Topological protection conserves non-locality — the measure is the distance exponent, and the frontier has an open corner

What is proved, and is therefore not the probe. Three no-go theorems, one shape:

• BHV (Bravyi–Hastings–Verstraete): no finite-depth local unitary circuit prepares a topologically ordered state; depth is Ω(L).
• BPT (Bravyi–Poulin–Terhal): for a geometrically local code in 2D, k·d² ≤ c·n; in D dimensions, k·d^{2/(D−1)} ≤ c·n.
• Baspin–Krishna 2022: beating BPT with a good qLDPC code costs a quantifiable amount of long-range connectivity that no finite dimension can absorb.

Each was tested against a scheme that seemed to escape it, and each escape relocated the non-locality rather than removing it: measurement-based preparation is constant-depth but needs global classical feedforward; good qLDPC codes have constant rate and linear distance but need expander connectivity; the 4D toric code is self-correcting and needs no active correction, having paid in spatial dimension. "Topological protection requires non-locality somewhere" is therefore a compression of theorems, not a conjecture, and by the map's rule earns no coordinate on its own — every absence it predicts was already known.

The honest resource inventory — five, not three. The first framing was "depth × density × connectivity," which put a payoff (density) on the cost side and missed two resources entirely:

Payoffs: rate k/n and distance d. The two missed cases were self-correction and dimension.

The conjecture — the live part. Every resource above must scale at least as the code distance d, and d is simultaneously the protection: the minimum weight of a logical operator, the minimum preparation depth (Lieb–Robinson across distance d), the reach the connectivity must span, and — via k d² ≤ n — the size of the object BPT is packing. Protection is not bought with non-locality; protection is non-locality, in the same units. If that is right, then the "single quantitative measure of which the three bounds are projections" is not a product of resources but one number, d, with each theorem a different resource's lower bound in terms of it. The strongest theorem-side anchor is NLTS (Anshu–Breuckmann–Nirkhe 2022): good codes admit no low-energy state of constant circuit depth, so circuit complexity of the code space already subsumes depth. Open: whether connectivity and correction time fold into circuit complexity, or whether d bounds each separately without a common invariant behind them.

How large the "somewhere" must be. At least d, and d is a vanishing fraction of the volume — for the 2D toric code a string of length L = n^{1/2} in area n. So the non-locality is codimension-(D−1): a string, not a region. It cannot be smaller than d because d is by definition the smallest support that touches the logical information.

No-communication is untouched, and the relation is worth stating because it is the opposite of a tension. The theorem forbids a local observer learning about a distant operation. A logical operator of support d on the code space produces no local change anywhere — that is what makes it logical. Topological codes are the extreme case where a local observer learns nothing even about a nearby operation. The non-locality is in the entanglement and in the access to it, never in signalling.

Pursued the same day — falsified as stated, sharpened into a frontier

What kind of number. d as computed is a count (v5 §1, first kind): the minimum weight of a logical operator, exact. But every bound above is asymptotic and uses d only through how it scales with n. The number doing the work is the distance exponent α in d ∝ n^α, and Bravyi–Terhal's d ≤ L^{D−1} with n = L^D gives

α_max(D) = (D − 1) / D :   0, 1/2, 2/3, 3/4, … → 1      (D = 1, 2, 3, 4, …, ∞)

an exact rational from the integer D — v5's second kind, a dimensional exponent. D = 1 gives zero, correctly. The 2D toric code saturates 1/2. Good qLDPC codes sit at α = 1, the D → ∞ limit, which is the precise sense in which dimension and connectivity are one resource — an expander graph is infinite dimension seen from inside a finite one. So the single measure, if there is one, is not d but α; not a count but an exact rational; and not complex or real-with- sigmas in any sense. The five resources collapse to four.

Counterexample from correction time — succeeded. "Every resource scales at least as d" is false as stated: the 4D toric code is self-correcting and needs zero active correction, ever. What survives is a trade-off frontier, and the D-versus-time frontier already has three points: 2D no (Bravyi–Terhal, no self-correction for 2D stabilizer codes); 3D partial (Haah's code: memory time grows polynomially in L at low temperature, not exponentially); 4D yes. The conjecture is now a frontier claim, not a per-resource bound.

Counterexample from dimension — became a named open problem. D → ∞ loosens every bound to nothing and is exactly what good qLDPC codes do; they pay in connectivity, which is the same resource, so no escape. But combine the two: the 4D toric code is self-correcting with k = O(1) (rate → 0); good codes have constant rate and are not known to be self-correcting. The corner that would break the trade-off is therefore self-correcting + constant rate + finite D. That is a recognised open problem with its own literature, and neither the construction nor the no-go exists.

What would promote it (revised): a self-correcting code of constant rate in finite dimension — two resources held small at once, no single invariant. Or a proof that none exists, which would be the invariant. Neither exists, and this is now the field's open problem rather than one invented here.

