Quadratic variation needs a partition convention

The identity $[B{]}_t=t$ established on this page is an almost-sure assertion along the fixed dyadic partition sequence of Quadratic variation along a partition sequence; the uniform form of that assertion is Uniform dyadic Brownian quadratic variation process. It is not a simultaneous assertion over all refining sequences, and the path-dependent or arbitrary refinements of a realized path are not covered.

What the quantifiers allow. The theorem cited above supplies almost-sure uniform convergence for the fixed dyadic sequence. For a general prescribed deterministic sequence whose mesh tends to zero, the standard conclusion without an additional summability or regularity hypothesis is convergence in probability, not almost-sure convergence along the whole sequence. In particular there is no single event on which every refining sequence simultaneously has the same limit, and the definition deliberately builds in no partition-independent object.

What fails without regularity. The convergence proofs use the independence of increments over a preselected mesh together with a summable mesh estimate. A refinement adapted to the oscillations of one realization destroys that independence and can change the sums; the deterministic partition dependence of quadratic sums is illustrated on the companion examples page, and the boundary is recorded here so that later semimartingale statements do not silently inherit a claim about arbitrary partitions.

No proof is attached to this remark: it records the quantifier boundary of the preceding definition and theorem rather than a new mathematical assertion.