Compatibility of the labeling system for constructing the Θ_d-colored (d+1)-operad (Conjecture 1)

Prove that the labeling system assigning, for every pair of leaves i and j with i less than j at level a in a pruned (d+1)-level tree, an upper bound on the parameters (μ,σ)(ω_{ij}) given by the canonical chains V^d and their shifted variants (as prescribed in the construction of the Θ_d-colored (d+1)-operad seq_d), satisfies the quasi-bijection compatibility condition across two-leaf trees required for operadic closure. Equivalently, establish that the system of inequalities specified in the construction of seq_d satisfies the quasi-bijection compatibility condition that ensures the family {seq_d(T)} forms a well-defined (d+1)-operad.

Background

The paper constructs Θ_d-colored (d+1)-operads seq_d by selecting, for each arity T, unions of ‘blocks’ inside the higher lattice path operad Ld subject to levelwise complexity bounds. To ensure these collections define operadic compositions, the bounds for two-leaf trees must be compatible under quasi-bijections (the ‘maincondition’).

In the general construction, the authors prescribe a canonical ordered set Vd of elements in the product poset K(2){×d} generated by elementary moves, along with shifted variants \tilde{V}ℓ, and impose levelwise bounds determined by these sets. Conjecture 1 asserts that these bounds satisfy the required compatibility for all d, a fact verified in the paper for d=2,3,4 but not proved in general.

References

Conjecture 1. The system conditionsd=gen satisfies maincondition.

conditionsd=gen:

 i<dj⇒(,)(ωij)≤(12)i<d−1j⇒(,)(ωij)≤(121)∣(12)…i<ℓj⇒(,)(ωij)≤ some ω~aℓ∈V~d−ℓ…i<1j⇒(,)(ωij)≤ some ω~ad−1∈V~d−1i<0j⇒(,)(ωij)≤ some ωad∈Vd\begin{aligned} \ & i<_{d}j\Rightarrow (,)(\omega_{ij})\le (12)\\ &i<_{d-1}j\Rightarrow (,)(\omega_{ij})\le (121)|(12)\\ &\dots\\ &i<_\ell j\Rightarrow (,)(\omega_{ij})\le \text{ some }\tilde{\omega}^\ell_a\in \tilde{V}^{d-\ell}\\ &\dots\\ &i<_1j\Rightarrow (,)(\omega_{ij})\le \text{ some }\tilde{\omega}_a^{d-1}\in \tilde{V}^{d-1}\\ &i<_0j\Rightarrow (,)(\omega_{ij})\le \text{ some } \omega^d_a\in V^d \end{aligned}

maincondition:

$\parbox{5,5in}{\rm For any quasi-bijection $(T_a^n,)\to (T^n_b,\rho)in in \mathcal{M}^n_2andany and any (_i(b),_i(b))thereexists there exists 1\le j\le s(a)suchthat such that (_j(a),_j(a))\ge (\rho^{-1}_i(b),\rho^{-1}_i(b)),where, where \geisthedominancerelationintheposet is the dominance relation in the poset (\mathcal{K}^n(2))^{\times d}$. } $

— Generalised Joyal disks and $Θ_d$-colored $(d+1)$-operads  (2510.05813 - Shoikhet, 7 Oct 2025) in Section 5.3, The general case (Conjecture 1)