Schur Functions at (1,-1,t,t^{-1}): A Closed Form on the Non-Identity Component of O(4) (Part IV)

For every partition with at most four rows, the Schur function sλ(1,−1,t,t⁻¹) is either zero or ± a product of exactly three SU(2) characters read off the 2-quotient of λ — with no hypothesis on the shape of λ. Part III (concept DOI 10.5281/zenodo.21438226) classified which SU(4) representations have identically-vanishing character on the non-identity component of O(4), and exported the underlying question about symmetric functions. This Part IV answers it with a closed form, of which that vanishing classification is exactly the zero locus. The alphabet {1,−1,t,t⁻¹} is the generic semisimple element of the determinant-−1 component of O(4); it sits between the reciprocal-pair factorisation programme of Ciucu–Krattenthaler and Ayyer–Behrend and the root-of-unity line of Littlewood, Prasad, Ayyer–Kumari and Karmakar, and belongs to neither. The obstruction that kept it untreated is one unstated hypothesis: the block argument needs every inversion-fixed point of the alphabet to contribute a constant row of the bialternant, which +1 does and −1 does not. Replacing the two fixed-point rows by even/odd column indicators repairs the argument at rank one, and not at rank two, which is where we stop and say so. The actual result is a reduction, not the identity. Conditionally on the closed form, every quantity that survives Part III's leading cancellation — the whole tower of even moments of the twist-imbalance distribution δ(m) — factors through three integers read off the 2-quotient: Φ = F ∘ Q. An arbitrarily large four-row partition is compressed to (p,q,r) with no loss of anything the physics can see, and weight enumeration is replaced by explicit polynomial formulae. The normalised moment tower lives in the invariant ring ℚ[Cₚ,Cₔ,C₟]^{S₃} of the three SU(2) Casimirs; in particular the variance is one third of the sum of the three Casimirs. Status, stated exactly. The closed form is an Observation: verified exhaustively by two independent implementations on all 3060 partitions with four parts ≤14 and all 4845 with parts ≤16, with no mismatch, against the bialternant in exact integer arithmetic — and not proved. The zero-locus statement is a Proposition, proved independently in Part III. Everything else is graded in the paper as Proposition, Observation, Inherited, or Open, and the grades are not blurred. Honesty ledger. Several ingredients are classical and we claim none of them: that log-concavity is preserved under convolution (Hoggar 1974) and gives a unimodal spectrum (Stanley 1989); that the coefficient sequence is a threefold discrete-uniform convolution whose moments are polynomial, because cumulants of a discrete uniform are polynomial in its width and add (Neuschel; Bradley–Gupta); and that the distribution has a piecewise-polynomial closed form by inclusion–exclusion (De Moivre; André 1876; Comtet; the object is a discrete box spline, Dahmen–Micchelli 1988). The involution character value reproduced along the way is Karmakar (arXiv:2412.17324); read in full for version 3, his Thm. 1A is the first branch of the zero locus and his Thm. 1B is the 2-quotient factorisation at t = 1, so only free t is ours there. The mechanism — characters of a disconnected group on a non-identity component, that is, twining characters — is Jantzen (1977) and Kumar–Lusztig–Prasad (2009), cited from version 3 onward; the surrounding programme is Nadimpalli–Pattanayak–Prasad (arXiv:2504.14684). Ours are: the closed form itself at free t, the fixed-point-pair repair, the second branch of the zero locus, the reduction Φ = F ∘ Q, and the Casimir reading of the moments. Rank two is open, and the shape of what is missing there is stated precisely enough to be attacked. Included: the paper in English and Spanish; the three figures with the scripts that regenerate them; and verify_secondfixed.py, a self-contained script that reproduces every displayed number and checks it against the bialternant in exact rational arithmetic, with a control built to fail.