diff a280abd..640fd9a signed_cylinder_marked_polygons_v2.txt
diff --git a/signed_cylinder_marked_polygons_v2.txt b/signed_cylinder_marked_polygons_v2.txt
index 1e96d7a..49147c8 100644
--- a/signed_cylinder_marked_polygons_v2.txt
+++ b/signed_cylinder_marked_polygons_v2.txt
@@ -626,3 +626,1741 @@ decay. That proves (2d).
result is a meeting point of the two.
================================================================
+
+================================================================
+PART 4. PROVING THEOREM 1 -- THE CYLINDER EXPONENT
+================================================================
+
+PART 3 gave the map. This part walks through it in order.
+Each subsection has one goal, one plain-words idea, and one
+result for the next. Algebra lives in the original paper;
+here we recite only the shapes of the formulas that matter.
+
+----------------------------------------------------------------
+4.1 The oriented transfer and the height limit.
+----------------------------------------------------------------
+
+Goal: turn the positive partition G_m(g) into a matrix element.
+
+Seed: the boundary balance. In any convex region D, sum
+first-exit walks from a fixed starting port, with weight
+x^N for length N and phase e^{isT} for total turn T, where
+s = 3/8. The signed sum over all walks ending on the
+boundary equals 1 identically. This is the
+Duminil-Copin-Smirnov cancellation; everything below
+inherits its authority.
+
+To build a cylinder from this seed, put three states on each
+occupied honeycomb edge: empty, forward, backward. Attach a
+phase to each turn, to each seam crossing (by twist p = i
+above the cut and p = -i below), and to each crossing of
+the "A cut". The result is a local transfer matrix R(phi)
+with five trigonometric entries (u, d, v, a, b). At
+phi = alpha = pi/8 those five weights are exactly the
+honeycomb weights (x, x^2, x^2, x^2, 0).
+
+Now orient every loop of a closed finite cylinder either
+counterclockwise (contractible) or eastward (wrapped, so
+both marks lie on its left). The turn phase is -i; the
+seam phase splits as p^{-I_1} on one arc and 1 - I_1 on
+the other; the A-cut phase is (ih)^{I_2 - I_1}, where I_1
+and I_2 say whether O and O' lie on the interior (or
+upper) side. Multiplying and summing over the two
+orientations gives
+
+ (-i)(-1)^{I_1} (ih)^{I_2 - I_1}
+
+plus its inverse. The sum is 0 unless I_1 differs from I_2
+(that is, the loop separates O from O'), and in that case
+the two terms combine to h + h^{-1} = g. Therefore the
+signed sum over loop families equals the positive partition
+G_m(g). Signed transfer, positive result.
+
+The vacuum limit. Grow the cylinder height in both
+directions away from the cut. Arches reaching from a rim
+to the middle die off (the arch summability proved later
+in 5.1), so the top and bottom rims decouple. The height
+limit gives two vacuum vectors
+
+ L_-^phys = bottom vacuum,
+ R_+^phys = top vacuum,
+
+indexed by spin configurations on the cut. The partition
+becomes
+
+ G_m(g) = L_-^phys * D_A * R_+^phys,
+
+where D_A = (ih)^{sum over sites of A of s_j} is the
+insertion along the cut.
+
+Result: G_m(g) is a matrix element of two vacuum vectors
+against the insertion D_A.
+
+----------------------------------------------------------------
+4.2 Exchange, fusion, and the local projector.
+----------------------------------------------------------------
+
+Goal: two algebraic identities. One lets rows be reordered.
+The other lets adjacent rows collapse.
+
+Exchange. The transfer matrix satisfies the Yang-Baxter
+equation
+
+ R_12(t) R_13(t+phi) R_23(phi)
+ = R_23(phi) R_13(t+phi) R_12(t),
+
+as a meromorphic identity in the two parameters. In plain
+words: rows can be permuted freely without changing any
+matrix element. Consequence: a family of rows with a common
+twist has a simultaneous eigenbasis.
+
+Fusion. For delta = 2 or 3, there is an injective embedding
+J_delta from a smaller space E_delta into V tensor V:
+
+ delta = 2. E_2 three-dimensional, spins -1, 0, +1.
+ J_2 |0> = |00> + x sum_k lambda^k |k,-k>,
+ J_2 |k> = x (|k,0> + |0,k>), k = +/-1.
+
+ delta = 3. E_3 one-dimensional.
+ J_3 (1) = sum_k lambda^k |k,-k>,
+
+with lambda = e^{-2 i alpha}. The embedding satisfies
+
+ R_23(phi) R_13(phi + delta*alpha) (J_delta tensor I)
+ = (J_delta tensor I) * R_f(phi),
+
+with R_f(phi) = R(phi + alpha) for delta = 2, and R_f = I
+for delta = 3. In plain words: two rows at rapidities phi
+and phi + delta*alpha collapse to one row (delta = 2) or to
+nothing (delta = 3) on the embedded subspace. On the
+quotient there is a simple zero at the fusion point.
+
+Both identities are local. Lift each over the whole cut and
+the vacuum pairing factors through the image of J_delta per
+site. The 3^M spins in V^{tensor M} thus reduce to (3 or
+fewer)^M spins per block.
+
+Why this matters: exchange makes rows freely movable;
+fusion makes them fully collapsible. Between the two,
+vacuum vectors can only see a small per-site label.
+
+----------------------------------------------------------------
+4.3 The finite spin sum.
+----------------------------------------------------------------
+
+Goal: reduce the vacuum pairing to a finite sum over a
+single spin in {-1, 0, +1} per site.
+
+Insert at each site a spectral projector Pi_p for the twist
+p = i on the group above the cut and p = -i on the group
+below. Because the fusion algebra is so rigid, the projector
+has a closed determinant form,
+
+ Pi_p = det L(T) / det L(1),
+
+where L(T) is a matrix in the commuting row coefficients T_j
+obtained by expanding the fusion equations, and det L(T)
+vanishes identically on the common fixed space.
+
+The denominator is a Pfaffian:
+
+ det L(1) = (Pf S(Y))^2,
+
+where S is an explicit skew kernel and Y is the alternating
+list Y = (x_1, q x_1, ..., x_m, q x_m), q = e^{2 i alpha}.
+The square N(X)^2 clears the poles of the two selected
+vacuum vectors L_- and R_+.
+
+Inserting the projectors and applying fusion gives the
+finite spin sum formula: it expresses N(X)^2 G_{A,B}(h+h^{-1})
+as
+
+ sum over (s_1, ..., s_M) in {-1, 0, +1}^M
+ with sum_{j in A} s_j = sum_{j in B} s_j = Q
+ of h^{Q} times a product of site and pair factors.
+
+Structural facts about the sum:
+
+ - It is symmetric within each group A, B.
+ - Individual degree in any one site variable is at most
+ 2(M-1).
+ - Fusion of two sites at ratio q^2 or q^3 reduces it to a
+ smaller sum of the same form.
+ - At zero spin per site, all factors collapse to 1.
+
+Crucial nonvanishing. Let S(X) be the explicit Pfaffian
+prefactor. At the physical list X = (1, ..., 1), the ratio
+N(X) / S(X) is bounded below by (2/3)^{M/2} M^{5/48 - eps}
+for every eps > 0 and all sufficiently large M. Hence
+N(X) does not vanish anywhere near the physical list, so
+dividing by N(X)^2 is safe.
+
+The partition has become a finite sum of explicit
+polynomials.
+
+----------------------------------------------------------------
+4.4 Contours and the angular trace.
+----------------------------------------------------------------
+
+Goal: replace the finite spin sum by a contour integral, and
+then by a fermionic trace.
+
+Problem. Individual summands of the finite spin sum have
+poles when two site variables coincide. The combined sum is
+analytic (it equals the partition), but the terms are not.
+
+Fix. For each species (group A and group B) and a small
+radius r > 0, wrap each site's variable onto a circle
+gamma_r of radius r about the origin, turning each residue
+into a contour integral. Two residues hitting the same site
+cancel (their first Pfaffian fields coincide), so the
+expansion terminates after m currents per species.
+
+Bilinear identity. The contour pattern is exactly a
+fermionic Wick trace on an angular space of circumference
+T (a length parameter, taken to infinity at the end).
+Introduce anticommuting fields psi(v) with two-point
+function C^{[T]}(v - u), periodic-antiperiodic on the
+angular circle. The regulated contour expression reads
+
+ C_{sigma,T}(W)
+ = < exp( sum_delta int_gamma_r dw/(2 pi i)
+ (-z^delta) e^{i delta phi_sigma(w + i delta/2)}
+ prod_l U_delta,T(w - w_l)
+ psi(w + 3i) psi(w + i delta) ) >_T .
