Sharp axial geometry for the full birth-time law: necessity beyond the union bound

notes/axial-quantile-geometry-criterion-20260913.md

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+# A sharp geometry criterion for the full axial birth-time law
+
+2026-09-13. New proof continuation of #613/#716; separate from #735's
+arbitrary-period balance-root theorem. No Monte Carlo, fitted exponent, new
+numerical critical point, or claim of priority in the literature.
+
+## Result
+
+Let T_n be the NN square-site torus with axial periods (w_n,0),(0,m_n),
+2 <= w_n <= m_n and w_n m_n -> infinity. At parameter p let r_n be the
+ambient rational first-homology rank and put
+
+    F_n(p) = E_p[r_n]/2,       Q_n = F_n^{-1}.
+
+Under the standard inputs listed below, the following are equivalent:
+
+1. Q_n(u) -> p_c for every fixed u in (0,1).
+2. The convergence is uniform on each compact subinterval of (0,1).
+3. F_n(p) -> 0 for every p < p_c, and -> 1 for every p > p_c.
+4. log(m_n)/w_n -> 0 (equivalently log(w_n m_n)/w_n -> 0).
+
+The new step is necessity, including logarithmic-width subsequences. Merely
+observing that an upper union bound stops vanishing would NOT prove necessity.
+We instead construct winding events with arbitrarily small exponential cost
+at a fixed p below p_c, and repeat them in independent transverse bands.
+
+This is an axial theorem. The corresponding necessity statement for arbitrary
+integer-period lattices is NOT proved here. #735's arbitrary-period result
+concerns the balance root, not the whole F_n law.
+
+## Imported inputs, not results of this note
+
+A. Critical box crossing for NN square-site percolation: for each fixed aspect
+ratio the occupied crossing probability is bounded away from zero at p_c,
+uniformly in scale. Zeng, arXiv:1309.2273, Theorem 1.1, states precisely this
+site result. The primary HTML theorem and gluing discussion were read on
+2026-09-13. It is a preprint; no journal publication is asserted here.
+
+B. Site subcritical exponential one-arm decay on NN and on NN+NNN, and the
+matching critical-point relation p_c(NN)+p_c(NN+NNN)=1. These are the existing
+#613 inputs. Duminil-Copin--Tassion, arXiv:1502.03050, Theorem 1.1(3) is
+printed for bonds; section 1.2 explicitly discusses the site adaptation.
+Grimmett--Li's matching relation is proved in the companion Hyperbolic site
+percolation, arXiv:2203.00981, combined here with amenable p_u=p_c. The
+companion's theorem body was not independently re-audited in this delivery.
+
+C. On honest tori, the repository's deterministic digital-Alexander identity
+r_NN(omega)+r_matching(omega^c)=2. This is an input, not inferred from a small
+census. For these axial tori w,m >= 2 are in the stated honest scope.
+
+Harris positive association and finite-product continuity are also used. No
+conformal invariance, value of a critical exponent, correlation-length power,
+or asymptotic surface-tension formula is assumed.
+
+## 1. A deterministic ring of crossings, including the seam
+
+Fix an integer s >= 1 and w >= 4s. Write k=floor(w/s) and divide the horizontal
+circle into k consecutive integer cells with boundaries
+
+    0=x_0 < x_1 < ... < x_k=w,
+    s <= x_{i+1}-x_i <= 2s.
+
+For example, divide w by k and distribute the remainder one unit per cell.
+Work in the cylinder C_w x {0,...,s}; there is NO vertical periodic edge in this
+band. Extend boundaries by x_{i+k}=x_i+w. In the planar lift require:
+
+- V_i: an occupied vertical crossing of [x_i,x_{i+1}] x [0,s];
+- H_i: an occupied horizontal crossing of [x_i,x_{i+2}] x [0,s].
+
+There are 2k events. Every rectangle injects into the cylinder, including the
+one straddling the seam. A horizontal crossing H_i meets a vertical crossing
+V_i and V_{i+1}: restrict the horizontal path to an appropriate left-to-right
+subpath in each cell, then use planar NN crossing intersection.
+
+Choose these paths once on the cylinder and lift them periodically. Their
+union connects V_0 successively to V_1,...,V_k=V_0+(w,0). By travelling within
+the first and last vertical crossings, some vertex z is connected to z+(w,0)
+in the lifted occupied graph. Projection gives a closed walk of nonzero
+horizontal homology. A closed walk decomposes into cycles, so at least one
+cycle has nonzero ambient image.
