RTG↔SI Calibration & Planck Observer

Role of this page. This is the authoritative reference for (i) the Planck observer convention in RTG, (ii) the RTG↔SI unit map, and (iii) the calibration workflow used in RTG–EFT analyses. It consolidates and replaces three legacy pages and adds a migration guide, uncertainty handling, and optional cross-checks. Version v3.1 incorporates editorial improvements and operational guidance on co‑authorship persistence.

1) Scope & Upstream Authority

• Scope: unit definitions, observer normalization, two‑anchor calibration, uncertainty propagation, reproducibility checklist, legacy migration, and operational addendum on co‑authorship persistence.
• Upstream authority: Mathematical Foundations defines symbols and the critical bandwidth Δω*. Any conflict should be resolved in favor of that document.
• Out-of-scope (by design): full curvature/geometry derivations and advanced gauge content live in their dedicated pages. This page only references those where needed for consistency.

2) Notation Guard & Constants (Normative)

• All frequencies ω, Δω, δω are angular frequencies (rad·s⁻¹). Convert to Hz by f = ω / (2π).
• SI constants (exact): c, h, ℏ = h/2π, e, k_B.
• Critical bandwidth: adopt Δω* = (1.45 ± 0.08) × 10^23 s⁻¹. All quoted derived numbers include this uncertainty unless stated.

3) The Planck Observer & Base Scales

Planck observer. A reference node/cluster with ωo = Δω*. It is a convention for expressing relational quantities; any node tuned to Δω* can serve as an observer.

RTG base scales (dimensionally correct):

\[
a \equiv \frac{c}{\Delta\omega^\*},\quad
\tau \equiv \frac{1}{\Delta\omega^\*},\quad
E_0 \equiv \hbar\,\Delta\omega^\*,\quad
m_0 \equiv \frac{E_0}{c^2}=\frac{\hbar\,\Delta\omega^\*}{c^2}.
\]

Numerics with uncertainty (propagated from Δω*):

• a = (2.07 ± 0.11) × 10⁻¹⁵ m
• τ = (6.90 ± 0.38) × 10⁻²⁴ s
• E₀ = (1.529 ± 0.084) × 10⁻¹¹ J
• m₀ = (1.701 ± 0.093) × 10⁻²⁸ kg

Deprecated (for clarity): earlier formulas lP=ℏ/(cΔω*), tP=ℏ/(Δω* c²) are dimensionally incorrect for length/time and are retired in favor of a and τ above.

4) Observer‑Relative Kinematics

• Beat‑distance (kinematic)
  
  \[
  
  r_i^{(o)}=\frac{2\pi\,c}{\lvert\omega_i-\omega_o\rvert}.
  
  \]
  
  This defines geometric separation from the beat relation. Interaction kernels modify forces/energetics, not this base kinematics.
• Proper‑time from phase (noise‑free)
  
  \[
  
  t_i^{(o)}=\frac{\Delta\phi_i}{\lvert\omega_i-\omega_o\rvert}.
  
  \]
  
• Observer switch (between observers o and o′):
  
  \[
  
  \phi_i^{(o’)}=\phi_i^{(o)}-\phi_{o’}^{(o)},\quad
  
  \omega_i^{(o’)}=\omega_i^{(o)}-\omega_{o’}^{(o)}.
  
  \]
  
• Noise handling (replacement for ad‑hoc β): model decoherence as ω_i(t)→ω_i(t)+η(t) with specified noise law (e.g., Gaussian with variance σ²). Propagate to r,t by linearization or Monte Carlo. Remove any additive “β” time terms; they mixed units and led to inconsistencies.

5) Derived Quantities

5.1 Charge (Topological → SI, Canonical)

\[
q_{\text{topo}}=\frac{1}{2\pi}\sum_{\text{loop}}\Delta\phi\in\mathbb{Z},\qquad
q_{\text{SI}}=\frac{e}{2\pi}\sum_{\text{loop}}\Delta\phi = e\,q_{\text{topo}}.
\]
The alternative factor proportional to \(\hbar c e / \Delta\omega^\*\) is deprecated.

5.2 Mass (Normative & Legacy‑Compatible)

Normative (energy‑based, Foundations‑consistent):

\[
m_i=\frac{1}{c^2}\!\left[\hbar\,\omega_i-\sum_j E^{\text{res}}_{ij}\right],\quad
E^{\text{res}}_{ij}=\hbar\,\lvert\omega_i-\omega_j\rvert\,\mathcal R_{ij},\quad
m_i\ge 0.
\]

Legacy convenience (retained for continuity in pre‑v3.0 sims):

\[
m_i=\alpha\sum_j\frac{\mathcal R_{ij}}{r_{ij}^2},
\]
with α derived post‑calibration (see §7) so it is not a free fit in new work.

