Applications / Foundation papers
What c is
In FDT c is derived, not given. It is the strand rate of the omnium: the rate at which every stroke runs, measured on the spot by clocks and rulers that are themselves closed strokes. It converts the time a pattern keeps into the length its strand runs and, squared, the matter reading of the one substance into its energy reading.
The record
What the constant’s record holds
The central invariant reads one identity four ways:
Its constant does two jobs. Read as d/t, it converts the time a pattern keeps into the length its thread runs: metres locked to seconds. Read as E/m, it converts the matter reading of the one substance into its energy reading. This paper derives it.
The constant entered physics as a rate before it had a name. Rømer’s timing of the eclipses of Io in 1676 showed that light runs at a finite rate. The letter came from a conversion factor: the c of Weber and Kohlrausch, in 1856, stood in Weber’s electrodynamic force law as a relative speed, and they measured it at 4.39 × 10⁸ m/s from the ratio of the electrodynamic and electrostatic units of charge, with charges and currents and no light anywhere in the apparatus. Weber’s law makes that constant √2 times the stroke rate. Drude gave the letter to electrodynamic waves in 1894, and Einstein took it up in 1907, in the review in which he set out the equivalence of mass and energy.
The modern record is a set of equalities, each measured to high precision:
| What the record shows | The measurement | Where the derivation returns it |
|---|---|---|
| The two responses set one rate | μ₀ = 1.256 637 061 27(20) × 10⁻⁶ N/A² gives ε₀ = 1/(μ₀c²) = 8.854 187 8188 × 10⁻¹² F/m | The stroke equation, c² = 1/(ε₀μ₀) |
| The gravitational and the electromagnetic stroke run at one rate | The gamma rays of GRB 170817A arrived 1.74 ± 0.05 s after GW170817: the two rates are equal within −3 × 10⁻¹⁵ and +7 × 10⁻¹⁶ | One carrier, one rate for every open stroke away from closed patterns (I-2) |
| Energy is mass times c² | The gamma rays emitted in neutron capture equal the mass change times c² to (−1.4 ± 4.4) × 10⁻⁷ | The inertia of a closed pattern, closure included: 1/c² per unit energy |
| The rate is a ratio of two patterns’ scales | From 1960 to 1983 the metre was 1 650 763.73 wavelengths of a krypton-86 line, and from 1967 the second was 9 192 631 770 periods of the caesium-133 hyperfine transition. Evenson et al. measured 299 792 456.2(1.1) m/s, which is νKr/νCs = 53 835.1286(2); the 1983 value lies 1.64σ from it | One rate on the spot, c = ΛC/ΛR |
| The units lock to the strand | Since 1983, c = 299 792 458 m/s by definition; since 2019, h as well, so c²/h = 1.356 392 489 652 × 10⁵⁰ Hz per kilogram | Metres per second of kept time; the bridge E = mc² = hν of 0.1a |
The FDT reading
0.1d The strand rate, I-2 one carrier, I-5 matter is closed stroke
One rate, measured against itself
The omnium carries one elementary disturbance, the stroke. Its two grades drive each other: the vector grade, with response coefficient ε₀, and the twist, with μ₀. In a region crossed by no closed pattern, their two circulation laws, with ∇·E = 0 = ∇·B, give every Cartesian component one equation, with one rate:
A clock is a closed pattern, and its tick is a length of strand run. A ruler is a closed pattern, and its length is one of its own scales. A stroke that passes them both reads
whatever rate strokes run at in any outside bookkeeping, and whatever common rescaling the region has undergone. The outside rate cancels because the clock is made of the thing it times. The uniform motions that carry strokes to strokes form the boost group with parameter c, and reciprocity with isotropy fixes the common rescaling at 1, so a clock and a ruler in uniform motion measure every stroke at c. The boost mixes the space reading and the time reading with the coefficient v/c², and c is the one number that makes the two readings commensurable.
