One substance, one carrier, one geometry, myriad forms

Energy, mass, frequency, the period and the length a thread runs in it are readings of one state, tied by the bridges of one chain. No transaction posts to one of them alone, and every cycle of exchanges closes. The paper proves that the books balance this way only when there is one substance, one carrier, one geometry and no fixed background, and sets each alternative against the record: one rate of mass to energy to 4 × 10⁻⁷, one weight for every energy channel to 10⁻¹⁵, one stroke speed to 10⁻¹⁵, and clocks that adjust across a single millimetre.

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The record

What a ledger posted by hand would show, and does not

The central invariant is one identity read four ways:

E/m = c² = 1/(ε₀μ₀) = (d/t)²

Its four terms adjust together, never one at a time (0.1a). A universe with a second substance, a second carrier speed, a second geometry or a background its clocks merely read would need an entry posted by hand somewhere in its books: a cross-rate for each process, a jump of a centre of inertia, a constraint on its metrics, energy from nothing in a closed cycle. The record shows none:

The record the derivation answers to, from the paper’s sources: Rainville et al. (2005), the MICROSCOPE collaboration (2022), the LIGO, Virgo, Fermi and INTEGRAL report on GW170817, Pound and Rebka (1960), Bothwell et al. (2022), the IAU 2015 nominal values and CODATA 2022.
What the record shows	The measurement	Where the derivation returns it
One rate of mass to energy	Nuclear gamma strokes against Penning-trap mass differences: 1 − Δ`m` `c`²/`E` = (−1.4 ± 4.4) × 10⁻⁷	One numeraire; every exchange cycle closes
One weight for every energy channel	Titanium and platinum, whose mass divides among rest-mass, nuclear and electromagnetic channels in different proportions, fall alike: η = [−1.5 ± 2.3 (stat) ± 1.5 (syst)] × 10⁻¹⁵	Every channel posts to one 𝓜 (I-⁠7)
One stroke speed	The gamma rays of GRB 170817A and the gravitational strokes of GW170817: −3 × 10⁻¹⁵ ≤ Δ`v`/`c` ≤ +7 × 10⁻¹⁶ after about 130 million years of travel	One carrier, one geometry
Clocks adjust with height	Over the 22.56 m tower the two-way shift is 1.05 ± 0.10 of the prediction; across one millimetre-scale atomic sample, −9.8(2.3) × 10⁻²⁰ per millimetre against −10.9 × 10⁻²⁰	No fixed background
Read-out differences, one ledger at every size	In `n` + `p` → `d` + γ, `Q` = 2.224566 MeV: the stroke carries 2.223249 MeV and the pattern recoils with 1.318 keV	The nested read-out at nuclear size

The FDT reading

0.1a One state, I-⁠3 one geometry, I-⁠5 integers of closures

The inventory

FDT has one substance, the omnium; one carrier, the spin-one stroke of the density field; one geometry, the density cone that is at once the light cone and the Einstein geometry, on an atlas with the shape of the three-sphere; and myriad forms, every one a configuration of the stroke. Unbound, the stroke runs straight at c and keeps no time, its grades turning about the run on a helix of radius c/ω (I-⁠2, 0.1d); it is read as light, gravity’s ripple, sound and the spin wave of a helical gradient, and standing inside a closed pattern’s own region it is read as the gluon. Bound, n strands of it close into braids: the n = 2 closure is read as the neutrino in the free room and as the neutron in the nuclear room, the n = 3 closure as the electron and as the proton (I-⁠5). The count of forms is open. The count of substances, carriers and geometries is one.

Every book is a reading of one state

A reading is a positive function on the admissible configurations, carrying a unit. A bridge is an identity between two readings with a constant of the medium:

m = E/c², ω = E/ħ,T = 2π/ω = h/E, ℓ = cT,R_s = 2G𝓜/c², α_σ(r_f) = (R_s/r_f)^σ

Here T is the period, which a closed pattern keeps as its time and an open stroke does not keep, and ℓ the length the thread runs in one period, for an open stroke its advance per cycle. Each bridge holds before and after any change of configuration, so every change posts to every reading at once, in the proportions the bridges fix: Δm = ΔE/c², ΔlnT = −ΔlnE, Δln ℓ = ΔlnT, ΔR_s = (2G/c²)Δ𝓜. No single-entry posting exists. A stroke read by two bookkeepers whose clocks differ by a factor 1 + z moves every reading and keeps two invariants:

