What F_max is

The central invariant of FDT is one identity read four ways (0.1, 0.1a):

Read numerator to numerator, it pairs energy with a squared length and mass with a squared time, and the factor of that pairing, E/d² = m/t², a force per length, differs from pattern to pattern. Taking one factor of length into the constant leaves a pairing that is linear, exact and the same for every pattern and body. Its constant is the ceiling force of core 0.2:

The CODATA 2022 values enter unchanged: c and ħ are exact, and G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻² is measured. The Einstein coupling is κ = 8πG/c⁴, with κF_max = 2π exactly. Over the strand rate the ceiling force is a mass rate, F_max/c = 1.0092 × 10³⁵ kg/s; times it, a power, F_maxc = 9.0706 × 10⁵¹ W; and over its square, a linear density, F_max/c² = 3.3665 × 10²⁶ kg/m.

With the quantum the ceiling fixes the floor. F_maxd_mint_min = ħ with d_min = ct_min gives d_min = 2ℓ_P = 3.2325 × 10⁻³⁵ m and t_min = 2t_P = 1.0782 × 10⁻⁴³ s, and the floor’s energy is E_floor = F_maxd_min = E_P/2 = 9.7804 × 10⁸ J, with E_floort_min = ħ (0.1a). The table reads two closed patterns and one body through both pairings, the ceiling’s and the quantum’s.

Energy is length, mass is time

Every pattern or body, of rest energy E₀ ≡ 𝓜c² = mc², has

E₀ is F_max times a length, and mc, the time component of the four-momentum at rest, is F_max times a time: the form of a work and of an impulse at the ceiling force, which no pattern reaches. Energy is length in the ratio F_max and mass is time in the ratio F_max/c, the same for every pattern and body because F_max is a constant of the medium. One joule is 3.3051 × 10⁻⁴⁴ m of Schwarzschild diameter, and one kilogram is 9.9084 × 10⁻³⁶ s and 2.9705 × 10⁻²⁷ m. In units with F_max = c = 1 the pairing reads E₀ = d_s and m = t_s with no factor left, and in those units G = ¼ and ħ = d_min².

The chain is the pairing’s quotient:

F_max cancels. The square in (d/t)² is two equal factors, the quotient E₀/(mc) = d_s/t_s of the two readings and the c that turns m into mc, and both are the strand rate. For any force F the pair (E₀/F, mc/F) lies on the stroke line d = ct. The chain fixes the line and not the point on it; the ceiling force fixes the point, by one constant for every pattern.

The energy triangle read at the ceiling tension

What c is reads time as circulated strand and space as translated strand. The ceiling force reads energy as strand length. A closed pattern moving at v = βc runs its strand at c on a helix, translating v and circulating u = c/γ, and keeps time by circulating (0.1d). Over the kept interval Δτ = t_s its strand runs, advances and circulates

and F_max times the three sides are its E, pc and mc²:

The energy–momentum relation is the strand’s right triangle, read at the ceiling tension. The run d = E/F_max is the bookkeeper’s energy in ceiling units, and the circulated length cΔτ = d_s is the Schwarzschild diameter of the rest energy, the same at every speed. The four-momentum is F_max/c times the four-displacement over the kept interval, p^μ = (F_max/c)Δx^μ, in every frame. In the strand plane the energy spinor is the ceiling force times the displacement spinor, P = F_maxD = pc + mc²Σ: pc is F_max times the translated grade and the rest energy F_max times the circulated grade, so a pattern at rest has P = F_maxd_sΣ, a pure bivector. An open stroke keeps no time: cΔτ = 0, and the triangle closes on its run, E = pc = F_maxd. That is the ceiling reading of its run, which no stroke attains.

