diff b2e687e..2d5618e signed_cylinder_marked_polygons_v2.txt
diff --git a/signed_cylinder_marked_polygons_v2.txt b/signed_cylinder_marked_polygons_v2.txt
index 0864837..ae2083d 100644
--- a/signed_cylinder_marked_polygons_v2.txt
+++ b/signed_cylinder_marked_polygons_v2.txt
@@ -871,3 +871,1217 @@ identified later: Z_R^chord = R^{3/4 + o(1)} and Z(R) = R^{1/12 + o(1)}.
Combining gives the mean E_R^chord N = R^{4/3 + o(1)} and bulk visit
probability R^{-2/3 + o(1)}. The proofs of the two identities occupy the
bulk of the paper.
+
+The finite spin sum.
+
+Place two groups A and B of m sites each on the horizontal cut of the
+cylinder, m even. A spin configuration assigns each site a value in
+{-1,0,1} and is retained only when the total charge in each group is zero.
+A site carries a variable x_a and a chosen square root t_a, x_a = t_a^2.
+The two-group spin sum F(X) is the sum over retained configurations of the
+product of the two-site factors, multiplied by the square-root normalisation
+prod_a t_a^{s_a^2}. Empty products have value one. The two-site factors
+factorise, for each ordered pair of spin values, into two pieces: a
+two-by-two Schur-type determinant contribution, antisymmetric in the pair,
+and a symmetric Gaussian kernel. This factorisation is what makes the sum
+reproducible by a Pfaffian contour integral below.
+
+Individual terms have simple poles when two site variables coincide. The
+sum is regular there. Residues cancel in pairs: a pair of sites with
+mismatched occupancy can exchange which coordinate carries the nonzero
+spin without changing the total charge content of the configuration, and the
+two residue coefficients are opposite. Iterating the exchange on a group of
+r coincident sites reduces the pole order by one at each step, so the sum is
+regular at simultaneous coincidences of any multiplicity.
+
+Contour replacement. For each fixed number of sites, the value of F at
+homogeneous rapidities is replaced by a periodic Pfaffian contour
+expression. Its limit as the horizontal period tends to infinity recovers
+F. Concretely, write w_j for the logarithmic site coordinates and let P be
+the period; the periodic contraction kernel is the sum over horizontal
+images of a fixed odd kernel, and the Pfaffian of the resulting matrix
+equals the finite spin sum term by term. Two inequalities control the
+passage to the limit: the contraction kernel decays exponentially in the
+imaginary direction, and the diagonal of the Pfaffian matrix is bounded by a
+constant multiple of the off-diagonal mass. The limit is therefore uniform
+on compact sets of site coordinates.
+
+Two kernels occur. With tilde B_beta(t) = sinh t / (cosh t + cos beta),
+put
+
+ B_0(w) = tilde B_{pi/4}(pi w / 4), C(w) = tilde B_{pi/2}(pi w / 6).
+
+Both are odd and analytic on |Im w| < 3, both are strictly increasing on the
+real line with B_0(w) > 0, C(w) > 0 for w > 0, and both are the ratios of an
+odd kernel to its own divided-difference normalisation. Their derivative
+transforms are
+
+ hat{B_0'}(k) = 8k cosh k / sinh 4k, hat{C'}(k) = 6k / sinh 3k,
+
+using hat h(k) = int_R e^{-ikw} h(w) dw. Both densities are strictly positive
+and even, all moments are finite, and
+
+ hat{B_0'}(k) / hat{C'}(k) = (2/3) e^{phi(k)},
+ phi(k) = log((1+t^2)(1-t^6)/(1-t^8)), t = e^{-|k|},
+
+continued by phi(0) = log(3/2). The identity follows from
+hat{tilde B_beta'}(xi) = 2 pi xi cosh(beta xi) / sinh(pi xi), obtained by
+applying cos beta + sin beta partial_beta to the transform
+int_R e^{i xi t} dt / (cosh t + cos beta) = 2 pi / (sinh(pi xi/2)) times
+the corresponding factor, and then scaling.
+
+Transfer matrix. Let V = C^3 with basis |0>, |+>, |->, let s act
+diagonally with eigenvalues 0, +1, -1, and consider a row of M rhombi with
+row parameters phi_j. Remove the telescoping phase exp(i theta (left -
+right)/4), and set q = e^{2i alpha}. The oriented row matrix with twist p is
+
+ T_p(w) = Tr_a p^{s_a} R_{a1}(xi - phi_1) ... R_{aM}(xi - phi_M),
+ p in {i,-i}, w = e^{2i xi}, x_j = e^{2i phi_j}.
+
+On the charge-zero subspace H_M of (C^3)^{tensor M}, the local limits of the
+sine weights give
+
+ z = w/x_j -> 0: (a,b,v) -> (q^{-3}, q^3, 1), (u,d) = O(z^{1/2}),
+ z -> infinity: (a,b,v) -> (q^3, q^{-3}, 1), (u,d) = O(z^{-1/2}),
+
+holomorphic in z^{1/2} and z^{-1/2} respectively. On a state with auxiliary
+spin r and site spin s the diagonal limit is q^{3rs} at infinity and q^{-3rs}
+at zero. The only surviving off-diagonal move at infinity changes the
+auxiliary spin from -1 to 1 and the site spin from 1 to -1, and at zero the
+reverse. A surviving move at an inactive site therefore preserves vacancy
+parity. Square roots occur only in the single turns, whose number in an
+auxiliary trace is even, so T_p is rational in w with possible simple poles
+at q^5 x_j and q^6 x_j. Changing a square-root sign conjugates T_p by the
+parity operator P_j = (-1)^{s_j^2}. At either endpoint of the row parameter
+a closed auxiliary trace cannot use a strictly one-directional off-diagonal
+move, and the remaining diagonal trace is
+sum_{r=-1}^{1} p^r q^{+-3r sum_j s_j} = 1 on H_M. Hence
+
+ T_p(0) = T_p(infinity) = I.
+
+With C_l(z,b) = z/(z - q^l b) this gives the partial-fraction form
+
+ H_p(w) = T_p(w) - I = sum_{j=1}^{M} sum_{t=0}^{1} C_{5+t}(w, x_j) U_{jt},
+
+where the coefficient operators commute, the basis functions vanish at zero
+and have distinct poles, and the condition at infinity additionally forces
+sum_{j,t} U_{jt} = 0.