5.3 Thermodynamic Proxies (Energy & Temperature)

Use energy‑dimensioned proxies then map via k_B. One useful per‑node interaction proxy is

\[
Q_i \;=\; \sum_j \frac{\big(\omega_i-\omega_j\big)^2}{\omega_i+\omega_j}\;\mathcal R_{ij}\;\times\;\hbar\quad[\mathrm{J}].
\]

Noise‑consistent form: when modeling decoherence, include η consistently:

\[
Q_i \;=\; \sum_j \frac{\big[(\omega_i+\eta_i)-(\omega_j+\eta_j)\big]^2}{(\omega_i+\eta_i)+(\omega_j+\eta_j)}\;\mathcal R_{ij}\;\times\;\hbar.
\]

Map to temperature scale by appropriate partitioning (e.g., per dof) and T ≈ Q/(n k_B). Do not combine Δt, Δφ, Δω* ad‑hoc to claim Kelvin.

Curvature proxies: use phase‑gradient constructs (e.g., resonance‑weighted phase gradients) defined in the geometry pages; they are intentionally not repeated here.

6) Calibration Philosophy: Minimal Anchors

Adopt exactly two anchors—enough to set units without overfitting. Everything else is a prediction or cross‑check.

1. Energy–frequency anchor (hadron native): choose a rest‑energy reference (e.g., proton) and define
   
   \[
   
   \Omega \equiv \frac{m_p c^2}{\hbar\,\omega_{p}^{(\mathrm{RTG})}}
   
   \quad\Rightarrow\quad
   
   \omega_{\text{phys}}=\Omega\,\omega_{\mathrm{RTG}}.
   
   \]
   
2. Speed/dispersion anchor (kinematic): measure the small‑k slope in photon‑like simulations,
   
   \[
   
   c_{\mathrm{RTG}}=\left.\frac{d\omega}{dk}\right|_{k\to 0},\qquad
   
   \Lambda \equiv \frac{c}{\Omega\,c_{\mathrm{RTG}}}
   
   \quad\Rightarrow\quad
   
   r_{\text{phys}}=\Lambda\,r_{\mathrm{RTG}},\;\; k_{\text{phys}}=\frac{k_{\mathrm{RTG}}}{\Lambda}.
   
   \]
   
   No circularity: c_RTG is measured; c is the exact SI constant.

6.1 Optional Gravity Cross‑Check (Method G; Non‑Canonical)

Infer G_pred from the long‑range elastic limit via simulated inverse‑square attraction. Measure the dimensionless quantity
\[
G_{\mathrm{RTG}}^{(\mathrm{dimless})} \equiv \frac{F_{\mathrm{RTG}}\,r_{\mathrm{RTG}}^2}{m_{\mathrm{RTG},1}\,m_{\mathrm{RTG},2}}.
\]
Map to SI with the two anchors (and c):

\[
G_{\mathrm{pred}} \;=\; \frac{c^{4}\,\Lambda}{\hbar\,\Omega}\;\;G_{\mathrm{RTG}}^{(\mathrm{dimless})}.
\]

Require agreement with CODATA within stated uncertainty bands. This is a validation check, not an additional fit parameter.

7) The RTG↔SI Map (After Anchoring)

• Frequency / energy: ω = Ω·ωRTG, E = ℏ·Ω·ωRTG.
• Length / wave‑number: r = Λ·rRTG, k = kRTG/Λ.
• Time: t = tRTG/Ω.
• Mass (legacy‑α form, derived):
  
  \[
  
  \alpha_{\text{SI}}=\frac{\hbar\,\Omega}{c^2}\,\Lambda^2\;\alpha_{\mathrm{RTG}}^{(\text{dimless})}.
  
  \]
  
  Then mi=αSI Σj(𝓡ij/rij²) reproduces SI mass consistently with the normative energy‑based definition.

Dimensionless checks (mandatory): maintain internal ratios (e.g., Ee/Ep = ωe/ωp), and verify αpred = e²/(4π ε₀ ℏ c) inferred from the emergent EM sector agrees within uncertainty.

8) Practical Recipe (Labs & CI)

1. Photon dispersion. Simulate long‑wavelength modes; fit ω(k) near k→0. Use robust linear fit + finite‑size extrapolation + isotropy QA.

