Gravity: two loops and the space between them, the full derivation

A photon is the smallest piece of light; this page follows one photon closed into a loop, the probe, standing in the space another loop has shaped, and derives what the high school page stated. A closed Void path removes a fixed transverse area, the deficit that is its inertial mass. That deficit does not stay local: it propagates outward as an inverse-square curvature of the surrounding space. A second loop, its action computed in that geometry, has its path deflected toward the first; in the weak-field, large-distance limit this is exactly Newton’s law. Gravity is an emergent property of the geometric deficits carried by the loops, not an imposed field. Left, the probe from inside: its hidden region and the path its photon runs. Right, the source from outside: the missing space, with the lattice drawn into it. The orange curve is the probe’s centre, computed, deflected toward the source. Each numbered circle is a panel below.

probe loop Γ′: inside view, its hidden region and its photon’s pathinertia: m′ ∝ SΓ′ = ∮ Ad dssource loop Γ: outside view, the missing space; the lattice drawn into itδ = R₀²/(R² + R₀²) → R₀²/R² (→ Newton)the probe’s centre, deflected toward Γ: computed, a = −GM r̂/r², started off the axis with a sideways velocity (deflection 53°); dashed, the same start with no sourceF(r) = G mΓ mΓ′ / r²1 lattice step (R₀ = 2.5)12345
The case drawn.
F(r) = G mΓ mΓ′ / r²,  δ = R₀²/(R² + R₀²) → R₀²/R²
(two loops, the probe’s geodesic deflected toward the source; Mathematical Bridge, section 3, step 6; the map, the drawing’s construction of the Bridge’s “fixing the missing volume” and “Δg(r) ~ 1/r²”)
In the textbook’s letters, the same line and its field and potential:
F = GMm/r²,  g = GM/r²,  Φ = −GM/r
(Newton’s law of gravitation; G an acceptance lock, κ = 8πG/c⁴ “fixed by comparison”, Bridge Math Appendix, Mass and Gravity 7)
Panels 1 to 4. The route picture and what it recovers, panels 5 and 6.
The general solution, and what it recovers, from the worldsheet action, the Einstein–Hilbert coupling and nothing else:
the two-loop setup, inertia of each looppanel 1the dent Tμν and Gμν = κTμνpanel 2the geodesic, elongation toward the source, no loop faster than cpanel 3∇²Φ = 4πGρ, Φ = −Gm/r, Newton’s lawpanel 4the cost map n(x): bending, delay, clock drift, precession, photon spherepanel 51.75″, 123.6 μs, 2.455 × 10⁻¹⁵, 43″/century, 3GM/c²panel 6
Named as the documents name them: the Einstein–Hilbert term is imported and coupled to the loop’s Tμν; the four weak-field numbers follow from one factor set once.

Published documents used, and nothing else: the Mathematical Bridge (section 3, steps 1 to 6) and its Math Appendix (Mass and Gravity from Closed Void Loops, sections 3 to 7 and 6A, both framings of the Newtonian limit); the Bridge Narrative, step 4; the Mechanics Narrative and Math Walk-Through (the cost map n(x), the ray equation, the bending, delay and clock-drift kernels, periapsis, the photon sphere and capture); the Mechanics Math Appendix (sections 4 to 8, the worked numbers). Only ratios enter; S₀ = ħ is the one scale; c and G are acceptance locks.

