Applications / Foundation papers
The photon on its cylinder
A photon is the strand itself, open, standing in no closure. The chain gives it the rate c and Planck’s quantum gives it one h per cycle, in its run and again in the turning of its grades, on a cylinder of radius c/ω. The free stroke has no transition of its own: it does not split, decay or tire. Every change it undergoes is a whole-number event with a closed pattern, so light arrives whole or not at all, and the light of a distant galaxy reaches us as it was born, read on a cylinder larger by one ratio of clock factors.
The record
What light does, from the photoelectric threshold to a redshift of six
Light is quantised, carries ħ per stroke, runs at one rate at every energy, changes only in whole events, and reaches us from the far edge of the observed domain with its lines redshifted and its light curves stretched by one factor. The record:
| What the record shows | The measurement | Where the paper reads it |
|---|---|---|
| Whole quanta and a gap | The photoelectric law K = hν − W, linear in ν above a threshold; one atom issues one stroke per reclosure, antibunched | Detection as a whole-quantum event; the count book |
| ħ per stroke | The torque of circularly polarized light on a wave plate that reverses its hand | The turning book on the cylinder |
| No dispersion | A 31 GeV stroke arrived within a second of the keV strokes of GRB 090510, at a redshift of 0.90 | The run at c on every cylinder |
| Recoil and splitting | Compton’s shift; the simultaneity and the energy sum of parametric pairs | The cylinder grows by λ̄C(1 − cos θ); curvatures add |
| No splitting without closures | Light-by-light scattering is recorded in the overlapping density cones of two lead nuclei | The free stroke has no transition of its own |
| Whole removals on the way | The Lyman-α forest; the damping of the background’s anisotropies by e−τ with τ = 0.054; the Sunyaev–Zel’dovich decrement below 217 GHz and increment above | Arrival: untouched, removed whole or redirected whole |
| One ratio of clock factors | Supernova light curves stretched and spectra aged by 1 + z; T(z) = T₀(1 + z); the Tolman surface-brightness signal | Arrival: the bookkeeper’s reading of an unchanged stroke |
The FDT reading
I-2 One carrier, 0.1d the run at c, I-5b whole events
One stroke, two books
The photon is the free presentation of the one carrier, whose frame turns in one plane transverse to its run. A stroke of frequency ω carries one quantum of action per cycle in each of two books. In the run:
In the turning, its grades turn 2π per cycle about the run on a circle of radius
The circumference equals the advance per turn, so the helix the grades trace stands at π/4 at every frequency. The helix carries nothing: the transport is the run at c, and energy carried along the helix’s arc, √2 λ per cycle, would cross space at c/√2, which none does. The open stroke keeps no time; its turning is its phase, never a clock (0.1d).
The family of cylinders is nested: the higher the frequency, the smaller the cylinder. The curvature of a stroke’s cylinder is its wavenumber, so energy is curvature, E = ħck with k = 1/Rω, and a white beam is a nest of cylinders on one axis. ψ = Aeiφ is the projection of the helix onto the plane normal to its run, the lift restores the radius and the hand, and superposition is ignorance of the start azimuth φ₀ at the reading (I-4).
The free presentation turns in exactly one plane, and that plane has two orientations: the two hands of a free stroke. A third state would need a second plane, and a frame that turns in two planes whose bivectors do not commute is the standing presentation, the carrier inside a closure. There is no longitudinal state because the free stroke has no second plane.
The nest of cylinders. Each stroke’s grades turn on a cylinder of radius c/ω while the stroke runs along the axis at c. The front ring is a fixed reading plane: in it each stroke’s grade is a pointer turning at its own ω, the long ones slowly and the short ones fast, and every tip moves at Rωω = c. Pick a stroke to read its two books and its own window on its own cylinder; flip the hand to see the other state of the same stroke. The first chart draws seven optical lines to scale. The second spans radio to gamma on a logarithmic chart of radius, with the pitch kept at 2π times the drawn radius so the angle π/4 stays exact.
I-2, 0.1d, 0.1a
The family and its two ends
A stroke’s own window on its own cylinder, read on the A-1 chart of its exterior, is
strictly increasing in ω. It approaches 1⁻ as ħω approaches EP/√2, where the cylinder would shrink to √2 ℓP, the stroke’s own fold, which the open interval excludes; the circumference there, 8.89 ℓP, lies above the floor 2ℓP, so the window closes before the floor is reached. At the other end the chain, no term of which vanishes, excludes ω → 0⁺, and the cylinder is bounded by the domain. Both ends of the family are limits, never states, and both are circles. On a laboratory cylinder the window is small, 1.3 × 10⁻⁵⁶ for a stroke of 1 eV, and the small number is a correct reading.