+
+Fixed vacuum lines. Let A be a generic linear combination of the finitely
+many probe deviations T_p(w_nu) - I, and let d_A be the scalar derivative
+partial_rho det(rho I - A) at rho = 0. Then zero is an exact eigenvalue of
+A, the projector is Pi_p = adj(-A) / d_A, and on the union of the three
+scales every diagonal block with a charged spectator is invertible. With all
+scales singleton and probes at the swap points, a diagonal block carries the
+eigenvalue p_j^{S_j}; odd total charge forces the exponent of q in p_j to be
+odd, so p_j^{S_j} is not 1, and only the all-empty block can carry the
+simultaneous character, with multiplicity one. A generic combination of the
+probe deviations therefore has a simple zero eigenvalue and, by rationality in
+square-root coordinates, a generically simple one, with no Jordan extension.
+
+Separated scales. Let positive scales rho_1, ..., rho_r satisfy
+rho_{k+1}/rho_k -> infinity, and let x_i/rho_{k_i} tend to a finite nonzero
+limit. With consistent local square roots and a probe w/rho_j tending to a
+finite nonzero limit at which the active local matrices are regular, the
+limiting row matrices are triangular in the grading sum_i k_i s_i with row
+grade at least column grade. Their diagonal grade blocks preserve every
+scale total S_k = sum_{i: k_i = k} s_i and act on the active scale j with
+twist
+
+ p_j = p q^{3 sum_{k != j} sgn(j-k) S_k},
+
+and each block entry is the corresponding active-transfer entry multiplied by
+F(s^{row})/F(s^{col}), with
+
+ F(s) = q^{(3/2) sum_{l<i} sgn(k_l - k_i) s_l s_i}
+
+restricted to the block. The proof is conservation of charge: with
+Delta s_i = s_i^{row} - s_i^{col}, an inactive site below the active scale
+gives Delta s_i <= 0 and one above gives Delta s_i >= 0, while total charge
+conservation gives Delta(sum_i k_i s_i) = sum_i (k_i - j) Delta s_i >= 0,
+strictly unless every inactive charge is unchanged. An inactive site
+contributes the diagonal factor q^{3 sgn(j - k_i) h_i s_i} with incoming
+horizontal spin h_i = h_0 + sum_{l<i} Delta s_l; the h_0 part changes the
+seam twist to p_j, and the remaining part cancels against F(s^{row})/F(s^{col})
+because inactive sites receive equal contributions from active sites before
+and after them.
+
+Selected vacuum vectors. Let Pi_i and Pi_{-i} be the projectors at the two
+twists, normalized by the empty vector: R_i = Pi_i |0>, L_{-i} = <0| Pi_{-i}.
+For M >= 1, generic nonzero X, and sigma in {-1,0,1}^M with zero total charge,
+put
+
+ V_sigma^R(X) = N(X) (R_i)_sigma / prod_j t_j^{sigma_j^2},
+ V_sigma^L(X) = N(X) (L_{-i})_sigma / prod_j t_j^{sigma_j^2}.
+
+The quotient by the normalizer N removes the common poles, and the functions
+are independent of the square-root signs. They extend to polynomials in
+C[x_1,...,x_M] with
+
+ deg_{x_j} V_sigma^epsilon <= M - 1 - sigma_j^2.
+
+With X_hat j the deletion of coordinate j and sigma_hat the corresponding
+deletion, for sigma_j = 0 and either superscript,
+
+ V_sigma(X)|_{x_j = 0} = (prod_{k != j} x_k) V_{sigma_hat}(X_hat j),
+ [x_j^{M-1}] V_sigma(X) = V_{sigma_hat}(X_hat j).
+
+Proof. Site sign covariance gives Pi_p(-t_j) =
+P_j Pi_p(t_j) P_j, and the empty vector has even vacancy parity, so
+(R_p)_sigma(-t_j) = (-1)^{sigma_j^2} (R_p)_sigma(t_j) and likewise for L_p.
+The determinant formula makes the quotients rational in the t_j and invariant
+under every sign change. The invariant rational field is generated by the
+squares, so both functions are rational in the x_j.
+
+For the pole removal, work with widely separated site magnitudes. One
+spectator on the same scale as the varied site forces every other site to be
+empty in a diagonal block carrying the fixed character, the active pair then
+has total zero and original twist p, and with coordinates u, v in
+left-to-right order and z = u/v the probe w = u is the twisted swap
+reduction R(phi_u - phi_v) diag(p^{s_first}). Its neutral block has
+eigenvalues 1, a, -a with
+
+ a = q^3 (z - q^2)(z - q^3) / ((z - q^6)(z - q^5)),
+
+the eigenvalue 1 coming from the vacuum character, trace 1, and trace of the
+square 1 + 2a^2. Normalizing the right and left fixed vectors r, l at the
+state (0,0), direct multiplication gives
+
+ r_- = i r_+, l_- = i l_+, l r = 1,
+
+ r_+ = d_tile lambda / (1 - i a / p), l_+ = d_tile p / (lambda (1 - i a p)),
+
+where d_tile is the nonzero local single-turn weight. At a pole of a the
+reverse probe w = v is regular with a(1/z) = 0, so with
+D(z) = (z - q^5)(z - q^6) the crossings are explicit:
+
+ a - 1 = (q^3 - 1)(z^2 - 1) / D(z),
+ a + 1 = (q^3 + 1)(z^2 + 2(1 + sqrt 2) z + 1) / D(z).
+
+At z = 1 use identical rapidity branches and also probe at w = q u; the
+twisted swap then has one eigenvalue 1 and one eigenvalue
+v_tile^2 + a_tile^2 + b_tile^2, which is not 1, so joint simplicity holds.
+
+Uniformity. Take a sequence of generic spectator lists whose scale gaps
+tend to infinity and a putative pole zeta not 0 or infinity. After passing to
+a subsequence, zeta lies either on a separated singleton scale or in a pair
+whose ratio tends to a finite nonzero limit, and the scale probes converge
+locally analytically in a square-root chart. Joint simplicity holds in the
+singleton case and at every simple pair ratio. The only excluded ratios are
+-1 and the two negative roots of z^2 + 2(1 + sqrt 2) z + 1. Around one of
+those, take a small complex disk and use the pair probe w = u together with
+the spectator singleton probes. The singleton tests keep every block with a
+charged spectator invertible, the pair eigenvalues cross transversely by the
+displayed factorisations, and the limiting scalar derivative has a simple
+zero at the central ratio. Convergence on the boundary of the disk together
+with the argument principle gives exactly one zero counted with multiplicity
+for large gaps, hence a unique simple zero; the projector has at most one
+simple pole in the disk.