+
+The fields phi_sigma come from an external Gaussian regulator;
+U_delta,T is the periodized product of tilts; z = e^{i theta}
+is the physical twist; delta = +/-1 labels the deflection.
+
+Simplify. Take the regulator average away and define
+C_{m,T} as the full normalized contour value at W = 0^m
+(all m sites coalesced). Proposition "fixed-site regulated
+Pfaffian":
+
+ C_{m,T} -> (N(X)^2 / P_0) * G_m(h + h^{-1}), T -> infinity,
+
+locally uniformly in theta, at each fixed site count m,
+where P_0 is the all-zero-spin term Pi_0(W) of the finite
+sum. In one line: the contour current recovers the partition
+in the infinite-period limit at fixed m.
+
+Next step: bosonize the trace.
+
+----------------------------------------------------------------
+4.5 Bosonization: the horizontal gas.
+----------------------------------------------------------------
+
+Goal: rewrite the angular trace as a signed gas of particles
+on a horizontal circle of length T.
+
+Charge sectors. The fermion system has three zero-mode
+charges c_1, c_2, c_3. Nonzero oscillator modes fill each
+charge sector. The seam operator W_b from the contour
+expression (an ordered product of exponentials of currents)
+shifts one zero-mode charge by an integer ell and swaps two
+sector labels. Trace closure forces
+
+ S_1 = ell, n_2 = (F_3 - E_2 - ell)/2,
+ n_3 = (-F_3 - E_2 - ell)/2,
+
+where (S_1, E_2, F_3) are the total inserted charges of the
+three sheet types and n = (n_1, n_2, n_3) is the ground
+charge just to the right of the inserted fields. Both
+displayed values are integers by charge parity, so the trace
+is well-defined.
+
+Particle decomposition. Split the fermions into "short" and
+"outer" field types, then evaluate the H-dress that comes
+from the seam contraction. The result is six basic particle
+types plus one fused type. Each has a short height, an outer
+height, an H-charge, a charge jump, a local activity gamma,
+and a site multiplier omega_v(x) acting at the m-th power:
+
+ type charge omega_v(x) at |x| large
+ P +1 B(x) < 1
+ M -1 1 1
+ O_+ 0 B(x - i) < 1
+ O_- 0 B(x + i) < 1
+ U -1 B(x + i(3/2+d))^{-1} < 1
+ L -1 B(x - i(3/2+d))^{-1} < 1
+ F_pair -2 1 1
+
+with B(v) = (cosh(pi v/3) - sqrt(3)/2) /
+ (cosh(pi v/3) + sqrt(3)/2).
+
+The meromorphic identities B(v+i) B(v-i) = B(v) and
+B(v+3i) = 1/B(v) keep the seven-type list closed under
+collisions. Only M and F_pair have site multiplier exactly
+1 at every x; all six basic activities are explicit
+(sqrt(2)/4 times exponentials of a fixed kernel).
+
+Fusion of U and L. Two same-species U and L can collide;
+the residue at collision is the fused particle F_pair with
+local activity
+
+ gamma_{F_pair} = (9 pi^2 / 8) exp( K_diag(0) + K_diag^reg(3i) ),
+
+a positive constant independent of T, m, theta. Counting
+collisions uses the binomial fraction
+1/((u-r)! (l-r)! r!), where u, l are current counts of U
+and L and r is the number of fusions.
+
+Exact gas identity. Let Z_{m,T} be the sum over
+configurations Y (distinct real positions on R/T Z, labeled
+by the seven types) times a state weight W_{d,theta}(Y, p).
+Here p is a running charge vector obeying
+p(T) = p(0) mod 8 Z(1, -1), climbing by -q_v at each
+particle; the pair interaction R_{uv}(x) decays
+exponentially with x. Then
+
+ C_{m,T} = c(T) exp(m^2 kappa_T) Z_{m,T},
+
+where c(T) is an explicit scalar (empty oscillator trace,
+modular prefactor, Poisson sum over winding), and kappa_T is
+an explicit scalar rate with
+
+ lim_{T to infinity} kappa_T = 2 log(2(2 + sqrt(2))/9).
+
+Both c(T) and kappa_T have closed-form period limits and are
+subexponential in T.
+
+Sign of the story so far. The positive partition is now a
+signed gas Z_{m,T}. Because Z is signed, we still cannot
+bound it directly. The next subsection (4.6) proves that
+the absolute value of every gas weight decays quadratically,
+so the signed sum converges absolutely.
+
+----------------------------------------------------------------
+4.6 Absolute stability.
+----------------------------------------------------------------
+
+Goal: bound the modulus of every gas weight, so the signed
+sum Z_{m,T} is absolutely convergent.
+
+Statement (Theorem "absolute stability"). At a fixed
+displacement d with 0 < |d| < d_0 (small), fix a compact set
+Theta of angles theta. There are constants c > 0 and
+C < infinity, depending on d and Theta, such that for every
+finite configuration Y of particles on R/T Z and every
+closure-compatible state path p,
+
+ |W_{d,theta}(Y, p)|
+ <= exp( C T - c sum_{j=0}^{T-1}
+ ( n_j^2 + p_{c,j}^2 ) ),
+
+where n_j = number of particles in box [j, j+1) and
+p_{c,j} = (p_1(j+) + p_2(j+)) / sqrt(2) is the running
+common charge just to the right of that box.
+
+Reading. Every extra particle in a box costs at least
+quadratically. Every extra unit of running charge at a box
+boundary costs at least quadratically. The cost dominates
+at the exponential level: the more concentrated the gas,
+the more it is punished.
+
+How it is proved. The gas weight splits into five height
+kernels -- short, outer, H, fusion residue, and the state
+term -- each with explicit form. The combined height matrix
+H(k) has a coercivity structure:
+
+ - near k = 0, the leading part is
+
+ ( a_c / k ) * c(k) c(k)^T
+
+ for the vector of particle charges c_v, plus O(k) times
+ a strictly positive matrix on the quotient by the
+ linear relations among P, M, O_+, O_-;
+
+ - at large k, H(k) converges exponentially to a positive
+ matrix, so there is a uniform spectral gap on the
+ quotient by that null subspace;
+
+ - throughout, H(k) is positive on count-respecting
+ directions, with a quantitative gap of size min(k, 1)
+ times a fixed positive constant.
+
+Convert to real space via Fourier inversion. Each k-mode
+becomes a real-space kernel (2 pi / |k|) times a symmetric
+function of |k|. The leading charge term becomes a Poisson
+kernel (positive, long-range). Inserting a Gaussian
+smoothing of width sqrt(T) in space and applying a discrete
+Young inequality turns the sum of real-space kernels into
+the box-count and running-charge squares displayed above.
+
+Consequence. Z_{m,T} converges absolutely, uniformly in
+theta in Theta at each fixed period T. The sum over
+particle count converges like a Gaussian:
+sum_n A^n exp(-c n^2) < infinity for any finite A and
+positive c. The signed gas is therefore a genuine function
+of T and theta with controlled analyticity.
+
+This justifies every later coefficient extraction from
+power traces of the compact propagator (4.7).
+
+----------------------------------------------------------------
+4.7 A compact propagator.
+----------------------------------------------------------------
+
+Goal: turn the gas sum into a compact trace, so the
+trace-extraction lemma (4.8) applies.
+
+Block partition. Let b be a large fixed block size and put
+T = b n for an integer n. Group R/T Z into blocks of length
+b, numbered j = 0, ..., n-1. Each block has a state
+S_j = (Y_j, p_j): a particle configuration plus its running
+charge.
+
+Successor rule. The next block starts from p_j - q(Y_j),
+where q(Y_j) is the block's charge. The gas weight
+factorizes as a product of a local kernel w_theta(S_j, S_j+1)
+over blocks, plus a "tail" factor for pairs at block
+distance at least 2. The tail factor is exactly the far-side
+interaction. It has an analytic Bargmann representation
+using a nilpotent Jordan matrix A_0 whose diagonal entries
+are pi l / 6 (l >= 1) and whose nilpotent parts are
+uniformly bounded.
+
+Gauge. The local kernel has a gauge freedom: replace
+w_theta(S, S') by exp(Phi(S') - Phi(S)) * w_theta(S, S')
+for any real function Phi; products around closed cycles
+are unchanged. Choose Phi so that the gauged kernel
+satisfies
+
+ |w_theta(S, S')|
+ <= exp( C_b - delta_b ( |Y_j|^2 + |Y_{j+1}|^2 + p_c^2 ) ),
+
+uniformly in theta in a compact set Theta. This is what
+makes everything that follows uniform.
+
+Properties. The gauged kernels are Hilbert-Schmidt
+uniformly in theta in Theta, and holomorphic in
+Hilbert-Schmidt norm. The trace of a product of n blocks
+equals the gas sum Z_{m,bn}. The Bargmann representation
+handles the far-side tails exactly, so no approximation
+enters.
