THE PAPERS BEHIND THE WORK.

01 · What it asks

The rule α(d, β) = 1 + d/(d + β) describes how two things couple. This paper treats the rule as a move on a curved surface, then repeats and resets the move to find the surface it builds. It asks whether that surface can ring like a bell or only support a sweep and return.

02 · How the mathematics works

The coupling law is a Möbius transformation. With β = 6/23 it acts as a hyperbolic boost, an orientation-preserving isometry of the hyperbolic plane. Paired with the reset x ↦ β²/x it generates a free group of rank two whose quotient is a one-holed torus, convex-cocompact, with the underlying quaternion algebra ramified at 3 and 23. The paper computes the spectrum of that surface.

• Möbius maps and PSL(2, ℝ)
• Fuchsian groups
• Selberg zeta function
• Patterson–Sullivan critical exponent

03 · Claims and limits

Derived: the group is free of rank two and convex-cocompact, not a modular level-23 group; discriminant 69; a continuous spectral floor at ¼ with no embedded eigenvalues; an infinite discrete set of decaying resonances. The proposed reading uses sweep and return on a surface which cannot ring, drawing on radar rather than resonance.

No physical identification of the surface is claimed. β = 6/23 is a scale choice, the level set that puts α₀ = 48/25 at d = 3; it is not a constant of nature.

01 · What it asks

Guevara, Lupsasca, Skinner, Strominger and Weil proved that certain gluon scattering amplitudes stay nonzero right at a boundary called the half-collinear regime, and wrote the amplitude there as a product of signs. This paper asks what happens on the way in: how the sign variables move as they approach that boundary, and why some systems commit to a sign pattern while others stall.

02 · How the mathematics works

Each discrete sign in their product formula is replaced by a smooth tanh of sharpness α on the cumulative spinor sums. The product becomes a continuous landscape whose fixed points sit exactly on the amplitude zeros, and gradient flow can run toward them.

• Spinor-helicity amplitudes
• tanh regularisation of sign functions
• Gradient flow on a potential landscape

03 · Claims and limits

Reported for n = 4 to 18: degenerate fixed points exist throughout; flat regions of near-zero force surround them whenever one factor stays unabsorbed by the cascade, shown constructively by adding a single uncoupled factor; commitment to a globally coherent sign pattern is universal at moderate perturbation; the flat regions intensify with n. Proposed name for the phase: the quarter-collinear regime.

An ansatz plus numerics, companion to arXiv:2602.12176. The α in this paper is the tanh sharpness, tested at 1, 3, 6 and 12. It is not the coupling α of C001 and C003; the shared letter does not make them the same parameter.

01 · What it asks

The document develops a proposed calculus around α(d, β) = 1 + d/(d + β), taking Hilbert’s twenty-third problem as its starting point. It explains the operations that make up the relation, compares its predictions with public data, and describes tests that could show where it fails.

02 · How the mathematics works

The functional form is classical: a unit-amplitude Michaelis–Menten, or Emax, saturation, equivalently the Möbius action of a fixed matrix. Its inversion is the odds transform. Its reset x ↦ β²/x is the involution in PGL(2, ℤ). The document combines these operations and lays out a concordance matrix: seven anchors, fifty-two fronts across five domains.

• Michaelis–Menten / Emax saturation
• Odds transform
• PGL(2, ℤ) involution
• Hilbert's twenty-third problem

03 · Claims and limits

Twenty-two predictions were laid against existing public data: twelve hit, none missed, the rest not yet scorable. The claim is the composition and the assignment of parameter roles. The underlying operations are classical. The proposed next work is to test an anchor, find where a connection fails, or work with research partners.

Retrospective coverage is a methodology record, not validation; there are no matched controls yet. A hypothesis with falsifiable predictions, not peer-reviewed. We propose; we do not prove.