+
+After dividing by the half powers the endpoint orders give no pole at zero,
+and at infinity deg_{x_j} N <= M - 1 gives degree at most M - 1 -
+sigma_j^2. The set of separated generic spectator values is Zariski dense,
+a relatively prime numerator and denominator in x_j stay relatively prime for
+generic specialisations of the spectators, so the one-variable conclusion
+holds over their coefficient field. Applying this successively to every j
+removes every irreducible denominator and proves joint polynomiality,
+including at simultaneous divisor intersections. The deletion values follow
+from the coordinate limits. QED
+
+Scalar characterisation. Write e_k(u,v) = v - q^k u, put
+F_{A,B} = N(X)^2 G(X; A, B) for the pairing G of the two vacuum families,
+and record the four properties that determine it. The empty-list value is 1.
+For M = |A| + |B| >= 1 each polynomial is symmetric within the two groups and
+has individual degree at most 2(M-1). Deleting a site j gives
+
+ F_{A,B}|_{x_j = 0} = (prod_{k != j} x_k^2) F_{hat j},
+ [x_j^{2(M-1)}] F_{A,B} = F_{hat j}.
+
+For two sites u, q^delta u in one group, delta = 2 or 3,
+
+ F_{A,B}(u, q^delta u, Z)
+ = (u + q^delta u)^2 (prod_{z in Z} ell_delta(u,z)^2) F_red(X_red),
+
+where X_red is the reduced list with the pair replaced by one site and
+ell_delta(u,z) = z^2 - (q^delta + q^{-delta}) u z + u^2 measures the shift.
+When |A| = |B| the quotient of F by N^2 has the paired-scale limit at
+generic fixed pair ratios. This family is the unique family with these
+properties.
+
+Proof of uniqueness. Induct on M and take the difference of two such
+families. If the larger group has size n > M/2, vary one of its sites.
+Induction gives zeros at the 4(n-1) fusion points q^{+-2}, q^{+-3} times the
+other sites of that group and at zero, and the top coefficient vanishes, so
+the degree is at most 2M - 3. Since 4(n-1) + 1 > 2M - 3, the difference is
+zero.
+
+If both groups have size n, then M = 2n, the vanished top coefficient reduces
+the degree in each variable to at most 2M - 3, and the zero at the origin
+together with the four zeros for each same-group partner gives degree
+1 + 4(n-1) = 2M - 3. Divisibility by these distinct linear factors shows that
+the difference is a constant in the site variables times
+
+ prod_i x_i prod_{same-group pairs i,j}
+ e_2(x_i,x_j) e_3(x_i,x_j) e_5(x_i,x_j) e_6(x_i,x_j).
+
+To determine the constant, put matched coordinates u_j, v_j on successive
+increasing scales rho_j. At generic ratios the normalizer has the limit
+
+ N(X) ~ prod_j (u_j + v_j) prod_{j<k} (u_k v_k)^2.
+
+Grouping the pairs in the Pfaffian, the two cross rows toward each external
+pair have identical limits, so the Pfaffian factors into its internal pair
+terms, and the divided-difference normalisation supplies the displayed
+cross-scale powers. If the coefficient of the possible difference were
+nonzero, its quotient by N^2 would have the nonzero limit given by that
+coefficient times prod_j u_j v_j / (u_j + v_j)^2. The common paired-scale
+normalisation forces the constant to vanish. QED
+
+Fusion multipliers. At v = q^delta u within a group the surviving terms fuse
+bijectively to reduced configurations. Write the fused spins as a, b, the
+reduced spin as gamma, and use * for a removed pair. For a spectator spin t at
+w, divide its two couplings by the reduced coupling when there is one; the
+result is the monic factor ell_delta(u,w)^2 times q^{nu t} for a same-group
+spectator and q^{-nu t} for a cross spectator. Divide the internal site and
+pair factors by the reduced site factor, when present, and by
+(u + q^delta u)^2; call the result I. The possibilities are
+
+ delta a b gamma nu I
+ 2 - 0 - 1 q^{-1}
+ 2 0 + + 1 q
+ 2 - + 0 -2 1
+ 3 - + * -1 1
+
+All other types have a zero e_delta. The spectator phases follow by adding the
+index multisets for (a,t) and the delta-shifted multiset for (b,t), counting
+denominator multiplicities negatively, and for delta = 2 subtracting the
+1-shifted multiset for (gamma,t); the remaining indices are 2[5] or
+2[5] + 2[6] respectively, and the kappa differences give the displayed
+phases. Internally,
+
+ p_{-0}(1,q^2) = p_{0+}(1,q^2) = (1+q^2)^2,
+ p_{-+}(1,q^2) = -2/c, p_{-+}(1,q^3) = -1,
+
+using (q-1)^2 = -c q. The spectator phases and I cancel by the charge
+constraint, and Q is unchanged.
+
+The two-colour braid-monoid formulation of dilute loop models and the fusion
+and current recursions for open-strip dilute O(1) models supply the
+integrable-model background for this structure, although neither identifies
+the marked vacuum pairing or the physical normalisation used below. A
+description of that background appears in
+
+ P. A. Pearce, J. L. Jacobsen and J. de Gier, The dilute O(1) loop model
+ on the honeycomb lattice, Journal of Physics A: Mathematical and General
+ 41 (2008), 335002.
+
+The fusion multiplicities p_{-+} with the two denominators (1 - i a/p) and
+(1 - i a p) are the standard Yang-Baxter fusion residues, and their unitarity
+relation p_{st} p_{ts} = 1 is the local inverse identity above.
+
+The angular fermionic trace.
+
+Fold the imaginary coordinate of the cylinder into six sheets indexed
+cyclically by j = 1, ..., 6, and write j* = j + 3. On the fermionic exterior
+space take
+
+ {Psi_{j,r}, Psi_{l,s}} = delta_{j*,l} delta_{r,-s},
+ Psi_{j,r}^* = Psi_{j*,-r}, r, s in 1/2 + Z,
+ Psi_j(v) = sum_r Psi_{j,r} y^{-r}, y = e^{2 pi i v / T}.
+
+Modes with r > 0 annihilate the vacuum. Let L_0 be the total creation
+energy, put qq = e^{-2 pi / T}, and let U fix the vacuum and act by
+Psi_j -> -Psi_{j-1} under conjugation. Set Gamma = qq^{L_0} U.