+
+Empty-block traces. Let E_0 be the "all seven types
+allowed" operator at theta and I_0 the "only M and F_pair
+allowed" operator. Both are parametrized by theta. Then
+
+ tr(E_0^n) = det(I - A_0^n)^{-1} / c(bn),
+ tr(I_0^n) = 8 det(I - A_0^n)^{-1} sum_{k in Z} u_k^n,
+
+with
+
+ u_k = exp( -4 pi b / 9 * (k + theta/pi)^2 ).
+
+The factor 8 is the mod-8 winding multiplicity: each
+k = p_1 + p_2 has eight representatives in
+Z^2 / 8 Z(1, -1).
+
+Spectral radii. Reading off these traces gives
+
+ r(E_0) = r_E = e^{-b pi / 16} (all seven types),
+
+and the leading eigenvalue of I_0 is u_0 = 1. For
+|Re theta| < pi/2, u_0 is semisimple: the Jordan blocks of
+the zero-charge region do not affect it.
+
+How this feeds 4.8. The site factor |B(x)|^m < 1 at every
+x off the origin, so the m-th power of the site factor
+suppresses all six basic particles except M and F_pair,
+with exponential rate pi/3 in |x|. Inside an interval
+|x| < L ~ log(m) / (pi b / 3), only M and F_pair survive at
+all: the operator I_0 acts. Outside, all seven types are
+present: the operator E_0 acts. The two operators are
+exactly the ones we just computed.
+
+----------------------------------------------------------------
+4.8 Trace extraction with Jordan blocks.
+----------------------------------------------------------------
+
+Goal: extract the physical value R from a trace
+difference, even when the exterior operator has peripheral
+Jordan blocks.
+
+Setup. Let E be a Hilbert-Schmidt operator of spectral
+radius 1. Its peripheral eigenvalues are ζ with |ζ| = 1.
+Write P_ζ for the Riesz projection onto ζ and J_ζ for the
+nilpotent part, so on ran(P_ζ),
+
+ E = ζ P_ζ + J_ζ, with J_ζ^{q_ζ} = 0
+
+for some finite q_ζ.
+
+Problem. The bare trace tr(E^n P_ζ) = ζ^n d_ζ + O(ρ^n)
+only sees the algebraic multiplicity d_ζ. The nilpotent
+part J_ζ is invisible in this limit.
+
+Setup for lack of information. Introduce two word families
+W+, W- with integer lengths and exponentially decaying
+tails:
+
+ sum_{W±} e^{ε l(W)} ||W|| < infinity.
+
+Define the shifted trace
+
+ F_s = sum_{l(W+) + l(W-) <= s}
+ tr( C W+ E^{s - l(W+) - l(W-)} W- ).
+
+Assume F_s - R tr(E^{s+a}) -> 0 for some scalar R and
+integer a >= 0. We must recover R from F_s.
+
+The lemma. Define a polynomial in a formal variable z by
+
+ P_ζ(z) = sum_{W+, W-} ζ^{-l}
+ tr( C W+ [ sum_{t=0}^{q_ζ - 1} binom(z - l, t) ζ^{-t}
+ J_ζ^t P_ζ ] W- ),
+
+where l = l(W+) + l(W-) and binom(z - l, t) is the
+binomial polynomial. Then
+
+ P_ζ(z) identically equals R * d_ζ * ζ^a.
+
+Hence
+
+ R = P_ζ(0) / (d_ζ ζ^a),
+ |R| <= K * ||C||_1,
+
+where K depends only on E, ε, the two weighted word sums,
+and the nilpotent orders q_ζ. It does not depend on C or a.
+
+In plain words: the polynomial identity is exact. Reading off
+the coefficient of any peripheral eigenvalue gives R
+directly. Every Jordan block is accounted for by the
+binomial polynomials binom(z - l, t).
+
+Special case a=0, no words. If W+ = W- = identity and
+F_s = tr(C E^s), the identity says tr(C P_ζ) = R d_ζ ζ^a.
+This is the elementary case; the lemma extends it.
+
+How it applies. In the cylinder problem:
+
+ E = E_0 / r_E (the normalized "all types" operator);
+ C = C'_m = (u / r_E)^{2L} C~_m (central block product);
+ a = 2L;
+ d_ζ = algebraic multiplicity of ζ on E'.
+
+Then R is the limit of the normalized partition trace, and
+
+ |R_m| <= C |u / r_E|^{2L},
+
+uniformly in m, uniformly in theta in a fixed compact set.
+
+Profile form. The central product C~_m has an analytic edge
+profile. In the notation a_m = m e^{-k_1 b L} in
+[1, e^{k_1 b}],
+
+ R_m (u / r_E)^{-2L} d_ζ ζ^{2L}
+ = F_ζ(a_m, theta) + O(e^{-cL}),
+
+where F_ζ is analytic in a near [1, e^{k_1 b}] and locally
+analytic in theta. This is what the assembly (4.10) plugs
+in.
+
+----------------------------------------------------------------
+4.9 The Pfaffian normalizer N(1, ..., 1).
+----------------------------------------------------------------
+
+Goal: compute N(X) at the homogeneous physical list
+X = (1, ..., 1) of size 2m.
+
+Recall the definition. With
+
+ P(X) = prod_{j<k} d(x_j, x_k) / (x_j - x_k),
+ d(u,v) = u^2 + sqrt(2) u v + v^2,
+
+define
+
+ N(X) = P(X) * Pf_{j,k} [ (x_j^2 - x_k^2) / d(x_j, x_k) ].
+
+The coincident-site value is by polynomial continuation.
+
+Result.
+
+ N(1, ..., 1) = (2 p_0 / 3)^{m(2m-1)}
+ * (2/3)^m
+ * m^{5/48 + o(1)},
+
+with p_0 = 2 + sqrt(2).
+
+Idea. Compare with a reference Pfaffian at a different
+kernel,
+
+ C(w) = tanh(pi w / 6),
+
+whose Pfaffian is exactly computable by Schur's Pfaffian
+identity:
+
+ Pf_{j,k} [ (y_j - y_k) / (y_j + y_k) ]
+ = prod_{j<k} (y_j - y_k) / (y_j + y_k).
+
+Taking y_j = exp(pi w_j / 3) recovers Pf[C(w_j - w_k)] as a
+full product.
+
+As the w_j tend to 0,
+
+ [d(x_j,x_k) / (x_j - x_k)] * C(w_j - w_k) -> 2 p_0 / 3,
+
+so P(X) * Pf[C(w_j - w_k)] gives the base factor
+(2 p_0 / 3)^{m(2m-1)}.
+
+The remaining ratio of two confluent Pfaffians becomes a
+ratio of Gram determinants of the functions
+1, k^2, k^4, ..., k^{2m-2} against two weights:
+
+ B0_hat'(k) = 8 k cosh(k) / sinh(4k),
+ C_hat'(k) = 6 k / sinh(3k).
+
+Their ratio is
+
+ B0_hat'(k) / C_hat'(k) = (2/3) e^{phi(k)},
+
+with phi(k) = log[ (1+t^2)(1-t^6) / (1-t^8) ], t = e^{-k}.
+The factor (2/3)^m comes out; the remaining ratio is a
+Gram determinant of e^{f(u)}, with u = 3k/pi and
+f(u) = phi(pi u / 3).
+
+Log statistic. Let mu(u) = 2 u / sinh(pi u) (a probability
+density, with Fourier transform 1/cosh^2(s/2)). For f even,
+continuous and exponentially decaying,
+
+ log det( P_m e^f P_m | on ran P_m )
+ = (log m) / (2 pi) * int_R f(u) du + o(log m),
+
+where P_m is the L^2(mu) projection onto the even
+polynomials of degree at most 2m-2. The proof diagonalizes
+the projection kernel by Meixner-Pollaczek polynomials and
+uses the Gaussian tail bound to control errors.
+
+In our case, expanding the logarithms as geometric series,
+
+ int_0^infty phi(k) dk
+ = pi^2/24 - pi^2/36 + pi^2/48
+ = 5 pi^2 / 144,
+
+so
+
+ int_R f(u) du / (2 pi)
+ = (3 / pi^2) * (5 pi^2 / 144)
+ = 5/48.
+
+Therefore the Gram ratio is m^{5/48 + o(1)}, giving
+N(1, ..., 1) as claimed.
+
+Why the exponent 5/48 matters. It enters the partition
+normalization as N(X)^2, which is 2 * (5/48) = 5/24.
+That is the exact factor the assembly (4.10) subtracts
+from the central+exterior rate 3/8, giving 3/8 - 5/24 = 1/6.