One modulus

Hooke’s law makes a load linear in a geometric quantity, and its constant is a modulus. The omnium is an elastic medium (P-⁠2), and F_max is the constant through which its geometry becomes its mechanics, linearly in each case: a length into energy, a product of two curvature ratios into force, a curvature into stress. The first law is the pairing, E₀ = F_max·2R_s. The second is the pair law of I-⁠1. For two bodies at separation r, each exterior input is the ratio of the body’s Schwarzschild diameter to the diameter of the sphere of radius r, and

The force is linear in each share, F = (F_maxα₂)α₁: the share of one body times the force scale F_maxα₂ that the other sets. The pair energy is the force times the separation, U = −F_maxα₁α₂r. On the pair domain F ≤ F_max/4, with equality only at equal saturated supports in contact, and the law is Newton’s exactly in model (I-⁠4, L1). The third law is the field equations. With κF_max = 2π, G_μν = κT_μν reads

Stress is linear in curvature with the modulus F_max/2π, and κ = 2π/F_max is the compliance. On a pattern the compliance turns rest energy into length, κE₀/2π = 2R_s, and in a static weak field the curvature integrated over a slice is ∫G₀₀ d³x = κE₀ = 2π·2R_s at leading order (I-⁠4, L2). G is not a constant of its own: G = c⁴/4F_max, I-⁠4 writes Einstein’s coupling as κ = 2π/F_max, and core 0.2 writes the Schwarzschild radius as R_s = 𝓜c²/(2F_max), which needs no numerical G.

The closure tension

The fourth law has F_max as its bound. A steady closed strand carries a constant energy e_ℓ per unit length and runs at the strand rate. Its momentum flux e_ℓ turns through the curvature k, which takes an inward force e_ℓk per unit length, and a tension 𝒯 supplies 𝒯k. Hence

and the strand’s stress and the closure’s cancel when integrated, as the inertia of a closed pattern requires (0.1d). With m_ℓ = e_ℓ/c² the inertia per unit length, 𝒯/m_ℓ = c²: c² is the closure’s tension over its strand’s inertia per unit length, as a string’s wave speed squared is its tension over its inertia per unit length. Over a strand of length L, E₀ = 𝒯L and m = m_ℓL, the closed pattern’s mass, closure included. Since 𝒯 < F_max, every closed strand is longer than its Schwarzschild diameter, L = E₀/𝒯 > 2R_s. F_max enters the closure only as its bound: in E₀ = F_max·2R_s, 2R_s is the length of strand that would carry E₀ at the tension F_max, which no strand reaches.

The floor: where the two pairings meet

The quantum pairs a pattern’s rest energy with a length the other way round. A pattern of rest energy E₀ keeps the period T_q = h/E₀, and its strand runs ℓ_q = cT_q in it; per radian of kept phase it runs the reduced Compton length

λ̄_C is a run of the strand, never the support R_box (I-⁠5). In the quantum pairing the rest energy grows as the length shrinks; in the ceiling pairing it grows with the length. The two multiply to the floor, for every pattern or body:

The floor length is the geometric mean of the gravitational diameter and the quantum length, and exactly one of the two lies above it unless both equal it: 2R_s/d_min = E₀/E_floor and λ̄_C/d_min = E_floor/E₀. The map ℓ ↦ d_min²/ℓ carries each length onto the other and is its own inverse, with the single fixed point d_min. It acts on the two lengths of one pattern or body and is neither the seam 𝔍 nor the branch exchange 𝔦 of 0.3a.

The quantum length is a one-action stroke of 0.1a. The run λ̄_C in the time λ̄_C/c under the mean force F_q = E₀/λ̄_C has action ħ, with force fraction

The ceiling pairing is the quantum pairing’s force fraction: b is the gravitational diameter in units of the quantum length, and the square of the rest energy in units of the floor’s. Under the premise Q8, which is to be derived (D-⁠F1), a closed pattern’s run per radian of kept phase is one physical stroke of action ħ; since F_max is never reached, b < 1: λ̄_C > d_min > 2R_s and E₀ < E_floor. For a closed pattern λ̄_C is a stroke length and 2R_s the length of no stroke. A body of rest energy above E_floor is therefore no single closed pattern, and since there are only patterns it is many: a laboratory body, a star, a compact object. A nucleus lies below E_floor and is one pattern (0.1b). The floor, where the two lengths would coincide at b = 1, is approached by the least closed pattern, the one with the shortest λ̄_C and the greatest rest energy, and occupied by none.

