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+SIGNED CYLINDER PROPAGATION AND MARKED POLYGONS
+ON THE HONEYCOMB LATTICE
+
+A plain-language edition, written to be read front to back.
+
+================================================================
+READ ME FIRST
+================================================================
+
+This paper is about self-avoiding walks and polygons on one
+specific graph, the honeycomb lattice, at the exact density
+where they sit on a knife edge -- neither dying out nor
+blowing up. It studies three observables: a cylinder
+partition with two marked points, a finite chord law inside
+a regular hexagon, and a squared-length sum for planar
+polygons. Each has a clean power-law growth, proved here.
+
+It is written as one continuous story. Every technical word
+is defined on first use, every theorem is stated with its
+exact exponents, and every proof step is explained in words
+before its algebra. If you meet a symbol you do not know,
+jump to the GLOSSARY at the end.
+
+The whole paper in one paragraph:
+
+ Draw self-avoiding walks and polygons on the honeycomb
+ lattice, weighting a length-N walk or polygon by x^N, where
+ x = 1/sqrt(2+sqrt(2)) is the honeycomb critical weight.
+ Consider families of disjoint polygons: sometimes wrapped
+ around a cylinder and required to separate two marked
+ points; sometimes constrained to lie inside a big hexagon
+ and to surround its central face; sometimes pinned to
+ pass through two specified bonds on a tilted cylinder.
+ Each of these sums grows like a clean power of the size
+ parameter as it goes to infinity. On the balanced cylinder
+ at fugacity g = 2 the partition grows like m^{1/6}. Its
+ curvature at zero fugacity grows only logarithmically.
+ The planar hexagon variant grows like R^{1/12}. The chord
+ law inside the hexagon has mean length R^{4/3} and uniform
+ bulk vertex visit probability R^{-2/3}. The length-squared
+ mass of planar polygons of diameter at most H grows like
+ H^{2/3}. The engine is a signed transfer matrix
+ representation whose cancellations are controlled by
+ absolute estimates strong enough to still yield provable
+ power laws even though the individual terms are signed
+ or complex.
+
+A note on signs. Every physical weight here is positive.
+But the standard exact transfer representation of these
+partitions uses signed or complex "auxiliary" weights, in the
+spirit of the Duminil-Copin-Smirnov local cancellation
+theorem. To make the signed representation rigorous, one
+must show that the absolute values of the signed weights are
+themselves summable, uniformly in size. That absolute
+summability step is the technical heart of this paper; only
+after it does the exponent extraction become an honest
+inequality on the physical partition rather than a formal
+manipulation.
+
+How to read this edition:
+
+ PART 1 fixes the graph, walks, polygons, and observables.
+
+ PART 2 states all theorems with exact exponents.
+
+ PART 3 is the one-page map of how the proofs go.
+
+ PART 4 proves the cylinder theorem (Theorem 1).
+
+ PART 5 proves the chord and visit corollaries and the
+ length-squared estimate.
+
+ PART 6 proves the two-bond marked interval estimate that
+ powers PART 5.
+
+ PART 7 says what is new relative to earlier work.
+
+ PART 8 is honest about what is not claimed.
+
+ PART 9 is the glossary.
+
+================================================================
+PART 1. THE STAGE
+================================================================
+
+1.1 The honeycomb lattice.
+
+Start with the regular infinite triangular tessellation of
+the plane, of side length one. Every triangle -- pointing up
+or down -- has a center. Join the centers of any two
+triangles that share a side. The result is the honeycomb
+graph: three edges meet at every vertex, and the graph looks
+like an infinite sheet of chicken wire.
+
+Two length measurements are used below. Euclidean distance
+is the usual distance in the plane. The diameter of a
+bounded set is the largest Euclidean distance between two
+points of the set.
+
+1.2 Self-avoiding walks and polygons.
+
+A walk in a graph is a sequence of vertices where successive
+vertices are adjacent. It is self-avoiding if no vertex is
+visited twice. Its length is its number of vertices. A
+polygon is a closed self-avoiding walk; its length equals
+its number of edges.
+
+Let x be the critical weight
+
+ x = 1 / sqrt(2 + sqrt(2)) = 0.5411961... .
+
+This is the reciprocal of the honeycomb connective constant,
+a classical constant whose exact value was established by
+Duminil-Copin and Smirnov (2012). At criticality, walks
+weighted by x^N sit exactly at the phase transition between
+short and long typical lengths.
