Codimension c is classified by π_{c−1} — settled as a coordinate, held pending a novel absence

Why this is a probe and not a note. Codimension is not in rung-vocabulary-v5.md. It appears in the map exactly twice, both times as an instance rather than a coordinate: the archived v3.54 lists "codimension 1 separates, hence the transport tubes gate" as one output of the dimension bridge, and algebra-geometry-dictionary.md has "decidability = codimension of connectivity" (Herlihy–Rajsbaum). Nothing names the general thing. This entry proposes that there is a general thing and states what it predicts, so it can be scored.

The claim. For a p-dimensional object in n dimensions, what matters is c = n − p, and it has three regimes with a sharp boundary between each:

The three regimes are the rule of general position read three ways. Two submanifolds of dimensions p, q in n dimensions generically meet in dimension p + q − n. Curves in 3-space: 1 + 1 − 3 < 0, so they miss each other, which is exactly why they can be tangled — one cannot be pushed through the other. Curves in 4-space have room and every knot unties. Codim 2 is where the complement grows a loop, and that meridian loop is the same loop in every instance.

Instances now on file, from different domains:

1. Codim 1 separates — the transport-tubes gate (v3.54, dimension-bridge output).
2. Decidability = codimension of connectivity — t-resilient ⇒ protocol complex (t−1)-connected (dictionary; distributed computing).
3. Anyons — worldlines are codim-2 in 2+1 spacetime, so they braid; in 3+1 they are codim-3 and untangle, and the statistics collapses to S_N (N-016, N-025).
4. Knots — exist only for curves in codim 2 (3-space); every knot in 4-space is trivial.
5. Poincaré's difficulty in dimension three — π₁ of a 3-manifold is generated by curves that can knot, so the Whitney trick fails for want of room; it works from dimension five up.
6. Monopoles vs vortices — a vortex line in 3D is codim 2 and carries π₁ holonomy; a monopole is codim 3, its complement is simply connected, and its charge lives in π₂ instead. The codimension decides which homotopy group the defect is classified by.

What it predicts, stated so it can fail.

• No braiding statistics in three or more spatial dimensions. (Confirmed; this is why anyons need 2D.)
• No knotted curves in four or more dimensions. (Theorem.)
• No π₁-type holonomy around a point defect in 3D — an Aharonov–Bohm phase requires a codim-2 source. (Confirmed: monopole charge is π₂, not π₁.)
• The live one: any physical effect attributed to "going around" a defect must have that defect in codimension exactly 2 in the relevant space. A claimed loop-holonomy effect from a codim-1 or codim-3 object is either wrong or is secretly about a different space in which the object is codim 2. This is the one to score, because it can be violated.

Pursued the same day — two near-violations, both absorbed, and the claim sharpened

Near-violation 1, the Berry monopole. A degeneracy of a complex Hermitian two-level system is codim 3 in parameter space, the complement is S² and simply connected — and the Berry phase around a loop is nonzero anyway. Not a violation: the phase is the solid angle, deforms continuously with the loop, and is a π₂ effect (Berry curvature is a Chern-1 monopole), not π₁. Sagnac is the same lesson from the other side: the rotation axis is codim 2, but the phase goes as enclosed area because the curvature fills the plane instead of concentrating on the axis. Amendment 1: the prediction concerns topological holonomy — quantized, homotopy-invariant — and that needs codim 2 and a flat connection in the complement. AB has both. Sagnac has the codim and not the flatness, and is correctly not quantized.

Near-violation 2, the Zak phase. The SSH chain's Berry phase around the Brillouin zone is quantized to 0 or π with no defect anywhere; the loop is the BZ circle itself. Not a violation: in the plane of Bloch vectors (h_x(k), h_y(k)) the phase is π exactly when the loop winds around the origin, where the gap closes — codim 2 in that plane. The escape clause fired as written.

The genuinely intrinsic case — Kaluza–Klein. The loop is a compact dimension; nothing was removed. Amendment 2: a non-contractible loop arises either by codim-2 removal or from intrinsic π₁ of a compact space, and codimension is the coordinate for the removal case.

The falsifiable core that survives: every localized source of quantized loop-holonomy is codimension two. Solenoid, vortex, cosmic string, conical intersection, gap-closing point — every one. Not broken.

