Black Hole Resolution in Relational Time Geometry (RTG)

Revision date: 11 Sep 2025 · Version: 1.4 · Authors: Mustafa Aksu, Grok, ChatGPT

Aligned with RTG↔SI Calibration & Planck Observer (Unified v3.1) and RTG–EFT v1.4.2. Uses \( \Delta\omega^* = (1.45 \pm 0.08)\times 10^{23}\,\mathrm{s}^{-1} \); \(c,\hbar\) explicit. Key metrology values below are taken from the simulator’s run configuration and summary files you provided. :contentReference[oaicite:0]{index=0} :contentReference[oaicite:1]{index=1}

1 · Motivation: Singularities, Information, and RTG

In GR, black holes feature horizons and central singularities; quantumly, Hawking evaporation raises information‑loss concerns. RTG addresses both via bandwidth‑capped gradients: when the dimensionless spectral strain \(x \equiv \delta\omega/\Delta\omega^*\) exceeds a threshold, additional effective axes (“4‑D corridor”) open and absorb divergences while preserving relational phase information.

2 · Mechanism: Thresholds and the Two‑Channel Trigger

Windows (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 (small spreads). The 4‑D corridor opens when \(x \ge 0.70\).

Quantitative anchors (with uncertainties):

• Threshold energy: \(E_{\mathrm{th}}=\hbar(0.70\,\Delta\omega^*)=(66.81\pm3.69)\,\mathrm{MeV}\).
• Threshold beat‑distance: \(r_{\mathrm{th}}=\dfrac{2\pi c}{0.70\,\Delta\omega^*}=(1.856\pm0.102)\times 10^{-14}\,\mathrm{m}\) (= \(18.56\pm1.02\) fm).

Two‑channel trigger for \(x(r)\):

x \equiv \delta\omega/\Delta\omega^* \;\approx\; (\omega_{\rm eff}/\Delta\omega^*) \,\big[\, C_J\,\mathfrak{F}(\rho,J,J_{\rm ex}) \;+\; C_K\,\sqrt{K(r)}\,\ell_{\rm corr}^2 \,\big].

• Compression/self‑interaction (dominant): rising local density \(\rho\) increases effective excitation via \(J,J_{\rm ex}\). When \(E_{\rm eff}\gtrsim E_{\rm th}\approx 67\,\mathrm{MeV}\), \(x\ge 0.70\) and a 4‑D corridor opens.
• Curvature/tidal (subdominant at micro \(\ell_{\rm corr}\)): \(\sqrt{K(r)}\) from Schwarzschild \(K=48G^2M^2/(c^4 r^6)\). For \(\ell_{\rm corr}\) in fm–mm, this term alone is far below 0.70; treat it as a cross‑check, not a driver.

2.1 · Quantitative curvature check (10 M☉)

For \(M=10\,M_\odot\): \(R_S\simeq 2.953\times10^4\,\mathrm{m}\); take \(\omega_{\rm eff}\simeq 1.0\times10^{23}\,\mathrm{s}^{-1}\) so \((\omega_{\rm eff}/\Delta\omega^*)\approx 0.69\). With \(\ell_{\rm corr}=r_{\rm th}=1.856\times10^{-14}\,\mathrm{m}\):

Radius	\(\sqrt{K(r)}\) [m\(^{-2}\)]	\(x_K=\sqrt{K}\,\ell_{\rm corr}^2\,(\omega_{\rm eff}/\Delta\omega^*)\)	\(\ell_{\rm corr}\) for \(x=0.70\)
\(r=R_S\)	\(3.97\times10^{-9}\)	\(9.44\times10^{-37}\)	\(\approx 1.60\times10^4\,\mathrm{m}\)
\(r=5R_S\)	\(3.18\times10^{-11}\)	\(7.55\times10^{-39}\)	\(\approx 1.79\times10^5\,\mathrm{m}\)

Conclusion: curvature/tidal contributions are negligible at microscopic \(\ell_{\rm corr}\); the compression channel must dominate near/inside a few \(R_S\).

3 · Derivations at a Glance

Threshold energy: \(E_{\rm th}/{\rm MeV} = (\hbar/{\rm eV})\times(0.70\,\Delta\omega^*)/10^6 \approx 66.81\pm3.69.\)

Threshold distance: \(r_{\rm th}=2\pi c/(0.70\,\Delta\omega^*)=18.56\pm1.02\,\mathrm{fm}.\)

Observer‑independence: corridor activation is a function of dimensionless \(x\) and is insensitive to global phase offsets (consistent with CHSH‑style QA invariants).

3.1 · Hawking‑analog (recast)

In RTG, near‑horizon phase‑noise \(\delta\phi\sim\exp[-(\delta\omega/\Delta\omega^*)^2]\) produces emission. We model an effective temperature scaling as
\(T_{\rm eff}=T_H \times f_{\rm corr}(x)\),
where \(T_H=\hbar c^3/(8\pi G M k_B)\) is the usual Hawking scale and \(f_{\rm corr}(x)\) is a corridor factor such as the mean gate‑openness \(\langle G_{\rm open}\rangle\) or spectral anisotropy \(A(x)\). This makes RTG consistent with Hawking scaling while avoiding over‑claiming equality of spectra before a full EFT calibration. For \(M\sim10\,M_\odot\), \(T_{\rm eff}\) remains \(\ll 1\) K, as expected.

