# Mercury perihelion 43"/century from QLF substrate **Scoping doc — Mercury's anomalous perihelion advance from QLF substrate Schwarzschild.** The 43"/century residue that Le Verrier identified in 1859 (and that Einstein computed from GR in 1915) falls out cleanly from the QLF substrate composition: 1. **Newton's law** `F = GMm/r²` from substrate event-counting ([`Gravity_From_Delay.md`](Gravity_From_Delay.md)). 2. **Schwarzschild metric** weak-field components `g_tt = -(1 - R_s/r)`, `g_rr = (1 - R_s/r)⁻¹` from Cross-Frequency Lorentz gravitational redshift + substrate radial-event scaling ([`GR_Schwarzschild.md`](GR_Schwarzschild.md)). 3. **Orbital effective potential** gains a `1/r³` correction term from `g_rr ≠ 1`, breaking the Bertrand-theorem closed-orbit condition. 4. **Per-orbit perihelion advance** `Δφ = 6πGM/(c²a(1-e²)) = 3π R_s/(a(1-e²))` from standard orbital mechanics applied to the corrected potential. For Mercury: `R_s(Sun) = 2.953 km`, `a = 5.79 × 10¹⁰ m`, `e = 0.2056`, **415 orbits/century → 43.0 arcseconds/century**, matching observation (Le Verrier residue = 42.98″, Einstein 1915 = 43.0″, modern measurement = 42.98 ± 0.04″) to **<0.1%**. --- ## §1 The Mercury residue In 1859 Le Verrier measured Mercury's perihelion precession (the gradual rotation of the orbit's long axis): **574"/century**. Of this, 532" was accounted for by Newtonian perturbations from other planets (Venus +278", Jupiter +154", Earth +90", others +10"). The remaining **43"/century was anomalous** under Newtonian gravity — Mercury's orbit was precessing faster than Newtonian theory could explain. Einstein 1915: the GR weak-field correction to Newton's law gives **exactly** this 43"/century shift, with no free parameters. This was the first observational success of GR (predating the 1919 light-bending measurement). Modern measurement (lunar laser ranging plus radar tracking of Mercury): `42.98 ± 0.04"/century`. QLF substrate aim: derive the 43"/century from substrate primitives that already compose Newton + Schwarzschild. --- ## §2 Schwarzschild effective potential for orbital motion For a test mass `m` in orbit around mass `M`, conservation of energy and angular momentum in the Schwarzschild metric give an **effective radial potential**: $$V_{\text{eff}}(r) \;=\; -\frac{GMm}{r} \;+\; \frac{L^2}{2mr^2} \;-\; \frac{GM L^2}{m c^2 r^3}$$ where the three terms are: - **`-GM m/r`** — Newton attraction (substrate-derived). - **`L²/(2mr²)`** — centrifugal barrier (standard orbital mechanics, no GR needed). - **`-GM L²/(mc² r³)`** — the **GR correction**, scaling as `1/r³`. The 1/r³ correction comes structurally from the `g_rr = (1 - R_s/r)⁻¹` factor in the radial part of the geodesic equation: - Energy conservation: `E² = (dr/dτ)²/(1-R_s/r) + (1-R_s/r)(1 + L²/(m²r²))` (Schwarzschild). - Solving for `(dr/dτ)²` and reading off the effective potential gives the three terms above. - The 1/r³ term arises from the product `(R_s/r) × (L²/m²r²) = R_s L²/(m²r³)`, which is the cross-term between the time-time and angular components. So the QLF substrate decomposition of the 1/r³ correction is: **`g_rr ≠ 1` from substrate radial event-scaling** × angular momentum centrifugal effect. --- ## §3 Perihelion advance from the 1/r³ correction For a perturbed Kepler orbit (Newton + small correction), perihelion advance per orbit is given by the integral: $$\Delta \phi \;=\; -\frac{\partial}{\partial L} \int_0^{2\pi} \delta H \, d\phi \;=\; \frac{6 \pi G M}{c^2 a (1 - e^2)}$$ where `δH = -GM L²/(mc²r³)` is the relativistic correction Hamiltonian. The derivation uses standard orbital perturbation theory (Bertrand-theorem analysis of how a 1/r³ term in V_eff shifts the perihelion). In dimensionless form using R_s = 2GM/c²: $$\boxed{\Delta \phi \;=\; \frac{6 \pi G M}{c^2 a (1-e^2)} \;=\; \frac{3 \pi R_s}{a (1 - e^2)}}$$ For circular orbits (e = 0): `Δφ = 3π R_s/a`. The eccentricity enters via `(1-e²)`, which slightly enhances the precession for non-circular orbits. --- ## §4 Numerical evaluation for Mercury | Quantity | Value | Source | |---|---|---| | `G` | `6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻²` | substrate primitive (= L_Planck² c³/ℏ in SI units) | | `M_Sun` | `1.989 × 10³⁰ kg` | astronomical measurement | | `c` | `2.998 × 10⁸ m/s` | substrate primitive (= L_Planck/τ_Planck) | | `R_s = 2GM/c²` | `2.953 km` | computed | | `a_Mercury` (semi-major axis) | `5.791 × 10¹⁰ m` | astronomical measurement | | `e_Mercury` (eccentricity) | `0.20563` | astronomical measurement | | `T_Mercury` (orbital period) | `87.969 d` | astronomical measurement | | Orbits per century | `100 × 365.25 / 87.969 = 415.2` | computed | **Per-orbit precession**: $$\Delta\phi_{\text{orbit}} \;=\; \frac{3\pi R_s}{a (1-e^2)} \;=\; \frac{3\pi \times 2953}{5.791 \times 10^{10} \times 0.9577} \;=\; 5.019 \times 10^{-7} \text{ rad}.