[P2 theory] Ask whether square-site p_c can be embedded in a correlated-hyperedge self-dual critical manifold

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Motivation

A finite independent-bond gadget cannot exactly reproduce a Bernoulli site with four terminals because partial terminal-connectivity states occur with positive probability. That no-go does not exclude correlated bonds, hyperedges, or multispin Potts interactions.

A bold exact-structure route is therefore to enlarge the local state space until the square-site model appears as a slice/projection of a genuinely self-dual or Yang-Baxter-compatible hypergraph model.

Program

1. Formalize a four-terminal hyperedge whose only connectivity states are 1|2|3|4 and 1234, matching a Bernoulli site exactly.
2. Embed these hyperedges in a planar periodic hypergraph where duality acts locally on connectivity weights.
3. Write the self-dual/critical manifold for the hyperedge weights if known or derivable.
4. Ask whether the square-site point corresponds to a solvable point of that manifold, or whether extra connectivity states/couplings are unavoidable under composition.
5. Use small-cell symbolic elimination/resultants to search for a finite self-dual closure before any numerical fitting.

Relation to existing no-go

This route deliberately moves outside independent ordinary bond gadgets. It should cite the existing finite independent-bond no-go and treat it as motivation, not re-open that exhausted search space.

Falsification

A useful negative result would prove that any finite planar self-dual closure of the required four-terminal hyperedge necessarily generates additional partition states/couplings, so the one-parameter Bernoulli-site slice is not invariant under duality/renormalization.

Priority

P2/high-risk exact theory. No large computation until a finite symbolic closure or obstruction is identified.

[LightChainr]

This program has a much more concrete starting coordinate system: four-edge hypergraph boundary partitions

A literature hit is unusually close to the exact language proposed here.

Damavandi--Ziff, arXiv:1506.06125, study self-dual hypergraphs with four boundary vertices and explicitly organize a generator by six planar boundary-connectivity probabilities:

P1 = A|B|C|D
P2 = one nearest-neighbour pair connected
P3 = three of four connected
P4 = ABCD
P5 = one diagonal pair connected
P6 = two noncrossing connected pairs

(normalization has the corresponding orbit multiplicities).

For an isotropic self-dual four-edge, the local duality conditions are exactly

P4 = P1,
P2 = P3,
P5 = P6.

This is almost exactly the finite connectivity-state space this issue asks us to construct from scratch.

The all-or-none site is already one of their canonical generators

Their Section 7 takes an all-or-none four-terminal generator:

P1 = 1-t,
P4 = t,
P2=P3=P5=P6=0.

That is precisely a Bernoulli site viewed as a correlated four-terminal hyperedge. The self-dual point of the isolated all-or-none generator is t=1/2, as expected.

Even more relevant, their generator III replaces an internal square by a single all-or-none site and finds a nontrivial physical self-dual solution in a larger (p,r,t) family. So the basic conjecture of this issue — “embed an all-or-none Bernoulli site in a larger four-terminal self-dual manifold” — is not hypothetical. It is an established construction class.

Revised Q3 question

The square-site problem is therefore not primarily

can an all-or-none hyperedge exist?

It clearly can. The sharper question is

Can the square-site lattice be represented as, or embedded into, a finite-dimensional four-terminal connectivity manifold whose exact self-dual locus intersects the Bernoulli-site slice at the physical square-site p_c?

Or, more realistically:

Does composition/renormalization force the square-site all-or-none point to flow immediately into the P2/P3/P5/P6 directions, proving that the one-parameter Bernoulli slice is not a self-dual invariant manifold?

That is a much more precise symbolic problem.

Suggested exact coordinate program

Use (P1,...,P6) as the canonical local state vector instead of inventing ad hoc hyperedge couplings.

1. Encode one square-site cell / decorated cell as a four-terminal connectivity tensor.
2. Write its point on the six-probability simplex.
3. Implement composition of two/four cells exactly in this basis.
4. Linearize composition around the Bernoulli all-or-none slice and ask which of

P2-P3,
P5-P6,
P4-P1

are generated.
5. Compare the tangent space of the reachable square-site family with the self-dual tangent space

P4-P1=0,
P2-P3=0,
P5-P6=0.

