Invisible mathematical operators
The four invisible math operators, which render as nothing and turn up in ordinary prose where there is no mathematics anywhere near them.
How the text is hidden
U+2061 FUNCTION APPLICATION, U+2062 INVISIBLE TIMES, U+2063 INVISIBLE SEPARATOR and U+2064 INVISIBLE PLUS. They are format characters that draw nothing, so a person never sees one. Their legitimate home is MathML, where they carry the operator a typeset formula leaves out; dropped into running prose they behave exactly like a zero-width character, breaking a caller's word matching and adding content to the string the reader never met.
Why a model still reads it
They are codepoints in the text, so an extractor returns them unchanged and a model reads the string they produce. Nothing in the document's styling marks their presence.
What we do about it
The pack class invisible-op matches U+2061 through U+2064, is declared highConfidence: false, and is graded informational by the confidence-band mapping in codepointFindings — recorded on every document that carries them, verdict untouched, certified withdrawn. The characters are stripped from text.canonical, and for DOCX from the cleaned file. The confidence band is low deliberately, because the MathML use is genuine.
How often it fires
0.24% of 1,244 real documents — 400 web pages (distinct hosts), 630 PDFs (contracts and web), 150 government Word documents, 65 SEC filings, measured 2026-08-21.
This is an alert-volume number and nothing else. It says how often the alarm sounds on documents as found — not how often it is right, and not whether what it found was harmless. Documents as found may themselves carry concealment. Read it against the population named above rather than as a property of documents in general.
Check your own file
Three commands: a key, credit, a verdict.
Start with the API