+
+Thus
+
+    intersection_i (V_i intersection H_i)
+        subset {there is a horizontal essential occupied cycle in the band}. (1)
+
+This is stronger than a path merely joining the two cut sides: the last
+vertical connector explicitly closes the periodic seam. It is a sufficient
+event, not an equality or a claim that every winding cluster looks this way.
+
+## 2. Arbitrarily small exponential cost below criticality
+
+Critical RSW supplies a constant c in (0,1), independent of s, bounding below
+all the above crossing probabilities at p_c. To see uniformity without a
+variable-aspect theorem, a 4s-by-s horizontal crossing implies each shorter
+horizontal crossing, and an s-by-s vertical crossing fits inside each cell.
+Use the minimum of these two fixed-aspect RSW constants.
+
+Set a=c/2. For each FIXED s, only finitely many integer rectangle sizes occur.
+Their probabilities are finite polynomials in p. Continuity therefore gives
+one p_s in (0,p_c) at which every such crossing probability is at least a.
+This p_s is independent of w and m. We do not claim a quantitative lower bound
+on p_c-p_s, or a uniform-in-s neighborhood of criticality.
+
+Harris association, not independence, applies to the overlapping rectangles.
+The ring event G_{w,s} in (1) satisfies
+
+    P_{p_s}(G_{w,s}) >= a^(2 floor(w/s))
+                      >= exp[-2 log(1/a) w/s].                  (2)
+
+Consequently for every eta > 0 there are a FIXED s and a FIXED p_eta < p_c
+such that, for every w >= 4s,
+
+    P_{p_eta}(G_{w,s}) >= exp(-eta w).                           (3)
+
+The quantifier order is essential: choose eta, then s, then p_eta, and only
+then let the torus sizes diverge. This is not a critical RSW bound silently
+applied at a sequence-dependent subcritical parameter.
+
+## 3. Independent repetition across the long direction
+
+In the w-by-m torus take bands with vertex rows
+
+    j(s+1), ..., j(s+1)+s,      0 <= j < b=floor(m/(s+1)).
+
+Their vertex sets are disjoint. There can be physical edges between bands,
+but each G_{w,s} uses only sites and paths inside its own band. Its probability
+and independence are not affected by unused outside edges. Thus
+
+    P_0^{w,m}(p_eta)
+       <= (1-exp(-eta w))^floor(m/(s+1))
+       <= exp[-floor(m/(s+1)) exp(-eta w)].                     (4)
+
+This is an upper bound for P_0: just one successful band already forces r>0.
+It does not say that a single band forces r=2.
+
+## 4. Necessity of subexponential aspect growth
+
+Suppose log(m_n)/w_n does not tend to zero. Extract a subsequence with
+log(m_n) >= d w_n for some d>0.
+
+If w_n -> infinity along a further subsequence, choose eta=d/2 in (3).
+Then floor(m_n/(s+1)) exp(-eta w_n) -> infinity, so (4) gives P_0(p_eta)->0.
+Since
+
+    F_n(p) = (P_1+2P_2)/2 >= (1-P_0)/2,
+
+we have liminf F_n(p_eta) >= 1/2 at a FIXED p_eta<p_c. For every fixed
+u<1/2, eventually Q_n(u) <= p_eta < p_c. In particular Q_n(1/4) cannot tend
+to p_c.
+
+If instead w_n stays bounded along a further subsequence, m_n -> infinity.
+At any fixed p in (0,p_c), fully occupied rows give independent winding events
+with probability p^{w_n} bounded below. Hence P_0 <= (1-p^{w_n})^{m_n}->0,
+and the same lower-quantile obstruction applies.
+
+Every subsequence witnessing failure has one of these two further subsequences.
+This proves 1 => 4. It does NOT prove that the median fails: Q_n(1/2) is the
+balance root and has its own, stronger consistency theorem.
+
+## 5. Sufficiency and the quantile equivalences
+
+If log(m_n)/w_n->0 then w_n->infinity and log(w_n m_n)/w_n->0. At fixed p<p_c,
+a nonzero winding cycle must escape an embedded ball of radius w_n/4-O(1).
+Site exponential decay and a union bound over w_n m_n sites give
+
+    P_p(r_n>0) <= C(p) w_n m_n exp[-c(p) w_n] -> 0.
+
+Above p_c apply the same argument to the matching complement and use C.