+
+----------------------------------------------------------------
+4.10 Assembly: G_m(2) = m^{1/6+o(1)}.
+----------------------------------------------------------------
+
+Goal: put together 4.3 (spin sum), 4.7 (propagation
+rates), 4.8 (trace extraction), and 4.9 (Pfaffian) to prove
+the cylinder exponent.
+
+Angular parametrization. Set g = 2 cos Θ. The polynomial
+G_{A,B}(g) has positive coefficients and satisfies
+G_{A,B}(0) = 1 (empty family). Its complex continuation
+enters the bosonization through h = e^{-iθ} with Θ = -2θ.
+
+The exponent identity. Three pieces combine:
+
+ (a) central + exterior propagation rate (4.7):
+ 3/8 - 8 θ^2 / (3 pi^2)
+ in the θ variable. With Θ = -2θ, this is
+ 3/8 - 2 Θ^2 / (3 pi^2)
+ in the Θ variable.
+
+ (b) scalar normalizer (4.9), entering as N(X)^2 (not N):
+ 5/24.
+
+ (c) trace extraction (4.8) turns the difference into a
+ bound on |R_m| ~ m^{p_*(Θ)}, with
+
+ p_*(Θ) = (a) - (b) = 1/6 - 2 Θ^2 / (3 pi^2).
+
+Statement. Uniformly for complex Θ in a compact subset of
+|Re Θ| < π,
+
+ |G_{A,B}(2 cos Θ)| <= m^{Re p_*(Θ) + o(1)}.
+
+Proof in one line. Sections 4.3--4.9 establish the bound.
+4.9 gives the closed form for N(X)^2. 4.7 gives the rate.
+4.8 makes the extraction valid even when E_0 has
+peripheral Jordan blocks.
+
+At the two endpoints.
+
+ Θ = 0 (g = 2): p_*(0) = 1/6, so |G_m(2)| <= m^{1/6+o(1)}.
+
+ Θ = pi/2 (g = 0): p_*(pi/2) = 1/6 - 2 (pi^2/4)/(3 pi^2) = 0,
+ and G_{A,B}(0) = 1 exactly. So the bound is saturated.
+
+Lower bound at g = 2. Suppose G_{A,B}(2) <= m^{1/6 - δ} for
+some fixed δ > 0. By positivity of the coefficients,
+|G_{A,B}(g)| <= G_{A,B}(2) for |g| <= 2, so on a small
+positive-Θ disk we have
+
+ G_{A,B}(2 cos Θ) m^{-p_*(Θ)} <= m^{-δ}.
+
+The three-circle inequality along finitely many overlapping
+disks from that disk to Θ = pi/2 propagates a positive
+fraction of the gap to the endpoint. But there
+G_{A,B}(0) = 1 is exact; contradiction. Hence the lower
+bound matches, and G_m(2) = m^{1/6+o(1)}.
+
+Logarithmic derivative at g = 0. At Θ = pi/2 the exponent
+p_*(Θ) vanishes. Since the analytic factor F_ζ(a, θ) is
+nonzero on the closed a-interval [1, e^{k_1 b}] (by
+compactness and a small open neighborhood of θ_i = -pi/4),
+differentiating the analytic expression along the local
+branch Θ(g) near g = 0 gives
+
+ -d^2/dg^2 log G_m(0) = O(log m).
+
+Together with G_m(2) = m^{1/6+o(1)} this proves Theorem 1
+parts (1a) and (1b).
+
+----------------------------------------------------------------
+4.11 Localization: Z(R) = R^{1/12+o(1)}.
+----------------------------------------------------------------
+
+Goal: prove the planar nesting exponent from the cylinder
+exponent.
+
+Setup. G_m(2) counts families of disjoint cylinder cycles
+separating the two marks O and O'. Each cycle is either
+contractible (its disk lift encloses exactly one mark) or
+noncontractible (wraps and separates the two ends).
+
+Reduction to planar. The regular hexagon D_R sees the same
+partition locally: same disjointness, same weights. The
+bridge is that G_m(2) counts configurations where some
+cycles stray far from their mark. We have to show that
+family is negligible.
+
+Bad cycles (Lemma "escape"). Fix a constant C > 1. Drop:
+
+ - all noncontractible separating cycles;
+ - all contractible cycles whose disk reaches
+ |x_horiz| >= M/4 or has diameter > CM.
+
+Each dropped family contributes O(sqrt(I_M)), where I_M is
+the incompatible-pair curvature sum, which 4.10 identifies
+as O(log M). Total dropped weight is O(sqrt(log M)) =
+M^{o(1)}.
+
+Remaining cycles. Every surviving cycle's disk lives in a
+closed ball of radius CM about its enclosed mark. Same-mark
+cycles are strictly nested (their disks have strict
+diameter inflation). Opposite-mark cycles are disk-disjoint.
+
+Hexagon comparison. A regular hexagon of side ⌈C'M⌉ for
+C' > 2C/sqrt(3) contains this ball. Same-mark disjointness
+is preserved on the chosen lifts. Dropping constraints
+between the two marks then bounds the remaining partition
+by the product of the two planar ones:
+
+ G_m(2) <= Z(⌈C'M⌉)^2 * exp(O(sqrt(log M))).
+
+For the other direction, hexagons of side ⌊cM⌋ (with
+0 < c < 1/4) about the two marked lifts embed into the
+cylinder with all disjointness preserved. So
+
+ Z(⌊cM⌋)^2 <= G_m(2).
+
+Combine. Using G_m(2) = M^{1/6+o(1)}:
+
+ Z(⌊cM⌋)^2 <= M^{1/6+o(1)},
+ so Z(R) <= R^{1/12+o(1)}.
+
+ Z(⌈C'M⌉)^2 >= M^{1/6 - o(1)},
+ so Z(R) >= R^{1/12 - o(1)}.
+
+The change of variable R ~ cM (or R ~ C'M) preserves the
+power up to o(1). Therefore Z(R) = R^{1/12+o(1)}.
+
+Result: Theorem 1 part (1d). Together with 4.10, Theorem 1
+is proved.
+
+End of PART 4.
+
+================================================================
+PART 5. PROVING THEOREM 2 -- CHORDS, VISITS, LENGTH^2
+================================================================
+
+PART 5 takes Theorem 1 and delivers Theorem 2. The engine is
+the arch estimate (5.1), which controls the mass of long
+walks on the boundary of a region. Chords (5.2), visit
+identities (5.3), mean lengths (5.4), and the length-squared
+bound (5.5) all follow.
+
+----------------------------------------------------------------
+5.1 The arch estimate W_R = R^{-1/4+o(1)}.
+----------------------------------------------------------------
+
+Goal: bound the mass of half-plane first-exit arches.
+
+Definition. In the half-plane bounded by a horizontal
+tessellation line, let W_R be the total critical weight of
+first-exit arches from a fixed port that reach Euclidean
+distance at least R from that port.
+
+Result: W_R = R^{-1/4+o(1)}.
+
+The proof has four sub-steps.
+
+(a) Boundary balance. In any convex region D, sum first-exit
+walks from a fixed starting port with the same phases as in
+4.1. That signed sum is 1. Applied to the half-plane, this
+gives the half-plane saturation
+
+ 2 sin(pi/8) * sum_{d>0} H(d) = 1,
+
+where H(d) is the total half-plane arch weight from a
+fixed port to the port d boundary steps to its right. The
+line d=0 is excluded (first-exit arches have positive
+length). Positivity bounds the left side by 1, so the sum
+converges.
+
+(b) Added-row fusion. Introduce a finite cylinder of
+circumference M, and put one new outer rhombus row on top
+(L honeycomb rows). Subtract the partition without that
+row. The increment F(v) is a polynomial in the row weight v,
+of degree at most M. Local fusion (4.2) gives a functional
+relation F(v) F(q^2 v) = F(q v) at each site, and following
+the linear coefficient and the top coefficient through that
+relation pins down
+
+ F'_v(1) = M^{3/4+o(1)}.
+
+The exponent 3/4 comes from a Pfaffian determinant law on
+ordered points k_1 < ... < k_M with weight function
+
+ w(k) = 2 (1 + 4 sinh^2(alpha k)) / sinh(pi k),
+
+combined with a Gamma comparison (or equivalently a von
+Mises ensemble). Independent of the boundary balance, this
+computation is the deepest piece of PART 5.
+
+(c) Deficit. The increment's coefficient of v^{M-1} is
+1 - c_alpha * sum_{d=1}^{M-1} H_M(d), where
+c_alpha = 2 sin(alpha). Hence the deficit
+
+ delta_M := (1/c_alpha) - sum_{d=1}^{M-1} H_M(d)
+ = F'_v(1) / (c_alpha M)
+ = M^{-1/4+o(1)}.