E T = h,ℓ/T = c by any bookkeeper

Every exchange cycle closes

Written in logarithms, each bridge is an affine relation of slope ±1 between two readings, and a closed cycle of bridges composes to an affine map. That map fixes every value its first reading takes, and unbound strokes carry a continuum of energies, so it fixes an interval: an affine map of the line that fixes two points is the identity. Every cycle has trivial holonomy, and every reading of a pattern read alone is a constant multiple of E or of 1/E:

m = E/c², ω = E/ħ, T = h/E,ℓ = hc/E, R_s = 2GE/c⁴ one numeraire

The cycle E → m → ℓ → T → E reads E/c², h/(mc), ℓ/c and returns h/T = E. The closing is a statement about readings, never about events: every step is a system-wide settlement of the state the system is in, events do not commute, and a process run backward is never the mirror of the forward one (0.1a).

This figure needs JavaScript. A stroke of 2.47968 eV reads 4.42044 × 10⁻³⁶ kg, a period of 1.66782 × 10⁻¹⁵ s and an advance of 500 nm per cycle; the cycle of readings returns 2.47968 eV. A kilogram lowered through 22.56 m yields 221.09 J; read at the top, its strokes carry 221.09 J less than mc², and the cycle posts zero.

One state, many readings. In the first view, pick a pattern or slide its energy and every reading moves at once; slide the bookkeeper and the readings move again while E T = h and ℓ/T = c stay. The token runs the cycle E → m → ℓ → T → E and comes home with the energy it left with. Give mass a rate of its own and the cycle returns a different energy: the entry a second substance posts by hand, against the record of one rate to 4 × 10⁻⁷. In the second view a kilogram is lowered, opened into strokes, sent up and closed again. With the clocks adjusting, the strokes arrive short by exactly the work taken out, and the cycle posts zero. With a fixed background they arrive whole, and every run posts mgΔh from nothing.

0.1a, 0.1c, I-⁠7

The four-momentum book

With G_μν = κT_μν and T_μν the one total source, every channel entering once, the conservation of the source follows from the contracted Bianchi identity, ∇_μG^μν ≡ 0, with no condition beyond the field equation. It is the geometric statement that the boundary of a boundary is empty, and FDT makes it a property of the one substance: the source is the omnium’s one stress tensor, the geometry is the adjustment of the omnium’s own patterns, and the read-out 𝓜 is the one account every channel posts to. The phases of the omnium exchange energy through rates Q_a^ν whose sum vanishes identically (I-⁠6). A configuration’s 𝓜, the mass aspect of its exterior solution, holds while nothing is radiated and falls by exactly the energy radiated outward.

The spacetime book

The support is the three-sphere, |z₁|² + |z₂|² = 1, with the share α_σ = |z₁|² and its complement 1 − α_σ = |z₂|² (0.3a). The share and its complement sum to one at every point, and the round measure is uniform in the share:

vol{a < |z₁|² < b} = 2π²(b − a) each solid torus carries π², half the support

The seam, the Clifford torus α = ½, divides the support into the spacetime phase and the matter phase. The jacket flip 𝔍 exchanges the two moduli at fixed phases: it carries α to 1 − α, exchanges the two solid tori, preserves the measure, carries Hopf fibres to Hopf fibres and fixes the seam point by point. It is not an isometry. The coordinate swap (z₁, z₂) ↦ (z₂, z₁) is 𝔍 composed with the exchange of the two phases: an isometry that also carries α to 1 − α, and on the seam it fixes only the great circle z₁ = z₂. Flipping a jacket inside out is the whole of it: nothing is added, nothing is taken away, and the seam reads the same garment either way.

The spacetime phase is decompressed omnium, and expansion is the flow of omnium from the matter phase to the spacetime phase across the seam (0.8). The effective cosmological term is the released stored potential accumulated over the domain’s history (0.7):

dΛ_eff/dt = (8πG/c²)(1/V_Ω) ∫ δ ṁ dV,Λ_eff = (8πG/c⁴) u_rel

Every increment of the spacetime phase is posted against a debit of the matter phase; nothing is posted from nothing.