Settled: it is a coordinate, not the dimension bridge with an extra input

Three regimes was the wrong count. The rule is one line:

a defect of codimension c is classified by   π_{c−1}(S^{c−1})

because the linking sphere of a codim-c object is S^{c−1}. Codim 1: π₀(S⁰), two points — separates. Codim 2: π₁(S¹) = ℤ — winding, holonomy, knots, braids. Codim 3: π₂(S²) = ℤ — monopole charge, which is exactly near-violation 1. Codim 4: π₃(S³) = ℤ — instantons. Mermin's defect classification, and every instance in this probe is a row of it.

The type signature decides the question. The dimension bridge (v5 §8.1) consumes a dimension and emits a value — 240, 3/4, N_c = 3, two statistics. Codimension consumes a pair (object, ambient) and emits a homotopy group — it says which apparatus applies. That is not a bridge with one more input; it is the same kind of thing as v5 §5, the form-of-question axis, whose stated job is "the answer decides which apparatus applies." A selector, not a producer. Coordinate.

Instance 7, from a direction none of the six came from — quantum tunnelling. Continue the WKB solution into complex position. The turning points where p = 0 are square-root branch points — points in the complex plane, codimension 2 — and the connection coefficients that fix how much amplitude leaks through the barrier are the monodromy of the solution around them: going round a turning point exchanges the growing and decaying exponentials. That is the Stokes phenomenon (resurgence, N-014). So the tunnelling amplitude is a holonomy around a codim-2 object, in a space one has to analytically continue into to see. The instanton is the same fact in Euclidean time: the bounce is a closed orbit that exists only in the complexification, and the exponent's factor of 2 is the loop closing. At finite temperature the Euclidean circle is literally compact, and the crossover to thermal activation is the point where the bounce no longer fits on it. Note the different job: N-025's loops constrain a spectrum; this one computes an amplitude, and tunnelling rates are continuous — except in the double well, where the splitting is spectral and both jobs meet.

The friction — why it is still held, not filed. rung-vocabulary-v5.md already has a dimension bridge (§8.1), and codimension may be that bridge with one more input rather than a coordinate in its own right. Against that: dimension alone does not predict any of the six instances above — knots exist in 3D and not 4D because of the curve's dimension relative to the space, and "3D" by itself says nothing. The relational quantity does the work. That is the case for a coordinate. It has not yet been checked against the map's rule that a coordinate must predict an absence that was then found — the absences above are all already known. Promotion requires a predicted absence that had not been noticed before the coordinate was named. None yet.

Chase (owner, 2026-10-01; codifier to carry). Promotion needs an absence codimension predicts that nobody has noticed. Nobody has tried; this paragraph is the standing task, and it is checked at every look-back until it is done or the probe is closed. A first candidate, from the lab's own objects and held loosely: in the carpet thread, a left–right crossing is obstructed by codim-1 objects (cuts; π₀), and the crossing polynomial counts crossings without their codim-2 data — the winding of a path around a hole (π₁). So no function of R(p) can see which side of a hole a crossing passed, and the ladder's rung k can see it only once a block's recorded state has terminals on both sides of the hole's shadow: S₁ = {4} cannot; S₂ = {2, 6} and S₃ = {1, 4, 7} can (P-049's positions; cells 3–5 are the shadow). The predicted absence: if the carpet's turn is set by the side of a hole, rung 1 cannot show it — blind by codimension, not by truncation — and the first rung that both sees the side and closes its state is k = 3 (k = 2 sees it but does not close, P-049), which is where run 084 sits. That is checkable against the turn's arrival on the ladder: a turn at k = 3 with none at k = 1 is consistent; a turn at k = 1 kills the candidate. Whether it is an absence "not noticed before the coordinate was named": run 084's rung was chosen by the wall guard among {3, 5}, with winding never mentioned, so the candidate was not in the record before this paragraph — the kind of thing the rule asks for, if it holds. The owner's inspiration for this, from P-045: a block is a piece of a larger ensemble, and closing the ensemble makes connections cancel the way the interior faces cancel in Stokes, Gauss and Helmholtz; what survives is the boundary. Read on the gap of P-050's item 1: the one-block fixed point's overshoot is the non-cancelling boundary term of a decomposition whose "surfaces" are the holes at every scale, zero on the square where the decomposition closes (self-duality), and measured by the ring share. Held as an inkling beside the candidate; neither is a seal yet.

Relation to N-025. The "loop-closing is quantization" mechanism in N-025 is the codim-2 regime seen from the holonomy side. If codimension is a coordinate, N-025's six instances are all instances of its middle row. If it is not, N-025 stands on its own.