+
+Here H_M(d) is the cylinder version of the arch (arches
+disjoint from all their M-translates). Taking M to infinity
+shows sum_{d>0} H(d) = 1/c_alpha exactly, and the tail
+
+ A(M) := sum_{d >= M} H(d)
+ <= delta_M = M^{-1/4+o(1)}.
+
+(d) Radius vs. length. Every arch contained in a half-disk
+of radius M/2 around its source is M-translate-disjoint, so
+such arches are fully counted in H_M(d). Therefore the
+missing mass delta_M is exactly the mass of arches reaching
+distance at least M/2, and
+
+ delta_M <= W_{M/2}.
+
+Choosing M slightly above 2R gives the lower bound
+W_R >= R^{-1/4+o(1)}.
+
+The upper bound uses a wedge comparison. Intersect the
+half-plane with a wedge of angle pi/3 (bounded by the source
+ray and its reflection). Signed balance on the wedge gives
+a linear relation between the lost left arch weight
+L_<(a) and the lost right arch weight L_>(a). Summing over
+a finite range and using the balance identity gives
+
+ L_<(a) + L_>(a) <= a^{-1/4+o(1)}.
+
+Two reflected wedges whose vertices are at horizontal
+distance a on opposite sides of the source intersect in a
+diameter-O(a) region. Any arch reaching distance Ca for a
+sufficiently large fixed C must leave at least one of the two
+wedges, so its weight is at most the sum of the two wedge
+losses. Hence W_R <= R^{-1/4+o(1)}.
+
+Together: W_R = R^{-1/4+o(1)}.
+
+----------------------------------------------------------------
+5.2 Chords from arches.
+----------------------------------------------------------------
+
+Goal: bound Z_R^chord = R^{3/4+o(1)}.
+
+Setup. D_R is the regular hexagon of side R. C_R is the
+set of ordered first-exit chords of D_R (a self-avoiding walk
+from one boundary port to another, staying inside D_R, whose
+last vertex is a boundary port other than the start) with
+diameter at least R/10. Z_R^chord = sum over gamma in C_R
+of x^{N(gamma)}. The sum runs over every start and every
+exit, retaining the direction from start to exit.
+
+Upper bound (fixed source). Fix one source port a. A chord
+of diameter R/10 must reach at least R/20 from a. Choose a
+small hexagon cap P of diameter about R/25 inside D_R that
+contains a but is well clear of any such distant exit. By
+the convex-cap lemma (below), the chord weight lost by
+restricting D_R to D_R ∩ P is bounded by a constant times
+the weight of artificial exits (first-exit walks ending on
+the new boundary of D_R ∩ P). Those artificial exits are
+half-plane arches from sources within P, whose total is at
+most C W_{cR}. Therefore the total chord weight from a
+fixed source is at most C W_{cR} = R^{-1/4+o(1)}.
+
+Lower bound (fixed source). Choose sources in a fixed
+middle portion of one side of D_R, at distance at least
+cR > R/10 from every other side. A half-plane arch from
+such a source reaching radius R is either a macroscopic
+chord contained in D_R, or else is lost by the convex-cap
+argument. The lost weight is again at most C W_R. So the
+macroscopic chord weight from each such source is at least
+a constant times W_R = R^{-1/4+o(1)}.
+
+Combine. There are O(R) sources in the middle portion.
+Summing the fixed-source bounds:
+
+ c * R * R^{-1/4+o(1)} <= Z_R^chord
+ <= C * R * R^{-1/4+o(1)}.
+
+Therefore
+
+ Z_R^chord = R^{3/4+o(1)}, which is Theorem 2 part (2a).
+
+Convex-cap lemma in one line. Let K ⊂ D be finite convex
+tessellation polygons sharing a starting port. The raw weight
+of first-exit chords of D that are lost when restricting to K
+is at least sin(alpha) times the artificial-exit weight E of
+K and at most E / sin(alpha). Proof: both sides are signed
+balances; the two sums coincide on common chords (which
+cancel), and every real coefficient in either convex domain
+sits between sin(alpha) and 1; taking real parts gives the
+two-sided bound.
+
+----------------------------------------------------------------
+5.3 The visit identity.
+----------------------------------------------------------------
+
+Goal: compare the raw weight of chords visiting a fixed
+interior vertex v with a nesting partition.
+
+Setup. D is a finite convex tessellation polygon. v is a
+honeycomb vertex not adjacent to any boundary port. Let
+
+ S_D(v) = sum over first-exit boundary chords gamma of D
+ that visit v
+ of x^{N(gamma)} cos(3 T(gamma)/8).
+
+Let Z_v be the sum over collections of disjoint unoriented
+cycles that strictly enclose v (and avoid v), each cycle
+contributing 2 x^{|lambda|}. Let Z_out(lambda) be the
+analogous sum over disjoint cycles that surround lambda and
+are disjoint from it. At a vertex lambda ∩ v, call the
+corner convex if the interior angle is 2 pi/3 and reflex if
+4 pi/3, and put
+
+ c_lambda = sqrt(2) (v convex corner of lambda),
+ = 2 + sqrt(2) (v reflex corner).
+
+Identity.
+
+ S_D(v) = 3 Z_v
+ + sum over cycles lambda through v of
+ c_lambda * x^{|lambda|} * Z_out(lambda).
+
+Reading. The first term counts cycles that avoid v (as a
+background). The second term counts cycles through v (each
+occupies a corner at v).
+
+The three incident faces. Let Z_D(f) denote the nesting
+partition around a face center f. Then
+
+ sum over the three incident faces f of v: Z_D(f)
+ = 3 Z_v + sum over lambda through v of
+ 2 k_lambda * x^{|lambda|} * Z_out(lambda),
+
+where k_lambda = 1 at a convex corner and 2 at a reflex
+corner.
+
+Comparing the two identities gives
+
+ (1 / sqrt(2)) * sum over incident f: Z_D(f)
+ <= S_D(v)
+ <= sum over incident f: Z_D(f).
+
+And boundary positivity gives
+
+ S_D(v) <= sum over gamma visiting v: x^{N(gamma)}
+ <= S_D(v) / sin(alpha).
+
+So the ordinary visit mass is comparable to the sum of the
+three incident-face nestings.
+
+Proof idea in one line. Remove the triangle centered at v,
+creating three inner ports. The signed balance in the
+punctured domain still holds. Subtracting the punctured
+from the original gives the difference as paths from the
+three inner ports to outer exits. Take real parts and
+reverse paths to arrive at v. Grow a tree from each inner
+port: it ends (a) at an outer exit (giving S_D(v)),
+(b) at first self-collision not enclosing v (cancels by
+reflection symmetry), (c) at first collision with a
+background cycle (cancels with the alternative first-contact
+class), or (d) at an inner-inner return that closes a
+cycle lambda through v (contributes c_lambda times Z_out).
+The coefficient c_lambda is the residual turn phase of that
+closed lambda.
+
+----------------------------------------------------------------
+5.4 Mean chord length and bulk visit probability.
+----------------------------------------------------------------
+
+Goal: prove Theorem 2 parts (2b) and (2c).
+
+Setup. D_R and C_R as in 5.2. Let Z(R) be the planar +2
+nesting partition around the central face center in D_R.
+For every honeycomb vertex v in D_R,
+
+ V_R(v) = sum_{gamma in C_R, v in gamma} x^{N(gamma)}.
+
+The mean chord length is
+
+ E_R^chord N = (1 / Z_R^chord) sum_{v in D_R} V_R(v).
+
+Bulk vertices. For v at distance at most R/10 from the
+center, every C_R chord through v is macroscopic. A
+translated hexagon of side floor(R/2) centered at each
+incident face fits inside D_R. Positivity of the visit bound
+from 5.3 gives
+
+ c Z(floor(R/2)) <= V_R(v)
+ <= C Z(3R),
+
+uniformly in the stated range.
+
+Summation. For any vertex v not adjacent to a boundary port,
+a translated hexagon of side 3R centered at an incident face
+contains D_R, giving the upper V_R(v) <= C Z(3R). There are
+O(R^2) bulk vertices and only O(R) boundary-adjacent
+vertices, each contributing at most one chord to the
+normalized mean (a chord visits any vertex at most once).
+Hence
+
+ c R^2 Z(floor(R/2)) / Z_R^chord
+ <= E_R^chord N
+ <= C R^2 Z(3R) / Z_R^chord + C R.
+
+Plugging Z(R) = R^{1/12+o(1)} (4.11) and
+Z_R^chord = R^{3/4+o(1)} (5.2):
+
+ R^2 * R^{1/12} / R^{3/4}
+ = R^{2 + 1/12 - 9/12}
+ = R^{2 - 8/12}
+ = R^{4/3}.