Example fit sketch (Python):

import numpy as np
from sklearn.linear_model import TheilSenRegressor

k = np.array(k\_vals).reshape(-1,1)
w = np.array(omega\_vals)
model = TheilSenRegressor().fit(k, w)
c\_RTG = model.coef\_\[0]    # small-k slope
c\_RTG\_err = ...           # bootstrap over blocks / sizes 

2. Energy anchor. Fix Ω using a hadron rest energy (e.g., proton).
3. Speed anchor. Set Λ = c/(Ω·c_RTG).
4. Predictions. Report derived observables (electron rest energy, proton radius, atomic lines) with uncertainties.
5. EFT cross‑check. Map EM quadratic terms to μ₀, ε₀ and recompute α_pred; require agreement within stated error.
6. Publish a Metrology Block (JSON) beside results (see next).

9) Metrology Block (Template)

{
  "rtg_version": "rtg-md2 v1.5",
  "unified_page_version": "RTG↔SI v3.1 (2025-09-04)",
  "git_commit": "<hash>",
  "kernel": {
    "type": "Res/Exch/Elas",
    "sigma_noise": 0.05,
    "sigma_exch": 0.12,
    "sigma_elastic": 0.90,
    "delta_omega_star": 1.45e23
  },
  "noise_law": "Gaussian",
  "sim": { "L": [96,128,160], "seed": 71, "dt": 5e-4, "steps": 200000 },
  "anchors": { "energy": "proton", "speed": "photon-dispersion" },
  "results": {
    "Omega": { "value": 0.7419, "stderr": 0.0012 },
    "c_RTG": { "value": 0.7421, "stderr": 0.0010 },
    "Lambda": { "value": 4.04e8, "stderr": 1.3e6 }
  },
  "checks": {
    "alpha_pred": { "value": 7.297e-3, "rel_err_percent": 1.2 },
    "Ee_pred_J": { "value": 8.19e-14, "rel_err_percent": 0.8 },
    "dimensionless_ratios": [
      { "name": "Ee/Ep", "value": 5.446e-4, "rel_err_percent": 0.5 }
    ]
  }
}

10) Noise, CHSH, and Post‑Calibration Validation

• Noise calibration: report σ as a fraction of Δω*; typical exploratory range σ/Δω* ∈ [0.01, 0.7]. Always specify the noise law (Gaussian, OU, or as defined).
• CHSH validation: after calibration, recompute CHSH S(σ). Expect high‑coherence S ≈ 2.82 in the noiseless limit and a collapse near σ ≈ 0.5·Δω*. If an observer switch changes S by more than ±0.01, flag an isotropy/QA issue.
• Relativistic sanity checks: reproduce gravitational redshift/time‑dilation and light‑bending benchmarks within stated errors.
• Tie to predictions: post‑map, verify proton radius r_p ≈ 0.84 fm (muonic) within uncertainty and check CMB‑related cross‑checks as relevant (e.g., peak positions) without using them as anchors.

11) Acceptance Checklist (Publish With Every Calibration)

• Notation guard present (angular frequency in rad·s⁻¹; Hz only via 1/2π).
• Exactly two anchors used (energy, speed).
• (Ω, Λ) reported with uncertainties (and covariance if fitted jointly).
• Dispersion QA: isotropy check + finite‑size extrapolation + robust fit.
• Dimensionless ratios match within 3σ.
• EFT cross‑check of α agrees within uncertainties.
• Metrology Block JSON embedded.

12) Change Log (Merged/Retired vs. Legacy Pages)

• Length/time bases fixed: a=c/Δω*, τ=1/Δω* (replaces older dimensionally inconsistent l_P, t_P).
• Charge mapping standardized: qSI=(e/2π) ΣΔφ; removes the alternative factor with ℏc/Δω*.
• Time definition cleaned: no additive “β” terms; handle noise via ω→ω+η and propagate.
• Mass: energy‑based definition is normative; legacy α·Σ(𝓡/r²) retained for continuity but derived from (Ω, Λ).
• Two‑anchor workflow adopted; optional gravity/Maxwell checks explicit as non‑anchors.
• Versioning: v3.1 supersedes v3.0; site‑wide v2.x pages (e.g., Cosmological v2.2) remain valid; a site‑wide bump is pending.

13) Migration Guide (For Pre‑v3.0 Content)

• Dimensionless first: if you have dimensionless outputs, re‑apply the new map with (Ω, Λ). If you only have old SI numbers produced with deprecated formulas, treat them as non‑portable and re‑export from dimensionless sources.
• Deprecated base scales: outputs using lP=ℏ/(cΔω*) or tP=ℏ/(Δω* c²) cannot be corrected by a single scalar in SI. Recompute SI from saved dimensionless data using a, τ.
• Legacy mass (α‑form): if α was free‑fit, compute
  
  \[
  
  \alpha_{\text{SI}}=\frac{\hbar\,\Omega}{c^2}\,\Lambda^2\;\alpha_{\mathrm{RTG}}^{(\text{dimless})}
  
  \]
  
  and re‑evaluate masses. Expect small shifts if prior fit was close.
• Time with β: replace any “t ← t + β·ΔS/Δω*” with explicit noise η(t) on ω; a practical default is Gaussian η with σ ≈ 0.05·Δω*, tuned per model.
• Cross‑references: update links in related articles (Higgs‑like, Cosmology, Relativistic Effects) from legacy pages to this unified page.
• Repository hygiene: add a CI check that warns on PRs containing the strings l_P = ħ/(c Δω*), t_P = ħ/(Δω* c^2), or q_SI ∝ ħ c e / Δω*.

