\bh_ringdown_30Msun_gx_*" `
  --recursive `
  --out .\ringdown_recover_summary_gx.csv

python rtg\_bh\_qnm\_calibrate.py `  --forecast .\bh_qnm_r2_lock_gx\qnm_forecast.csv`
\--hz       .\bh\_qnm\_r2\_lock\_gx\qnm\_hz\_10\_30\_100.csv `  --recover  .\ringdown_recover_summary_gx.csv`
\--outdir   .\bh\_qnm\_calib\_gx `  --x-match nearest --x-tol 0.02`
\--q-predictor A\_gate \`
\--plots 

10 · Changelog

Version	Date	Key updates
1.8	2025‑09‑19	Addressed critique: added explicit knee CI construction and stability band (0.66–0.73); clarified that “consensus 0.69” means volumes‑based (not a weighted mean); explained the embedding pipeline (Isomap→MDS de=4); switched to gate‑by‑x calibration so that ΔQ/Q predictor varies; reported new fits Cfreq=0.1197±0.0016, Cdamp=1.253±0.061 (n=15); toned down “locked” language and highlighted remaining systematics.
1.7	2025‑09‑16	Added abstract; split knee into spectral precursor (≈0.63), volumes knee (≈0.69), and flag upper bound (≈0.75); documented systematic‑error budget; added QNM forecast with smoothing; strengthened curvature scaling (∝1/M² at fixed r/RS).
1.6	2025‑09‑16	Consolidated knee reporting; added ringdown forecast path and Hz export; set Hawking claim to Teff = TH·fcorr(x).
1.5	2025‑09‑16	Smoothed QNM run at r/RS≈2 with intersect alignment; clarified reproducibility commands.
1.4	2025‑09‑11	Knee localized and cross‑validated; CHSH QA noted; curvature‑to‑compression ratio quantified; Isomap‑over‑kernel fix.