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The digital-Alexander theorem gives the finite matching function a new exact meaning:
M_N(p)=E_p[q]
=E_p[r]-1
=P_2(p)-P_0(p),
where
r = rank im[H1(occupied black complex) -> H1(T^2)],
P_j = Prob(r=j).
Therefore every finite matching-polynomial/root estimate is solving
boxed:
P_2(p_N)=P_0(p_N)
or equivalently
E_{p_N}[r]=1.
This is a much stronger interpretation than “the finite matching polynomial happens to have a root near p_c”. It is the point at which trivial ambient topology and full ambient topology have equal probability.
Call p_N the homological balance point.
Main theorem target
Prove, for the declared square-site torus sequence,
p_N -> p_c(square-site)
without using CFT, a guessed correction exponent, or numerical knowledge of p_c.
The proof should use only monotonicity / sharpness / torus crossing-homology arguments plus the exact 4/8 matching duality.
This would give the finite matching root a rigorous probability-theory role independently of the operator-identification program.
Phase A — finite-volume monotonicity and uniqueness
The ambient rank is monotone under adding occupied sites:
omega subset omega' => r(omega) <= r(omega').
Hence
M_N(p)=E_p[r]-1
is monotone increasing in p.
Strengthen this to strict monotonicity on (0,1) for every nondegenerate torus by either:
an explicit rank-pivotal configuration at every site class / a Russo formula; or
the already-established matching Russo identity.
With
M_N(0)=-1,
M_N(1)=+1,
this gives a unique balance root p_N.
Record the exact relation
M_N'(p)=d/dp E_p[r]
as total expected rank-birth influence. This is the natural topological version of the pivotal mass.
Phase B — prove localization around p_c
For every fixed epsilon>0, target
P_0(p_c-epsilon) -> 1,
P_2(p_c+epsilon) -> 1
on growing square tori.
Then
M_N(p_c-epsilon) -> -1,
M_N(p_c+epsilon) -> +1,
and monotonicity immediately implies
p_c-epsilon < p_N < p_c+epsilon
for sufficiently large N.
This is the entire convergence theorem.
The probabilistic input should be stated cleanly in terms of standard subcritical exponential decay and supercritical torus wrapping / uniqueness, or an RSW + sharpness formulation. Do not invoke a finite-size scaling ansatz to prove convergence.
Relation to homological-percolation literature
Duncan--Kahle--Schweinhart, arXiv:2011.11903, prove sharp transitions and threshold convergence for ambient-homology percolation on growing tori, including self-dual middle-dimensional cases.
Our square-site problem is not literally their plaquette model, but after PR #271 the observable is exactly of the same ambient-image type:
rank im[H1(P)->H1(T^2)].
Use their proof architecture as a template and identify precisely which steps transfer to site percolation with the 4/8 digital complex.
The point is not to reprove the full percolation phase transition. It is to deduce convergence of this specific median-like homology balance statistic from known sharpness facts.
Phase C — define rank-pivotal events geometrically
The derivative
d/dp E[r]
has a more canonical local meaning than a generic wrapping pivotal.
For a site v, define
Delta_v r = r(omega with v=1)-r(omega with v=0).
Classify the possible rank births:
0 -> 1,
1 -> 2,
possibly 0 -> 2 on finite degenerate geometries if allowed.
and retain the new ambient homology line/subspace created at the insertion.
This gives a homology-marked pivotal measure whose unmarked total is exactly M_N'(p).
It should become the preferred local bridge to the global matching observable, because it differentiates the exact quantity r itself rather than a proxy wrapping event.
A later spin-4 projection can ask which continuum sector carries rank birth.
Phase D — only after convergence, study the shift
Once p_N -> p_c is established, write
p_N-p_c = - M_N(p_c)/M_N'(xi_N)
for some intermediate xi_N, or use the local implicit-function expansion when justified.
This factorization is conceptually useful:
numerator = finite-size violation of the critical homology balance identity,
denominator = rank-birth pivotal susceptibility.
It separates two pieces that the current root fits mix together.
Then compare fixed mechanisms for the numerator:
singlet spin-4 lattice defect / Q4-Jordan,
ordinary analytic correction,
other topology-sector correction,
while the denominator is controlled by the thermal/pivotal exponent.
Do not fit a free root-shift exponent before deriving these two ingredients.
Phase E — continuum balance at criticality
Arguin, arXiv:hep-th/0111193, gives the critical FK torus relation
Z_Q(cross)=Q Z_Q(trivial).
At Q=1 this becomes
pi_1(rank2)=pi_1(rank0)
for the continuum critical theory.
Thus the finite balance root has a natural continuum target: it is the value of p where the lattice rank contrast most directly enforces the same topological equality that holds exactly in the Q=1 continuum limit.