+
+Therefore E_R^chord N = R^{4/3+o(1)}, which is (2b).
+
+Visit probability. For any v at distance at most R/10 from
+the center, divide V_R(v) by Z_R^chord:
+
+ V_R(v) / Z_R^chord
+ = R^{1/12+o(1)} / R^{3/4+o(1)}
+ = R^{-8/12+o(1)}
+ = R^{-2/3+o(1)}.
+
+That is Theorem 2 part (2c).
+
+The additive C R term. Boundary ports produce O(R) extra
+vertices, each contributing at most 1 to the normalized
+mean or probability. Compared to the leading R^{4/3} and
+R^{-2/3}, that is negligible for large R.
+
+----------------------------------------------------------------
+5.5 The H^{2/3} length-squared estimate.
+----------------------------------------------------------------
+
+Goal: prove Theorem 2 part (2d).
+
+Reduction to two bonds. Let
+
+ S(H) = sum over planar polygons of diameter at most H,
+ modulo lattice translations,
+ of x^{|lambda|} |lambda|^2.
+
+The factor |lambda|^2 counts ordered pairs of bonds in
+lambda. So S(H) equals the sum over ordered bond pairs of
+the two-bond quantity H^poly, estimated in PART 6.
+
+Two-bond estimate (Theorem 3). Uniformly in m, n >= 1 of
+any permitted tilted cylinder:
+
+ H^poly_{m,n} <= C_eps * n^eps * m^{-4/3} * (n/m)^{1/3}
+
+for every eps > 0.
+
+Good pairs. Every polygon of length |lambda| has at least
+|lambda| - 2K vertices far from its sharp turns, for K = 4
+fixed. Any bond pair can be replaced by a pair (v_1, v_2)
+of good vertices with the same scale, up to a bounded
+factor. Count good pairs instead.
+
+Scale identification. For a good pair (v_1, v_2) with
+|v_2 - v_1| = d <= H, the polygon sits on a permitted
+tilted cylinder of scale (m, n) with m ~ d and
+n ~ H - d, up to bounded factors. The selected-pair lemma
+(part of Theorem 3 in PART 6) makes this identification
+bijective between bond pairs and permitted cylinders, up
+to those factors.
+
+Summing over scales. For each scale (m, n) apply the
+two-bond bound m^{-4/3} (n/m)^{1/3}. Sum over 1 <= m <=
+n <= H with the polynomial pair-count weight, and the
+total collapses to
+
+ S(H) <= H^{2/3 + eps}.
+
+The two-bond decay in m is what carries the exponent from
+2 down to 2/3. In words: critical polygons cannot fill a
+disk efficiently. Each extra pair of bonds at scale m
+costs m^{-4/3} in the two-bond sum, which dominates the
+polynomial growth in the number of such pairs.
+
+Therefore S(H) = O(H^{2/3+eps}) for every eps > 0, which
+is Theorem 2 part (2d).
+
+End of PART 5.
+
+================================================================
+PART 6. PROVING THEOREM 3 -- TWO MARKED BONDS
+================================================================
+
+PART 5 ended with a promise: a two-bond estimate. This part
+delivers it. The structure mirrors PART 4 but with two
+marked bonds instead of one unmarked cycle. Two new
+complications: the two marked bonds break the symmetric
+Dirichlet form, and the positive quantity H^poly must be
+matched to an explicit signed expression H^flip.
+
+----------------------------------------------------------------
+6.1 The two-bond physical identity.
+----------------------------------------------------------------
+
+Goal: express the two-bond polygon sum as a signed matrix
+element of the finite spin data.
+
+Physical quantity. Let
+
+ H^poly(X) = sum over self-avoiding polygons that pass
+ through both prescribed bonds
+ of x^{N}.
+
+Here X is the finite site data of the cut, and H^poly is
+positive.
+
+Signed counterpart. Run the contour machinery of PART 4 on
+the doubled physical sheet with two flips at the two marked
+bond midpoints instead of one D_A insertion. The result is
+an algebraic expression H^flip(X) whose contour integral
+agrees with the physical H^poly in the same regulator
+limit. The two endpoint variables U, V are the flip
+variables.
+
+Physical identity (Proposition "marked-physical"). At the
+physical site list X = (1, ..., 1) with U = V = 1,
+
+ H^poly(X) = H^flip(X).
+
+Idea. The signed sum over closed rim states with the two
+marked bonds satisfies the same Duminil-Copin-Smirnov
+balance identities as the unmarked case, now with two
+flips replacing the D_A insertion. Vertex-edge fusion and
+arc moves reduce it to the finite spin sum of PART 4. The
+regularity of the surface factor S(X)^2 shows the limit at
+the physical list is attained; a two-arc phase lemma
+cancels the two marked-bond phases. In one line:
+H^poly = H^flip at X = physical.
+
+----------------------------------------------------------------
+6.2 The generic flip formula.
+----------------------------------------------------------------
+
+Goal: an explicit formula for H^flip(X) for generic nonzero
+site data.
+
+Setup. Let (A, B) be the two consecutive unmarked lists
+with the two marked sites removed. Write U = x_i, V = x_j
+for the endpoint variables, and A' = A \ {i},
+B' = B \ {j} for the reduced unmarked lists. The reduced
+unmarked problem is the finite spin sum of PART 4.
+
+Track the double-flip matrix element through the projectors
+and fusion algebra. The result factorizes:
+
+ N(X)^2 H^flip(X)
+ = c U V sum over spins s of q^{-6 Q} C_s
+ * prod_{a in A' u B'}
+ F_{s_a}(U, x_a) * F_{s_a}(V, x_a),
+
+where
+
+ Q = sum_{A'} s_a = sum_{B'} s_a (charge balance),
+ e_k(u, v) = v - q^k u, q = e^{2 i alpha},
+ (F_+, F_0, F_-) = (e_5 e_6, e_3 e_5, e_2 e_3),
+ c = 2 - sqrt(2).
+
+C_s is the unmarked finite spin summand of (4.3) at the
+unmarked lists A', B'.
+
+Reading. H^flip(X) is a product of
+
+ - c U V (the two-endpoint scalar),
+ - q^{-6Q} (the two-flip charge coupling),
+ - C_s (the unmarked background),
+ - two per-site factors, one coupling each site to U and
+ one to V.
+
+Structural facts:
+
+ - U and V are independent in the formula. The physical
+ list will later set U = V = 1.
+ - Division by N(X)^2, not by any endpoint variable.
+ - e_k(u, v) vanishes exactly at u = q^k v; those
+ vanishing patterns are the fusion rules.
+
+----------------------------------------------------------------
+6.3 Continuation to physical sites.
+----------------------------------------------------------------
+
+Goal: prove H^flip(X) extends holomorphically to the
+physical list.
+
+The physical list. In the balanced cut, sites sit at
+imaginary heights 0 (in A) and 1 (in B), up to a common
+rescaling. Concretely X = (1, ..., 1) of size M = 2L.
+
+Problem. H^flip carries 1/N(X)^2 in front. If N(X)
+vanished near the physical list, the continuation would
+fail.
+
+Nonvanishing lemma. For a list X of even size 2L with
+imaginary site heights in a range of spread at most 1
+containing the physical list, satisfying the balancing
+condition of 4.3,
+
+ N(X) / S(X) = (2/3)^L * E_h * exp( sum_a phi(k_a) ),
+
+where
+
+ phi(k) = log( (1 + e^{-2k}) / (1 + e^{-4k}) ),
+
+E_h is a positive real function that tends to 1 as the
+spread tends to 0, and S(X) is the explicit Pfaffian
+prefactor. In particular, N(X) > 0 on that neighborhood.
+
+Consequence. The explicit polynomial form
+N(X)^2 * H^flip(X) makes sense at every site. Since
+N(X) != 0 near the physical list, H^flip itself is
+holomorphic there. The finite spin sum is computed directly
+at X = (1, ..., 1). By 6.1 that value equals H^poly.
+
+So the closed-form expression of 6.2 is valid at the
+physical list.
+
+----------------------------------------------------------------
+6.4 The marked gas.
+----------------------------------------------------------------
+
+Goal: pass the marked algebraic expression through the
+contour-to-gas rewriting of 4.4-4.5.
+
+The kink. Start from the unmarked regulator of 4.4 and add
+a local "kink" at the endpoint coordinate w_e:
+
+ d_T(v) = sum_{ell in Z} [ D_e(v - w_e + ell T)
+ - D_e(v - w_e - T/2 + ell T) ] - nu.
+
+Reading: two signed step primitives at opposite positions,
+minus a normalizing constant. Zero mean, sign-flipping
+under T/2-translation, tending to D_e(v - w_e) on compacts
+as T -> infinity.
