The math page
Checked by proof software
The idea
Five checks that all copy each other are really one check.
Say five people check an answer and all five agree. That feels safe. But if all five learned it from the same place, their agreement is not five opinions. It is one opinion repeated five times. The number of checks looks like five. The real number is one.
This matters for AI. An AI system can run many checks on its own reasoning and still be fooled, if those checks share the same blind spot. Coherence Collapse Analysis is how CIRIS tells the difference between real checks and echoes.
The one formula
Counting the checks you really have.
There is a single short formula at the heart of it. It comes from survey statistics, where it is called the Kish design effect. CIRIS was the first to use it for AI alignment.
real checks = checks / (1 + copying × (checks − 1))
“Checks” is how many checks you ran. “Copying” is how much they overlap, from 0 (all independent) to 1 (all the same). The result is how many checks you really have.
No copying: ten checks count as ten. Every check earns its place.
Total copying: ten checks count as one. It does not matter how many you add.
The healthy band
Not too samey, not too scattered.
The same shape shows up here that shows up everywhere CIRIS looks. If the checks copy each other too much, the system is too rigid: one voice repeated, easy to fool. If they have nothing in common at all, it is too scattered: they cannot agree on anything. Healthy checking lives in the band between, the same corridor the rest of CIRIS is built around.
The exact edges of that band depend on the system being measured. There is no single magic number that works everywhere, and the research is honest about that. The corridor idea, in full, is on the vision page.
On real traffic
This was measured, not just argued.
CIRIS measured the real-check count on its own live agent traffic, across thousands of recorded decisions. On healthy traffic it has run in the range of about seven to nine genuinely independent checks. That measurement, and how it was done, is the Constrained Reasoning Chains study. You can watch the live numbers on the research page.
The honest ceiling
What this math cannot do.
The paper proves a hard limit on itself. Some kinds of harm do not come from one dishonest part. They come from honest parts that add up to a bad result, and roughly forty percent of that kind of harm cannot be caught by any checker, no matter how good. CIRIS says so plainly rather than pretending the math catches everything.
What the math does say is about cost over time: running real, honest reasoning for weeks on end is steadier and cheaper than keeping a lie consistent across thousands of recorded decisions. It tilts the ground toward honesty. It does not promise to catch every single lie.
How it fits together
The math is one part of a larger whole.
This page is the measurement. The Coherence Ratchet is how the measurement is put to work. The Federation is how it becomes something many systems share. And the full proofs, written so a computer can check them line by line, are in the paper and the RATCHET repository.