+
+Conditional angular trace. Fix T > 0, m >= 0, theta in C, and a homogeneous
+analytic sample of the Gaussian and opposite-uniform fields. The normalising
+trace is
+
+ Z_Gamma = Tr Gamma = prod_{r in 1/2 + N_0} (1 - qq^{6r}) > 0.
+
+For s >= 1, if the folded coordinates have strictly increasing imaginary
+parts with total range less than 1, then
+
+ (-2 pi i / (a_* T))^s Tr(Gamma Psi_{j_1}(v_1) ... Psi_{j_{2s}}(v_{2s}))
+ / Z_Gamma
+ = Pf_{1 <= a,b <= 2s} C^{[T]}(v_b - v_a + i (j_b - j_a)).
+
+The trace denotes its absolutely convergent ordered mode sum; for s = 0 both
+sides are 1; odd products vanish; the formula at other nonsingular
+coordinates is its meromorphic continuation.
+
+Proof. The operator qq^{L_0} is trace class because
+prod_{r>0} (1 + qq^r)^6 < infinity, and U is unitary. At each positive energy
+r the exterior trace is the determinant of the identity plus the one-particle
+action, which is -qq^r times a six-cycle, so its determinant is 1 - qq^{6r};
+the sum of qq^{6r} converges, giving the product and its positivity.
+
+Cyclicity for bounded individual CAR modes gives, for r > 0,
+
+ C_{jl} := Tr(Gamma Psi_{j,r} Psi_{l,-r}) / Z_Gamma
+ = sum_{n>=0} qq^{rn} delta_{l-n, j*},
+
+from the recurrence C_{jl} = delta_{l,j*} + qq^r C_{j+1,l}. The CAR then give
+-sum_{n>=1} qq^{-rn} delta_{l+n,j*} for r < 0, and the other energy pairings
+vanish. Summing the half-integer modes for 0 < Im(v-u) < 1 gives
+
+ Tr(Gamma Psi_j(u) Psi_l(v)) / Z_Gamma
+ = (i/2) sum_{n in Z, n = l-j-3 mod 6} 1/sin(pi (v-u+in)/T),
+
+the two geometric sums converging on the two sides of the annulus.
+Multiplication by -2 pi i/(a_* T) gives C^{[T]}(v-u+i(l-j)). Moving one mode
+cyclically in a higher product, contracting it by the CAR, and solving the
+same six-step recurrence gives the Pfaffian rule. For ordered heights
+eta_a < eta_b, direct contractions decay from eta_b - eta_a and contractions
+wrapping through Gamma decay from 1 - (eta_b - eta_a); both are positive in
+the stated strip, which makes finite products of mode sums absolute. Graded
+alternation gives the meromorphic continuation. QED
+
+Graded current bound. Let P_N project onto the energy-N subspace,
+N in (1/2) N_0. For a fixed finite word Y of exponentials of the currents
+X_{bc}(f) with periodic coefficients analytic on a common strip of positive
+width, there are C, c > 0 with
+
+ ||P_{N'} Y P_N|| <= C e^{C sqrt(N+1)} e^{-c |N'-N|},
+
+uniformly in bounded numerical multiples of the currents in the
+exponentials. The energy-level trace with Gamma is therefore absolutely
+convergent, as is every exponential series of scalar trace coefficients.
+
+Four-current trace. For the periodic current monomial with inputs b, b^{-1},
+b_+^{-1}, b_- and the fixed analytic sample,
+
+ C_sigma = (1/Z_Gamma) Tr_gr(Gamma e^{X_{23}(b_+^{-1})} e^{X_{21}(b)}
+ e^{-X_{12}(b^{-1})} e^{-X_{16}(b_-)}),
+
+where Tr_gr(Gamma Y) = sum_N qq^N tr_{P_N F}(U P_N Y P_N) with F the fermionic
+exterior space.
+
+Fermion contraction identity. For X = x + i a and (Phi_1, Phi_2) =
+(U, V) or (V, U), there are constants c_1, c, C > 0 with
+
+ |G_Phi(X)| <= c_1 a^{-3/2},
+ |G_Phi(X) - c_2 (2 pi / a)^{1/2} e^{-X^* X / (2a)}| <= C (b/a)^{3/2},
+
+where c_2 involves Phi_1(0) Phi_2(0) and their first moments, and
+X^* X = x^2 + (a - b)^2. The derivation separates the two theta groups:
+with w = exp(i Theta) and h = w^2,
+
+ g_{Phi_1,0} = e^{i r X},
+ g_{0,Phi_2} = e^{i s X} / [2 sinh(pi (X + i beta)/T)] with beta = 3 or 4,
+
+and each factor is pairing-symmetric. With
+w^{-2} = e^{-i Theta} h^{-1/2} = e^{-i Theta} y^{1/2} / (1 + y) and q = e^{2 i
+alpha}, the variable change q -> q e^{i pi/2} passes to diagonal theta
+differences. The short-time expansion of h^{-u} is u-specific, and the terms
+with u not in {n/2} vanish by pairing antisymmetry, leaving the displayed
+estimate after the constant, linear, and quadratic exponents are gathered.
+
+The conditional trace identities hold for each fixed T. Passing the Gaussian
+expectation through the reorganized expansion uses the absolute estimate of
+the next section, and the gas identity is stated at sufficiently large integer
+periods to which that estimate applies.
+
+Bosonization to a horizontal gas.
+
+The vertical coordinate of the cylinder compresses into a horizontal circle
+of length T. Each spin configuration of the finite sum becomes an ordered
+list of Weyl currents at the real projections of the sites, with an
+oscillator trace over the auxiliary direction and a common state carried by
+the running charge.
+
+Three-root form. Fold the six sheets of the fermionic trace around the
+horizontal cut into three ordered roots 1, 2, 3. For a normalised current
+W_b = W_{b_w} exp(A) with A of zero spatial mean and
+b_w = b_0 y^{-ell}, the action on a ground-charge state |c> = |c_1, c_2, c_3>
+is
+
+ W_{b_w} |c> = (-1)^{c_1 + c_1 c_2} b_0^{ell + c_2 - c_1}
+ |c_2 + ell, c_1 - ell, c_3>,
+
+together with interchange of the oscillator labels 1 and 2.
+
+Verification. In the three-root word the modes with r + s = ell form
+commuting blocks (1+F)(1+E')(1+F) with
+
+ F = b_0^{-1} Psi_{4,r} Psi_{2,s}, E' = -b_0 Psi_{5,-s} Psi_{1,-r}.
