The grid Upsilon function agrees with the original Upsilon invariant

Given: A knot grid $G$ with grid number $n$, its full filtered grid complex, the diagonal quotient, and the two Upsilon conventions just defined. Countable Choice is assumed for the knot-level constructions and geometric comparisons; finite algebra on the stated supplied complexes and maps requires no further choice.

Basepoint variables act chain homotopically on a knot grid: for consecutive markings the rectangle homotopy $H_{\mathrm{\ell }}$ satisfies $\mathrm{\partial }{H}_{\mathrm{\ell }}+H_{\mathrm{\ell }}\mathrm{\partial }=(U_i+U_j)\mathrm{id}$ and is Alexander-filtered; following the one-cycle marking route identifies any two variable actions by filtered chain homotopy.

Grid Upsilon from the diagonal grid complex: the grid invariant is the maximal grading of a homogeneous non-torsion class in the t-modified diagonal quotient on $[0,1]$, with reflection for $t>1$.

The Upsilon function from the filtered knot Floer complex: the original invariant is $-2$ times the minimum $F_t$ filtration of a Maslov-zero cycle representing the normalized generator.

Knot Floer duality and exchange of the two filtrations: exchanging the two knot Floer filtrations identifies the original Upsilon values at $t$ and $2-t$. Its actual reflected/conjugate diagram maps preserve the normalized Maslov grading; no grid equality is a premise of that analytic symmetry.

Grid homology is independent of the grid and computes knot Floer homology now has an authored Proof section. Its Statement gives the chosen-variable filtered comparison, the simultaneous-collapse $CF{K}^{-}(K)\otimes V^{\otimes (n-1)}$ factor, and the normalized Maslov and Alexander origins. MOS Proposition 2.3 states the chosen-variable filtered comparison and tilde-homology $V$-factor, and Theorem 3.3 identifies the grid filtered complex with the knot filtration; the current in-run proof supplies the chain-level ordinary-collapse factor. Its absolute shifts are fixed in steps 4.1–6.1 under that item's listed [F1]–[F5] prerequisites. We use exactly this stated comparison, not an unproved claim that the citation alone fixes the absolute shifts.

Filtered tower splitting for the Upsilon minimax proves the filtered projection and the exact regrading map identifying the maximal non-torsion degree with $-2{\nu }_t$, using $U=v^2$.

Basepoint cones, a single variable action, and ordinary collapse gives an explicit filtered deformation retraction from the $U_i+U_j$ cone to the ordinary quotient, including a polynomial-dependent differential. Its null-homotopy comparison supplies exactly one $W$ factor per marking identification over the surviving chosen-variable action.

The t-modification preserves graded filtered maps, homotopies and tensor products proves the exact coefficient reweighting for maps and homotopies, including column-finite polynomial modules and tensor products.