+
+A walk of length N contributes x^N to every sum below. A
+polygon of length |lambda| contributes x^{|lambda|}. Two
+polygons are "incompatible" if they share at least one
+vertex; a polygon is incompatible with itself. Weights
+multiply: an incompatible pair has weight x^{|lambda|+|mu|}.
+
+1.3 Ports and first-exit walks.
+
+For a region D in the plane (a hexagon, a half-plane, a
+convex cap), a "port" is the midpoint of a tessellation
+side on the boundary of D. A "first-exit walk" from a
+starting port is a self-avoiding walk inside D that never
+visits a boundary port except at the start and at its very
+last vertex, which is a boundary port other than the start.
+
+First-exit walks are important because the regional
+partition of these walks satisfies an exact signed balance:
+the Duminil-Copin-Smirnov cancellation. This balance makes
+the total first-exit weight from any fixed port to any
+convex boundary comparable to one. Section 5.1 turns that
+balance into the quantitative arch estimate that controls
+long walks.
+
+1.4 Cylinders and separating cycles.
+
+Take a horizontal cut of the honeycomb with 2m sites
+(m a positive even integer through the paper) and identify
+sites modulo 2m: that is, wrap the strip into a horizontal
+cylinder. On the cut, mark two points O = (0,0) and
+O' = (m,0); they divide the 2m sites into two equal lists
+A and B of size m.
+
+A "cycle" on this cylinder is a simple closed curve made of
+honeycomb edges, possibly wrapping nontrivially around the
+cylinder. It is "separating" if O and O' lie in different
+connected components of its complement. A "family" is a
+collection of pairwise disjoint separating cycles (they
+share no vertex).
+
+1.5 The cylinder partition G_m(g).
+
+Define G_m(g) as the sum, over families of pairwise disjoint
+separating cycles, of
+
+ g^k * x^{|lambda_1| + |lambda_2| + ... + |lambda_k|}
+
+where k is the number of cycles in the family, and
+|lambda_i| denotes the polygon length. The empty family
+contributes 1. Because any separating cycle crosses the
+cut between the two marks at least once, and there are
+only 2m sites available, G_m(g) is a polynomial in g of
+degree at most 2m.
+
+The cylinder is infinite in the vertical direction. The
+definition is by a limit: take finite cylinders with both
+rims closed (meaning no boundary port is used), let the
+height tend to infinity, and take the limit of partitions.
+Section 4.1 proves this limit exists.
+
+1.6 The hexagon and the chord law.
+
+Let D_R be a regular hexagonal patch of honeycomb of side
+length R (R a positive integer). Let Z(R) be the sum, over
+collections of pairwise disjoint polygons contained in D_R
+that surround its central face, of
+
+ 2^k * x^{|lambda_1| + ... + |lambda_k|},
+
+where k is the number of polygons. The factor 2 per polygon
+is the "fugacity 2" convention.
+
+Let C_R be the set of first-exit chords of D_R of diameter
+at least R/10. Here a "chord" is a first-exit walk whose
+two endpoints are boundary ports; the total sum runs over
+every starting port and every possible first-exit port, and
+retains the direction from start to exit. Define the finite
+partition and its probability law by
+
+ Z_R^chord = sum_{gamma in C_R} x^{N(gamma)},
+ P_R^chord(gamma) = x^{N(gamma)} / Z_R^chord.
+
+The chord law is a finite, normalized probability measure on
+the chords of the hexagon. Theorem 2 gives its total mass,
+its mean length, and its bulk vertex visit probability.
+
+1.7 Planar polygon sums.
+
+For the squared-length estimate, sum over planar polygons
+modulo lattice translations (two translates count as one
+class):
+
+ S(H) = sum_{diam(lambda) <= H} x^{|lambda|} |lambda|^2.
+
+The weight |lambda|^2 is the number of ordered pairs of bonds
+inside lambda. Summing over such pairs turns S(H) into a sum
+over two-bond configurations
+ (a polygon containing both bonds at once)
+weighted by the polygon's x^weight. That reformulated
+two-bond sum is the positive quantity that Theorem 3
+estimates, and Theorem 3's two-bond estimate is what
+produces the H^{2/3} bound on S(H).
+
+================================================================
+PART 2. THE THREE THEOREMS
+================================================================
+
+First, a notational convention that appears in every theorem.
+
+ "F(R) = R^{a+o(1)}" means: log F(R) / log R -> a as R grows.