+
+At opposite signs of r and s the block fixes the vacuum, at both positive it
+contributes E', at both negative it contributes F. Grouping creators by
+species and reversing the second energy list leaves
+W_{b_w}|0> = b_0^ell |ell, -ell, 0>. Conjugation sends
+Psi_1 -> -b_0^{-1} y^ell Psi_2 and Psi_2 -> b_0 y^{-ell} Psi_1, so
+increasing c_1 multiplies the ground coefficient by
+(-1)^{1+c_2} b_0^{-1}, increasing c_2 by (-1)^{c_1} b_0, and c_3 is inert.
+These ratios together with the vacuum value give the displayed action.
+
+Rotation of the ground charges sends d -> (d_2, d_3, -d_1) with sign
+(-1)^{sum d_j + d_1(d_2 + d_3)}, counting the creator signs and the move of
+the first species list past the other two. Applying rotation to the action
+above gives the seam-twisted form
+
+ Gamma W_{b_w} |c> = qq^{|c''|^2 / 2} (-1)^{c_2 + c_3 + c_2 c_3}
+ b_0^{ell + c_2 - c_1} |c''>, c'' = (c_1 - ell, c_3, -c_2 - ell),
+
+with qq = e^{-2 pi / T}. On creation oscillators of frequency n > 0 it acts,
+in addition to the ground action, by B_n = qq^n diag(1, R) with
+R = [[0,1],[-1,0]].
+
+Oscillator contractions. For normalised creators with coefficient vector u
+followed by annihilators with vector v', the oscillator trace with
+propagator B_n, divided by its empty trace, is
+
+ exp( v'^t B_n (I - B_n)^{-1} u ).
+
+On one oscillator with multiplier t, cyclicity and the commutator give
+<(a^*)^l a^l> = l! (t/(1-t))^l and zero for unbalanced moments;
+diagonalising the unitary part gives the vector formula. Reordering two
+exponentials adds the contraction of the earlier annihilator with the later
+creator; an unsplit exponential adds half its internal contraction.
+
+Applied to the zero-mean source exp(sum_n A_n (-h_{1,n} + h_{2,n})) with
+k = 2 pi n / T > 0 and s = (-1, 1, 0)^t, the Fock identity gives
+
+ s^t ( (1/2) + B_n (I - B_n)^{-1} ) s
+ = (1/2) coth(k/2) + (1/2) tanh k
+ = cosh(3k/2) / (2 sinh(k/2) cosh k) =: E(k),
+
+and with D(k) = k E(k)/(2 pi) the source self-contraction is
+exp( (1/2) int_T A D A ). Cross contractions dress a positive short and a
+positive outer field by
+
+ f_s^osc(w) = -sum_{k != 0} A_n e^{ik w} / (2 sinh(k/2)),
+ f_o^osc(w) = sum_{k != 0} A_n e^{ik w} / (2 cosh k),
+
+a negative field receiving the negative exponent. In the short coordinate
+y^n = qq^{n/2} e^{ik w}; the two outer coordinates carry y^n = qq^n e^{ik w}
+and e^{ik w}. The empty oscillator trace is
+
+ D_W = prod_{n>=1} (1 - qq^n)^{-1} (1 + qq^{2n})^{-1}.
+
+Signs. With N_j the number of fields of type j and n_j those at each root,
+the Klein exponent apart from inversions is n_1(N_2 + N_3) + n_2 N_3, and
+adding the seam sign and substituting the charge closure
+n_1 = ell, n_2 = (F_3 - E_2 - ell)/2, n_3 = (-F_3 - E_2 - ell)/2 leaves,
+modulo two,
+
+ (F_3 + E_2)/2 + #(3 before 2) + #(outer before short).
+
+Each such phase is absorbed by the coordinate powers and the modular pair
+factors; after the modular self factors remove (E_2+F_3)^2/4 and the pairs
+contribute tanh(pi d'/4) with d' = w_l - w_i, the surviving sign is
+(-1)^{#(outer before short)}, which combines with the odd like-family bases
+to make the coordinate sum alternating under exchange of neighbouring
+fields. Whole even current pairs may therefore be reordered without a sign,
+and the total winding scalar per species is
+
+ exp( -pi T ell^2/12 - 2 pi ell^2/(3T) ).
+
+Rationality and residues. Put Delta(X) = prod_{j<k}(x_j - x_k). The kernel
+P_* is analytic off the four points 1, -1, 1+sqrt2, 1/(1+sqrt2), where its
+denominator and numerator zeros are simple, and the product
+P_*(u,v)(u-v)^2 is analytic and nonzero near u = v = b != 0. Hence
+det L(1)/Delta(X)^4 is a nonvanishing analytic factor near the homogeneous
+list, and the divided differences of det L(1) exist there.
+
+Particles. A short or outer entry (a, epsilon) denotes a field at
+w = x + i a of sign epsilon; an H entry (a, r) denotes the full factor
+H(x+ia)^r. All entries in one row belong to one species. The four rows
+P, M, O_+, O_- come from roots 12, 15, 13, 16 in the horizontal weight; U
+and L come from the rightmost and the cycled leftmost root. The rows are
+
+ P shorts (1/2, +1) outers (1, +1) sign + H (0, +1)
+ M shorts (1/2, -1) outers (1, +1) sign + H (0,+1),(1,-1),(-1,-1)
+ O_+ shorts empty outers (1,+1),(0,+1) sign + H (0,+1),(1,-1)
+ O_- shorts empty outers (1,+1),(0,-1) sign - H (0,+1),(-1,-1)
+ U shorts (d, -1) outers (1/2+d, +1) sign - H (3/2+d, -1)
+ L shorts (-d, -1) outers (-1/2-d, -1) sign + H (-3/2-d, -1)
+
+The internal order puts the height-1 outer first in each D row and the short
+first in U and L. A cycled D pair carries Psi_3 Psi_{j-1} at x+i and its
+coefficient at x; the recurrences give the four dresses
+
+ H(x), H(x) / (H(x+i) H(x-i)), H(x)/H(x+i), -H(x)/H(x-i),
+
+and for U at folded coordinate v = x + i(1/2+d) the dress is -H(v+i)^{-1},
+for L at its pre-cycling coordinate v = x - i(1/2+d) the dress is
+H(v-i)^{-1}. These are exactly the listed heights and signs.
