Chapter 04
OPH in five sentences.
- 01Patches.
Reality is described by many local observers, each with their own algebra of observables and a local state.
- 02Overlaps.
Two patches that share a region must agree on the observables they share. That's an overlap constraint.
- 03Loops.
If three or more patches form a cycle, pairwise agreement does not automatically give global agreement. The cycle can carry a holonomy.
- 04Repair.
A schedule-independent repair process drives the patch net toward a unique normal form, lowering a Lyapunov functional.
- 05Reality.
The physical observable is the normal-form data on the gauge quotient. Holonomies that survive are real; everything else was bookkeeping.
Touch it.
Three patches, three overlap edges. Drag a patch to rotate its phase, or change an edge's transition. The amber ring shows the loop holonomy — the obstruction to global gluing. Hit Repair to find the unique normal form.
Drag patches to rotate them, or change edge transitions. The amber ring grows when the loop fails to close. Repair drives patches to the unique gluable normal form on the gauge quotient — the OPH solution.
Why this dissolves the hard problems.
The "three-body problem" and the "Aharonov–Bohm effect" both look hard because they insist on local language. OPH starts with overlap data, not local fields. The thing that was obstructing a closed-form answer is exactly the thing OPH names and computes: loop holonomy. Once that's the primary object, both phenomena are special cases of the same procedure.
Where to read more.
- The OPH paper (GitHub) — core definitions, finite theorems.
- learn.floatingpragma.io — Book 1 (gravity from observers), Book 2 (Standard Model).
- oph-book.floatingpragma.io
- Repository