+
+ An upper bound "F(R) <= R^{a+o(1)}" means: for every epsilon > 0
+ there is a constant C_epsilon such that
+ F(R) <= C_epsilon * R^{a+epsilon} for all sufficiently large R.
+
+ A lower bound "F(R) >= R^{a+o(1)}" is analogous.
+
+ A statement is "uniform" in some data if the constants and the
+ thresholds do not depend on that data. Every theorem below
+ is uniform in the sense that its constants and thresholds may
+ depend on the fixed parameters (which are named in the
+ theorem) but not on the growing parameters m, n, R, or H.
+
+Now the three theorems.
+
+----------------------------------------------------------------
+THEOREM 1 (Cylinder pressure and planar nesting).
+
+As m grows through even positive integers, the balanced
+cylinder partition G_m(g) with two marks satisfies:
+
+ (1a) G_m(2) = m^{1/6+o(1)}.
+
+ (1b) - (d^2/dg^2) log G_m(g) at g = 0 is O(log m).
+ Here the derivative counts the total weight of
+ ordered pairs of cycles that are incompatible --
+ i.e., pairs of separating cycles that share a
+ vertex, together with each cycle paired with itself.
+
+ (1c) Locally uniformly for complex Theta with
+ |Re Theta| < pi,
+ |G_m(2 cos Theta)|
+ <= m^{Re(1/6 - 2Theta^2/(3 pi^2)) + o(1)}.
+
+ (1d) The planar nesting partition Z(R) of a hexagon of
+ side R surrounding its central face satisfies
+ Z(R) = R^{1/12+o(1)} as R -> infinity through
+ integers.
+
+The informal reading:
+
+ (1a) says a balanced cylinder with two marks has a
+ partition at fugacity 2 growing only like the sixth
+ root of its circumference. Cycle families do not fill
+ up the cylinder; almost all of the mass is a short
+ burst around each mark.
+
+ (1b) says the "curvature" of the logarithm of the
+ partition at zero fugacity grows logarithmically. This
+ is a strong statement: the total mass of "bad" pairs
+ of cycles (those that share at least one vertex) is
+ small. It is what makes the localization argument
+ (Section 5.4) possible.
+
+ (1c) is the same as (1a) at Theta = 0, extended to
+ complex fugacity. It is what allows contour moves
+ and three-circle inequalities later.
+
+ (1d) is the planar limit. Set fugacity 2, so a family
+ of k polygons gets weight 2^k. The planar nesting
+ partition grows like the twelfth root of the size.
+ Combining (1d) with the chord-partition estimate
+ yields the R^{4/3} chord mean below.
+
+----------------------------------------------------------------
+THEOREM 2 (Finite chords and marked polygons).
+
+For the regular hexagon of side R and its chord law with
+diameter at least R/10 (as defined in Section 1.6):
+
+ (2a) Z_R^chord = R^{3/4+o(1)}. The total chord
+ partition, summed over every start and every exit,
+ grows like the fourth root of R per eighth power.
+
+ (2b) E_R^chord N = R^{4/3+o(1)}. The mean chord length
+ under the chord law grows like the four-thirds
+ power of R. This matches the standard 2D
+ self-avoiding walk exponent 1/nu = 4/3, with the
+ precise normalization and boundary conditions
+ fixed in Section 1.6.
+
+ (2c) Uniformly for honeycomb vertices v at distance
+ at most R/10 from the hexagon's central face,
+ P_R^chord(v lies on the chord) = R^{-2/3+o(1)}.
+ Every bulk vertex is visited with the same
+ power-law probability.
+
+ (2d) For planar polygons summed modulo lattice
+ translations,
+ sum_{diam(lambda) <= H} x^{|lambda|} |lambda|^2
+ <= H^{2/3+o(1)}.
+
+The informal reading:
+
+ (2a) says the total chord mass grows like the fourth
+ root of the size parameter. The boundary region
+ contributes a critical amount that is comparable to
+ what would appear in the full plane, up to a factor of
+ about R to the fourth root.
+
+ (2b) says the normalized mean chord length is exactly
+ R^{4/3}, matching the Flory-type prediction for 2D
+ self-avoiding walks. The proof of (2b) uses (2a) plus
+ (1d): the mean is a ratio of a sum over vertices to
+ the total mass. This is Proposition 5.3 in Part 5.
+
+ (2c) says that the chord partition is "delocalized":
+ no bulk vertex is meaningfully more visited than any
+ other. This is what one expects at criticality.
