Arbitrary-period balance-root consistency and Jordan visibility (do not merge)

notes/arbitrary-period-balance-consistency-20260912.md

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+# Matching balance on arbitrary integer-period tori
+
+2026-09-12. Completed proof continuation of #718/#732 and #276/#613.
+The result concerns **balance roots and conditional rank odds**, not the full
+birth-time mixture. It does not assert a new value, a near-critical rate, an
+all-width transfer representation, or publication novelty.
+
+## 1. Statement and external inputs
+
+Let Lambda be any rank-two sublattice of Z^2. Put
+
+    N = [Z^2:Lambda],     ell = min{|u|_2: 0 != u in Lambda}.
+
+On the occupied square-site NN quotient, let r be the rational rank of its
+ambient H1 image, P_j=Pr_p(r=j), and M=P_2-P_0. All sufficiently large ell
+are honest square-cell tori. The matching graph has the eight steps
+(+-1,0),(0,+-1),(+-1,+-1); parallel lifted edges are not collapsed.
+
+**Theorem A.** If p_Lambda is the unique zero of M, then
+
+    lim_(L->infinity) sup_{Lambda: ell(Lambda)>=L}
+              |p_Lambda - p_c^site(Z^2)| = 0.                     (1)
+
+There is no restriction on area N relative to ell, aspect, shear, orientation,
+Smith class, or primitivity of a Gaussian representative. The local site
+product law and actual period lattice are essential. No arbitrary graph family
+or dependent random-cluster law is included.
+
+More quantitatively, for each fixed p<p_c(NN), there are L(p), kappa(p)>0,
+independent of Lambda, such that
+
+    P_2^Lambda(p)/P_0^Lambda(p) <= exp[-kappa(p) N/ell], ell>=L(p). (2)
+
+Above p_c the inverse ratio obeys the analogous estimate, with matching-side
+constants. The conditional CDF H=P_2/(P_0+P_2) therefore has all its fixed
+interior quantiles tending to p_c uniformly in Lambda. H is NOT F=(1+M)/2.
+
+The imported probability inputs are the same as in #718:
+
+* Subcritical **site** one-arm exponential decay on both infinite finite-range
+  transitive graphs. Duminil-Copin--Tassion, arXiv:1502.03050v3, Thm 1.1(3)
+  is printed for bonds; the explicit site-adaptation discussion in section 1.2
+  is part of the provenance. We do not relabel the printed bond theorem as a
+  printed site theorem, or substitute square-bond p_c=1/2.
+* p_c^site(NN)+p_c^site(NN+NNN)=1. Grimmett--Li,
+  arXiv:2205.02734v3, introduction Eq (1.3), together with amenability
+  p_u=p_c, states the relation and identifies the companion proof.
+* The repository's configurationwise honest-torus digital-Alexander identity
+  r_NN(omega)+r_matching(omega^c)=2. It is an input, not inferred from this
+  delivery's finite checks.
+
+The new step beyond #718 is the oblique, integer-period slab construction.
+
+## 2. Reduce the period lattice, not the physical interaction
+
+Choose a shortest nonzero u in Lambda. It is primitive **in Lambda**: a proper
+integer multiple would contradict shortness. Complete it to a basis (u,v),
+orient det(u,v)=N>0, and replace v by v-ku so that
+
+    |u dot v| <= ell^2/2,    |v|>=ell.
+
+Let n=(-u_y,u_x)/ell and h=n dot v=N/ell. Pythagoras gives
+
+    h >= sqrt(3) ell/2.                                         (3)
+
+This is a change of period basis only. Physical edges remain the original NN
+or NN+NNN steps, not rotated nearest-neighbour edges on a new grid.
+Neither u nor v is required to be primitive in ambient Z^2. In particular an
+axis period u=(w,0) is completely legitimate.
+
+The map theta(x)=n dot x modulo h is well-defined on the continuous torus.
+Its fibres have length ell. On lattice vertices it can be implemented exactly
+as q(x)=det(u,x) modulo N; each local edge e changes its lifted q by det(u,e).
+Every edge has physical length and projected displacement at most sqrt(2).
+An ambient rank-two cycle space necessarily contains a cycle whose theta
+winding is nonzero. Repeating a lift traverses every transverse band.
+
+## 3. A finite-support local arm gives a lower bound for P_0
+
+Set r=ell/64. On the infinite graph let a_r(p) be the probability that the
+occupied origin has an occupied path reaching Euclidean distance at least r.
+Stop the path on first exit. The event is measurable in radius r+sqrt(2).
+For ell>=64, twice that support radius is smaller than ell, so it injects
+into every quotient under consideration. Define the translated local event A_x
+at each of the N torus vertices. Its probability is exactly a_r(p).
+
+A nonzero ambient cycle has a lift escaping such a ball. Thus
+
+    intersection_x A_x^c subset {r=0}.
+
+These are overlapping, decreasing events. Harris association for independent
+sites, iterated over the N indicators, gives
+
+    P_0 >= (1-a_r)^N.                                           (4)
+
+No independence of overlapping local balls is assumed. Association itself
+follows by induction over Bernoulli coordinates: the conditional covariance
+is nonnegative, and so is the covariance of the two monotone conditional means.
+
+## 4. Vertex-disjoint oblique slabs give an upper bound for P_2
+
+Choose physical slab width b=ell/8 and
+
+    k=floor(h/b)=floor(8N/ell^2).
+
+Translate their boundaries by a generic common offset to avoid vertices.
