From framed points
to the sphere spectrum

Work over k = ℂ and let . For an integer d ≥ 2 and n ≥ 1, define

Set p = id and Z = V(φ). The monic equation gives

The last isomorphism uses the basis 1,z,…,z^d−1. Finiteness holds over the entire algebraic parameter line, not merely the drawn interval. Thus Φ is an explicit level-n framed correspondence from X to pt. The definition requires finite support and an étale neighbourhood; it does not require the zero scheme to be étale over X. We retain its natural scheme structure to track length.

Restriction to t = 0 and t = 1 gives the faces of an algebraic 1-simplex. For d = 2 the two real roots become two imaginary roots through a double zero. Point labels are visual tracking labels, not a canonical labelling across the multiple zero.

In the first view, the exact visualization map on the chosen real slice is

The horizontal sheet is the current complex z-plane; the curves are solutions of z^d = 1−2t. At framing level n > 1 the picture is the slice w₁ = ⋯ = w_n−1 = 0. The extra complex directions are omitted, not represented as extra visible real axes.

At a simple root a, the complex derivative λ = d a^d−1 is invertible. The displayed arrows have the directions of λ⁻¹ and iλ⁻¹. Their lengths are normalized for readability; the inspector gives the actual inverse-Jacobian norm. In real coordinates,

At the multiple root these arrows disappear because the inverse derivative does not exist. The algebraic framing does survive: the regular sequence φ identifies . Its conormal basis must not be confused with an invertible differential.

The second view uses the inverse stereographic map

The polynomial extends to a based map . Both spheres are exact geometric models of ℙ¹(ℂ). Surface colours sample arg P_t(z) on the domain and arg w on the target; the finite mesh is a numerical rendering. Equal phase does not determine a point without its modulus.

For a regular value q, all d preimages have local degree +1, so the global degree is d. At q = 0 and t = ½, the single zero has local degree d. On a small circle, z^d winds d times. The compactified maps form a continuous homotopy and retain the same degree through the collision.

“Loop target” sets . Its inverse images are . After θ runs from 0 to 2π, a_j reaches the initial position of a_j+1. This is analytic monodromy over the complement of the critical value. The circular θ-path is not presented as a polynomial algebraic 1-simplex.

Via a Pontryagin–Thom collapse around the simple zeros, the local complex frames give a based degree-d map S² → S². The compactified polynomial represents the same based homotopy class, since based maps S² → S² are classified by degree. After stabilization and desuspension, this gives d ∈ π₀𝕊 ≅ ℤ. For n > 1 the corresponding class is represented by the 2(n−1)-fold suspension; the componentwise proper map (P_t,w₁,…,w_n−1) has that one-point compactification.

The framed motive is the Segal S¹-spectrum built from the Γ-space

Its r-th space is , with the diagonal understood. Here S^r is a simplicial sphere. Γ-structure organizes labelled supports and additivity; higher simplices provide coherence that a finite collection of point configurations cannot display.

The proof of Theorem 11.9 combines the framed reconstruction of the motivic sphere (Theorem 11.1), cancellation and fibrant replacement of the framed bispectrum, and Levine’s fully faithful constant-spectrum functor over an algebraically closed field of characteristic zero embedded in ℂ. Its derived right adjoint is evaluation of the zeroth bispectrum level at pt. The resulting ordinary spectrum is stably equivalent to M_fr(pt)(pt).

Theorem 11.10 then identifies positive spaces with QS^r, r > 0, and the group completion of the r = 0 space with QS⁰. The diagram explorer records these exact categorical steps. Its arrows are statements of constructions and theorems, not geometric deformations of an ordinary sphere.

1. Higher algebraic simplices. Vary several coefficients over Δ² and inspect all three face restrictions. This would display compatibility among homotopies, rather than only one path.
2. Labelled disjoint supports. Add finite pointed-set labels, permutation actions, and Segal maps. This would expose the Γ-space structure and distinguish coherent group completion from the subtraction of degrees.
3. Higher stable classes. Construct explicit parameterized framed cycles and compare their realization with representatives of stable homotopy classes. A braid in the present root picture alone must not be identified with η without this additional construction.