Relation to N-025 and P-024. N-025's loop-closing is the mechanism by which the code space is non-local (Wilson loops around non-contractible cycles); this probe is about the cost of that non-locality. P-024's codimension gives the shape of the minimal logical operator: a string in 2D, a membrane or a string in 3D, and — for fractons — fractals, which is where the loop account stops and this probe's accounting still has to hold.

Pursued further, same day — the open corner was closed by BPT, and a six-cell grid replaced it

Correction. The corner named above — self-correcting + constant rate + finite D — is not open. Apply BPT: k·d^{2/(D−1)} ≤ cn with k = cn forces d = O(1). Constant rate in any finite Euclidean dimension kills the distance outright. Rate and distance are structurally conjugate in Euclidean geometry, so this "two resources small at once" was Heisenberg-shaped, not Shannon-shaped, and the theorem was already cited above; it had not been applied to the rate axis. Recorded rather than silently corrected.

Why "finite D" needed sharpening. Guth–Lubotzky's 4D hyperbolic codes have constant rate and growing distance in D = 4; they evade BPT because it assumes polynomial volume growth and hyperbolic space grows exponentially — as an expander does. Curvature, dimension and connectivity are three faces of one resource: volume growth.

The honest enumeration. Volume growth is classified — Gromov: polynomial (virtually nilpotent, integer degree D) / intermediate (Grigorchuk) / exponential — and the instances already in hand were split by a second axis, manifold or not. Six cells:

The structurally empty cell is evidence the grid is the right grid: a cell the classification produces and then rules out. The theorems' hypotheses say which cells are closed — BPT assumes integer D and polynomial growth, Baspin–Krishna assumes an embedding in ℝ^D — which is the map's own rule, check the tools' hypotheses, not just the objects.

The 3D cell — updated against the literature, and the update is recent. The two routes named in session (translation-invariant stabilizer → log L; self-similar non-TQFT → unknown) were both superseded by a third that neither anticipated. Layer codes (Williamson–Baspin 2023; self- correction analysis, October 2025) are 3D, geometrically local, not translation-invariant, built by replacing each qubit and check of a good quantum Tanner code with a surface-code patch joined by defect lines. Parameters [[Θ(L³), Θ(L), Θ(L²)]] — which saturates both Bravyi–Terhal (d = L^{D−1}) and BPT (k·d = n) in 3D — and energy barrier Θ(L), linear. The log L ceiling was a ceiling on translation invariance, not on three dimensions.

But the free-energy obstruction named above is exactly what stops them. Memory time is Ω(e^{acβL}) only up to a temperature-dependent maximum size L ≤ O(e^{(1−a)β/3}); beyond it the entropy factor (1 + e^{−(1−a)β})^N wins (Baspin). The paper calls this partial self-correction and states plainly that layer codes "fall short of achieving strict self-correction in the thermodynamic limit." The self-similar route has a candidate too — cored product codes on the aperiodic pinwheel tiling (October 2025), with numerical evidence of memory lifetime growing with size to 60,000 qubits and no theorem.

So the 3D cell now reads: energy barrier — closed, linear achieved. Free-energy barrier in the thermodynamic limit — open, with the obstruction identified as entropic and a partial result on each side of it. The precise open question stands as written, and it is the only thing left in the cell: is the entropic obstruction a theorem for all 3D local codes, or a property of the constructions so far?

The entropic obstruction — theorem or construction? Answered, September 2025. Baspin's Free Energy Barrier paper proves an Eyring–Polanyi bound t_mix ≤ O(β n² e^{βf̄}) with f̄ the free-energy barrier — a strictly weaker quantity than the energy barrier, because it subtracts (1/β)·log(number of low-energy paths). It then proves layer codes have constant f̄ despite their linear energy barrier: the defect structure that gives them the barrier also gives them exponentially many paths over it, and the entropy term eats the height. That is the no-go for the one construction. And the paper poses the general case as open in so many words: "Can the free energy barrier bound for Layer codes be generalised to other 3D local codes with high energy barriers?" So: theorem for layer codes, open for three dimensions, and the question in this probe is the field's own open question as of last month.

What a 3D code would have to do is now stated exactly: linear energy barrier and a number of paths over it that does not grow exponentially in the barrier — rigid walls, not merely tall ones. The 4D toric code has that (a membrane grows only at its perimeter, polynomially many ways). Layer codes do not. Nothing in 3D is known to.

Count of open corners, honestly. Not three. Six cells from a theorem: one structurally empty, one partially closed along D and updated against October 2025 results, four open, one of which (N-026) was examined for the first time in this session — and found to have a homological floor of log n under a separator ceiling of n/polylog, with the gap between them as its open problem.