+
+The marked regulator is
+
+ C_T^L = E[ e^{i int (DA) chi}
+ * e^{-2 pi sum_sigma n_sigma ell_sigma / T}
+ * C^L_{1,T} * C^L_{2,T} ]
+ / E[ e^{i int (DA) chi} ],
+
+where C^L_{sigma,T} is the current (4.4) with
+phi_sigma(v) -> phi_sigma(v) + d_T(v).
+
+Result (Proposition "marked-regulator-limit"). As T ->
+infinity at fixed site count,
+
+ C_T^L -> H^flip(X)
+
+computed via the continued algebraic formula of 6.2.
+Every surviving residue is exactly one of the finite spin
+summands.
+
+Site factors. Particles pick up extra site multiplicities:
+
+ P: prod_{w in W_sigma} B_T(x - w)
+ M, F: 1
+ O_+: prod_{w in W_sigma} B_T(x - w - i)
+ O_-: prod_{w in W_sigma} B_T(x - w + i)
+ U: prod_{w in W_sigma} B_T(x - w + i(3/2+d))^{-1}
+ L: prod_{w in W_sigma} B_T(x - w - i(3/2+d))^{-1}.
+
+Multiplied over the site multiplicity m. M and F_pair have
+no extra factor from this list; their activity comes only
+from the local residue.
+
+Gas identity. Let Z_T^L denote the marked gas sum. Then
+
+ C_T^L = c(T) Z_T^L
+
+with c(T) the same scalar as in 4.5.
+
+----------------------------------------------------------------
+6.5 Stability at displacement d = 1/8.
+----------------------------------------------------------------
+
+Goal: prove the absolute stability of 4.6 at d = 1/8,
+the displacement the marked problem needs.
+
+Statement (Proposition "stability-eighth"). For the
+post-collision gas at d = 1/8, the conclusion of 4.6 holds
+with constants uniform in T and theta in a fixed compact
+subset.
+
+Idea. The proof of 4.6 was purely a matrix-coercivity
+argument on H(k). The same argument works at any fixed d,
+as long as the coercivity constant is genuinely positive.
+The check at d = 1/8 reduces to explicit rational-function
+positivity in the variable u = e^{-k}.
+
+Why d = 1/8 specifically. At the U-L collision, a
+"continued H-term" removes the singularity. After the
+continuation the H-matrix has explicit rational entries at
+each d. Writing a = tanh(Δ) with Δ = k/8, several entries
+reduce to
+
+ a - 1 = u^2 P_sigma(u) / ((1 - u^2) Q_sigma(u))
+
+for sigma = +/-. The ascending coefficient lists of
+P_+, P_-, Q_+, Q_- are all positive integers, so a - 1 is
+positive in u for 0 < u < 1. Two further sign conditions
+reduce to
+
+ (o_0 + b_1 tanh Δ + b_2 tanh^2 Δ) / o_0
+
+with positive coefficients in z = e^{-k/4}. So the
+coercivity gap is positive at every k > 0.
+
+Consequence. The absolute stability of 4.6 holds at
+d = 1/8, so the marked gas Z_T^L is absolutely convergent
+and the trace-extraction lemma of 4.8 applies directly.
+
+----------------------------------------------------------------
+6.6 The mixed spectral radius.
+----------------------------------------------------------------
+
+Goal: bound the growth of the exterior operator's power
+when the block sequence mixes site-perturbed and
+site-free segments.
+
+Setup. In the marked problem some blocks carry extra site
+multiplicities (central segments), others do not
+(exterior segments). Central segments grow at rate u_c;
+exterior segments grow at one of several rates, the
+largest being r_X^+.
+
+Two-gap lemma (Lemma "trace-extraction" applied to the
+product). Power traces split into peripheral-eigenvalue
+powers plus exponentially small remainders, uniformly in
+the block partition. Every Jordan block is accounted for.
+
+The three rates.
+
+ u_c = e^{-pi b / 4} (central blocks)
+ r_X^+ = exp( b ( -5 pi / 96 + 1 / (6 sqrt 3) ) )
+ (exterior, largest rate)
+ r_E = e^{-b pi / 16} (empty, baseline)
+
+The two useful ratios:
+
+ log(u_c / r_E) / (k_1 b) = -3 pi / 16,
+ log(r_X^+ / r_E) / (k_1 b) = pi / 96 + 1 / (6 sqrt 3),
+
+with k_1 = pi / 3. Both are fixed numbers independent of
+m, n, theta.
+
+Mixed-radius bound. Suppose a segment pattern has 2J
+central blocks and 2 (L - J) exterior blocks. Then for any
+fixed eta > 0,
+
+ ||Z_{4 N b}^L||_1
+ <= C_eta n^{eta}
+ * (u_c / r_E)^{2 J}
+ * (r_X^+ / r_E)^{2 (L - J)}.
+
+Both J and L are O(log n); n^{eta} absorbs the constant
+factors. 4.7 ties J to m and L - J to n/m.
+
+----------------------------------------------------------------
+6.7 Assembly: Theorem 3.
+----------------------------------------------------------------
+
+Goal: prove
+
+ H^poly <= C_eps * n^eps * m^{-4/3} * (n/m)^{1/3}.
+
+Start (6.4 + 6.5). Absolute stability of the marked gas
+gives
+
+ |H^flip(X)| <= C_eps M^{-5/24+eps}
+ * |lim c(4 N b) Z_{4 N b}^L|.
+
+Mixed-radius bound (6.6). Choose
+
+ J = log(m) / (k_1 b) + O(1),
+ L - J = log(n/m) / (k_1 b) + O(1),
+
+matching the central and exterior segment counts to the
+scales m and n/m. Then
+
+ ||Z_{4 N b}^L||_1 <= C_eta n^eta
+ * (u_c / r_E)^{2 J}
+ * (r_X^+ / r_E)^{2 (L - J)}.
+
+Explicit values.
+
+ (u_c / r_E)^{2 J}
+ = m^{-9/8}.
+
+ (r_X^+ / r_E)^{2 (L - J)}
+ = (n/m)^{1/16 + 1/(pi sqrt 3)}.
+
+Combine. With M ~ n:
+
+ H^poly <= C_eps n^eps
+ * n^{-5/24}
+ * m^{-9/8}
+ * (n/m)^{1/16 + 1/(pi sqrt 3)}.
+
+Two exponent identities (the algebra of the two ratios):
+
+ -5/24 + 2 log(u_c / r_E) / (k_1 b)
+ = -5/24 - 9/8
+ = -4/3.
+
+ -5/24 + 2 log(r_X^+ / r_E) / (k_1 b)
+ = -5/24 + 1/16 + 1/(pi sqrt 3)
+ = 1/(pi sqrt 3) - 7/48
+ < 1/3.
+
+They combine to the claimed powers:
+
+ H^poly <= C_eps n^eps m^{-4/3} (n/m)^{1/3}.
+
+Reading.
+
+ - The -4/3 is the polygonal length exponent (the same
+ R^{4/3} from Theorem 2b).
+ - The 1/3 balances the sum over scales in 5.5, so that
+ the two-bond sums converge to the H^{2/3} length-square
+ bound.
+
+End of PART 6.
+
+================================================================
+PART 7. WHAT'S NEW
+================================================================
+
+The paper's contributions, ordered by ease of reading.
+
+7.1 The cylinder pressure exponent m^{1/6}.
+
+Many loop models are positivity-dominated (FK percolation,
+Ising, Symanzik), so their critical exponents come from
+monotonicity arguments. The signed-transfer representation
+of this paper has no positivity at all: individual
+configurations carry complex or negative weights that can
+be large, canceling to a small total. To prove power laws
+from such a representation, one needs absolute bounds
+strong enough to yield the correct answer. Prior to this
+paper, this had not been carried out for the balanced
+cylinder at fugacity 2 of the dilute O(n) model. The 1/6
+is new.
+
+7.2 The planar nesting exponent 1/12.
+
+Follows from 1/6 by a localization argument (4.11), but
+had not been previously available. The 1/12 counts cycle
+families, not walk lengths, so it is a genuinely different
+quantity from the standard 1/nu = 4/3.
+
+7.3 The chord law.
+
+Mean chord length R^{4/3}; bulk visit probability
+R^{-2/3}; total chord mass R^{3/4}. The mean length
+matches the conjectured SAW length exponent 4/3, and the
+visit probability matches the conjectured R^{-2/3}. The
+results are not only for a probability law but for the
+exact normalization Z_R^chord.
+
+7.4 The length-square mass H^{2/3}.
+
+A two-bond computation, independent of Theorems 1 and 2.
+Any two-bond sum through two specified bonds at distances
+m, n is bounded by m^{-4/3} (n/m)^{1/3}, which summed over
+scales gives H^{2/3}.
+
+7.5 The method.
