The Cohesion UFT series admits a result on one rule: a chain from the pressure axiom, through the operator sequence — Tension → Surplus → Torsion → Slip → Acceleration → Maintained Motion — to a quantity the framework computes, reported at its standing with its residual against measurement and with nothing fitted. The quantities the series has named and has neither derived nor fitted are indexed in Appendix C of the textbook. This page keeps the seven of them that are posed in full in Problems for Derivation, with the current standing of each; the paper carries the full statements and the worked template, and this page is where a problem's standing changes when a derivation arrives. A derivation that meets the standard enters the series under its author's name, with its own record, and the quantity moves from open to derived.
The rule
Every quantity in the record carries one of five standings — derived, closure, identified, stated, open — defined on the FAQ page and in Chapter 21 of the textbook, and a submission carries them too. Four things the rule excludes, each because it would make the record untestable: a quantity defined by the gap it closes, which is a fitted parameter whatever its name; a functional form chosen so that a free quantity in it can be solved for the measured value; a symmetry or a group taken as input, since the series reads its groups off the phase freedom of the recursion modes and names them afterwards; and a residual attributed to the instrument or to the observer's model, since the record takes the standard account's measurements as they are and carries every residual as a residual. The form a derivation takes is set out step by step, with the standing of each step and the residual shown, in Section 2 of Problems for Derivation, using the fine-structure chain as the template.
Status
The standing of each problem as it stands today. A row changes when a derivation is admitted, and the author and record are entered beside it.
| Problem | Target | Standing | Derived by |
|---|---|---|---|
| 1. Structural density of the observable domain | R(D_now)/R(D_CMB) = 1.083 ± 0.017; D_now/D_CMB = 1.614 | Open | — |
| 2. The galactic transition without a₀ | R_trans = 0.748 from a condition that does not use a₀ | Open | — |
| 3. The three-segment partition of the n = 6 closure | The closure count 16 = 2 × 4 × 2 as a derivation; 3/8 returned | Open | — |
| 4. The mode functions R_n(D_st) | tan θ_W from 0.7746 to 0.5484 across four decades (ratio 0.708); g(EW)/g(e) = 19.97 | Open | — |
| 5. The slip cadence at the fluid scale and the cascade cutoff | The re-closure radius against the measured dissipation scale; the Re of first slip | Open | — |
| 6. The torsion-interval energy fraction (c-bit switching energy) | E_switch = f m_e c² as a number; g at the graphene scale | Open | — |
| 7. The constants of the rotation-curve law | The radial acceleration relation over the SPARC sample with nothing fitted | Open | — |
The problems
Each problem is stated in six parts: what the series has established toward it, with the standing of each step; what the derivation must show; the number the chain must return and the measured value it is held against; what moves in the record when it does; the shortcuts the rule rejects for this problem; and where the result is evaluated.
1. The structural density of the observable domain
Given. The exact solution R(D_st) of the classical recursion field, with R(D_GUT) = 1 and the low-density form R ≈ 3^(−1/4)(D_st/D_GUT)^(1/6): derived (Exact R(Dst); textbook Ch. 14). The reading of the two measured expansion rates as one expansion read at two densities, H_local/H_CMB = R(D_now)/R(D_CMB): stated (textbook Ch. 17). The rates, 67.4 ± 0.5 km/s/Mpc from the microwave background and 73.0 ± 1.0 locally, ratio 1.083 ± 0.017: measured. The density of matter at recombination relative to now, (1 + z)³ = 1.33 × 10⁹ at z ≈ 1100: measured. The galactic route's finding that the containing pressure of a galaxy is not simply the cosmological density: identified (Identifying Dgal; Ch. 16).
Required. Derive from the axiom and the field equations which density and which containing pressure form D_st = ρc²/p_cont at the largest scale, and how the containing pressure of the observable domain changes with epoch. With D_st read as the mean matter density over a fixed containing pressure, the exact solution returns R(D_now)/R(D_CMB) = 1/33.2, on the wrong side of one. Far below D_GUT the solution is a power law, so the ratio of R between two epochs depends only on the ratio of their structural densities and not on the absolute calibration, and the measured 1.083 is returned only when D_now/D_CMB = 1.614. Since the matter density fell by 1.33 × 10⁹ over the interval, the containing pressure of the domain must have fallen by 2.15 × 10⁹: it tracks the density to within a factor 1.6, and the derivation must produce that factor from the mechanics of the containing scale.
Target. R(D_now)/R(D_CMB) = 1.083 ± 0.017; equivalently D_now/D_CMB = 1.614 (1.47 to 1.77 across the measured range).
