Same form, one substance

The chart prints the wave equation in one and two dimensions, the string, the harmonic and the damped oscillator, radioactive decay, the heat equation, Poisson’s equation, the Maxwell–Faraday law, the Schrödinger equation, the Navier–Stokes equations and Einstein’s field equations, and closes with the line that each describes how things change from one instant, or one point, to the next. The record behind them is accepted in full:

Same form, one substance

FDT has one substance, the omnium: matter = energy = spacetime. The chain is one identity read by quantum mechanics, electromagnetism and general relativity, and its four terms adjust together, never one at a time; every step is a system-wide settlement of the state the system is in (0.1, 0.1a). A carrier is a disturbance of a medium at that medium’s speed, the chain gives one speed to every reading, and there is one medium, so there is one carrier: photon = graviton = gluon = phonon = magnon, one stroke read in different regimes (I-⁠2).

The chart’s worlds are therefore one. Each equation is a reading of the chain on the one medium, and the shared form is the chain’s d/t held in a regime. The map places the twelve, with generalised Snell’s law as a thirteenth, on four trunks of the core: the stroke, the closed stroke on its cylinder, whole events, and the geometry with the medium that runs on it.

The stroke

The omnium’s two response coefficients make the two grades of its stroke drive each other. With ∇·E = 0 the curl of the curl turns the two responses into one equation:

This is the chart’s wave equation with v = c. Clocks and rulers are closed strokes, so the rate is one on the spot: every outside rate and every common rescaling cancel, because the clock is made of the thing it times (0.1d). Where the chart’s v is less than c, v is the transport of the one stroke in a regime: among the clamps of a vibrating volume a loaded line carries its long wave below c while the stroke between the clamps runs at c. The regime is geometry, never a second kind of wave.

The equation in two dimensions takes a plate’s boundary data. A rectangle clamped along its edges carries the modes sin(jπx/L_x) sin(lπy/L_y) cos ω_jlt, which vanish on all four edges: the clamps set the mode geometry, not the medium, and excitations collect on the nodal lines as sand collects on a vibrating plate (P-⁠2, P-⁠3). A standing mode is two counter-running strokes, and one nodal architecture organises every level up to the merged domain (0.8).

A string under tension 𝒯 with inertia μ per unit length carries waves at v = √(𝒯/μ). A closed strand is the string the chain fixes: a closure of energy E_s and radius R is held by the tension E_s/2πR, its strand’s energy per unit length e_ℓ, and its inertia per unit length is e_ℓ/c², so

c² is tension over inertia per unit length. A wire under 100 N with 1 g per metre carries its waves at 316.2 m/s, its tension 1.11 × 10⁻¹² of its rest energy per unit length, and every tension is below F_max = c⁴/4G = 3.0256 × 10⁴³ N, the force ceiling (0.2).

Faraday’s law is one of the two responses, and with the other it makes the stroke. In the geometric algebra of space F = E + cI₃B joins the two grades, and Maxwell’s four equations are the grades of one:

A plane stroke satisfies all four and is null, F² = 0. The magnetic field is a helical density gradient of the omnium, the bivector grade of the density field aligned along an axis (P-⁠5): Faraday’s law says that a twist changing in time drives a circulation of the electric grade about it, with no carrier exchanged.

The closed stroke on its cylinder

The projection x = A cos(ωt + φ₀) of a point running a circle satisfies ẍ = −ω²x, and with its partner A sin(ωt + φ₀) it is the circle itself. The oscillator is the shadow of a circle, and the circle run in time is a helix on a cylinder: the oscillator is the cross-section of a helix, as ψ is (I-⁠4). A closed pattern keeps time by circulating, so every clock is this oscillator, and one quantum of it keeps the period of the bridge, T_q = 2π/ω = h/E. Its ground state has the profile ρ₀ ∝ e^{−mωx²/2ħ}, and the potential and the profile term add to ħω/2 at every x: the zero-point energy is the confinement cost of the ground state’s profile, carried by an omnium that is never empty.

The damped oscillator’s two projections lie on a circle of radius Ae^−γt, with γ = b/2m: the helix runs on a cone. The energy obeys dE/dt = −bẋ², and the mode posts what it loses to the strokes and phonons of the medium. Run backward, the motion solves mẍ − bẋ + k_sx = 0, not the forward equation: the oscillator and the medium settle together at every step, and a process run backward is never the mirror of the forward event (0.1a).