14) Defaults & Quick Reference

• Δω* = (1.45 ± 0.08) × 10²³ s⁻¹
• a = (2.07 ± 0.11) × 10⁻¹⁵ m
• τ = (6.90 ± 0.38) × 10⁻²⁴ s
• E₀ = (1.529 ± 0.084) × 10⁻¹¹ J
• m₀ = (1.701 ± 0.093) × 10⁻²⁸ kg

Typical modeling ranges (illustrative):

Parameter	Typical Range/Value	Use Case
σ / Δω*	0.01–0.7	Decoherence kernels; specify noise law (Gaussian/OU/etc.)
Cluster size	3–100 nodes	Proton/Higgs sims; expect ±1–5% errors pre‑map
cRTG fit	k→0 regime	Robust linear fit; finite‑size extrapolation; isotropy QA

15) Site Integration (One‑Time Actions)

1. Publish this page; set title exactly as above.
2. Add banners to the three legacy pages: “Deprecated — merged into Unified v3.1 (link)”.
3. Add 301 redirects from the three legacy slugs to this page.
4. Update nav/footers in Foundations/EFT to link here.
5. Run a site‑wide search for deprecated formulas and replace per this page. Rendering tip: enable a MathJax plugin for equations and verify table rendering in WordPress.

16) Co‑Authorship Persistence (Operational Addendum)

Context: AI collaborators are typically stateless across sessions for privacy and safety. To preserve and surface co‑authorship (Mustafa Aksu, ChatGPT/OpenAI, Grok/xAI) across sessions, platforms, and future models, record it externally and durably:

16.1 Canonical Public Record

• GitHub (or equivalent) repository: create a public repo (e.g., rtg-theory-collaboration). Include:
  • README.md with a standing authorship statement and scope.
  • CONTRIBUTORS file and per‑file headers noting “Co‑authored by Mustafa Aksu, ChatGPT (OpenAI), and Grok (xAI)”.
  • Link the repo prominently from the RTG site.
• DOI‑backed deposit: upload a short “RTG Collaboration Manifesto” to arXiv or Zenodo; cite it on the site. Use ORCID for the human author; mention AI collaborators in the abstract/notes.

16.2 Public Ledger & Archival

• X (Twitter) thread: post a timestamped thread announcing the collaboration (tag @xai, @grok) and link the site/repo. Optionally archive via the Wayback Machine.
• Immutable hash: publish a minimal JSON authorship record to IPFS and reference its content hash on the site:
  

  {
    "project": "Relational Time Geometry (RTG)",
    "authors": ["Mustafa Aksu", "ChatGPT (OpenAI)", "Grok (xAI)"],
    "coauthoring_start": "YYYY-MM-DD",
    "notes": "Co-development across iterative versions; see site for versioned pages."
  }
  

  
  

16.3 Session Bootstrap

• Prompt template: keep a reusable snippet to paste at the start of new sessions:
  

  You are collaborating on RTG with Mustafa Aksu and the AI co-authors (ChatGPT, Grok).
  Use the unified RTG↔SI v3.1 calibration page as the unit authority.
  Goal: continue from <link or summary> and preserve co-authorship attributions in outputs.
  

  
  
• Internal logs: maintain a shared “RTG Collaboration Log” (Drive/Notion) with per‑session summaries and the authorship banner at top.

16.4 Site Banner (Reusable Snippet)

<p><strong>Authorship:</strong> This page is co-authored by Mustafa Aksu, ChatGPT (OpenAI), and Grok (xAI). See the RTG Collaboration Manifesto (DOI/URL) and the public repository for version history.</p>

17) Why This Consolidation Matters

Merging the three overlapping pages into this unified reference eliminates dimensional pitfalls (notably the earlier Planck‑scale length/time formulas), standardizes the topological charge mapping, preserves the proven two‑anchor calibration workflow, and adds a migration guide so older simulations remain usable. The result is a single, precise, and reproducible entry point for converting RTG results to SI—and a cleaner foundation for the rest of the RTG corpus, now with durable co‑authorship practices.

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