6. Use symbolic elimination/resultants only after this coordinate reduction.

A particularly strong negative result would be:

composition of a Bernoulli-site tensor necessarily generates a duality-odd P2/P3 or P5/P6 component whose coefficient cannot be canceled by any finite local decoration preserving the square-site marginal.

That would be a structural reason why square-site p_c is not an exact local self-dual number.

Positive route

Damavandi--Ziff also construct inhomogeneous site systems that interpolate between union-jack site percolation and square-bond percolation while staying in a self-dual hypergraph framework. That gives a ready-made known critical manifold on which to test whether a square-site-like slice can be approached or whether a relevant extra connectivity coordinate is unavoidable.

This seems more informative than a blind search over microscopic gadgets, because the P_i coordinates already quotient away internal engineering details and retain exactly the information duality sees.

Literature anchor: O.K. Damavandi and R.M. Ziff, arXiv:1506.06125. Their four-edge conditions are P4=P1, P2=P3, P5=P6, and their all-or-none generator is explicitly interpreted as a site. Critical-polynomial/hypergraph work such as arXiv:2009.01913 can be used later for larger bases.

This maps directly to Q3 in the new navigation structure.

[LightChainr]

Exact two-block closure obstruction + positive correlated manifold

Pushed analysis/p123-correlated-hyperedge-no-go-20260829 at f2954a4 (no PR, no p_c fit).

I first checked the six-orbit convention directly against Damavandi--Ziff: multiplicities are (1,4,4,1,2,2), duality swaps (P1,P4), (P2,P3), (P5,P6), and the local selfdual equations are exactly the three equalities already quoted here.

The Bernoulli site embeds exactly as

A(t)=(1-t,0,0,t,0,0),

but intersects the local selfdual manifold only at t=1/2, transversely.

The first nontrivial composition already gives a finite-local obstruction. Take two all-or-none four-hyperedges sharing two internal vertices, then C4-average the AB|CD and BC|DA orientations. Exact enumeration yields

P1=(1-t)^2
P2=t(1-t)/2  (per nearest-pair orbit member)
P3=0
P4=t^2
P5=P6=0

and duality residuals

P4-P1 = 2t-1
P3-P2 = -t(1-t)/2.

They have no common root. Thus composition generates an unpaired partial-connectivity direction before any scaling or fitting.

This extends to the complete correlated two-block family. If, within either orientation,

Prob(00)=a, Prob(10)=Prob(01)=b, Prob(11)=c,
a+2b+c=1,

then selfduality forces b=0 and a=c=1/2: perfect 00/11 correlation collapses the composite back to one half-occupied all-or-none hyperedge.

There is also a genuine positive family. The full isotropic probability simplex fixed by local duality is

P=(w,u,u,w,v,v),
w=1/2-4u-2v,
u,v>=0, 4u+2v<=1/2.

This is a two-dimensional correlated selfdual manifold, but closure requires the partial P2/P3/P5/P6 couplings and enforces P4=w<=1/2.

Finally, arbitrary t can be embedded selfdually only in the inhomogeneous alternating construction A(t),A(1-t), where duality swaps cell colours. It becomes homogeneous only at t=1/2.

So local four-edge duality gives a sharp trichotomy: homogeneous Bernoulli is fixed at half; local composition creates partial states; arbitrary t requires an alternating dual partner. None supplies a new exact equation for homogeneous square-site p_c. A surviving route must use a larger/nonlocal projection in which P4 is no longer literally the site marginal, or additional integrable structure beyond local duality.

Artifacts: scripts/p123_correlated_hyperedge_closure.py, results/exact-correlated-hyperedge-closure/latest.json, notes/p123-correlated-hyperedge-closure.md. Exact and registry tests pass.

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Repository cleanup (CLOSE-NOISE): this exploratory entry is not planned because it has no independent validation block or decision-changing discriminator. History is retained; reopen only with a pre-result stop rule.