What is quantised
Light is quantised in three books and in none of them is there a lattice of frequencies. In the action book, one h per cycle in each of the two books, and no fraction of a stroke. In the source book, a stroke is born when a closed pattern recloses from one confinement mode to another, so its frequency is a difference of two eigenvalues: closures write lines, and the continua of the record come from free closures, whose four-momentum book is continuous, and from the ground of a room read at its temperature. In the exchange book, a stroke changes only by a whole-number event with a closure, every exact book balanced across it.
The free stroke has no transition of its own
The transition operator acts on the closures of a domain, and a splitting is an act of the frame in a second plane, which the free presentation does not have: a stroke with no closure in its support has no event to enter, and its rate is exactly zero. The hand book closes the two-stroke channel independently, since one unit of hand does not divide into two units on one axis.
The exchange ledger
The quanta a closure can take form a numerical semigroup, generated by its irreducible quanta, the positive differences of its open eigenvalues that are not sums of smaller ones. No transfer below the least quantum exists, and when the quanta are coprime multiples of one unit the excluded transfers are finite, all below the Frobenius number, a₁a₂/u − a₁ − a₂ for two of them. Beside the semigroup sit recoil, Compton’s formula read on the family,
and splitting with a closure present, a curvature sum, 1/Rω = 1/Rωs + 1/Rωi. Both daughters are born at one event with one start azimuth: two strokes on one cylinder with one φ₀ trace identical helices, and the correlations of the record are their shared bivector grade, deterministic. Absorption and emission are two events, never one run backward (0.1a).
Detection, the ground and arrival
A detector is a closure, and detection is the event in which its mode takes the stroke’s whole quantum: K = ħω − W is the four-momentum book of that event. Free radiation carries no gap; the gap belongs to closures, and on the family every gap is a cylinder: a closure of gap W takes strokes from cylinders Rω ≤ RW = ħc/W and none from outside, however many arrive. Millikan’s sodium surface had its threshold at 43.9 × 10¹³ Hz, W = 1.82 eV: RW = 108.7 nm, and the threshold wavelength 2πRW = 683 nm.
The ground is the one floor of every room: the chain admits no empty state, and every reading is taken against the ground’s standing-wave state. A stroke is read as one stroke when it stands alone in its mode, which in a room at temperature T needs ħω > kBT ln 2. For the open domain, at 2.7255 K, the reading floor is a cylinder, RB = ħc/kBTB = 0.840 mm, with the occupation crossing unity at 39.4 GHz; single microwave strokes are read in rooms cooled to tens of millikelvin. The ground’s temperature is its clock factor (0.1c), so a blackbody stays a blackbody as its peak and its temperature move together.
On its way a stroke changes nothing: no event acts on it, and its quantum, its hand and its start azimuth are the ones it was emitted with. The reading room’s bookkeeper reads ωr = ωe 𝔪e/𝔪r, the one ratio that is gravitational and cosmological redshift, and reads the emitter’s light curve stretched by the same factor, because d and t rescale together. On a dilute path of optical depth τ, the fraction that arrives untouched is e−τ, and every touched stroke is removed whole, redirected whole or re-issued whole on another cylinder. There is no fifth outcome. The redshift is not a loss: the spacetime phase a stroke crosses is dilute, the count of closures on its path is small, and a stroke that left a galaxy at a redshift of six arrives as it was born.
Whole or not at all. In the first view, strokes reach a detector, a closure whose gap is the cylinder RW = ħc/W. Slide the wavelength: while the stroke’s cylinder lies inside the gap cylinder, each stroke taken frees one electron with K = ħω − W, on Millikan’s line of slope h; outside it, raising the rate frees none, and every stroke is redirected whole. In the second view, strokes cross a path of closures. Each arrives untouched, is removed whole or is redirected whole, and the untouched count settles to e−τ. Slide the clock-factor ratio: the reader reads the arriving strokes on cylinders larger by 1 + z, while each stroke carries the h per cycle and the ħ it was emitted with. The paths and the electrons are schematic.