The strand’s right triangle
A closed pattern moving at v runs its strand at c on a helix. It translates v along its axis and circulates u around it:
A helix is a geodesic of its cylinder. Unrolled, the cylinder is a plane and the helix a straight line, the hypotenuse of a right triangle whose legs are the translated length v dt and the circulated length u dt. A pattern keeps time by its circulation, one tick per circulated arc, and the arc is transverse, the same in every frame. So c dτ = u dt, and the triangle is the interval:
Time is circulated strand and space is translated strand, and c converts the one into the other. Dilation and contraction are the boost read on a clock and on a ruler. Every massive pattern is its own stroke clock and its own stroke ruler: matter keeps time and length because matter is closed stroke.
The strand of a moving pattern. The ring is a closed pattern sliding along its axis at v; a point of its strand circulates at u = c/γ and leaves a helix. Beside it the cylinder is unrolled: the strand runs the hypotenuse c dt, translates v dt and circulates u dt = c dτ, the time the pattern keeps. The squares on the three sides carry c² = v² + u². Set the speed or pick one. At 0.6c the triangle is the 3 : 4 : 5 triangle; at 45° the two readings are equal and γ = √2. The ring’s radius and the drawing’s rate are schematic; the proportions are exact.
0.1d, I-5
The open stroke stands at π/4
An open stroke’s run does not circulate. It carries the whole rate as translation, v = c and u = 0, keeps no time, and runs at c in every frame, since the boost carries x = ct onto x′ = ct′. Its grades turn about the run. A stroke carries momentum 1/c per unit energy, and its momentum E/c carries its angular momentum, one quantum ħ, at the radius
On that cylinder the turning grades advance one wavelength per turn around a circumference of one wavelength, so the helix stands at π/4 at every frequency, and higher frequencies stand on inner cylinders. In the bookkeeper’s time the turning sweeps the cylinder at ωλ̄ = ω/k = c while the run advances at c: the two legs of the helix are equal because the stroke equation is non-dispersive. The phase kx − ωt = k(x − ct) is constant along the run, so the helix stands at π/4 in space at a fixed time, in time at a fixed place, and as the line x = ct in the plane of x and ct.
The angle belongs to the open stroke. A closed pattern’s phase has two rates, its phase rate ω/k and its run dω/dk, which multiply to c². Their ratio,
exceeds the open stroke’s 1, so its phase helix stands steeper than π/4; only the open stroke has equal rates at every frequency. Masslessness is openness: an open stroke has no frame in which it rests, and its turning is its phase, not a clock it keeps. Closedness is mass: a closed stroke circulates, keeps time, and has a frame in which its whole rate is circulation.
The stroke’s helix at π/4. Three open strokes an octave apart stand on concentric cylinders of radius λ̄ = c/ω. The higher the frequency, the inner the cylinder, and every helix advances one wavelength per turn around a circumference of one wavelength. The run advances at c while the grades turn, each at c around its own cylinder. Move the frequency from radio waves to X-rays: the radii change and the angle does not. The second view sets a closed pattern’s phase beside it. Its two rates, the phase rate ω/k = c²/v and the run dω/dk = v, have the ratio tan Θ = c²/v² > 1, so its phase helix stands steeper than π/4, and approaches it only as v → c; the small triangle draws the two rates. The cylinders are drawn in proportion to one another; the readouts give their sizes.
I-2, 0.1d
The chain is one rate, squared
A closed pattern at rest with energy E₀, set moving at v, carries
with inertia 1/c² per unit energy and m = E₀/c². The closure is what makes it so. The pattern is one pattern of the omnium, closure included, and its stress–energy is conserved, so its integrated rest-frame stress vanishes (von Laue). The strands of a ring circulating in a plane that contains v would carry, alone, pc²/(Ev) = 3/(2 + β²), with β = v/c; the tension that holds them restores 1. For every pattern, then,
The matter–energy conversion is the square of the space–time conversion because two factors of the rate enter the inertia: the stroke’s momentum per unit energy, 1/c, and the strand’s translation per unit speed, v/c per v. In general both are the coefficient v/c² with which the boost mixes energy into momentum. Mass is the circulating share of the energy, E₀ = E sin ϑ, and momentum the translating share. Read through the chain, the product ε₀μ₀ is the inertia per unit energy of every pattern: the electromagnetic reading of the chain is the matter–energy reading, not a coincidence beside it.