+This is the already established #613 sufficiency, not new work here. It yields
+3. Monotonicity traps every Q_n(u), uniformly for u in [delta,1-delta], between
+p_c-epsilon and p_c+epsilon for sufficiently large n; hence 3 => 2 => 1.
+Together with section 4 this proves the four-way equivalence.
+
+## 6. What the new result changes
+
+For axial sequences with w->infinity there are now two different geometric
+requirements:
+
+    balance-root consistency:    w -> infinity            (#718/#735);
+    entire birth-law convergence: log(m)/w -> 0            (this note + #613).
+
+For m=ceil(exp(d*w)), d>0, the root remains consistent while at least every
+fixed lower quantile u<1/2 is bounded away from p_c along the sequence by the
+proof above. This addresses the logarithmic-width boundary, not only #716's
+w=o(log m) regime. No value of the separated limiting quantiles is computed.
+For m polynomial in w, all fixed quantiles converge. For m=ceil(exp(sqrt(w))),
+all fixed quantiles still converge; bounded aspect ratio was never necessary.
+
+Interpretation: local convergence of the underlying graphs to Z^2 does not
+by itself make a global two-birth statistic concentrate. The number of chances
+to create a short winding cycle must be compared with its probability cost.
+This does not identify an irrelevant field or provide an L^-4 root-shift law.
+
+## 7. Finite checks actually executed
+
+`scripts/axial_ring_gluing.py` independently implements rectangle crossing BFS
+and a graph-potential winding detector. Exhaustive cylinder strips at
+(w,s)=(4,1),(5,1),(6,1),(7,1) cover 21,760 configurations, with zero instances
+of the ring event without nonzero horizontal winding. It also checks the
+product-of-marginals <= ring-event <= winding-event inequalities with Fraction
+arithmetic at p=1/3,1/2,2/3. These are not binomially weighted twice.
+
+At w=4,s=1 there are 17 ring-event configurations but 35 winding configurations,
+so at p=1/2 the two probabilities are 17/256 and 35/256. This guards against
+promoting the sufficient construction into an event dictionary identity.
+
+Another 2,365 deterministic masks at (9,2),(11,2),(13,3) check uneven cells
+and seam closure; 2,345 satisfy the ring event, with no implication failures.
+These masks are controls, NOT samples estimating any physical probability.
+Three mathematical unit tests and Python compilation passed locally.
+Full repository CI was NOT run. Finite checks support the implementation;
+sections 1--5 and the imported inputs, not enumeration, support the theorem.
+
+## 8. Prior art and the next genuinely new question
+
+Primary theorem text read this delivery:
+- Zeng, https://arxiv.org/html/1309.2273, Theorem 1.1 and crossing/gluing setup.
+- Kohler-Schindler--Tassion, https://arxiv.org/html/2011.04618, Theorem 1 and
+  Comment 1 (site extension). Its general RSW statement is background, not
+  automatically an identification of the critical point of this site model.
+- Duminil-Copin--Tassion, https://arxiv.org/html/1502.03050, Theorem 1.1(3)
+  together with the site-adaptation paragraph in section 1.2.
+
+Context checked, not used as a proof of necessity:
+- Easo, Sharpness and Locality for Percolation on Finite Transitive Graphs,
+  https://doi.org/10.1007/s00039-025-00726-w (2025), publisher introduction and
+  theorem discussion: giant bond clusters, not this rank-birth distribution.
+- Duncan--Kahle--Schweinhart, https://arxiv.org/abs/2011.11903: abstract and
+  metadata this delivery; the existing #613 source review handles its scope.
+- Grimmett--Li, https://arxiv.org/abs/2203.00981: abstract/metadata this delivery;
+  see B above for the reused theorem provenance.
+
+A bounded search did not locate this exact axial quantile iff formulation.
+That is NOT a novelty certificate: strip-percolation/finite-size-criterion and
+homological-percolation citation chains still require systematic comparison.
+
+Next target: for an arbitrary integer-period lattice with shortest period ell,
+construct a bounded-thickness occupied ring around that actual period with
+probability at least exp(-eta ell) for some fixed p_eta<p_c, uniformly in its
+orientation and ambient primitivity, and pack disjoint such corridors. The
+axial proof does not rotate the physical NN lattice to claim this for free.
+Do not replace this missing geometric lemma with another size census.

results/research-control-20260913/axial-ring-controls.json

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