+
+Fused type. The meromorphic identities B(v+i)B(v-i) = B(v) and
+B(v+3i) = 1/B(v) for
+
+ B(v) = (cosh(pi v/3) - sqrt 3/2) / (cosh(pi v/3) + sqrt 3/2)
+
+keep the seven-type list closed under collisions, and the two-particle
+boundary state of U and L carries charge -2 in one species with the listed
+weights. The fused row F_pair has shorts empty, outers empty, sign + and H
+weights -delta_{3/2} - delta_{-3/2}, with the physical count 1/2 and the
+physical charge -1 relative to the count-one row R_d = (U+L)/2.
+
+Site factors. The deterministic factor multiplying the m-th power of each
+site is omega_v(x)^m, with omega_v read off the B factors and the signed H
+powers. The scalar charges, height moments, and site multipliers are
+
+ v q_v s_v b_v omega_v(x)
+ P 1 1/2 0 B(x)
+ M -1 -1/2 0 1
+ O_+ 0 0 -1 B(x-i)
+ O_- 0 0 1 B(x+i)
+ U -1 -d -3/2-d B(x + i(3/2+d))^{-1}
+ L -1 d 3/2+d B(x - i(3/2+d))^{-1}
+ F_pair -2 0 0 1
+
+The periodised product is omega_{v,T}, with value 1 in the two constant rows.
+The original deterministic factor is exactly omega_{v,T}(x)^m. The displayed
+simplified forms also prescribe the value when separate factors in a ratio
+have cancelling zeros and poles. Every basic activity is sqrt 2/4 times an
+exponential of a fixed kernel; M and F_pair have site multiplier exactly 1.
+
+The horizontal gas.
+
+Constant modes. Write theta = vartheta + i tau, X = Y + i Z, and
+alpha_p = a_p + i tau 1 with a_p real. Each heat summand of the completed
+contour current has modulus
+
+ |e^{-T alpha_p^2/(4 pi) - alpha_p . X}|
+ = e^{-T a_p^2/(4 pi) - a_p . Y} e^{T tau^2 |1|^2/(4 pi) + tau 1 . Z}.
+
+The second factor costs e^{O_theta(T+N)} with N the particle count, because
+the height lists are bounded. At fixed k_0 = p_1 + p_2 the opposite uniform
+mode and the remaining integer sums unfold to du/(2 pi) on (u, -u). The
+common Gaussian has covariance pi P_+/(4T) with P_+ the projector onto
+(1,1)^t, and the real completion, including e^{-pi Y^t S Y/T}, gives
+
+ (sqrt(8/9) / sqrt(2T)) sum_{k_0 in Z}
+ e^{-T beta_vartheta^t L beta_vartheta/(4 pi) - beta_vartheta^t L Y},
+ beta_vartheta = (vartheta + pi k_0) 1.
+
+Here L is the fixed real symmetric two-by-two symbol of the common channel:
+L kills the difference vector (1,-1) and is positive on the diagonal, so the
+common channel is massless and the opposite channel is massive. The opposite
+integral contributes (2T)^{-1/2} e^{pi Y_-^2/T} and the common integral
+contributes sqrt(8/9) e^{pi Y_+^2/(9T)}, with Y_+ = 1.Y/sqrt2 and
+Y_- = (1,-1).Y/sqrt2; the two exponentials cancel the corresponding parts of
+-pi Y^t S Y/T.
+
+Discrete phase check. Choose p_0 in Z^2 with p_{0,1} + p_{0,2} = k_0 and put
+p(x) = p_0 - Q(x). Expanding the completed square, using
+int_T Q dx = T ell - sum_j q_j x_j and L ell = 0 with ell = (1,-1)/sqrt2,
+the real linear term, the heat term and the mean term combine into
+
+ -pi int_T (p + (theta/(2 pi)) 1)^t L (p + (theta/(2 pi)) 1) dx.
+
+The imaginary terms become
+
+ -i pi sum_j (s_j - S b_j) . (p(x_j-) + p(x_j+)) - i theta 1^t L sum_j b_j.
+
+The particle data give
+
+ s_v - b_v = (1/2, -1/2, 1, -1, 3/2, -3/2, 0)
+
+for P, M, O_+, O_-, U, L, F_pair. This vector is congruent to q_v/2 modulo
+integers, so its total is integral in each species by the gas charge closure.
+The phase left by the substitution is
+
+ 2 pi p_0 . sum_j (s_j - b_j) - pi ell . sum_j (s_j - b_j),
+
+a multiple of 2 pi. Shifting p_0 by 8(1,-1) leaves every local weight
+unchanged, and for fixed k_0 the eight classes modulo that shift carry the
+same full-tour weight, because shifting between them by (1,-1) changes the
+product phase by 2 pi times the same integral total. This supplies the factor
+1/8 in the normalisation below.
+
+Fused local phase. If U precedes L and s = z_L - z_U = x - 2 i d, the full
+short and H factors divided by their residual factors give
+
+ e^{pi s - pi S_{sigma sigma}(s - 3i)}
+ = e^{pi L_{sigma sigma} x} e^{i pi (3 S_{sigma sigma} - 2 d L_{sigma sigma})}.
+
+The first factor is the pair term in the state square. The second is the
+product of the local phases for successive states p, p + e_sigma,
+p + 2 e_sigma, and the theta phases cancel because b_U + b_L = 0. At
+collision this phase is already part of the full fused germ K_0. The fused
+row has q = -2 and s = b = 0 and needs no further local phase.
+
+The gas. Let
+
+ V = {1, 2} x {P, M, O_+, O_-, U, L, F_pair}.
+
+A configuration Y = ((x_j, v_j))_{j=1}^N consists of finitely many particles
+with 0 <= x_j < T and v_j in V; coincident real positions form a null set and
+are discarded. For [p_0] in Z^2 / 8Z(1,-1) let p have the jumps -q_j in
+increasing order, using a consistent lift during the period, and impose
+p(T) = p(0) in the quotient. Define
+
+ Q_theta(Y, p) = exp[ -pi int_T (p + (theta/(2 pi)) 1)^t L
+ (p + (theta/(2 pi)) 1) dx ]
+ x prod_j exp[ -i pi (s_j - S b_j) . (p(x_j-) + p(x_j+))
+ - i theta 1^t L b_j ],
+
+and
+
+ W_{d,theta}(Y, p) = Q_theta(Y, p) prod_j gamma_{v_j}
+ prod_{1 <= j,k <= N, n in Z, x_k + nT > x_j} R_{v_j v_k}(x_k + nT - x_j),
+
+with the case j = k, n = 0 excluded and every j = k, n >= 1 included. The
+constants gamma_v are the basic or fused activities, independently of the
+species. For the empty configuration the particle and residual products are
+one and the state heat factor remains.