A point of a helix has cylindrical coordinates (z, ρ, φ), and its projection onto the plane normal to the axis is ρe^Σφ, with Σ = e₁e₂ the unit bivector of that plane, Σ² = −1. The complex numbers are the even algebra of the plane, so every complex equation of quantum mechanics is a pair of real equations. With ψ = ψ₁ + iψ₂, the Schrödinger equation is a real rotation:

In the cylinder’s own coordinates, ψ = ρe^iφ, it is two real equations and a closure:

with the profile term Q_ρ = −(ħ²/2m)∇²ρ/ρ, the confinement cost of I-⁠5 written on the radius. (C) and (H) are Madelung’s equations of 1927; (W) says that φ is an angle, so a helix closes on its cylinder only after a whole number of turns. The equation is the slow reading of the closed strand: a closed pattern’s phase obeys ω² = c²k² + ω_m², and with its rest turning ω_m = mc²/ħ removed the leading term is ħk²/2m. Read on the cylinder the quantum face is deterministic: the start azimuth φ₀ fixes the helix, superposition is ignorance of φ₀ at the reading, and a closed pattern seats whole in one port with the frequency of its share ρ² (I-⁠4).

Whole events

Decay is loop unwinding through the one channel left open, every exact book balancing across the event: four-momentum, charge, fermion parity and angular momentum, with m_ic² = Σ_kE_f,k in the rest frame (I-⁠5b). Each closure unwinds whole, at a time fixed by its own configuration and the settlement of its domain, and the decay law counts such events among closures whose phases are unread:

A million closures, each decaying at a time fixed by its own phase, return the exponential to 0.3 per cent at every time sampled. The same count governs a beam of strokes on a dilute path, whose untouched fraction is the exponential of minus its optical depth: both count whole events. The rate of a closure comes from its decay kernel, to be derived. Unwinding releases the stored potential of compression to the spacetime phase, and the effective cosmological term is the running total of those releases (0.7).

Heat is carried by strokes: phonons, the medium’s own stroke among its clamps, and radiation, the open stroke. A stroke runs between whole events, removed whole, redirected whole or re-issued whole, and each event sets it on a new axis. Strokes running at a transport speed v, redirected whole at mean interval τ, obey the telegraph equation, and read at times long against τ it is the heat equation:

The fronts move at √(D/τ) ≤ v, and no signal outruns the strokes: the heat equation’s spread to every distance at once is the slow reading of a law whose signals run at the speed of the strokes that carry it. Temperature is the medium’s clock factor, the reading of its flow and never the rate of a clock it carries; held static in a static geometry the medium settles with T√(−g_tt) constant, Tolman and Ehrenfest’s relation (0.1c, I-⁠7).

The static response and the geometry

With no change in time Faraday’s law makes E a gradient, and the scalar grade of the one equation gives Poisson’s, ∇²ϕ_e = −ρ_e/ε₀: ε₀ is a response coefficient of the omnium, and ρ_e counts charges of closures, each a reading of an integral strand ledger (I-⁠5a). For gravity, Poisson’s equation is the static weak-field limit of the field equations at leading order in α_ext:

The potential is the bookkeeper’s reading of the lapse √(1 − α_ext), and its gradient is the free fall of a pattern moving at constant speed, on its own clocks and rulers, through the density gradient (0.1c).

The field equations stand unchanged, G_μν = κT_μν with the one total source and κF_max = 2π exactly. The exterior of a spherical source written with α_ext = R_s/r_f is Ricci-flat: α parameterises the solution and is downstream of it (I-⁠3). FDT has no cosmological-constant term. The chart’s Λg_μν is read as Λ_eff, the released stored potential of compressed omnium:

a running total that grows with cumulative unwinding, its release function to be derived (0.7). Dividing the exterior by its lapse squared leaves the exact optical metric, and generalised Snell’s law is its Fermat jump law:

This is Bouguer’s form of Snell’s law for a stratified medium, here exact: θ is the angle to the radius and r_opt the radius of the sphere’s circumference, both measured in the optical metric. A stroke grazing the solar limb is bent by 4G𝓜_⊙/(R_⊙c²) = 1.7516″, and the exact null path returns the same. A lone index (1 − α_ext)^−½, with no radial term, bends it by half, 0.8758″: the radial factor (1 − α_ext)⁻¹ of the exact metric supplies the other half. A slow pattern reads the lapse and a stroke reads the optical metric, so the factor of two is the extended equivalence principle read on a stroke. α is the density coordinate, it parameterises the exterior solution of the field equations, and that exterior is the optics of the exact index: FDT = the Einstein field equations = generalised Snell’s law, the title of I-⁠3.

