Multidimensional cost geometry

Referee: The framing as the 'canonical' multidimensional extension overstates what §2 proves. Lemma 2.1 plus the all-equal-reduction fixes only αᵢ = ±1/n given the assumed ansatz J = ½(R+R⁻¹)−1 with R = Πxᵢ^{αᵢ}; it does not select that ansatz among the infinitely many f(α·t)−1 yielding the same rank-one t-geometry. Either weaken 'canonical' to 'natural', or state an explicit theorem of the form 'Among ansätze ½(F(R)+F(R⁻¹)), conditions X,Y,Z single out F=id and αᵢ=1/n.'

Authors: We agree. The §2 argument is conditional on the multiplicative ansatz R = Πxᵢ^{αᵢ} and on the d'Alembert composition law in [20] supplying F = id; given that input, Lemma 2.1 plus the diagonal-reduction step fixes αᵢ = ±1/n, but as stated this does not single out F = id among arbitrary f(α·t). We will adopt option (i) and rephrase: the title and abstract will be revised to 'a natural permutation-symmetric extension', and the introduction wording 'canonical multidimensional extension' will be replaced by 'natural extension'. We will also add a sentence at the end of §2 explicitly noting that, by Remark 3.1, the rank-one t-geometry is shared by every J = f(α·t), so what §2 fixes is the parameter vector α (up to sign) within the multiplicative-cosh ansatz, not the ansatz itself. We considered option (ii) — formulating a uniqueness theorem within the family ½(F(R)+F(R⁻¹)) — but the natural conditions (composition law, unit log-curvature) reduce to the 1-D theorem of [19,20] applied to F, and we feel a re-statement here would duplicate that result without new content. We therefore prefer the honest weakening of language. revision: yes

Referee: Example 3.1 does not illustrate what the surrounding text claims. Φ:ℝ²→ℝ⁸ has 2-D image, J₈ = cosh(S₈(r,s))−1 lives on the (r,s)-domain, and since S₈ is nonlinear in (r,s), ∂ᵢ∂ⱼ(cosh∘g) = cosh(g)∂ᵢg∂ⱼg + sinh(g)∂ᵢ∂ⱼg has a non-rank-one term. 'Depends on a single scalar' ≠ 'rank-one Hessian'. Reformulate so S is genuinely linear in independent coordinates, or remove.

Authors: The referee is correct, and we thank them for catching this. As written, Example 3.1 conflates two distinct properties: dependence on a single scalar (which holds), and rank-one Hessian (which does not hold in (r,s) because S₈ is nonlinear in (r,s); the sinh(S₈)·∂²S₈ term spoils the rank-one structure). Our intended point — that the rank-one phenomenon survives in higher ambient dimension — is correctly captured only when S is a linear function of independent affine coordinates. We will replace Example 3.1 with a corrected version in which the eight components of Φ are themselves treated as independent affine coordinates (uᵢ) on an open subset of ℝ⁸, and S₈ = (1/√8)Σ aᵢuᵢ is a genuine linear form, so that ∇²J₈ = cosh(S₈)aaᵀ has rank one on ℝ⁸. The (r,s)-pulled-back picture will be removed, since it carries the misleading non-rank-one Hessian the referee points out. The surrounding sentence will be rewritten accordingly to refer to a linear functional on the ambient space rather than a nonlinearly embedded 2-D image. revision: yes

Referee: The Christoffel symbols (4.18)–(4.23) in (s,t) involve coth(as+bt) factors individually singular at the zero-cost locus q=0, while the metric (4.9) is itself degenerate there. Please (i) state explicitly the open set on which (4.18)–(4.23) are valid, and (ii) reconcile these formulas with the regular-looking coefficients in (q,r)-coordinates (4.27)–(4.28), or note where the simplification comes from.

Authors: Agreed. The Levi-Civita connection on Mₓ in (s,t)-coordinates is defined only on the open set where the metric (4.9) is nondegenerate, namely D_{st} = {(s,t) : q := as+bt ≠ 0 and (a+b)coth(q) ≠ 1}, the latter being the (s,t)-image of the secondary singular locus ∆ = 0. The coth(q) factors in (4.18)–(4.23) reflect precisely the q=0 boundary of D_{st}; they are not artifacts. We will add an explicit statement of D_{st} immediately before (4.18). Regarding reconciliation with (4.27)–(4.28): the (q,r)-form is obtained by the linear change of variables (4.24)–(4.25), which is global and nonsingular (Jacobian a²+b² ≠ 0). The apparent regularization is cosmetic — multiplying through by ∆ (or equivalently by 2((a+b)cosh q − sinh q)) clears the denominators, so (4.27)–(4.28) are the geodesic equations in the form ∆·(LHS) = 0 rather than (LHS) = 0. We will add a remark stating this explicitly: the (q,r) equations have the same singular set, encoded now as the vanishing of the leading coefficient of q'' and r'', and a one-line algebraic identity showing that pulling back (4.27)–(4.28) through (4.24)–(4.25) reproduces (4.18)–(4.23) on D_{st}. revision: yes

Referee: The 'formal expansion' near R=1 in §4.6.2 only extracts a leading-order constraint. Please clarify (a) whether x₁ = y₁ z₀^{1/a+1/b} and x₁ = −(b/a)y₁ z₀^{1/a+1/b} correspond to extendible C¹ geodesics or only to formal power series, (b) whether higher-order coefficients can be solved recursively (Fuchsian indicial conditions consistent at all orders), (c) what this implies geometrically — finite affine parameter to R=1, continuability through it.

Authors: We agree the present statement is suggestive rather than sharp, and we will moderate the language and add what we can substantiate. (a) At present we have established only formal-series solvability at leading order; we have not proved C¹ extendibility of either branch through R=1, and we will say so explicitly, removing any implication that the two values of x₁ correspond to bona fide geodesic tangent directions at q=0. (b) For the second branch x₁ = −(b/a)y₁ z₀^{1/a+1/b} the system is genuinely Fuchsian at q=0 and we have verified that the recursion for (xₙ,yₙ) is solvable to order n=4 with no obstruction; we will add this as a remark, while noting we do not have an all-orders proof. The first branch x₁ = y₁ z₀^{1/a+1/b} corresponds (via (4.30)) to r₁ = ((a−b)/(a+b))q₁, which is the direction tangent to the q=0 locus and is degenerate at this order. (c) Numerically (Figures 2–4) Levi-Civita geodesics reach R=1 in finite affine parameter and the residual blows up there, consistent with non-extendibility through the curvature singularity, but we do not have an analytic proof of finite-time arrival or non-continuability. We will rewrite §4.6.2 to (i) call the result a 'formal indicial constraint', (ii) report the order-4 recursive check, and (iii) state the finite-parameter arrival and non-continuation only as a numerical observation supported by Figure 4, leaving the sharp claim as an open question. revision: partial