The case drawn (panels 1 to 4), the route picture and what it recovers (panels 5 and 6)

1In this case: two loopsThe single-loop case (the Mass page) gives inertia. “To derive a force law,however, we must consider two loops: a source loop Γ that generates acurvature deficit, and a probe loop Γ′ whose action is computed within thedistorted geometry created by Γ.” Each carries its own loop action,mΓ ∝ ∮Γ Ad ds,  SΓ′ ∝ ∮Γ′ Ad dsand the probe’s is evaluated in the curved geometry Γ generates: “The geodesicof Γ′ is deflected toward Γ, reflecting gravitational attraction. Evaluating theweak-field, large-R limit yields a mutual force law F(r) = G mΓ mΓ′/r².”In the drawing the probe is shown from inside (its hidden sphere, its photon’spath) and the source from outside (the missing space the lattice leans into).Mathematical Bridge, section 3 (Mass and Gravity), introduction and step 6.R₀ = 2.5 lattice steps for both loops; the electron would set R₀ = ƛₑ = 386.16 fm.
2The dent, and its coupling to geometryThe source loop’s stress–energy is the variation of its worldsheet actionwith respect to the metric (the Mass page, panel 6); it is localized on theworldsheet and conserved. The total action is the Einstein–Hilbert actionplus the Void action:Stotal[g, W] = (1/2κ) ∫ R √(−g) d⁴x + Svoid[W; g]and varying with respect to gμν gives the Einstein field equations,Gμν = κ Tμν“the precise mathematical statement that a Void loop curves spacetime.”The Einstein–Hilbert term is imported here, with the loop’s Tμν as its source;κ = 8πG/c⁴ is “fixed by comparison” in the Newtonian limit (panel 4): G is anacceptance lock, not an input and not a derivation.Bridge Math Appendix, Mass and Gravity, sections 3 and 5.
3The probe’s motion: geodesic, elongation, no v > cThe second loop’s action is Svoid[W′; g]; its equations of motion are theminimal-surface equations in the curved g. In the small-loop limit itscentre-of-energy worldline obeys the geodesic equation,D²xμ/Dτ² + Γμαβ (dxα/dτ)(dxβ/dτ) = 0“the geometry telling the second loop how to move.” Each segment of the loopruns at c along its closed path (null: gμνuμuν = 0); curvature tilts “straightahead” inward, so to a distant observer the loop’s circle is deflected and“elongates toward the curvature source”, Leff = ∮ √(gij dxi dxj) > Lflat, andmobs = (σs/c²) Leff rises with it. A loop can never reach c: “if the loopattempted v > c, the path would fail to geometrically close”; in flat-spacelanguage acoord = aproper/γ³. The drawing’s probe path is that centre-of-energymotion in the Newtonian limit of panel 4, integrated from an off-axis start.Bridge Math Appendix, Mass and Gravity, sections 6 and 6A.
4The Newtonian limit: Newton’s law recoveredLinearize gμν = ημν + hμν, |h| ≪ 1, in harmonic gauge:□ h̄μν = −2κ Tμν,  T00 ≈ ρc² (static source)Defining h00 = −2Φ/c² reduces this to the Poisson equation, and for a pointlikeloop of inertial mass m (panel 1) the solution and the force on m′ are∇²Φ = 4πGρ,  Φ(r) = −Gm/r,  F = −∇Φ = G m m′/r²“with κ = 8πG/c⁴ fixed by comparison. Thus, the exact Newtonian law isrecovered from the Void loop picture.” Φ “is simply the mathematical shorthandfor that elongation as seen by a distant observer.” Kinematics split: segmentsof the front propagate at c (null); the centre of energy follows a timelikeworldline; the static slice is a synchronous gauge valid for |h| ≪ 1.Bridge Math Appendix, Mass and Gravity, section 7 (both framings); MathematicalBridge section 3, steps 3, 5 and 6; Mechanics Math Appendix 4 (imports these).
