The tau level is independent of the normalized Heegaard model

Apply [L1] to the two analytic doubly pointed diagrams in the Statement. Its maps and inverse homotopies are finite polynomial identities preserving their Seifert-normalized Alexander filtration. Specialize those identities at U=0; no flatness assertion about homology is required. The resulting actual finite hat models are filtered chain homotopy equivalent and carry the nonzero total generators to each other. Therefore [F1] gives equality of their thresholds. This proves the analytic-diagram comparison in the first sentence.

Fix a grid $G$ with $n$ marking pairs and choose one marking variable $U_i$. Use the actual normalized grid-to-analytic comparison in [L3], together with the pointed-move comparison in [L1], to convert its full filtered minus complex to the finite singly pointed model for $(K,F)$. In this comparison first restrict scalars along $U\mapsto U_i$ and retain the other $U_j$ as independent module variables. The supplied chain maps and inverse homotopies preserve the Alexander filtration and the Seifert-normalized origin. Specializing these actual identities at the chosen $U=0$ gives the corresponding filtered equivalence to the singly pointed hat endpoint. The restricted grid module can still have infinite rank over ${\mathbb{F}}_2$ after this specialization, so it is not itself an input to the finite-complex definition of $\tau$; the grid value in the Statement is the threshold on the finite normalized singly pointed endpoint obtained by this conversion.

The separate simultaneous ordinary collapse sets all marking variables equal before taking the one-variable quotient. By [L3] and [L1] its filtered type is the singly pointed minus model tensored with $V^{\otimes (n-1)}$, with each $V$ carrying bidegrees $(0,0)$ and $(-1,-1)$. After quotienting the remaining variable by zero, this collapsed model has the extra $V$ homology factors and is not the finite complex with one-dimensional total homology required in the definition of $\tau$. Those factors are accounted for by the actual basepoint-pair cone and retraction comparison in [L1, L3]: use the retained-variable equivalence to the singly pointed model when defining the grid's value, and do not apply $\tau$ to the unreduced simultaneous-collapse complex. Thus no extra filtration shift from the $V$ factors is included in the claimed value.

For any two grid presentations, [L3] supplies this conversion for each grid to a normalized singly pointed model. Compare those finite endpoints by the actual pointed-move equivalence [L1], specialize its maps and filtered inverse homotopies at $U=0$, and apply [F1]. This gives the same threshold as in step 1.1. The grid-move and grid-isotopy suppliers [L2, L4] describe additional combinatorial comparisons; they do not replace the filtered grid-to-analytic conversion or alter the variable-collapse distinction above.