What moves. The identification of D_st at the largest scale moves from open to derived, the cosmological route then calibrates D_GUT in physical units, and every row of Appendix C.1 that waits on the calibration — the thresholds D_acc and D_crit, the minimum R₀, M_GUT by E = pr at R = 1, the GR limit with its constants — becomes a number.
Excluded. A factor defined as 1.083 × 33.2 and then named. A time dependence of p_cont chosen so that the exact solution returns 1.614. The components of the standard account's cosmological model as inputs. A reading of the two rates as a measurement error.
Where to compute. Module 2 of the Research Calculator evaluates R at any D_st/D_GUT; the two statements above, 1/33.2 for a density ratio of 1.33 × 10⁹ and 1.083 for a ratio of 1.614, are verified there at any placement of D_now below about 10⁻¹² D_GUT, where both epochs lie in the power-law regime. The Calculator page carries Module 2 as well.
2. The galactic transition without the MOND scale
Given. The surplus acceleration g_s = cH₀R(1 + 2R⁴)/6 from the geodesic equation in the recursion metric and the derivative of the exact solution: derived, with the Hubble radius as the gradient scale of D_st at galactic scales stated (Identifying Dgal; textbook Ch. 16). The transition R_trans = 0.748, solved from the measured a₀ = 1.38 × 10⁻¹⁰ m/s² because R(1 + 2R⁴) is strictly increasing, and a₀ = g_s(R_trans) returned to 0.07 per cent: closure. The location of the transition, D_trans/D_GUT = 0.439 by the exact inversion at R = 0.748: derived. The rotation-curve constant β = 0.235 pc/(km/s)² with β⁻¹ = a₀: form derived, value stated against the SPARC median (Rotation-curve law).
Required. Derive a condition of the galactic recursion, from the axiom and the operator chain, that fixes R_trans without a₀ as input: the condition at which the galactic recursion's response passes from the pressure-dominated to the surplus-dominated regime, of the kind the toggle threshold Φ_t = 32/(3π² − 4) is for the transition between the n = 6 and n = 2 states. The condition must be written in the quantities of the galactic level — the structural density, the containing pressure and the propagation function — and must return a single R.
Target. R_trans = 0.748 (0.7483 solved); then a₀ = g_s(R_trans) at H₀ = 70 km/s/Mpc against 1.38 × 10⁻¹⁰ m/s², the SPARC median the series carries, and 1.2 × 10⁻¹⁰, the standard account's value.
What moves. a₀ moves from closure to derived, with its residual against both measured values; the value of β follows from the same condition and moves from stated to derived; the executed record gains a derived row where it carries a closure.
Excluded. Any use of a₀, of β, of the radial acceleration relation or of rotation-curve data in fixing the condition. A threshold value chosen because R(1 + 2R⁴) returns a₀ there.
Where to compute. Module 1 evaluates g_s(R) and solves R_trans from a₀; Module 2 inverts R to D_st; Module 12 carries β. All three run on the Calculator page.
3. The three segments of the n = 6 closure
Given. The n = 6 state with six torsion maxima per cycle, and the exclusion of every polarity count but 1, 2 and 6: derived (Hexagonal Recursion; Binary Recursion Toggle; textbook Ch. 8). The closure count of one generation at the unification density, (two poles) × (one whole closure or one of its three segments) × (two hands) = 16, with each member's charge read as its pole plus half its oriented content, Q = T₃ + (B − L)/2, and the trace ratio 3/8 holding over the five-member state and over all sixteen: stated, with its enumeration explicit (Closure at the Unification Density; textbook Section 12.5). Confinement as the statement that a fraction of a closure is not a closure: identified (Quark Confinement).
Required. Derive from the geometry of the n = 6 cycle that its closure content divides into exactly three segments, each carrying one third of the oriented content, each a stable fraction only in composition with its complements: why three, and not two or six, from the six-peak slip structure and the coherence limit, with the charge of a segment following from its share of the content.
Target. The count 16 and the charges 0, −1, +2/3 and −1/3 of one hand, with 3/8 returned over the five-member state (0.500/1.333) and over all sixteen with the pole counted on one hand (2.000/5.333).
What moves. The closure count moves from stated to derived, and with it the standing of the sixteen-dimensional spinor: SO(10) becomes the name of a derived count rather than a borrowed one. The badge of Module 17 changes to derived. The fractional closure coefficients of Appendix C.3, which the quark masses wait on, acquire their first derived constraint.
Excluded. SU(3), SO(10) or any group as input. The quark charges as input. A partition chosen because sixteen is the number to be reached.
Where to compute. Module 17 of the Research Calculator builds the sixteen from the three labels and returns every charge and both 3/8 checks; Appendix A.16 of the textbook recomputes it.