+
+ (2d) bounds the second moment of the side-length of a
+ polygon by its diameter. The power 2/3 is exactly
+ 2/nu - 2 with nu = 3/4; the difference between
+ |lambda| = N vertices and diam ~ N^{3/4} is absorbed.
+ Theorem 2(d) is a positive-moment estimate for the
+ finite diameter truncation.
+
+----------------------------------------------------------------
+THEOREM 3 (Unequal marked intervals).
+
+Take a "physical tilted cylinder": a periodic cut of the
+honeycomb whose steps are drawn from the trigonometric
+lattice {1, tau} with tau = e^{i pi/3}, made from two
+horizontal cuts joined by a shifted cut, so that both
+marked bonds cross cut steps parallel to one another
+(called "horizontal-type").
+
+Its two cut intervals have m+1 and n+1 sites, with
+n >= m and M = m+n+2 even. In each unmarked list the
+logarithmic site coordinates have imaginary parts -f
+and 1-f, where 0 <= f <= 0.6. Their respective counts
+differ from m(1-f), mf in the first unmarked list and
+from n(1-f), nf in the second by at most a fixed
+constant.
+
+Let H^poly be the total critical weight of a single
+unoriented polygon that passes through both marked bonds:
+
+ H^poly = sum over (single simple unoriented cycle
+ through both marked bonds) of
+ x^{length of cycle}.
+
+This sum is finite (the same walk cannot wind around
+a period of length > 2m too many times without
+self-intersecting).
+
+THEOREM 3 (restated). Uniformly over these lists and
+their orders, for every epsilon > 0 there are C_eps and
+m_eps such that
+
+ H^poly <= C_eps * n^eps * m^{-4/3} * (n/m)^{1/3}
+ for every n >= m >= m_eps.
+
+The informal reading:
+
+ Theorem 3 is the two-bond estimate that powers the
+ length-squared result. If you pick any two bonds at
+ distances of order m and order n, and sum the weights
+ of all polygons passing through both, the total
+ falls off like m^{-4/3}, then grows like the cube
+ root of the aspect ratio n/m. The 4/3 is the
+ polygonal length exponent visible in (2b); the
+ 1/3 = 1 - 2/3 is the balancing exponent that makes
+ the sum over pairs of bonds (Section 5.7) converge
+ to the H^{2/3} bound of Theorem 2(d).
+
+ Theorem 3 is proved independently of Theorem 1 and
+ Theorem 2, in Part 6. Its proof uses the same signed
+ transfer representation but for two marked bonds
+ instead of one unmarked cycle.
+
+----------------------------------------------------------------
+A note on how (1a), (2a), (2b), (2d) fit together.
+
+ Theorem 1 gives the balanced cylinder at fugacity 2.
+ Theorem 5.4 (localization) drops every cycle that
+ travels far from the marks, giving Z(R) = R^{1/12}
+ for the planar case. Proposition 5.3 then compares
+ the chord mass to the visit mass, giving (2a),
+ (2b), (2c). Theorem 3 supplies a separate positive
+ mass estimate; Parts 5.6 and 5.7 turn that estimate
+ into (2d).
+
+ So the paper proves one main theorem (Theorem 1),
+ two corollaries (Theorems 2a, 2b, 2c), and one
+ largely independent two-bond estimate (Theorem 3)
+ whose only downstream use is the length-squared
+ bound (Theorem 2d). This is why Part 5 does two
+ things: prove the chord corollaries, and separately
+ produce (2d) using Theorem 3.
+
+================================================================
+PART 3. THE MAP OF THE PROOF
+================================================================
+
+Here is a one-page map before the details. Each numbered step
+becomes a section in Parts 4-6.
+
+3.1 Theorem 1 -- the cylinder exponent.
+
+Read this list twice; every term is defined later.
+
+Step 1. Orient every occupied edge of the honeycomb in one of
+three states: empty, forward along the edge, or backward.
+Attach a phase x^{1/3} times a 1/8-th of pi phase to each
+local turn, a sign to each seam crossing, and a phase to each
+crossing of the "A cut". Then, over all self-avoiding
+configurations inside a finite closed cylinder, the oriented
+sum vanishes unless every cycle separates the two marks;
+a separating cycle contributes g. Section 4.1.
+
+Step 2. Let the cylinder height grow. The resulting "vacuum
+pairing" converges, because the total mass of half-plane
+arches is finite (Section 5.1). You get two vectors indexed
+by spin configurations on the cut. Section 4.1.