+For an exact implementation, raw q-width is ell^2/8 and raw offset 1/17;
+q-values are integers, so no boundary passes through a vertex. The k half-open
+bands have disjoint vertex sets. Leave any residual strip unused.
+
+In each band let B_j be existence of an occupied path contained in that band,
+from its lower sqrt(2)-layer to its upper sqrt(2)-layer. Edges crossing a band
+boundary are not part of that band's event. The last-entry/first-exit portion
+of any nonzero theta-winding cycle provides a B_j crossing in every band.
+This remains true for a cycle that backtracks. Therefore
+
+    {r=2} subset intersection_j B_j.                            (5)
+
+Each B_j is measurable using just that slab's site variables. These events ARE
+independent. Diagonal edges or horizontal periodic winding do not create shared
+site variables between bands.
+
+**Uniform bound on possible entry vertices.** Centre one unit square at each
+lattice vertex; these squares tile the torus and have total area N. Squares
+centred in a band of physical width sqrt(2) are contained in its enlargement
+of width 2sqrt(2). Since fibres have length ell, the number of entry vertices
+is at most 2sqrt(2) ell, hence at most M=4 ceil(ell). This area argument handles
+arbitrary orientation, nonprimitive u in Z^2, and twisted longitudinal seam.
+
+A crossing's endpoint separation in the theta direction is at least
+ell/8-2sqrt(2)>ell/64=r, for ell>=64. Its initial lift therefore witnesses A_x
+at one of the entry vertices (possibly exiting the local ball sideways first).
+By a union bound, Pr(B_j)<=M a_r. Consequently, when M a_r<1,
+
+    P_2 <= (M a_r)^k.                                           (6)
+
+Equation (3) implies 8N/ell^2>=4sqrt(3)>2, so
+
+    k >= 4N/ell^2.                                             (7)
+
+The new proof depends on the shortest period and transverse area, not on an
+axis-aligned slicing of a chosen HNF display. A long HNF basis vector is not
+itself evidence of a large systole.
+
+## 5. Uniform comparison of rates
+
+For fixed subcritical p, write a_r<=C exp(-c r)=C exp(-c ell/64).
+For all sufficiently large ell, uniformly over orientation and N,
+
+    log(M a_r)<=-c ell/128,   a_r<=1/2,
+    2a_r<=c/(64ell).
+
+Combining (4),(6),(7), using -log(1-a)<=2a,
+
+    log(P_2/P_0)
+       <= k log(M a_r) - N log(1-a_r)
+       <= -c N/(32ell) + 2N a_r
+       <= -c N/(64ell).
+
+This proves (2). The negative logarithmic contribution in (4) is retained;
+no exponentially small probability has been approximated by zero.
+
+For p>p_c(NN), set p*=1-p<p_c(matching). The same geometry and argument apply
+to the eight-step graph. Configurationwise duality exchanges rank 0 and rank 2
+and yields the opposite odds estimate.
+
+Finite r is monotone under adding occupied sites. Both increasing events
+{r>0} and {r=2} are nonconstant. A positive pivotal configuration along an
+empty-to-full chain has positive product probability at every interior p, so
+M'=dPr(r>0)/dp+dPr(r=2)/dp>0. With endpoint values -1 and +1, the root is unique.
+The two fixed p values p_c+-epsilon trap it for all lattices with sufficiently
+large ell. This proves (1), without choosing a numerical p_c.
+
+For H, H<=P_2/P_0 below p_c and 1-H<=P_0/P_2 above p_c. This traps its compact
+interior quantiles as well. No claim about the behaviour exactly at p_c is used.
+
+## 6. What does not follow
+
+The theorem does NOT imply concentration of the birth-time mixture F. The
+axis sequence w=j, m=ceil(exp(j^2)) is a special case of (1), but #716's
+full/empty-row argument still gives L(T)=>0.5 delta_0+0.5 delta_1. Root consistency
+and full-law concentration remain different statements in the SAME site model.
+
+No L^-4 rate, critical exponent, sharpness constant near p_c, numerical interval,
+or proof that a particular all-width pTL eigenvalue crossing equals a finite
+balance root is supplied. An inner cylinder limit, if separately known to exist,
+inherits consistency by this uniform theorem; existence of that limit is not
+silently assumed for arbitrary widths.
+
+## 7. Exact finite controls
+
+The script uses integer Lagrange reduction, verifies shortness against ambient
+lattice membership on every HNF with determinant <=50, and verifies height and
+period identities. That is 2,080 period bases, not 2,080 percolation productions.
+
+On HNFs (10,3,1), (13,5,1), (5,2,3), (4,1,4), every configuration is traversed
+as a physical lifted graph for NN and matching. That gives 215,040 graph/config
+checks and 107,520 complementary pair checks. Slab crossings are checked by a
+separate boundary-restricted reachability search in the determinant coordinate.
+Tiny controls use wider raw slabs than the asymptotic choice, retaining the
+explicit separation and injectivity inequalities; they are not claimed to have
+ell>=64. With r<1 their arm probability is exactly p[1-(1-p)^degree]. Rational
+P_0 lower, P_2 upper and slab union bounds are checked at three p values.
+
+Larger fixed masks include true oblique Gaussian ideals (63+16i) and (32+57i),
+plus a twisted rectangle. Every chosen coarse slab is explicitly vertex-disjoint.
+Two distant-subcritical sign examples use a certified simple-path union bound;
+a Bernoulli-inequality check avoids exponentially large integer powers. They are
+not critical-point enclosures. The proof above, not an extrapolation of controls,
+is the reason the conclusion covers all integer period lattices.