c² is an area
The chain’s constant enters squared, and a square is an area. In the geometric algebra of the unrolled strand plane, with Σ = e∥e⊥ and Σ² = −1, the square on the strand rate is the sum of the translation square and the circulation square, and the strand’s own spinor carries c² as its area factor:
Momentum is the scalar grade and rest energy the bivector grade. A pattern at rest has P = mc²Σ: rest energy is circulation, an oriented area. An open stroke has P = pc, a pure scalar. The rectangle of the two readings, vu = (c²/2) sin 2ϑ, is largest at π/4, where the translation square, the circulation square and the rectangle all equal c²/2. Rest energy and kept time are quadratures,
each the side of the square equal to the rectangle spanned by two strokes: a uniform motion squeezes the rectangle and keeps its area. The stroke itself is the null multivector. With F = E + cI₃B, F² = (|E|² − c²|B|²) + 2cI₃ E·B = 0: its vector grade and its twist balance under c.
The spinor flip, 𝔣(w) = Σw̃ = u + vΣ, exchanges translation and circulation, keeps c² and fixes the 45° spinor. It carries the open run, w = c, onto the pattern at rest, cΣ: closing a stroke turns its run into circulation. The stroke’s helix, run c and turning c, is c + cΣ, and the flip leaves it as it is.
The velocity sphere, and the support of a held pattern
A strand’s state is its pair of rates together with two phases: its axial position within a turn, φ₁, and its position around the axis, φ₂. With ζ₁ = (v/c)eiφ₁ and ζ₂ = (u/c)eiφ₂, |ζ₁|² + |ζ₂|² = 1: the strand states form a three-sphere of radius c, the velocity sphere, one radius at every share because the stroke has one rate. The strands are its Hopf fibres. Its tori |ζ₁|² = 𝔰 have area 4π²√(𝔰(1 − 𝔰)) in the unit sphere, largest, 2π², at 𝔰 = ½, the torus the 45° strand spans; the spinor flip exchanges the two moduli at fixed phases and fixes that torus pointwise. The translation share 𝔰 = v²/c² is kinematic. It is never an α, with one exception.
A pattern held at field radius rf on the exterior branch moves against the omnium’s inflow at v = c√αext and keeps time at u/c = √(1 − αext). For that pattern
and the velocity sphere is the support of 0.3a. Its 45° torus is the Clifford torus, at rf = 2Rs, the seam; the spinor flip is the seam 𝔍: α ↦ 1 − α, which fixes the Clifford torus pointwise; and the open interval reads on the strand as u > 0 and v > 0. The branch exchange 𝔦: rf ↦ Rs²/rf keeps α, fixes the fold and swaps the two seams; it is a different involution. The hold that keeps the pattern at rf is the I-1 pair, Fmaxα₁α₂ = G𝓜m/rf².
The quarter circle of strand rates. Every strand state is a point (v, u) at distance c from the origin: the meridian of the velocity sphere. The shaded rectangle is the bivector vuΣ, largest for the 45° strand. The open run (c, 0) keeps no time; a pattern at rest, (0, c), is all circulation; the corner (c, c) is the stroke’s helix, which the flip leaves fixed. Drag the point or flip it: the spinor flip exchanges translation and circulation, keeps c², and carries the open run onto the pattern at rest. Switch to a pattern held on the exterior branch: its share is αext, the scale takes the colour of the regimes, the 45° strand sits at the seam rf = 2Rs, and the flip is the seam 𝔍. The ends of the held scale are limits, never states.
0.1d, 0.3a, I-6
Finite, nonzero, and what the number records
Strokes approaching the floor run at c, so c = dmin/tmin, a ratio of two nonzero, finite extents. With Fmax held, c → 0 sends dmin to zero and c → ∞ sends tmin to zero, a vanishing variable of the chain either way. The matter phase says the same: with Fmax or G held, either limit sends every share to 0⁺ and empties the α > ½ solid torus. Matter exists, so 0 < c < ∞.
The number counts metres of strand run per second of kept time. From 1967 to 1983 it was a ratio of two transition frequencies, each a difference of confinement eigenvalues of a closed pattern, multiplied by two defined numbers:
In 1983 the metre became “the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second”, read in FDT as the length a stroke runs in that time, and the number became a definition: metres locked to seconds through the strand, the d/t = c exactly of 0.1a. In 2019 the Planck constant was fixed as well, which locks the kilogram to the second through c²/h: the bridge E = mc² = hν, built into the units. The SI is the central invariant written as a system of measurement.