The floor also carries the modulus of the field equations. Let the n strands of a braid (n ≥ 2, I-⁠5) each close once per kept period. Each strand has length ℓ_q = 2πλ̄_C and carries E₀/n, so its tension is its energy per unit length, and the braid’s closure tension summed over its strands is the same for every n:

The modulus of the field equations is the floor limit of a braid’s summed closure tension, and κ = 2π/F_max is its compliance. In the same limit the compliance relation κE₀ = 2π·2R_s tends to κE_floor = 2πd_min.

One rate, one modulus, one quantum

Three constants carry the conversions:

The rate converts time into length. The modulus converts energy into length and momentum into time, for every pattern and body. The quantum converts energy into time for every pattern, and at the floor. Their combinations carry the rest: c² = 𝒯/m_ℓ = 1/(ε₀μ₀), the rate squared, a stiffness over an inertia, and E/m is that ratio taken over a whole closed strand; 4G/c⁴ = 1/F_max and κ = 2π/F_max, the compliances; and the floor, d_min = √(ħc/F_max), t_min = d_min/c and E_floor = √(ħcF_max). G is not a fourth constant. The floor is fixed by all three: it is the length at which a pattern’s two lengths would coincide, approached by the least closed pattern and occupied by none.

The paper supplies what F_max is, and every observable it returns is already in the record. Its commitments are the ones the core makes through it, each with the result that ends it, and its open items are named.

1. E_floor
   No closed pattern at the floor’s energy. Every closed pattern has rest energy below E_floor = 9.7804 × 10⁸ J, which is 6.1045 × 10²⁷ eV, because its run per radian is a one-action stroke the floor bounds (Q8). A single closed pattern at or above that energy ends this reading; a body above it is many patterns.
2. I-⁠1
   Newton’s law, exactly. Between two separate spherical bodies F = F_maxα₁α₂ = G𝓜₁𝓜₂/r² holds exactly in model, and the force stays at or below F_max/4. A measured departure from G𝓜₁𝓜₂/r² between static, nonrotating spherical bodies in the weak field (I-⁠4 L2), with every energy counted once in 𝓜, ends the pair reading.
3. I-⁠3
   The field equations, unchanged. G_μν = κT_μν with κ = 8πG/c⁴ = 2π/F_max, and their Newtonian form in the weak field. For the same source and data FDT reads the same curvature; a measured curvature that differs from κT_μν for a source counted in full ends the modulus reading.
4. D-⁠F1
   The run per radian. That a closed pattern’s run per radian of kept phase is one physical stroke of action ħ, bounded by the floor (Q8), is to be derived (D-⁠F1). The bound E₀ < E_floor and the floor limit of the braid tension rest on it.
5. O-⁠9
   The braid’s strands and the mass ratio. How many times each strand of a braid closes per kept period, and how it winds, are to be derived (O-⁠9); the braid tension above takes one closure per kept period. The mass ratio of the electron and the proton, the one n = 3 braid read from two rooms, is a held-out target (O-⁠9): the confinement operator must return it without taking it as an input; it is carried as a test and never quoted as an FDT output.

For the same sources and data FDT returns Newton’s law of gravitation on the pair domain, exactly in model (I-⁠1, I-⁠4 L1); E = mc² and E² = (pc)² + (mc²)²; Einstein’s field equations with κ = 8πG/c⁴, and their weak-field form at leading order (I-⁠4 L2); and the reduced Compton lengths ħ/(mc). It uses the CODATA 2022 values unchanged, c and ħ exact and G measured.

No divergence is claimed: the paper supplies what F_max is, not a changed observable. F_max is the modulus of the omnium. The rest energy of a pattern or body is F_max times its Schwarzschild diameter, and mc is F_max times the time the strand rate takes to cover that diameter; in motion, E, pc and mc² are F_max times the length the strand runs, the length it advances and the length it circulates. The same constant is linear in the pair law and, as F_max/2π, in the field equations; it bounds the tension that holds every closed pattern; and with the quantum it fixes the floor.