+
+First-contour identity. The conditional trace and oscillator calculation give
+each fixed current coefficient, with two empty oscillator factors D_W/Z_Gamma
+and two Poisson factors sqrt(T). Source removal gives D_g e^{m^2 kappa_T}.
+The pointwise bound below justifies integrating and summing these
+coefficients on the first contours. Signed Wick and constant-mode
+integrations give the real heat completion, with prefactor
+sqrt(8/9)/sqrt(2T), and cancel every winding-dependent exponential, leaving
+D_ell^{-1}. The finite periodic contour identities give exactly the basic and
+fused terms with the activities below; endpoint absolute bounds and the
+fusion factorial justify summing them, and the signed algebra of the state
+weight identifies the sum with Z_{m,T}. The eight equal full-tour
+representatives supply 1/8.
+
+Winding constant. With t = e^{-k},
+
+ lim_{T->infinity} kappa_T
+ = -2 int_0^inf t(1-t)(1-t^3)^2 / ((1+t^2)(1+t^3)(1+t^4)) dk/k
+ = 2 log( (2 (2 + sqrt 2)) / 9 ).
+
+Evaluation. Multiplying the rational function R(t) of t by 1 - t^{24} gives
+coefficients d_j = d_{24-j} for 1 <= j <= 23, whose first twelve values are
+
+ (1, -1, -1, -2, 3, 3, 1, -4, -3, -1, 3, 2),
+
+and whose total after reflection is zero. The grouped partial sum through
+period N is R(e^{-k})(1 - e^{-24(N+1)k}), whose quotient by k is dominated by
+R(e^{-k})/k, integrable because R(e^{-k}) = O(k^3) at zero and O(e^{-k}) at
+infinity; dominated convergence justifies the period expansion. Frullani's
+log difference on each block makes its integral -sum_j d_j log(24n + j).
+Pairing j with 24 - j, with exponent d_{12}/2 at j = 12, and using
+
+ prod_{n=0}^N (n+a)(n+1-a) / (N! (N+1)!) -> a prod_{n>=1}(1 - a^2/n^2)
+ = sin(pi a)/pi,
+
+the exponentiated sum of paired logarithms becomes prod_{j=1}^{11}
+sin(pi j/24)^{d_j}, and the sine product follows from finite odd sine
+multiplication in the limit N' -> infinity. Collecting the resulting log
+gives 2 log(2(2+sqrt2)/9).
+
+Signed height measures and the coercive matrix.
+
+Write mu_i^s, mu_i^o, mu_i^H for the signed measures on real heights in the
+short, outer, and H columns of the particle table, for a label i in one
+species. For example mu_P^s = delta_{1/2}, mu_P^o = delta_1,
+mu_P^H = delta_0. The separate coefficient sign in the particle table belongs
+to the local factor and does not multiply the measures. The fused measures
+are the union of the U and L measures at zero displacement:
+
+ mu_F^s = -2 delta_0,
+ mu_F^o = delta_{1/2} - delta_{-1/2},
+ mu_F^H = -delta_{3/2} - delta_{-3/2}.
+
+In the order P, M, O_+, O_-, U, L, F_pair the totals of both the short
+measure and the H-measure give
+
+ bar q = (1, -1, 0, 0, -1, -1, -2),
+
+so the physical charge vector is q_{(alpha,i)} = bar q_i e_alpha for species
+alpha. For a label iota = (alpha, i) put c_iota = bar q_i/sqrt2,
+c = (c_iota)_iota, a_c = 8/9. The jump of the common charge at label iota is
+-c_iota.
+
+Green kernels. For k > 0 the short and outer height kernels are
+
+ p(a) = cosh((1/2 - |a|) k) / (2 sinh(k/2)), |a| <= 1, period 1,
+ o(a) = sinh((1 - |a|) k) / (2 cosh k), |a| <= 2, antiperiod 2.
+
+Both are positive covariance kernels, with Green series
+
+ p(a) = sum_{n in Z} k e^{2 pi i n a} / (k^2 + (2 pi n)^2),
+ o(a) = (1/2) sum_{n in Z} k e^{pi i (n+1/2) a} / (k^2 + pi^2 (n+1/2)^2).
+
+Strict positivity is what makes the quadratic forms below positive definite
+whenever their signed measures do not vanish identically. The Fourier
+exponentials of the extreme H charges are
+
+ h_+(k) = (1/9) / k + O(k), h_-(k) = 1/k + O(k) (k -> 0),
+ h_+(k) = O(e^{-4k}), h_-(k) = 4 e^{-3k} + O(e^{-4k}) (k -> inf).
+
+Set C_i(k) = int cosh(k a) d mu_i^H(a) and S_i(k) = int sinh(k a)
+d mu_i^H(a), so C_i and S_i are the even and odd Fourier exponentials of
+the H-measures. Define the dimensionless matrix H by
+
+ H_{alpha i, beta j}(k) = delta_{alpha beta}
+ [ int int p(a-b) d mu_i^s(a) d mu_j^s(b)
+ + int int o(a-b) d mu_i^o(a) d mu_j^o(b) ]
+ + sum_{sigma = +-} (P_sigma)_{alpha beta} Phi^sigma_{ij}(k),
+
+with
+
+ Phi^{sigma, pre}_{ij} = -h_sigma C_i C_j,
+
+ Phi^{sigma, post}_{ij} = -h_sigma (C_i C_j - S_i S_j)
+ + 4 * 1_{sigma = -} int int sinh(k (|a-b| - 3)_+)
+ d mu_i^H(a) d mu_j^H(b).
+
+The pre-collision formula is the Gaussian absolute moment: reality of the
+field on the real line gives Re xi(x+ia) = (xi(x+ia) + xi(x-ia))/2, which
+replaces each signed charge by two signed halves without changing the sign of
+its magnitude. The post-collision formula uses the original charges together
+with cosh(k(a-b)) = cosh(ka)cosh(kb) - sinh(ka)sinh(kb), and the last term is
+the continued half of the two extreme H-charges, nonzero only in the minus
+channel and only after the collision.