The number carries no geometric content of its own. The geometric content of c is what the derivation supplies: that it exists and is one; that it is finite and nonzero; that it is the rate of every open stroke away from closed patterns and the circulation of every closed one; that its square is the area of the strand square; and that it is the radius of the velocity sphere, which for a pattern held on the exterior branch is the support.
What c is
Every pattern runs at c. An open stroke runs it straight and keeps no time, while its grades turn at the same rate on a helix standing at π/4. A closed pattern at rest runs it around and keeps time by doing so. A closed pattern in motion runs it on a helix, translating v and circulating u with v² + u² = c². Space is translated strand, time is circulated strand, and c converts the one into the other. Mass is the share of energy that circulates and momentum the share that translates, and c² is the square they share: the area factor of the strand spinor, and the squared radius of the velocity sphere. The name records which open stroke was measured first.
The tests
Use 2 What would end each derived result
The paper states its convergence with the record. Each line below is the direct converse of a derived result: the measurement that would end it.
- T-1One rate for light and the gravitational stroke. Away from closed patterns the two run at one rate, as one carrier requires; GW170817 with GRB 170817A holds them equal to parts in 10¹⁵. A measured difference between those rates ends the one-carrier reading.
- T-2Dilation γ, contraction 1/γ. The boost group returns both in uniform motion. A measured dilation other than γ, or contraction other than 1/γ, in uniform motion ends the derivation of the boost group.
- T-3Inertia 1/c² per unit energy. E = mc², with the precision of Rainville et al., (−1.4 ± 4.4) × 10⁻⁷. A measured departure of the emitted energy from the mass change times c² ends the inertia derivation.
Three pieces sit outside the derivation and leave every result in it unchanged:
- O-9The number’s dimensionless content. The eigenvalue ratio νKr/νCs = 53 835.1286(2), and every strand scale Λq of a closed pattern, come from the confinement operator and are to be derived (O-9).
- O-16The share on the interior chart. The kinematic reading of the translation share inside a bound configuration is to be derived (O-16); the flow reading stops at the fold.
- O-8The aperture. The stroke read across closed patterns, through the medium’s aperture, is to be derived (O-8).
Readings declined
Use 6 Data yes, interpretation not necessarily
Standard reading
c is the speed of light in vacuum: a property of one kind of radiation crossing an empty background.
FDT reading
c is the strand rate of the omnium: the conversion between its space and time readings and, squared, between its matter and energy readings. Every open stroke runs it straight where it crosses no closed pattern, and every closed pattern runs it in circulation. Light is the first open stroke that was measured at that rate, and there is no vacuum: ε₀ and μ₀ are response coefficients of the omnium (0.1, 0.1d, I-2).
The constancy of c for every observer is a postulate.
One rate on the spot follows from the instruments. Clocks and rulers are closed strokes, so a stroke measured on any pattern’s own clocks and rulers returns the ratio of two of their scales, and the boost group follows with its rescaling fixed at 1 (0.1a, 0.1d).
Mass and energy are equivalent through a constant of nature, with no mechanism behind the square.
Mass is the circulating share of a pattern’s energy and momentum the translating share. The conversion is the square of the space–time conversion because two factors of the strand rate enter the inertia, and the closure supplies the stress that makes the inertia exactly E/c² (0.1d, I-5).
Clocks and rulers sit in a spacetime background whose geometry they read.
Clocks and rulers are closed patterns of the omnium. Time is circulated strand and space is translated strand, and the interval is the right triangle of a pattern’s strand (0.1a, 0.1d).
Convergence
Use 2 Same source, same data
For the same sources and data FDT returns the boost group with invariant speed c, and with it every measured time dilation and length contraction; local null cones with d/t = c on the spot in every gravitational field, with the curvature of general relativity unchanged (I-3); c² = 1/(ε₀μ₀) with the measured responses of CODATA 2022; E = mc² and E² = (pc)² + (mc²)², with the precision of Rainville et al.; equal rates for the gravitational and the electromagnetic stroke, with GW170817 and GRB 170817A; and the SI value.
No divergence is claimed: the derivation supplies what c is, not a changed observable.