Issue #275 asks which irrelevant singlet field controls the residual at fixed critical p. This issue asks the complementary probability-theory question: where must the zero of that residual converge?
High-risk stronger conjecture
The matching-polynomial root may be best regarded as a topological phenomenological renormalization condition:
E[r]=1.
It is analogous to fixing a dimensionless crossing probability at a universal value, except the target value follows from Alexander duality and the exact torus homology relation rather than being chosen empirically.
If so, root sequences on different torus moduli should share the same thermal scaling field but have shape-dependent irrelevant shifts. This could make the matching root a particularly clean coordinate for finite-size RG.
Trigger
The digital-Alexander theorem gives the finite matching function a new exact meaning:
where
Therefore every finite matching-polynomial/root estimate is solving
or equivalently
This is a much stronger interpretation than “the finite matching polynomial happens to have a root near p_c”. It is the point at which trivial ambient topology and full ambient topology have equal probability.
Call
p_Nthe homological balance point.Main theorem target
Prove, for the declared square-site torus sequence,
without using CFT, a guessed correction exponent, or numerical knowledge of
p_c.The proof should use only monotonicity / sharpness / torus crossing-homology arguments plus the exact 4/8 matching duality.
This would give the finite matching root a rigorous probability-theory role independently of the operator-identification program.
Phase A — finite-volume monotonicity and uniqueness
The ambient rank is monotone under adding occupied sites:
Hence
is monotone increasing in
p.Strengthen this to strict monotonicity on
(0,1)for every nondegenerate torus by either:With
this gives a unique balance root
p_N.Record the exact relation
as total expected rank-birth influence. This is the natural topological version of the pivotal mass.
Phase B — prove localization around p_c
For every fixed epsilon>0, target
on growing square tori.
Then
and monotonicity immediately implies
for sufficiently large N.
This is the entire convergence theorem.
The probabilistic input should be stated cleanly in terms of standard subcritical exponential decay and supercritical torus wrapping / uniqueness, or an RSW + sharpness formulation. Do not invoke a finite-size scaling ansatz to prove convergence.
Relation to homological-percolation literature
Duncan--Kahle--Schweinhart, arXiv:2011.11903, prove sharp transitions and threshold convergence for ambient-homology percolation on growing tori, including self-dual middle-dimensional cases.
Our square-site problem is not literally their plaquette model, but after PR #271 the observable is exactly of the same ambient-image type:
Use their proof architecture as a template and identify precisely which steps transfer to site percolation with the 4/8 digital complex.
The point is not to reprove the full percolation phase transition. It is to deduce convergence of this specific median-like homology balance statistic from known sharpness facts.
Phase C — define rank-pivotal events geometrically
The derivative
has a more canonical local meaning than a generic wrapping pivotal.
For a site
v, defineClassify the possible rank births:
and retain the new ambient homology line/subspace created at the insertion.
This gives a homology-marked pivotal measure whose unmarked total is exactly
M_N'(p).It should become the preferred local bridge to the global matching observable, because it differentiates the exact quantity
ritself rather than a proxy wrapping event.A later spin-4 projection can ask which continuum sector carries rank birth.
Phase D — only after convergence, study the shift
Once
p_N -> p_cis established, writefor some intermediate
xi_N, or use the local implicit-function expansion when justified.This factorization is conceptually useful:
It separates two pieces that the current root fits mix together.
Then compare fixed mechanisms for the numerator:
while the denominator is controlled by the thermal/pivotal exponent.
Do not fit a free root-shift exponent before deriving these two ingredients.
Phase E — continuum balance at criticality
Arguin, arXiv:hep-th/0111193, gives the critical FK torus relation
At Q=1 this becomes
for the continuum critical theory.
Thus the finite balance root has a natural continuum target: it is the value of
pwhere the lattice rank contrast most directly enforces the same topological equality that holds exactly in the Q=1 continuum limit.Issue #275 asks which irrelevant singlet field controls the residual at fixed critical p. This issue asks the complementary probability-theory question: where must the zero of that residual converge?
High-risk stronger conjecture
The matching-polynomial root may be best regarded as a topological phenomenological renormalization condition:
It is analogous to fixing a dimensionless crossing probability at a universal value, except the target value follows from Alexander duality and the exact torus homology relation rather than being chosen empirically.
If so, root sequences on different torus moduli should share the same thermal scaling field but have shape-dependent irrelevant shifts. This could make the matching root a particularly clean coordinate for finite-size RG.
Falsification / boundary
p_c.p_c; this is a structural statement first.Literature anchors
Related: #3, #47, #54, #114, #118, #123, #269, #275, PR #270, PR #271.