Positive wedge-toggle kinetics give a dense condensate

Summary

If you want spacetime to emerge from relations between things, you need a rule that makes relations sparse. A manifold looks locally like a low-dimensional lattice: each point has few neighbours. So the question is whether some natural, purely local, positive rule for switching relations on and off settles into a sparse phase on its own, without being told to.

For one such rule — the positive wedge-toggle generator, in which a relation may switch when its two endpoints share a common neighbour — the answer is no, and the failure is quantitative. The ground phase is asymptotically dense, with each possible relation active with probability 5/6, giving a mean microscopic degree of about (5/6)N. That is about as far from a low-dimensional lattice as a graph can get.

Positive wedge-toggle kinetics do not select sparse microscopic geometry. The unrestricted ground phase of H_N = -λ A_N is asymptotically dense with one-edge density 5/6.

This is a negative result about one declared rule, and it is the useful kind: it closes off a natural architecture rather than adding another speculative one.

1. The model

For N ≥ 3, every unordered pair of nodes ij carries a relation qubit with number projector n_ij and Pauli flip X_ij. The generator is

A_N   = sum over i<j<k of T_ijk
T_ijk = n_ij n_jk X_ik + n_ij n_ik X_jk + n_ik n_jk X_ij
H_N   = -λ A_N,   λ > 0

In words: an edge ik may coherently toggle exactly when i and k have a common neighbour j — the "wedge" condition. The operator is real symmetric and non-negative in the graph basis.

What matters as much as what is in the Hamiltonian is what is not. It contains no target dimension, no target valence, no graph-distance potential, no connectivity reward, no loop target and no preferred graph. Nothing tells it to be sparse, and nothing tells it to be dense. This is deliberate: a rule that produces a lattice because a lattice was written into its potential has explained nothing.

2. What is established

For the declared generator, the accepted theorem establishes four things:

• the unrestricted global ground sector is connected — so the result is about H_N itself and not about a sector chosen by hand;
• normalised ground energy tends to 25√5 / 36;
• mean edge density tends to 5/6;
• every fixed triangle marginal tends to the corresponding productised p = 5/6 state.

The connected-sector step is what makes the statement unconditional. An earlier version of this work conditioned the density on a choice of sector, which would have left open the objection that the sparse phase simply lives somewhere else in the spectrum. It does not: the global Perron sector is the connected sector, and disconnected sectors are strictly energetically disfavoured.

3. Claim ceiling

Every result here is stated with an explicit ceiling — the point past which the evidence does not reach. For this theorem the ceiling is unusually important, because a density result about a relational model is easy to over-read as a result about spacetime. It is not one.

The theorem does not establish any of:

global product-state convergence     arbitrary propagation of chaos
the exact first 1/N coefficient      dynamics from disconnected initial data
quantum gravity                      novelty

In particular the global state is not proved to be a product state; only fixed marginals converge. And overall coupling rescaling cannot change the pure-kinetic eigenvectors, so it cannot cure the density problem either — the result is not an artefact of a badly chosen coupling constant.

4. Prior art and novelty

The substrate is Quantum Graphity — the all-pairs relation-qubit substrate with full S_N symmetry is Konopka–Markopoulou–Smolin (hep-th/0611197) and Konopka–Markopoulou–Severini (Phys. Rev. D 77, 104029, 2008). Graphity's high-energy phase is already known to be highly connected, and it reaches a low-dimensional lattice only by adding valence and loop potentials — exactly the target-fitting this programme forbids. So this theorem sharpens a known qualitative fact into an exact constant for one specific generator. It does not discover the phenomenon.

Nor is the mean-field behaviour surprising. A_N is a 3-local Hamiltonian on the complete 3-uniform hypergraph of triangles — maximally dense — and "product states are exact for dense k-local Hamiltonians" is a known theorem class (Brandão–Harrow, STOC 2013 / CMP 2016). The expected outcome for a generator of this shape is precisely that the ground energy density equals the separable optimum.

What is left over, at most, is the exact constant 25√5/36, the density 5/6, and the explicit rank-7 decomposable witness. These are recorded as NOVELTY_UNRESOLVED rather than as new: structurally the model is a kinetically-constrained (PXP-type) model with a wedge constraint on E(K_N), and that literature has not been searched exhaustively.

5. Why it matters

The value of the result is that it closes a door, and says which doors remain open. If you want a relational substrate to produce sparse geometry, a positive local kinetic rule of this shape will not do it for you. The honest escapes are:

• a physically justified conserved relational resource that restricts accessible link number;
• non-stoquastic or frustrated phase structure that defeats Perron-positive path multiplicity;
• independently derived diagonal physics, rather than target-valence fitting;
• matter-supported geometry, labelled honestly as such;
• a different microscopic factorisation or substrate.

Each of those is a real option. None of them is free, and the point of stating the constant exactly is that it makes the cost of the remaining options legible.

6. How it was reviewed

This result passed a fresh review by an agent that did not author it, working from the declared model rather than from the author's scripts. The reviewer independently reconstructed the unique triangle-product optimum p = 5/6, the exact constant 25√5/36, the PPT witness decomposition with identically zero residual, rank Q_A = 7 with unique product kernel, exact positive-semidefiniteness via characteristic-polynomial coefficients, and the operator-norm bound via leading principal minors.

At banking time, 9 of 10 committed verifiers exited zero, and 32 of 32 load-bearing files were byte-identical between the accepted head and the integrated tree. The single non-zero exit is a toolchain issue, not a scientific one: a convenience line at the end of one script asks a computer-algebra system to order four eigenvalues returned in radical form carrying imaginary dust of order 10⁻⁵⁸, and it refuses to order non-real expressions. Every scientific assertion in that script precedes that line and passes, and the same bound is proved exactly by a second script that exits zero.

7. Known imperfections

The reviewer recorded eight non-blocking observations. Repairing them would mutate the accepted commit and void the acceptance, so the disposition is no change. They are published rather than quietly carried:

NB5 is the one worth reading twice. A margin of 4×10⁻⁵ on a bound that is otherwise proved exactly by a second route is a reminder of how little slack the witness construction has.