The medium in motion

The omnium is a medium, and a fluid is the flow of its patterns. For a stress tensor of perfect form at fixed entropy per count, the four-momentum book of the field equations is the medium’s equation of motion, and 0.1c reads it as free fall:

The medium’s patterns move free of every force in their clock factor, the enthalpy per count, and the pressure force is the bookkeeper’s reading of their common rescaling. For a liquid of constant density the force per unit mass is −∇P/ρ_m at leading order in 1/c², and the chart’s equation follows:

up to corrections of relative order |v|²/c². Euler’s part is free fall on the medium’s own clocks and rulers. The viscous term is the first-order stress of a causal fluid, the transfer of momentum between neighbouring layers, its coefficient η from the omnium’s constitutive law to be derived. The body force carries gravity, the motion of the fluid’s patterns through a density gradient, and the medium’s vorticity is the swirl of its flow: mechanical vorticity and magnetic moments are two grade projections of one density-gradient topography (P-⁠5). The resolved transport is the normalised average of strand velocities that each run at c, so it never exceeds c, and on a fixed flat domain with positive viscosity that maintained bound carries every strong solution forward for all time.

The common form

The chart’s equations share a form: a time derivative of first or second order equals a spatial operator. The shared form is the chain’s d/t held in a regime, and its order says what the equation counts. The second-order equations, the waves, the string and the oscillator, are the stroke and its closure, even in time because a free stroke has no event of its own: every change it undergoes is a whole event with a closure, where the settlement enters. The Schrödinger equation is first order because its i is a turn, and running it backward turns the helix the other way. The heat and decay equations are first order for the opposite reason: they count whole events of a settlement, and events do not commute. The damped oscillator stands between, a closed stroke posting its energy to the medium. Poisson’s equation is the stationary limit, the field equations and generalised Snell’s law are the one geometry every other equation runs on, and the Navier–Stokes equations are the medium’s own motion on it.

The chart closes with the line that each equation describes how things change from one instant, or one point, to the next. In FDT that change is a settlement: the omnium solves its one optimisation again at every step, from the state the system is in (0.1a).

Each line below is the converse of a derived result: the measurement that would end it.

1. T-⁠1
   One rate. An energy-dependent speed of free strokes from one source, away from closures, ends the stroke equation with v = c on the spot.
2. T-⁠2
   Strokes carry heat. A thermal signal that outruns the strokes of its medium ends the reading of heat as strokes run between whole redirections.
3. T-⁠3
   The factor of two. A bend of a stroke other than twice the slow reading at leading order, γ_PPN ≠ 1, ends FDT = the field equations = generalised Snell’s law.
4. T-⁠4
   Released potential. A decrease of the effective cosmological term over the domain’s history ends its reading as the running total of released stored potential.
5. T-⁠5
   Single seatings. A frequency of single seatings other than the share ρ² ends the seating law with which the cylinder’s determinism meets the record.

Six pieces sit outside the derivation and leave every result in it unchanged:

1. D-⁠01
   Decay rates. The decay kernel of a closure and its rate λ are to be derived (core O-⁠12).
2. D-⁠02
   Charge to force. The map from a charge to an input of the pair law is to be derived (core O-⁠6).
3. D-⁠03
   Viscosity. The coefficient η of the first-order stress of the medium, from the omnium’s constitutive law, is to be derived.
4. D-⁠04
   The material index. The index of a medium, the aperture of P-⁠4 that carries light through matter below c, is to be derived (core O-⁠8).
5. D-⁠05
   The release function. The mass-loss rate density and the compression contrast that feed Λ_eff are to be derived (core O-⁠14).
6. D-⁠06
   Spectra. The confinement operator, whose eigenvalues are the levels of closed patterns and the lines they emit, is to be derived (core O-⁠9).

For the same source and data FDT returns every equation of the chart as it stands. The stroke equation, the plate’s modes, the string, Faraday’s law and the oscillators are recovered exactly. The Schrödinger equation is exactly equivalent to (C), (H) and (W), its predictions unchanged, and the matter waves of Davisson and Germer and of C₆₀ are its record. The decay law and the heat equation are the counting laws of whole events, the latter at times long against the interval between redirections, and Norman’s 45 half-lives of ⁵⁶Mn are the exponential’s record. Poisson’s equation is exact for electrostatics and the leading weak-field order for gravity. The field equations are unchanged, and generalised Snell’s law is their exterior’s optics, with the solar-limb bend of general relativity, 1.7516″, and γ_PPN = 1. The Navier–Stokes equations are the slow limit at the orders stated.

FDT supplies an additional source in one place: the term the chart writes as Λ is the released stored potential of unwinding matter, a running total, its release function to be derived. Where a premise departs from nature, an empty vacuum or a probability amplitude, FDT is under no obligation to reproduce it.