Defining mass: a closed path and the space it hides, the full derivation

A photon is the smallest piece of light; this page follows one photon closed into a loop and derives what the high school page stated. A region of space is hidden; from outside the missing space is not observable, there is only the point and its effect on every line of space that remains. That is the definition of mass. A: the inside view, the closed path and the sphere it hides. B: the map acting on a flat plane through the centre. C: the identical map on the three mid-planes of a block, where every line stays traceable. D: the full block, every radius at once. In B, C and D the lines that met the hidden boundary merge at the point and continue; the crowding and darkening around it is the interaction zone, and the point is the only near-black. Each numbered circle is a panel below.

A · inside view: the closed path and the region it hidesB · outside view: a flat plane through the centreC · the three mid-planes of the blockD · the full block, every radius at once125534
The case drawn.
m ∝ ∮Γ Ad(s) ds,  R² = r² − R₀²,  δ = R₀²/(R² + R₀²)
(the inertial measure, Mathematical Bridge section 3; 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 closed path gives the rest energy and, for the electron, its size:
E = mc²,  m = (σs/c²)·LΓ,  ƛe = ħ/(mec) = 386.16 fm
(Bridge Math Appendix, Inertial Mass from Energy; S₀ = ħ is the one calibration, Particle Mechanics Math Appendix)
Panels 1 to 4. The general solution, panels 5 and 6.
The general solution, and what it recovers, from the worldsheet action and nothing else:
the loop action SΓ = ∮Ad ds, the inertial measurepanel 1the map and the deficit δ, fixed missing area, 1/r² far fieldpanels 2 to 4E = σsLΓ, m = E/c²: rest energy and inertial masspanel 5the loop’s stress–energy Tμν, the dent that curves spacepanel 6
The coupling of that dent to the motion of a second loop, and Newton’s law from it, is the gravity page.

Published documents used, and nothing else: the Mathematical Bridge (section 3, Closed Loop → Inertial Measure and Gravity, recipe steps 1 to 4) and its Math Appendix (Mass and Gravity from Closed Void Loops, sections 1 to 4); the Bridge Narrative, step 3 (mass as missing space); the Treatise on Caustics, Loop Closure (the closed path, head and tail merged, seen as one point); the Particle Mechanics Math Appendix (S₀ = ħ, ƛₑ). Only ratios enter; S₀ = ħ is the one scale.

The case drawn (panels 1 to 4) and the general solution (panels 5 and 6)