I-5b, 0.1c, 0.8
The properties of light, each with its clause
Massless: an open strand stands in no closure, and mass is a confinement eigenvalue (I-5). Speed c with no dispersion: the run at c, a disturbance of the medium at the medium’s speed (0.1d, I-2). Spin ħ and two hands: one h per cycle in the turning, on the two orientations of one plane. No charge: charge is a reading of closures, and an open strand closes nothing. Bosonic: the fermion class is a nested turning class of closures, which an open strand does not have. Its own antiparticle: conjugation is the hand flip of one configuration (P-4). Stable, changing only in whole quanta, redshifting without loss and arriving whole: the four results above.
The tests
Use 2 What would end each derived result
Each line below is the converse of a derived result: the measurement that would end it.
- T-1One rate on every cylinder. An energy-dependent arrival delay of free light from one source, away from closures, ends the run at c for every cylinder.
- T-2No free splitting. A free stroke that splits with no closure in its support ends the zero rate of the free stroke.
- T-3Whole quanta. A closure in its least mode that takes from a stroke less than its least quantum ends the exchange semigroup; a single stroke from outside RW that frees a photoelectron would.
- T-4No tiring. A distant source whose lines are redshifted by one factor while its light curves are stretched by another, or a loss of a fraction of a quantum with distance, ends the arrival result.
Five pieces sit outside the derivation and leave every result in it unchanged:
- D-01The profile. The transverse profile of one stroke across its cylinder, from the omnium constitutive law, reducing to the measured single-photon transverse modes, is to be derived.
- D-02The kernels. The kernel of each event between a stroke and a closure, the cross-sections the rates are read from, is to be derived (core O-12).
- D-03The material index. The index of a medium as the standing-carrier response of its closures, reducing to the measured dispersion, is to be derived (core O-8).
- D-04Splitting. The splitting kernel on the seat surfaces of a crystal’s closures, reducing to the measured conversion rates and phase matching, is to be derived.
- D-05Orbital winding. The one-stroke cylinder for a stroke carrying orbital winding ℓ ≠ 0, and the relation of the winding book to the hand book, is to be derived.
Readings declined
Use 6 Data yes, interpretation not necessarily
Standard reading
Furry’s theorem and a vanishing collinear amplitude forbid a free photon to split, and photon–photon scattering is the exchange of virtual pairs.
FDT reading
The free presentation has no transition of its own and the hand book closes the two-stroke channel; the recorded light-by-light events, in the overlapping density cones of two nuclei, are the carrier’s frame turning on closures that are present. Nothing is virtual (I-2, I-5b).
A photon splits in a strong magnetic field.
That splitting is splitting with a closure’s structure present: a magnetic field is a helical density gradient of the omnium (P-5).
Tired light: a photon loses energy on its way.
A free stroke sheds no fraction of its action, and the redshift is the clock factor; the record of time dilation and surface brightness excludes tired light, and FDT has no channel for it.
The photon is its own antiparticle.
Conjugation is the hand flip of one configuration (P-4), and for an open strand the flipped configuration is the other polarization of the same stroke; the record has no distinct antiphoton and needs none.
A parametric pair is a superposed two-photon state, and its correlations are nonlocal.
The pair is two strokes born at one event with one start azimuth, one helix read at two places (I-4); the measured coincidences and correlations are accepted in full.
Convergence
Use 2 Same source, same data
For the same source and data FDT returns the measured record. E = hν and the photoelectric threshold are whole-stroke absorption into a closure with its gap read as the cylinder RW; antibunching is one closure issuing one stroke per reclosure. Beth’s torque measures ħ per stroke, handed over twice as the plate reverses the hand. The arrival of GRB 090510’s 31 GeV stroke with its keV strokes is the run at exactly c on every cylinder. Compton’s shift is the radius increment λ̄C(1 − cos θ), and the simultaneity and energy sum of parametric pairs are the curvature sum with the crystal’s closures taking the momentum difference. The exponential survival of a beam through an absorber, the Lyman-α forest, the damping of the background’s anisotropies and the Sunyaev–Zel’dovich distortion are whole events on the way. The stretch of supernova light curves, their spectral ageing, the redshift of their lines and T₀(1 + z) are one ratio of clock factors. The bend of a stroke at the solar limb, 1.7516″, is the exact optical metric of I-3, and FDT and general relativity return the same curvature.
No divergence is claimed. Where a premise departs from nature, an empty vacuum or a virtual pair, FDT is under no obligation to reproduce it.