Run 030 — the scale-dependent-dimension speculation measured, classically

The speculation raised at the end of the session — that a host whose effective dimension is low at small scales and high at large ones might decouple the entropy term in Baspin's free-energy barrier from the energy term, a fourth self-correction mechanism — was put to the k = 1 memory (the Ising model, whose logicals are cuts) on Cay(G/St(4)) at n = 4096, T = 1, against the 2D torus, the 3D torus and a random 4-regular graph (lab/runs/030, R23_ising_barrier/). The decomposition came out Euclidean in every respect: the entropy fraction is local, and the host's is 2D's (ρ = 0.29–0.37 vs 0.34; the cubic lattice's is 0.004); the barrier is the minimum cut, 326 against 223 (2D) and 1020 (3D), growing with exponent 0.76 in n between the two quotients. Rigid walls come from membranes at the smallest scale, i.e. high local dimension; the speculation had the sign of the local term backwards. Nothing here is a fourth mechanism, and the classical proxy already says why: on this host a cut-like logical would inherit 2D wall entropy — survivable, as 2D Ising survives — with a barrier that at this size a 3D lattice beats 3:1. What the host has that the lattices lack is glassy walls: near-minimal cuts in metastable families 25% apart, which slow the wall's formation and its removal alike. The one remaining question is whether the cut's exponent climbs toward 1 with n; the minimum bisection of G/St(5) decides it, and no sampling is needed.

Same evening — a symmetry-group axis, and two more readings of the expander cell

Symmetry axis. On an expander the discrete landscape is a glass and the continuous one is trivial: minimum bisection of a sparse random graph has exponentially many uncorrelated minima (Fu–Anderson; run 034's overlaps), while the Kuramoto flow has a single stable equilibrium (expanders are globally synchronising). Z₂ sectors proliferate, U(1) sectors vanish. A torus has both (strips and windings). The intermediate host: Z₂ rugged (run 034's T = 0 barriers, ~100 J between families at N = 1024); U(1) sealed as run 035. Where a host sits on this axis is a property of the symmetry of the memory as much as of the host, which the grid did not have.

Expander codes are this cell's mechanism made explicit. Sipser–Spielman and Tanner codes, and the quantum Tanner codes on left-right Cayley complexes, rest on unique-neighbour expansion of small sets — the "explicit randomness" of expanders. That is why the intermediate cell was empty: every construction requires what amenability forbids, and amenability forbids it only asymptotically, at rate 1/log N here (the separator). Read through the code theorems that gives the cell's ceiling from the other side — distance ≲ expansion × N ≈ N/log N — and the live question: does the combinatorial guarantee degrade gracefully when small-set expansion decays like 1/log N? Sealed as run 036 (the expansion profile of the coset graph). The construction to seal after it: a Tanner code on the coset graph, then the left-right Cayley complex of the Grigorchuk group with two generating sets — preferred over the layer-code import.

Arithmetic instance of the same mechanism. Helfgott–Radziwiłł (arXiv:2103.06853) prove logarithmic Chowla by the expansion of a divisibility graph on the integers: the exponential cell's one fact — mixing from a spectral gap — used as an instrument on primes. P-008 records why it stops at the logarithmic average (N-006's Haar weight). N-028 chases and breaks the N/log N coincidence between the host's separator and the prime number theorems.

Back-links (2026-09-29)

Dated pointers added by the codifier from lab/backlinks.py; nothing above this heading was changed.

Its closing question — whether the cut exponent climbs toward 1 — was answered by run 031 (confirmed: b₅ = 90,112 = (11/16)·131,072, level-4→5 exponent 0.900). Two levels of the same constant is a prediction for level 6, not a result (N-038 §4).

• Run 030 — unanticipated (2026-09-27). lab/runs/030-ising-barrier-intermediate-growth.md: UNANTICIPATED — scored 2026-09-27 by the human's ruling, the codifier having declined to score its own run at the boundary.
• Run 031 — confirmed (2026-09-27), via the builds_on thread. lab/runs/031-grig5-bisection.md: CONFIRMED — scored 2026-09-27 (codifier).
• Run 032 — falsified (2026-09-27, measured), via the builds_on thread. lab/runs/032-glassy-wall-anatomy.md: FALSIFIED on both clauses — L_min = 0.128, Δpos = 10.
• Run 033 — confirmed (2026-09-27, measured), via the builds_on thread. lab/runs/033-host-primes-cogrowth.md: CONFIRMED — κ = 0.665.
• Run 035 — falsified (2026-09-27, measured), via the builds_on thread. lab/runs/035-kuramoto-host-sync.md: FALSIFIED — f = 1.000: all 200 random starts on Cay(G/St(4)) end in non-synchronised equilibria (199 converged to max|θ̇| < 10⁻⁶, one still moving at t = 2000), with r from 0.02 to 0.9 (histogram over [0,.2,.4,.6,.8,.9,.99,.999]: 61, 94, 34, 10, 0, 1, 0, 0).
• Run 036 — confirmed (2026-09-27, measured), via the builds_on thread. lab/runs/036-expansion-profile-coset-graph.md: CONFIRMED — ρ_Φ = 104/134 = 0.776.
• Run 043 — falsified (2026-09-28, measured), via the builds_on thread. lab/runs/043-growth-axis-on-one-tree.md: FALSIFIED — the relator clause's falsifier fires: Basilica's unique-neighbour minimum at N/8 is 1 of its control's 101 at best-of-20 and 0 of 94 at best-of-40, u/u_random ≤ 0.10, so the square obstruction of N-034 is not specific to the Klein-four contraction;