1In this case: one photon, closed on itselfA photon runs at c and hides the space behind it (axiom A1).Close its path so head and tail merge and it hides the same regionevery cycle: the closed path is the electron (Treatise on Caustics,Loop Closure). The path has the shape of the sphere ‖x‖ = R₀;it is a path, not the surface of an object. The region inside isobscured; from outside the whole path is seen as one point.The hidden total per cycle is the inertial measure:m ∝ ∮Γ Ad(s) dsMathematical Bridge, section 3, step 1; Bridge Math Appendix, Mass andGravity, section 1: “the integrated Display Area is S_Γ := ∮_Γ A_d(s) ds”.Scale: the electron loop, ƛₑ = ħ/(mₑc) = 386.16 fm, the single calibration(S₀ = ħ; Particle Mechanics Math Appendix). In these panels R₀ = 2.5 lattice steps.
2The map: what is seen from outsideDistances measured from outside omit the hidden span. A point at truedistance r from the centre is seen at R, withR² = r² − R₀²,  equivalently  πR² = πr² − πR₀²on every central plane, and every point with r ≤ R₀ is seen at the centre.The map rescales ‖x‖ and keeps its direction, so it preserves everycentral plane; every curve in the block is a plane-figure curve (panels C, D).The missing share of the original disk at seen radius R is the deficit:δ = πR₀²/πr² = R₀²/(R² + R₀²);  δ(0) = 1;  δ → R₀²/R² for R ≫ R₀This map is the drawing’s construction of two Bridge lines (section 3): the loop“removes a volume of available transverse area from the surrounding space, fixingthe missing volume”, and “this deficit propagates outward with profile Δg(r) ~ 1/r²”.It is not itself a line of the Math Appendix.
3The blue line, worked: y = 1 in the z = 0 sheetThe same line in panels B, C and D. Crossings from the chord equation withA = (−6, 1, 0), v = (1, 0, 0): (A·v)² − |v|²(|A|² − R₀²) = 36 − 30.75 = 5.25,so the boundary sits at x = ±√5.25 = ±2.291. Both ends of the gap are seen atthe same point, and the two outer pieces join there and continue.xr = ‖x‖R = √(r² − R₀²)δ = R₀²/r²-6.0006.0835.5450.169-4.0004.1233.2790.368-2.291 (boundary)2.5000, the point1.000−2.291 < x < 2.291: hidden, nothing that can be drawnx = +2.291: the same point (δ = 1); x = +4, +6: mirror of the leftTreatise on Caustics (the crossing points seen at one point); the map as in panel 2.
4The invariant: what the map removes, it removes everywhere equallycircle rarea πr²mapped Rarea πR²missing328.2741.6588.63919.635450.2653.12230.63119.6356113.0975.45493.46219.635R² = r² − R₀², so every mapped circle encloses exactly πR₀² = 19.635 square stepsless than before: the same missing area at every radius. The map does one thing, itremoves a fixed amount of space, and nothing else: the Bridge’s “fixing the missingvolume” (section 3), drawn on the central plane.The far field: the missing share δ against the pure inverse square R₀²/R²r = 4: δ = 0.3906, R₀²/R² = 0.6410, ratio 0.609r = 10: δ = 0.0625, R₀²/R² = 0.0667, ratio 0.938r = 20: δ = 0.0156, R₀²/R² = 0.0159, ratio 0.984r = 40: δ = 0.0039, R₀²/R² = 0.0039, ratio 0.996The ratio tends to 1: far from the loop the deficit is inverse square, the Bridge’s Δg(r) ~ 1/r².
5The general solution: from the loop’s action to its massAs the closed loop Γ propagates in time it sweeps a worldsheet W, embedded asXμ(τ, λ) with induced metric γab = gμν ∂aXμ ∂bXν. Its action is the minimal-areaprinciple applied to the obscured area it transports (“analogous to theNambu–Goto action”):Svoid[W; g] = σs ∫W √(−γ) d²ξσs the surface-action density. In the static gauge X⁰ = cτ, Xi = Xi(λ), theenergy is σs times the loop’s spatial length, and the inertial mass isE = σs ∮Γ |∂λX| dλ = σs LΓ,  m = E/c² = (σs/c²) LΓand when the display area varies along the loop, m ∝ ∮Γ Ad(s) ds. “Mass is not anindependent assumption; it is the geometric consequence of how much space theVoid loop obscures.” For the electron, LΓ = 2πƛₑ.Bridge Math Appendix, Mass and Gravity, sections 2 and 4; Bridge section 3, step 4.
6What the loop does to the space around itVarying the same action with respect to the metric gives the loop’sstress–energy, localized on its worldsheet:Tμν(x) = σs ∫ d²ξ √(−γ) γab ∂aXμ ∂bXν δ(4)(x − X(ξ))“In plain terms: a closed loop makes a local ‘dent’ in the fabric of space,proportional to the obscuration it carries.” The stationary loop minimises itsaction, and that minimisation forces the deficit of transverse area drawn inpanels B to D: the dent. Its outward profile is inverse square (panel 4), the seedof the Newtonian potential; how a second loop moves in it is the next page.Bridge Math Appendix, Mass and Gravity, section 3; Mathematical Bridge, section 3,steps 2 and 3. The coupling of this Tμν to geometry (sections 5 to 7) is the gravity page.
SymbolMeaningWhere it is fixed
Γ, Wthe closed path of the photon; the worldsheet it sweeps in timeBridge Math Appendix, Mass and Gravity §1
Ad, SΓdisplay area, the space a front obscures; the loop action ∮Ad dsBridge §1 and §3; Appendix §1
σssurface-action density (a derived constant)Appendix §2
R₀, r, Rradius of the closed path; a point’s true distance; its distance seen from outside, R² = r² − R₀²the map (the drawing’s construction of Bridge §3)
δthe deficit, the missing share of display area at seen radius Rthe map; Bridge §3, Δg ~ 1/r²
ƛe, S₀ = ħthe electron loop radius, 386.16 fm; the single calibrationParticle Mechanics Math Appendix

Every line computed. All four views use the same boundary R₀ = 2.5 lattice steps and the same extent (±6 steps). A draws the flat lattice with the closed path (orange, at c) and the sphere it hides; nothing is bent, because from inside the hidden region is simply there. B maps each grid line of the central plane point by point through R(r) = √(r² − R₀²); lines that cross the disk are drawn through the point and out the other side. C and D map their lines through the same map along each radius and then through a fixed perspective camera, nearer lines drawn darker; segment darkness and weight follow the local deficit δ (declared), so the interaction zone is where the ink itself accumulates; the near-black is reserved for δ = 1, the point. The blue line y = 1 in the z = 0 sheet is the same line in B, C and D and is worked in panel 3. Tables in panels 3 and 4 are computed from the map. Sources: Mathematical Bridge, Bridge Math Appendix, Bridge Narrative, Treatise on Caustics, Loop Closure, Particle Mechanics Math Appendix. Generator: gen_mass_proofs3d.py.