$$ **Per-century precession**: $$\Delta\phi_{\text{century}} \;=\; 415.2 \times 5.019 \times 10^{-7} \;=\; 2.083 \times 10^{-4} \text{ rad} \;\times\; 206265 \text{ arcsec/rad} \;\approx\; \mathbf{42.98 \text{ arcsec/century}}.$$ **Match to observation**: `42.98 ± 0.04 arcsec/century` (modern lunar laser ranging). QLF substrate reproduces this to **<0.1%** with no free parameters. Numerical verification in [`mercury_perihelion_demo.py`](mercury_perihelion_demo.py). --- ## §5 What the substrate composition delivers **Tier 1 (structural decomposition).** The 43"/century derives from: - Newton's law from substrate event-counting ([`Gravity_From_Delay.md`](Gravity_From_Delay.md)) - Schwarzschild weak-field metric from substrate Cross-Frequency-Lorentz + radial event scaling ([`GR_Schwarzschild.md`](GR_Schwarzschild.md)) - Orbital mechanics 1/r³ perturbation analysis (standard mechanics, no QLF input needed) **Tier 2 (numerical, h + observables).** Substrate primitives `G`, `c` (from `L_Planck`, `τ_Planck`, `ℏ`) + astronomical observables `M_Sun`, `a_Mercury`, `e_Mercury`, `T_Mercury`. Result: 42.98"/century, **<0.1% match**. **Tier 3 (open).** - Substrate derivation of orbital mechanics 1/r³ perturbation analysis. Currently uses standard classical mechanics; substrate origin of the perturbation integral is a separate Tier-3 piece. - Strong-field GR effects: deflection of light by Sun (1.75"), Shapiro time delay, frame-dragging (Lense-Thirring). Same substrate primitives extend. - Substrate derivation of full Einstein equations (curvature side), needed for strong-field and cosmological tests. - Closed-form Mercury perihelion via substrate-event-counting at the orbital scale (currently the demo uses the standard analytic formula). A truly substrate-native computation would count Markov-blanket events directly. --- ## §6 What this is NOT - **Not a new GR claim.** Einstein 1915 already gave the formula `6πGM/(c²a(1-e²))`. The QLF contribution is the **substrate decomposition** of every input: Newton via holographic event count, Schwarzschild via Cross-Frequency Lorentz, G via Planck primitives. - **Not a full GR test suite.** Only the static-weak-field perihelion advance. Light bending, Shapiro delay, GW emission, gravitational redshift in clocks — same substrate framework would extend, but not done here. - **Not a derivation of cosmological constant.** The Λ in `R_μν − (1/2)R g_μν + Λ g_μν = 8πG T_μν` is a separate substrate question, possibly tied to vacuum-alignment ([`VacuumEnergy.md`](VacuumEnergy.md) §6). - **Not strong-field/black-hole physics.** Weak field `R_s/r ≈ 10⁻⁸` for Mercury; strong-field substrate treatment of singularities is open work. --- ## §7 References ### Internal - [`Gravity_From_Delay.md`](Gravity_From_Delay.md) — Newton's law from substrate event-counting. - [`GR_Schwarzschild.md`](GR_Schwarzschild.md) — Schwarzschild metric from substrate (the metric that defines the orbit). - [`Cross_Frequency_Lorentz.md`](Cross_Frequency_Lorentz.md) — gravitational redshift in substrate language. - [`Kitada_Local_Time_GR.md`](Kitada_Local_Time_GR.md) — Markov-blanket / substrate GR programme. - [`Per_Qubit_Mass_Quantum.md`](Per_Qubit_Mass_Quantum.md) — `R = E_Planck/(Mc²)` substrate identity. - [`Experimental_Consistency.md`](Experimental_Consistency.md) §10 — Mercury perihelion Class-B falsifier. - [`lean/QLF_GravityFromDelay.lean`](lean/QLF_GravityFromDelay.lean) — Newton substrate. - [`lean/QLF_MercuryPerihelion.lean`](lean/QLF_MercuryPerihelion.lean) — Lean anchor for this module. - [`mercury_perihelion_demo.py`](mercury_perihelion_demo.py) — numerical companion. ### External - Le Verrier, U. J. J. (1859). *Lettre de M. Le Verrier à M. Faye sur la théorie de Mercure et sur le mouvement du périhélie de cette planète*. Comptes Rendus de l'Académie des Sciences 49, 379 — identification of the 43" residue. - Einstein, A. (1915). *Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie*. Sitzungsber. Preuss. Akad. Wiss. 47, 831 — Einstein's 1915 derivation. - Will, C. M. (2014). *The Confrontation between General Relativity and Experiment*. Living Rev. Relativity 17, 4 — modern review. - Park, R. S., et al. (2017). *Precession of Mercury's perihelion from ranging to the MESSENGER spacecraft*. Astron. J. 153, 121 — modern measurement (42.98 ± 0.04"/century). - Weinberg, S. (1972). *Gravitation and Cosmology*. Wiley. Ch. 8 — standard textbook derivation of the perihelion shift.