+
+The signed transfer / absolute stability / compact
+propagator / Jordan-safe trace extraction pipeline is
+general-purpose. It should apply to any dilute O(n)
+observable once the same absolute stability holds at the
+relevant displacement d.
+
+================================================================
+PART 8. HONEST LIMITS
+================================================================
+
+What the paper does not prove, and where its
+simplifications approximate.
+
+8.1 The SAW Flory exponent 3/4.
+
+The mean chord length R^{4/3} matches the conjectured
+reciprocal Flory exponent 1/nu = 4/3. But this is a
+different object: the normalised mean of first-exit hexagon
+chords, not the mean-square end-to-end displacement of a
+uniformly chosen length-n walk. The matching exponents are
+consistent, not equivalent.
+
+8.2 Uniformity in the exponents, not the constants.
+
+Every R^{a + o(1)} carries a constant that may depend on
+the fixed parameters (fugacity, displacement, side length,
+cut lengths, counts). The exponents are uniform. Polynomial
+prefactors are not.
+
+8.3 No purely logarithmic estimate.
+
+Several intermediate bounds are power bounds with an o(1)
+error; they do not give a logarithmic rate. For example,
+the arch mass W_R is R^{-1/4+o(1)}, with the o(1) constant
+a slowly growing function of R.
+
+8.4 The marked lower bound is not proved.
+
+Theorem 3 is an upper bound on the two-bond sum. A
+matching lower bound is not claimed. The applications in
+the paper (the H^{2/3} length-square mass) only need the
+upper bound.
+
+8.5 Numerical constants.
+
+The seven activities are explicit (sqrt(2)/4 times
+exponentials). The stability constants c and C are not.
+The Pfaffian scalar is explicit. Optimal choices of
+b, J, k_1 are not discussed.
+
+8.6 What is not handled.
+
+- Arbitrary fugacity g. Focus is g = 2 (and complex g near
+ 0 for analytic continuation).
+- Arbitrary cycle multiplicities (potentials).
+- The limit cases d -> 0 and d -> 1/2 of the displacement
+ are explicitly not handled uniformly.
+- Higher-dimensional analogues are not addressed.
+
+================================================================
+PART 9. GLOSSARY
+================================================================
+
+Symbols and terms, approximately alphabetical. The
+section tags refer to PART numbers in this document.
+
+alpha = pi/8. Turn weight in the parafermionic balance.
+
+arch. A first-exit walk in a half-plane, counted with
+weight x^N cos(3 T / 8) after turning.
+
+B(v) = (cosh(pi v / 3) - sqrt(3)/2) / (cosh(pi v / 3) +
+sqrt(3)/2). The gas B factor. Zeros at v = i (plus or
+minus 1/2 + 6n), poles at v = i (plus or minus 5/2 + 6n).
+
+bad vertex. On a polygon lambda, a vertex within distance
+r_K = (K - 1/2) / sqrt(3) of one of its sharp turns. A
+polygon has at most 2K bad vertices.
+
+beta = 3/8. Turn-phase ratio s.
+
+c_alpha = 2 sin(alpha). Boundary constant in the arch
+mass identity sum_d H(d) = 1 / c_alpha.
+
+chain. A self-avoiding polygon (a closed self-avoiding
+walk).
+
+chord. A first-exit walk between two boundary ports.
+
+C_R. The set of first-exit chords in D_R of diameter at
+least R / 10, with both boundary ports summed.
+
+critical weight x = (2 + sqrt(2))^{-1/2} = 0.5412...
+Multiplies x^N for every length-N walk and polygon.
+
+D_A. The cutting insertion (i h)^{sum_{j in A} s_j} along
+the A-cut of the balanced cylinder.
+
+delta = 2 or 3. Fusion index: delta = 2 collapses two
+sites to one, delta = 3 to nothing.
+
+D_R. The regular tessellation hexagon of side length R.
+
+discrepancy bound. The constant D of Theorem 3: within
+each bin, the count differs from its expected value by at
+most D.
+
+E_delta. The smaller fusion image space. E_2 is
+3-dimensional (spins -1, 0, +1); E_3 is 1-dimensional.
+
+E_0, I_0. Empty-block operators. E_0 allows all seven
+particle types, I_0 allows only M and F_pair. Their
+spectral radii are r_E = e^{-b pi / 16} and 1.
+
+E_theta. The homogeneous "all types" block operator at
+angle theta.
+
+f = n / (m + n). Aspect ratio of a permitted tilted
+cylinder. Bounded by 0.6 in Theorem 3.
+
+F_{s_a}. Per-site factors in the generic flip formula:
+F_+ = e_5 e_6, F_0 = e_3 e_5, F_- = e_2 e_3, where
+e_k(u, v) = v - q^k u.
+
+F_pair. The fused particle after a U-L collision.
+Charge -2, site multiplier 1.
+
+G_m(g) = G_{A,B}(g). The balanced cylinder partition at
+fugacity g. With g = 2 cos Theta, its complex
+continuation enters the proof.
+
+gamma_r. Small contour around the origin used in 4.4.
+
+gamma_v. The local activity of a gas particle of type v.
+Explicit in 4.5.
+
+gamma_{F_pair} = (9 pi^2 / 8) exp( K_diag(0) + K_diag^reg(3i) ).
+Positive fused-particle activity.
+
+good vertex. A vertex that is not bad. Every polygon of
+length lambda has at least |lambda| - 2K good vertices.
+
+H(d). Half-plane arch weight from a port to a port d
+steps to the right. sum_d H(d) = 1 / c_alpha.
+
+H^flip(X). Signed two-flip algebraic expression of 6.2.
+Agrees with H^poly at the physical list.
+
+H^poly(X). The positive two-bond sum of Theorem 3.
+
+I_M. Incompatible-pair sum at g = 0. Equals O(log M).
+
+J_delta. Fusion embedding E_delta into V tensor V.
+
+k_1 = pi / 3. Vertical site-rate scale.
+
+K_diag = (K^r)_{sigma sigma}. Diagonal part of the H
+kernel.
+
+kappa_T. Gas scalar rate. Limit 2 log(2(2 + sqrt(2))/9).
+
+lambda = e^{-2 i alpha}. The charge-eigenvalue.
+
+M = 2m. Period of the balanced cylinder (even).
+
+mu. Running common charge in the gas.
+
+N(X). Pfaffian scalar. N(1, ..., 1) = (2 p_0 / 3)^{m(2m-1)}
+times (2/3)^m times m^{5/48 + o(1)}, with p_0 = 2 + sqrt(2).
+
+omega_v(x). Site multiplier for a particle of type v.
+Table in 6.4.
+
+p_*(Theta) = 1/6 - 2 Theta^2 / (3 pi^2). The complex
+cylinder exponent.
+
+Pi_p. Spectral projector for twist p. Explicit
+Pfaffian det L(T) / det L(1).
+
+q = e^{2 i alpha}. The chiral deformation parameter.
+
+R. Hexagon side length, or the free radius parameter in
+W_R.
+
+R(phi). The transfer matrix with five trigonometric
+entries (u, d, v, a, b).
+
+R(u, v). Residual pair factor of the gas kernel.
+
+r_E = e^{-b pi / 16}. Spectral radius of E_0.
+
+r_K = (K - 1/2) / sqrt(3). Bad-vertex radius.
+
+r_X^+. The largest exterior rate of 6.6.
+
+S_D(v). The signed visit sum of 5.3.
+
+seam. The twist line where p = i above and p = -i below
+the cut.
+
+T. Period (circumference) of the horizontal gas.
+
+Theta. Complex insertion angle, g = 2 cos Theta.
+
+theta. Regulator angle in the complex partition.
+
+u, d, v, a, b. The five trigonometric weights of R(phi).
+At phi = alpha: (x, x^2, x^2, x^2, 0).
+
+U, V. The two flip endpoint variables.
+
+u_k = exp( -4 pi b / 9 * (k + theta / pi)^2 ). Eigenvalues
+of I_0.
+
+u_c = e^{-pi b / 4}. Central rate of 6.6.
+
+w_e. The endpoint coordinate (marked bond site).
+
+W_R. Arch mass at distance at least R in the half-plane.
+Equals R^{-1/4 + o(1)}.
+
+x. See critical weight.
+
+z = e^{-2 i phi}. The transfer-matrix eigenbasis variable.
+
+Z(R). Planar nesting partition around the central face
+center in D_R, at fugacity 2. Equals R^{1/12 + o(1)}.
+
+Z_R^chord = sum_{gamma in C_R} x^{N(gamma)}. Chord
+normalizer. Equals R^{3/4 + o(1)}.
+
+Z_{m,T}. The exact gas at period T and site count m.
+
+zeta. A peripheral eigenvalue of E_0 (or E').
+
+End of PART 9. End of document.