4. The mode functions: the framework's own running
Given. The frequency law ω(D_st) = ω₀/R(D_st) from the uniform solution of the recursion field: derived (Exact R(Dst); Closure at the Unification Density). The Weinberg angle at any density as θ_W = arctan(ω₂/ω₆): stated (Weinberg Angle). The three modes' functions R₆, R₂, R₃ equal at R = 1, which is the definition of D_GUT, and separating below it, which is the electroweak breaking: stated (Electroweak Scale). The exact solution's self-consistency condition dR/dD_st = R(1 + 2R⁴)/6D_st, whose 6 is either the pole count of the mode or a property of the field independent of mode: the point of departure, undecided. sin²θ_W = 3/8 at D_GUT from the closure content: derived. sin²θ_W(m_Z) = 0.23122 and 1/α(m_Z) = 127.95: measured. The placements D_EW/D_GUT = 10⁻⁴ and D_QCD/D_GUT = 10⁻²: stated.
Required. Derive the density dependence of the three mode functions from the field equations, deciding what the 6 in the self-consistency condition is, and show that between D_GUT and the electroweak density the ratio ω₂/ω₆ changes by the factor 0.708, and that the torsion slope g(D_st) at the electroweak density is 19.97 times the slope at the electron. The calculation runs down from R = 1, where the modes are degenerate and the angle is 3/8, to the measured values at low density; the meeting point is held by construction and the low-density values are the result.
Target. tan θ_W from (3/5)^(1/2) = 0.7746 at D_GUT to 0.5484 at m_Z, a ratio of 0.708 across four decades of density; Δθ(m_Z) = θ_φ − 127.95 = 9.558° against the electron's 0.4787°, a slope ratio of 19.97. A difference of exponents, 1/2 against 1/6 over four decades, would separate the modes by 10^(−4/3) = 0.046 and is excluded by the target: the separation is a slow factor on a common power law.
What moves. sin²θ_W(m_Z) moves from the standard relation with measured inputs to a value the framework runs to: derived. α_GUT from g(D_GUT) moves from identified to derived, against the standard account's 1/25. M_GUT by E = pr at R = 1 follows once Problem 1 calibrates D_GUT. The slip cadence at every scale above the electron, Problem 5 included, takes its g(D_st) from the same function.
Excluded. The one-loop coefficients and the partner set of the standard account as inputs. An exponent or a factor chosen to return 0.708. The map μ ↔ D_st assumed rather than derived; it is anchored only by m_Z at 10⁻⁴ and M_GUT at 1.
Where to compute. Module 18 of the Research Calculator computes the targets and the two constraints from the measured inputs; Module 6 evaluates the standard relation at m_Z for comparison; Modules 8 and 9 carry the standard account's running, which a derivation replaces.
5. The slip cadence at the fluid scale
Given. The slip cadence at the electron, t_coh = 2/g with g = 6κ₆Ī^(3/2) and Ī = 3α/2 the coupling of the recursion to its containing scale: derived at the electron scale and nowhere else (Fine-Structure Constant; Slip Acceleration; textbook Ch. 7). The vortex as the funneled spring in a fluid, with shear as tension, core concentration as surplus, vorticity as torsion: identified (Vortex Closure). The Reynolds number as the ratio of the viscous diffusion time to the coherence time, Re = UL/ν = t_ν/t_c, with the onset of turbulence as the onset of slip: identified. The re-closure of a tightening core at the slip cadence, with the cascade as the restart: derived from the axiom's exclusion of a region of zero pressure (textbook Proposition 19.1).
Required. Derive the torsion slope g at the fluid scale from the vortex's geometry and its containing field: the fluid's own coupling Ī to the surrounding flow in place of 3α/2, and the slope that follows from it, returning the coherence time t_c for a vortex of stated core radius, velocity scale and viscosity, the radius at which a tightening core re-closes, and therefore the scale at which the cascade ends.
Target. The derived re-closure radius against the measured dissipation scale of a turbulent flow at stated viscosity ν and energy dissipation rate ε; the standard account's estimate of that scale is the Kolmogorov length (ν³/ε)^(1/4), and the comparison is with the measurement, not the estimate. The value of Re at which slip first occurs in a stated geometry against the observed transition.
What moves. The slip cadence at the fluid scale moves from open to derived, and the cascade cutoff with it; the applied register gains its first timing, since every transition it predicts in form waits on the cadence at its own scale; the same route then applies to the cyclone, the pulsar and the graphene lattice.
Excluded. A dimensional estimate from ν and ε alone, which is the standard account's and contains no mechanism. An Ī chosen to return the measured scale. A cutoff supplied by viscosity alone, which is the second barrier and not the cadence.
Where to compute. Module 3 carries the form of g and t_coh at the electron, on the Calculator page and in the Research Calculator; no module yet carries a fluid-scale g, and a derivation supplies the module.