This figure needs JavaScript. The torus |z₁|² = α of the three-sphere, drawn in stereographic projection, has major radius 1/√α and minor radius √((1 − α)/α); at α = ½ it is the Clifford torus, with radii √2 and 1. The jacket flip carries it to the torus at 1 − α and fixes the Clifford torus point by point; the coordinate swap fixes there only the circle z₁ = z₂.

The jacket flip. The three-sphere is drawn in stereographic projection: each torus |z₁|² = α is a torus of revolution, the α → 1 core circle is the ring at the centre and the α → 0 core circle is the vertical axis through infinity. Slide the share and the marked points ride their torus; flip them and they move to the torus at 1 − α. With 𝔍 the phases stay, and at the seam every point stays where it is. With the coordinate swap the phases exchange, and at the seam only the points on the circle z₁ = z₂ stay. The bar below reads the measure: the support between the two tori, and half of it on either side of the seam. Cobalt marks the spacetime phase, brass the matter phase.

0.3a, I-⁠6

The helix books

A closed strand K of a bound form, with declared framing Γ, satisfies the Călugăreanu–White–Fuller theorem:

Lk(K, Γ) = Tw(K, U) + Wr(K),Lk ∈ ℤ

Along every evolution that passes no strand through a strand, its ribbon or the axis, the linking number is constant while twist and writhe trade continuously, ΔTw = −ΔWr. The books of I-⁠5 built on such linking numbers, the charge sign sgn(ν_oν_i) and the helicity sign sgn ν_i, and the charge magnitude n mod 2, are constant along every such evolution and change only at a reclosure, where every exact book balances: four-momentum, charge, fermion parity and angular momentum (I-⁠5b). A closure coils, precesses, supercoils and deforms through myriad shapes, and the integer never moves. No law is consulted: the balance is the theorem.

One ledger at every size

A composite’s total is read from its own exterior solution, every channel entering once and binding never added as a local term (0.5). For two parts on the I-⁠1 domain, at the leading weak-field, slow-motion order, with exterior inputs α_i = R_s,i/d and kinetic content K:

R_s,C = R_s,1 + R_s,2 − ½ α₁α₂d + 2GK/c⁴,U(d) = −F_maxα₁α₂d the pair entry, the force read twice

No bridge refers to the size of the configuration containing a pattern, so the ledger has one form at every size, from nucleus to domain. For the Sun and Jupiter on a Kepler orbit the pair entry is −¼ α_⊙α_Jd = −2.67 nm of fold radius; at nuclear size the deuteron’s rest read-out falls short of its two closures’ by Q/c² = 3.9656 × 10⁻³⁰ kg, 0.118 per cent. The repetition is bounded below by the floor, d_min = 2ℓ_P = 3.2325 × 10⁻³⁵ m and t_min = 2t_P = 1.0782 × 10⁻⁴³ s, a minimum and never a staircase (0.1a).

Why nothing else balances

A connected ledger balances by itself when every rate is a constant and every cycle of conversions returns each reading; when the conservation of the source is an identity of the geometry; when every structure entering a balance is posted by the configuration; when every conserved label is fixed under continuous deformation; when no entry depends on grouping; and when every reading is defined, positive and bounded. The paper proves what each condition demands:

A second substance needs a cross-rate for each conversion process, and an arbitrage cycle wherever two processes disagree. Two substances with one universal rate are one substance with two names.
A second carrier speed lets an internal relabelling move an isolated configuration’s centre of inertia, by sE(c₂⁻² − c₁⁻²)/M′, and lets a lowering-and-raising cycle deliver work every run.
A second geometry leaves N − 1 vector constraints on its metrics that hold on solutions and not identically.
A fixed background lets a kilogram lowered and returned as light post mgΔh from nothing. The cycle closes only if the kept time of patterns adjusts with height by ΔT/T = gΔh/c².
A size-selecting chain changes an entry when a pattern is regrouped, with no transaction.
A vanishing term, a divergent reading or a singular interior leaves an account unpriced or unaudited.

The parsimony theorem states both directions: a connected ledger balances by itself if and only if it has one substance, one carrier, one geometry obeying G_μν = κT_μν, no fixed background, discrete labels, a chain of ratios, and bounded readings with a floor. FDT is that configuration.