5The route picture: the cost map and its three kernelsSeen from far away the stationary loop is a smooth cost map n(x): “higher nmeans this step hides a bit more display-area than average”. Routes followthe eikonal |∇S|² = n², d/ds(n t) = ∇n, withn(x) = 1 + η₀φ,  φ = Φ/c²,  Φ = −GM/r,  |η₀| = 2bending α ≈ ∫ ∇⊥ ln n ds → 4GM/(bc²) (|η₀| = 2 set once here, toward the mass);delay Δt = (1/c)∫(n − 1) ds → (2GM/c³) ln((rS + rR + D)/(rS + rR − D));clock drift dτ ≈ dt √(1 + 2φ), Δν/ν ≈ −Δφ; periapsis Δϖ ≈ 6πGM/(a(1 − e²)c²).Steep profiles, same kernel: closed null routes at rph = 3GM/c², capture belowbc = 3√3 GM/c². “One ontology, one scale (S₀ = ħ), one cost map n(x). Theonce-set factor that matches solar deflection also fixes delay and clock-shift.”This is the same content as panels 2 to 4 in route language: the Bridge’s “routechoice in a shaped background set by another loop’s missing-space circulation”.Mechanics Math Walk-Through, conventions, sections 1 to 5 and 1A; Bridge Narrative step 4.
6What it recovers, and the far field of the drawingresultvaluewhereNewton’s law, F = Gmm′/r²exact, weak fieldAppendix 7solar limb deflection1.75 arcsecMech. Appendix 5Shapiro delay, Earth–Mars123.6 μsMech. Appendix 6Pound–Rebka, 22.5 m2.455 × 10⁻¹⁵Mech. Appendix 7Mercury’s perihelion43.0″ per centuryMech. Appendix 8photon sphere, capture3GM/c², 3√3 GM/c²W.-T. 1AThe drawing’s far field: the source’s missing share δ against the pure inverse squarerR = √(r² − R₀²)δ = R₀²/(R² + R₀²)R₀²/R²ratio31.6580.69442.27270.30643.1220.39060.64100.60965.4540.17360.21010.826109.6820.06250.06670.9382019.8430.01560.01590.984The ratio → 1: the deficit is inverse square far out, the seed of Φ = −Gm/r (Bridge §3, step 3).
SymbolMeaningWhere it is fixed
Γ, Γ′the source loop; the probe loop (both closed Void paths)Bridge §3
Ad, SΓ, SΓ′display area; the loop actions ∮Ad ds (source, probe)Bridge §1, §3; Appendix §1
Tμν, Gμν, κthe loop’s stress–energy; the Einstein tensor; κ = 8πG/c⁴, fixed by comparisonAppendix §3, §5, §7
Φ, hμνthe Newtonian potential, −Gm/r; the metric perturbation, h₀₀ = −2Φ/c²Appendix §7
n(x), η₀the cost map, 1 + η₀Φ/c²; |η₀| = 2 set once by the deflectionMechanics Walk-Through, conventions and §1
R₀, r, R, δloop radius; true distance; seen distance R² = r² − R₀²; the missing sharethe map (the drawing’s construction of Bridge §3)
G, c, S₀ = ħacceptance locks; the single scaleMechanics Narrative, language and locks

Every line computed. The lattice is drawn through the map R(r) = √(r² − R₀²) about the source, R₀ = 2.5 lattice steps, the same map and scale as the Mass page; ink darkens with the local deficit δ (declared). The source loop removes a fixed transverse area πR₀² from every circle about it on the central plane (the Mass page, panel 4) and its deficit falls as 1/r² (panel 6 table), the seed of the Newtonian potential. The probe loop is drawn from inside: its hidden sphere and its photon’s path, the great circle in the loop’s plane (front half solid, back half dashed), its lattice lines clipped at the sphere. The probe’s centre (orange) is integrated under a = −GM r̂/r² from a start off the axis with a transverse velocity, GM a visibility choice, then pushed through the map; the arrowheads sit at equal-time positions; the dashed line is the same start with no source. All relations are from the published Mathematical Bridge, section 3, its Math Appendix and the Mechanics documents; G is an acceptance lock and is not fit. Sources: Mathematical Bridge, Bridge Math Appendix, Bridge Narrative, Mechanics Math Walk-Through, Mechanics Math Appendix. Generator: gen_gravity_proofs3d.py.