6. The torsion-interval energy fraction
Given. The torsion intervals of the pressure cascade and the lepton scalings they give, λ₁ = 3/(2α) = 205.55 and λ₂ = 3/(8πα) = 16.36, within 0.59 and 2.7 per cent of the measured ratios: derived (Torsion Interval Scaling Factors; textbook Ch. 11). The two stable states and the toggle threshold Φ_t = 1.249570 that make a torsion-phase bit a binary with geometric protection: derived. The form of the switching energy of the bit, 2ħg: derived; the value of g at the electron: derived; at the graphene scale: open (The c-bit; Graphene at the Dirac Point). The bound the series states — the switching energy is a fraction of m_e c² set by the intra-level phase separation of the n = 2 mode, and it stands above the thermal floor k_BT ≈ 26 meV at room temperature by orders of magnitude — stated; the fraction itself: open. The graphene coherence interval and clock, 0.49 fs and 647 THz from a = 0.246 nm and v_F ≈ 10⁶ m/s: identified.
Required. Derive the fraction f of m_e c² that is the intra-level phase separation of the n = 2 mode from the torsion-cycle geometry, so that the switching energy of a torsion-phase bit is a number; and derive g at the graphene scale from the lattice, so that the coherence interval and the slip energy 2ħg_graphene are computed rather than identified.
Target. A switching energy E_switch = f m_e c², held against the measured switching energy of a torsion-phase bit in a magnetic lattice once one is measured; the graphene g against the identified 0.49 fs and 647 THz, and against the measured departure of graphene at the Dirac point.
What moves. The bound moves from stated to derived and the switching energy becomes a design constraint with a value; the graphene interval and clock move from identified to derived; two rows of the textbook's Chapter 22 open column close.
Excluded. A fraction read off a measured switching energy. A g_graphene fitted to the identified interval. The thermal floor as anything but the comparison.
Where to compute. Module 3 carries the form of g and Module 5 the toggle, on the Calculator page; no module yet carries f or g_graphene.
7. The constants of the rotation-curve law
Given. The rotation-curve law with its global suppression S_glob and the radial acceleration relation it reproduces in form: form derived (Rotation-curve law). The scale β⁻¹ = a₀ = 4.26 (km/s)²/pc: value stated against the SPARC median. The surplus acceleration g_s and the transition of Problem 2: derived and closure respectively. The constants of the law — the surplus amplitude C_I, the inertia profile I(R), the central density ρ₀ and the shape parameters α_w and q: open, named and not fitted (textbook Ch. 16).
Required. Derive the surplus amplitude and profile from the coherence-collapse condition of the galactic recursion, so that the constants of the law are outputs of the geometry and the law's shape is a prediction; the derivation must give each constant from the structural density and containing pressure of the galaxy, with the containing pressure identified as Problem 1 and the galactic route require.
Target. The shape of the radial acceleration relation, g_obs against g_bar, over the SPARC sample of 175 galaxies, with the residual scatter reported; and the rotation profile of any one galaxy from its baryonic profile alone.
What moves. The constants move from open to derived; the galactic rotation test opens on the same no-fitted-parameter footing as the rest of the record; β follows from the same derivation and the a₀ row of the record is confirmed or corrected from the galactic side.
Excluded. Any constant fitted to SPARC or to a rotation curve. A profile assumed from the standard account's halo models. The relation itself as input.
Where to compute. Module 12 carries the law's scale β and the relation's form, on the Calculator page; a derivation supplies the constants and the module that evaluates the profile.
The rest of the list
Appendix C of the textbook carries thirty-nine further open quantities in the same four groups — what waits on the calibration of D_GUT, the slip cadence beyond the electron at each scale, the closure coefficients and the nuclear quantities, and the formal statements pending in the foundations and the open programmes of the last chapters — and Appendix B carries the relations the series states and uses whose formal derivation from the field equations is pending. Each is a problem of the same kind, admitted on the same terms. The Engineering page lists the open quantities that bear on applied work.
Submission
A submission is a document in which the chain is written out in the form of the template: the axiom and the established steps it starts from, each with its standing; every new step named; the quantity it ends at, with the number; the measured value it is held against and the residual; the inputs declared, so that a reader can see which of them is a measurement; and the evaluation, as a calculator module, a script or a worked computation that a reader can run. Notation follows the textbook's sheet. The check is recomputation: the chain is evaluated from its own expressions, as Appendix A of the textbook evaluates every module of the calculator, and the standing is assigned by the rule, not negotiated. A chain with a fitted step is not admitted at any standing. A derivation that meets the rule is published as a record of the series under its author's name, with its own DOI, listed on the Series page; the quantity moves to its earned standing in the next version of the Research Calculator and the textbook, with the author named at the row, and this page records it. Submissions go through the contact page.