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-⁠1
One rate. A conversion of matter into strokes at a rate other than c², a measured 1 − Δm c²/E ≠ 0, ends the one numeraire.
T-⁠2
One weight. A composition-dependent free fall of bodies whose energy divides among rest-mass, nuclear and electromagnetic channels in different proportions ends the one account of I-⁠7.
T-⁠3
One stroke speed. A measured difference between the speeds of the gravitational and the electromagnetic strokes of one source ends the one carrier and the one geometry.
T-⁠4
No fixed background. A gravitational shift of the kept time other than gΔh/c² at first order ends the closing of the kilogram cycle.
T-⁠5
Integers. A charge or a helicity sign that changes along a deformation with no reclosure ends the helix books.

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

D-⁠01
Framing to strokes. The map from a closure’s framing integers to the helicities and angular momenta of the strokes it opens into, and the fermion class on the exact framed closures, are to be derived (core O-⁠11, O-⁠12).
D-⁠02
The phase book’s rates. The exchange rates across the seam, and the release function that feeds Λ_eff, are to be derived (core O-⁠12, O-⁠14).
D-⁠03
The domain ledger. Each universal domain’s 𝓜 read on its own boundary, what a merger of domains conserves, and the ledger that receives the energy strokes give up crossing a decompressing domain are to be derived (core O-⁠15).
D-⁠04
The regular core. The regular core chart, with its finite-core radius, is to be derived (core O-⁠1).
D-⁠05
Nuclear inputs. The force inputs at nuclear size that return the measured read-out differences of reclosures such as n + p → d + γ are to be derived (core O-⁠6).

Readings declined

Use 6 Data yes, interpretation not necessarily

Standard reading

An interaction is the exchange of virtual quanta that borrow energy for a time of order ħ/ΔE, the books reopened and closed inside each exchange.

FDT reading

Every interaction is curvature plus geodesic motion of one carrier with no mediation (I-⁠2), and every configuration keeps the chain at every instant with its terms adjusting together (0.1a): the books are never open.

A reaction and its reverse share one matrix element, detailed balance, and a response is one-to-one between particles.

Every step is a system-wide settlement solved from the state the system is in, many-to-many, with events that do not commute (0.1a). The closing of an exchange cycle is an identity among the readings of one state, never a forward event mirrored by its reverse.

The energy of the empty state is set to zero by subtracting a zero-point sum that diverges without a cutoff.

There is no empty state and no divergent sum: no term of the chain vanishes, the floor bounds every stroke, and the lowest state of every region is a pattern of positive, finite energy that enters 𝓜 once (0.1, 0.1a, 0.5).

Dark energy is a cosmological constant, a vacuum energy fixed for all time.

The acceleration is the released stored potential of compressed omnium, with no Λ term and Λ_eff the running total of releases: every increment of the spacetime phase is posted against a debit of the matter phase (0.7).

Charge and the fermion property are intrinsic labels, each conserved by a symmetry postulated for it.

They are integers of framed closures of the one carrier, held by Lk = Tw + Wr through every deformation and changed only at reclosures where every exact book balances (I-⁠5, I-⁠5b).

Fields live on a fixed spacetime background whose energy book closes only along the background’s symmetries.

There is no background: clocks and rulers are patterns whose adjustment is the geometry (0.1a), and the geometry posts every change through G_μν = κT_μν (I-⁠3).

Interiors exist where curvature and density diverge, regions whose content leaves every book.

None exist: no term of the chain vanishes, force is capped by F_max, and the density coordinate lives on the open interval with a regular core; a compact object is a hollow, double-sided density boundary, dilute at the centre and densest at the skin (0.6, 0.8).

Convergence

Use 2 Same source, same data

For the same source and data FDT and general relativity solve the same equation and return the same curvature (I-⁠3). The local identity ∇_μT^μν = 0, the mass aspect and its loss by radiated flux, the gravitational shift of strokes and kept times, the composite read-out and every read-out difference of the worked ledgers are the consensus results, recovered exactly at the orders stated. The mass–energy rate, the universality of free fall, the equal speeds of gravitational and electromagnetic strokes, the shift over the Harvard tower and across one atomic sample, the solar and Jovian nominal values and the CODATA masses are accepted in full, and each shows the absence of an entry a second substance, a second carrier, a second geometry or a fixed background would require.

FDT supplies one additional source, in the spacetime book: the spacetime phase receives the released stored potential of unwinding matter, and Λ_eff is the running total of those releases, its release function to be derived (D-⁠02).