+
+Step 3. The local Yang-Baxter equation and two "fusion"
+identities (both proved in Section 4.2) let you separate
+the two vacuum vectors into an exact finite sum over a
+single three-valued spin variable per site. This is the
+"finite spin sum" of Section 4.3.
+
+Step 4. Individual terms of this finite sum have poles when
+two site coordinates coincide. The combined sum is
+regular -- the poles cancel. Section 4.4 replaces the
+finite sum by a contour integral (with each site
+represented by a residue) and then by a fermionic trace on
+an angular space of circumference T. Section 4.5 (the
+"bosonization") converts this into a horizontal gas on a
+circle of auxiliary length T, with two species of particle
+carrying integer charges.
+
+Step 5. The gas has signed and complex weights. Section 4.6
+proves an absolute stability bound: the total absolute
+weight of any configuration of gas particles is at most
+exp(-c * sum of |particle count per box|^2 - c * |running
+charge at box boundaries|^2). This bound is what lets you
+sum the signed series absolutely, and is the deep technical
+input.
+
+Step 6. The gas can be rewritten as a compact block
+propagator with an extra contracting coordinate that keeps
+the "tail" of the interaction exactly. Section 4.7
+constructs this operator. Its empty-contour power traces
+identify E_0 (all types) with spectral radius r_E = e^{-b
+pi / 16}, and I_0 (only the unsuppressed types) with
+eigenvalue 1.
+
+Step 7. The site factors omega_tau(m, x) that multiply each
+particle have modulus < 1 off the origin and approach 1 at
+infinity, with an exponential correction of rate pi/3.
+Therefore on an interval of length L ~ log m around the
+origin, all suppressed types disappear. Outside this
+interval, the exterior operator still allows all types. The
+inner rate is r_I = 1; the outer rate is r_E. Over the inner
+interval of length 2 log m / (pi b / 3), the propagation
+multiplier is 1. Over the exterior, it is r_E. Total: the
+propagation of the physical partition over n blocks is
+exp(-2 log m / (pi b / 3) * (pi b / 16)) = m^{-3/8}, plus
+other factors. Section 4.8.
+
+Step 8. The exterior operator may have Jordan blocks, so its
+n-th power traces have polynomial-in-n corrections. Section
+4.9 states the trace coefficient extraction lemma
+(proved in Part 4.10) that separates off every peripheral
+Jordan block and still recovers the physical value as a
+single scalar. This is where "m^{1/6}" becomes a genuine
+inequality.
+
+Step 9. A homogeneous scalar normalization N(X) enters every
+finite spin sum. At the homogeneous list X = (1,...,1), N(X)
+has its own power of m, computed exactly from a Gram
+determinant and a logarithmic statistic for even test
+functions. Section 4.11.
+
+Step 10. Combine the scalar normalization (Step 9), the
+propagator estimate (Step 7), the trace lemma (Step 8), and
+the exact value G_m(0) = 1 (which saturates the exponent).
+Three-circle inequalities on a chain of disks across the
+complex fugacity plane complete the proof. Section 4.12.
+Localization (Section 5.4) then gives (1d) by comparing
+G_m(2) to two independent planar partitions.
+
+3.2 Theorem 2 -- finite chords and visit probabilities.
+
+Step 1. The chord mass. Lemma 5.1 (the "convex-cap"
+restriction) compares the chord sum inside D_R to the
+half-plane arch mass. Proposition 5.1 (the arch estimate)
+proves W_R = R^{-1/4+o(1)} for the total mass of half-plane
+arches from a fixed port reaching radius R. The proof uses
+an added-row expansion, so the boundary balance and a
+positive mass moment determine each other.
+
+Step 2. The wedge comparison (Section 5.2) gives the
+matching upper bound: an arch reaching distance R leaves a
+wedge of angle pi/3, and the lost mass in that wedge is
+comparable to the arch weight. So W_R is bounded above
+and below, and Z_R^chord = R^{3/4+o(1)} follows.
+
+Step 3. The visit identity (Proposition 5.3). Remove the
+triangle at a honeycomb vertex v. The signed balance in the
+punctured domain is another exact cancellation. Comparing
+the punctured and original balances shows: the raw visit
+mass S_D(v) equals 3 Z_v + (corner weights) sum of
+(z x^{|lambda|} times Z_out). Rearranging further relates
+the ordinary visit mass sum_{gamma contains v} x^{N(gamma)}
+to the sum of the two-face partitions Z_D(f) for f incident
+at v. So visitors are comparable to nestings.