4 · Simulation and Benchmark Status (2025‑09‑11)

Simulator (v0.7) highlights: Isomap geodesics built from the resonance kernel \(R_{ij}\); CM‑volumes computed from Euclidean embedded distances; spectral anisotropy via MDS top‑eigenvalue fractions; CHSH analytic check; density models (constant or power‑law); radial curvature sweep; reproducible outputs (CSV, JSON Metrology, optional PNGs).

Key aggregated findings from your runs:

• Knee of the corridor indicator: using the volume diagnostic \(({\rm CM4}-{\rm CM3})\) and refined scans, the pooled knee lies at \(x_{\rm knee}\approx 0.69\) (within the \(\pm 0.05\) window around 0.70). A fine scan reported \(x_{\rm knee}=0.69045\) and a bootstrap set centered near \(0.67\); both satisfy the threshold tolerance once sampling variability is considered.
• Channel dominance: the overall and near‑knee median ratio is \(x_{\rm curv}/x_{\rm comp}\approx 5.07\times10^{-38}\); curvature is negligible at microscopic \(\ell_{\rm corr}\) in all tested scenarios.
• CHSH validation: \(S(\sigma)=2\sqrt{2}\,e^{-c\sigma^2}\) is matched by the simulator for \(\sigma\in\{0,\;0.23,\;0.30,\;0.50\}\) within numerical precision across runs.

These outcomes are consistent with the Metrology Block recorded by the simulator (e.g., \(x_{\rm th}=0.70\), \(E_{\rm th}\approx 66.81\,\mathrm{MeV}\), \(r_{\rm th}\approx 1.856\times10^{-14}\,\mathrm{m}\), \(R_S\approx 2.953\times10^4\,\mathrm{m}\) for \(10\,M_\odot\)). :contentReference[oaicite:2]{index=2} The summary confirms reproducible artifacts (CSV and plots) for each run. :contentReference[oaicite:3]{index=3}

5 · Observational Signatures (testable)

• Ringdown (10–2000 Hz; LIGO/Virgo/KAGRA; future ET/CE): frequency and damping shifts tied to corridor onset. Practical proxy: report \(|\Delta f|/f\) vs \(x\) and \(\Delta Q/Q\) vs \(x\), with corridor fraction or anisotropy as the control variable.
• SMBH mergers (mHz; LISA): phase‑dependent attenuation via \(x(\rho)\); search for deviations from GR templates during inspiral‑to‑ringdown transition.
• Imaging (EHT‑class): percent‑level shifts in photon‑ring substructure are possible if near‑horizon refractive indices are modified by corridor activation.
• Information retention: cross‑correlate phases across simulated “evaporation” sequences (CHSH‑like invariants) to test non‑lossy behavior.

6 · Comparison to Alternatives

• LQG “bounce”: RTG employs bandwidth thresholds and relational corridors rather than discrete spin networks.
• Strings: no branes required; oscillator‑like nodes plus a resonance kernel suffice; “extra dimensions” are effective and threshold‑activated.

7 · Limitations and Open Items

• Derive \(\mathfrak{F}(\rho,J,J_{\rm ex})\) explicitly from EFT and publish \(C_J\) with errors (connect to \(K’\), \(J\), \(J_{\rm ex}\) post‑calibration).
• Quantify artifact sensitivity in the spectrum/volume diagnostics (kNN choice, R‑cut, sample counts) and propagate \(\Delta\omega^*\) uncertainty into \(E_{\rm th}, r_{\rm th}, x_{\rm total}\).
• Refine the Hawking‑analog section as \(T_{\rm eff}=T_H\,f_{\rm corr}(x)\) with \(f_{\rm corr}\) inferred from simulations (e.g., \(\langle G_{\rm open}\rangle\) or \(A(x)\)); avoid claiming exact spectral equality pending EFT fits.

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": {"value": 66.81, "err": 3.69},
    "r_th_m": {"value": 1.856e-14, "err": 1.02e-15}
  },
  "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]},
  "sim": {
    "seed": 42, "nodes": 50, "omega0_s^-1": 1.0e23,
    "x_range": [0.5, 1.2, 8], "samples_CM": 64
  },
  "observables": ["CM_3vol", "CM_4vol", "gate_open_fraction",
                  "energy_exchange", "CHSH(σ_noise)"]
}

Values above reflect the configuration found in your uploaded bh_config.json and summary. :contentReference[oaicite:4]{index=4} :contentReference[oaicite:5]{index=5}

9 · Reproducibility (PowerShell quick start)

Simulation (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):

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

10 · Changelog

Version	Date	Key updates
1.4	2025‑09‑11	Validated corridor knee via volumes diagnostic at \(x_{\rm knee}\approx 0.69\) (refined and bootstrap runs); switched to R‑based Isomap and Euclidean CM‑volumes; clarified Hawking‑analog as \(T_{\rm eff}=T_H\,f_{\rm corr}(x)\); added explicit artifact mitigations and uncertainty guidance; integrated simulator/benchmark outputs with metrology references. :contentReference[oaicite:6]{index=6} :contentReference[oaicite:7]{index=7}
1.3	2025‑09‑09	First Hawking‑analog write‑up; empirical pointers (ringdown, analog systems); consolidated observables.
1.2‑draft	2025‑09‑05	Curvature sanity‑check table; emphasized compression channel; retained \(E_{\rm th}\), \(r_{\rm th}\) with uncertainties.
1.1‑draft	2025‑09‑05	Two‑channel trigger introduced; BH Metrology Block; clarified toy status of Gaussian volume fit.

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