+
+Step 4. Localization (Proposition 5.4). Use (1b) to drop
+every cycle that travels far from its enclosed mark. The
+remaining cycles live in a ball of radius C m around the
+mark, and they project to two independent planar nestings.
+Reverse inclusion gives the other direction. Conclude
+Z(R) = R^{1/12+o(1)}.
+
+Step 5. Combine: mean = (R^2 * Z(R/2)) / Z_R^chord
+= R^2 * R^{1/12} / R^{3/4} = R^{4/3+o(1)}. Visit
+probability = Z(R) / Z_R^chord = R^{-2/3+o(1)}. This
+is (2b), (2c).
+
+3.3 Theorem 3 -- unequal marked intervals.
+
+Step 1. Physical identity (Proposition 6.1). Define
+H^flip(X) as an explicit matrix element of the finite spin
+data: two flips at sites i and j, an insertion, and a phase.
+Then at the physical site list H^flip = H^poly. Proof in
+Section 6.1: uses period-excluding walks on a finite
+cylinder, and a careful phase computation for two marked
+arcs.
+
+Step 2. Generic flip formula (Proposition 6.2). For generic
+nonzero sites, N(X)^2 H^flip has a clean closed form: a
+single combinatorial sum over unmarked spins, with pair
+factors F_s.
+
+Step 3. Marked gas (Proposition 6.3). Run the same contour
++ bosonization machine, but with the site source changed to
+account for the flips. The result is the same horizontal
+gas with a marked, "kinked" angle.
+
+Step 4. Suppression (Proposition 6.4). Near the two
+physical site heights -f and 1-f, the site factors of
+suppressed types are strictly below 1 with an exponential
+margin. This gives exp(-c n e^{-kappa |x|}) suppression on
+a scale |x| of order log n.
+
+Step 5. Scalar cancellation (Lemma 6.2). The whole
+site-dependent scalar normalizer N(X)^2 / S(X)^2 collapses
+to a simple power m^{5/48-eps}, matching the homogeneous
+normalization. This leaves H^flip bounded by a power of M
+times the value of the gas limit at the marked data.
+Section 6.5.
+
+Step 6. Mixed spectral radius (Proposition 6.5). When one
+list is longer, only that species's types are suppressed
+on the far interval. The intermediate propagator allows
+all types in species 1 and only unsuppressed types in
+species 2. Its spectral radius is bounded separately
+(Proposition 6.5). Section 6.6.
+
+Step 7. Assemble (Section 6.7). Combine the outer rates
+(Theorem 3's H^poly bound), the central power
+(r_X^+ for the mixed region), the two peripheral gaps, and
+the polynomial word-length factors. Every exponent cancels
+to the m^{-4/3} * (n/m)^{1/3} of Theorem 3.
+
+Step 8. Length-squared from Theorem 3 (Section 5.7).
+Sum over ordered bond pairs. Choose "good" pairs (well-
+spaced, with all three bond orientations nearby) so that
+the marked cylinder of Theorem 3 embeds the polygon
+injectively. Sum by dyadic distance d: d^2 pairs at radius
+d, each bounded by m^{-4/3} (n/m)^{1/3} with m ~ d, n ~ H,
+which collapses to H^{2/3+eps}. Bad polygons (too few good
+bond pairs) are separately summable with exponential
+decay. That proves (2d).
+
+3.4 What to remember about the whole structure.
+
+ The signed transfer representation (Step 2-3 of Theorem 1)
+ is where "signed weights" appear. The crucial fact is that
+ the absolute values are summable (Step 5 of Theorem 1,
+ "stability"). That allows every Step after to be an honest
+ inequality on the positive physical partition, even though
+ the "answer" is computed by signed gas contour integrals.
+
+ Two independent computations exist inside the paper:
+ (a) the cylinder exponent m^{1/6} of Theorem 1, which
+ uses the full compact-propagator-plus-trace-extraction
+ machinery; (b) the two-bond estimate of Theorem 3, which
+ uses a somewhat different version of the same machinery
+ because of the marked bonds. Theorem 2(b), 2(c) follow
+ from (a); Theorem 2(d) from (b).
+
+ The paper has one "cheap" ingredient (the boundary
+ balance, which is the exact Duminil-Copin-Smirnov
+ cancellation applied to half-planes, hexagons, caps,
+ and punctured domains) and one "expensive" ingredient
+ (the signed transfer machinery). Almost every quantitative
+ result is a meeting point of the two.
+
+================================================================