+
+Normalised coordinates. In this calculation only, replace F_pair by
+F = F_pair/2, halving all three signed measures. The auxiliary coordinate can
+be included in the pre-collision calculation and removed afterwards. Put
+
+ D_0 = diag(1, 1, 1, 1, 1, 1, 2), D_F = diag(D_0, D_0),
+
+on P, M, O_+, O_-, U, L, F in each species. Then
+H_phys = D_F H_norm D_F, and a physical coefficient vector v_phys has
+normalised coordinates D_F v_phys, so the physical particle count is
+represented by
+
+ n_norm = D_0^{-1} 1 = (1, 1, 1, 1, 1, 1, 1/2),
+
+and the normalised charge row is (1, -1, 0, 0, -1, -1, -1). The fused label
+F therefore has count 1/2 and charge -1, while R_d = (U+L)/2 has count 1 and
+charge -1.
+
+This difference is why the displacement must stay fixed. At d = 0 the height
+measures of R_d and F agree but their count values do not, so the count bound
+cannot be uniform through d = 0. The same-species U, L factor also carries
+the simple collision pole of the boson section, with magnitude proportional
+to 1/|x +- 2 i d| near collision, so pointwise constants depend on d. The
+contour argument sums the series at fixed endpoints on the two sides of the
+collision, which is precisely the fixed-displacement hypothesis.
+
+Even core. In the quotient, the even direction (O_+ + O_-)/2 equals
+(P+M)/2. On P, M, F the short and outer measures give
+
+ A_sigma = M_0 - h_sigma X X^t - sigma lambda V V^t,
+
+ M_0 = [[f+g, -f+g, -l+j_0],
+ [-f+g, f+g, l+j_0],
+ [-l+j_0, l+j_0, f+g/2]],
+
+with
+
+ f = p(0), g = o(0), l = p(1/2), j_0 = o(1/2),
+ X = (1, 1 - 2 cosh k, -cosh(3k/2))^t, V = (0, 1, 1)^t.
+
+For the third row, the normalised F short is -delta_0 and its two outers have
+weights 1/2 and -1/2. The three even H-moments give X. The same core
+submatrix appears for nonzero d. Strict Green positivity gives M_0 > 0:
+cancelling its short measure forces equal P and M coefficients and a zero F
+coefficient, and the remaining outer measure then forces both coefficients to
+vanish.
+
+Coercivity inequality. Let u = e^{-k/2} and set
+
+ x = X^t M_0^{-1} X, y = X^t M_0^{-1} V, v = V^t M_0^{-1} V.
+
+Direct inversion gives
+
+ x = (1-u^2)(u^4+1)(u^8+u^6-u^4+u^2+1) / (3 u^6 (u^2+1)),
+ y = 2(u^2-1)(u^4+1) / (3 u^3),
+ v = (7u^2 - 12u + 7) / (3(1-u^2)).
+
+The signs of 1 - h_+ x and 1 - h_- x are respectively positive and negative.
+After positive denominators are cleared the numerators are
+
+ (u^4+1)(3u^8 + 2u^6 + 2u^4 + 2u^2 + 3)
+
+and
+
+ -(1-u^2)^2 (u^4+1)(u^8 + 2u^6 - 2u^4 + 2u^2 + 1),
+
+and the last parenthesis is positive because with v_0 = u^2 it equals
+(1-v_0^2)^2 + 2 v_0 (1+v_0^2). Put
+
+ w_sigma = v + h_sigma y^2 / (1 - h_sigma x).
+
+Substituting the rational formulas for h_sigma at t = u^2 and
+lambda = 3(1-u^8)/(8(1+u^8)) gives
+
+ (1/w_sigma) - sigma lambda = ((1-u^2)/u) (N_sigma(z)/D_sigma(z)) > 0,
+ z = u + u^{-1} - 2,
+
+with coefficients in ascending order
+
+ N_+ : (96, 1280, 5496, 9312, 7696, 3360, 772, 84, 3),
+ D_+ : (128, 2240, 13888, 38304, 55392, 45584, 21984, 6128, 912, 56),
+ N_- : (32, 256, 424, 1024, 1384, 832, 232, 28, 1),
+ D_- : (128, 2240, 13760, 36704, 49120, 36176, 15264, 3664, 464, 24).
+
+All coefficients of N_+, D_+, N_-, D_- are positive for z >= 0, so the
+right side is finite and positive, which prevents w_sigma from vanishing. In
+the plus channel M_0 - h_+ X X^t > 0, so w_+ > 0 and the displayed inequality
+is 1 - lambda w_+ > 0; both negative rank-one updates then preserve
+positivity. In the minus channel the first update has exactly one negative
+direction, and since w_- -> -3 as u -> 0 it is negative throughout, so the
+inequality gives 1 + lambda w_- < 0. The determinant lemma makes the
+determinant positive after adding lambda V V^t, and a positive rank-one
+update cannot add negative directions, so it removes the unique negative
+direction. Hence A_sigma > 0 for every k > 0.
+
+The displaced and odd directions, and the absolute estimate.
+
+The remaining even direction is R_d = (U+L)/2; put Z_d = R_d - F and
+D_* = |d| k. The H-moment of R_d is -cosh((3/2+d)k). The odd directions are
+
+ O_odd = (O_+ - O_-)/2, W_d = (U - L)/2,
+
+whose H-sinh moments are -sinh k and -sinh((3/2+d)k). Write T for the
+continued correction in the post-collision minus channel. Summing the
+opposite extreme charges with their actual half weights gives exactly
+
+ T_{R_d R_d} = 2 sinh(2D_*), T_{R_d F} = 2 sinh D_*,
+ T_{W_d W_d} = -2 sinh(2D_*),
+
+and no other continued entries. These terms are absent before collision.
+
+For small D_* and small |d| the square of Z_d is D_*(1 + O(D_* + |d|)) and
+its crosses with the core are O(D_*), uniformly. For the field part the
+short square is exactly
+
+ (3/2) p(0) + (1/2) p(2d) - 2 p(d)
+ = f(cosh D_* - 1)^2 + sinh D_* - (1/4) sinh(2D_*),
+
+and the outer square is
+
+ o(0) - o(d) + o(1+d) - (1/2) o(1+2d).
+
+Each has leading term D_*/2, and the Green formulas also give the stated
+cross bounds. At small k the possibly larger second-order bounds
+O(D_*^2/k) are still O(|d| D_*). The uncontinued H-square difference is
+O(D_*^2) with crosses O(D_*). The continued contribution to the Z_d square
+is 2 sinh(2D_*) - 4 sinh D_* = O(D_*^3), and the V-terms obey the same
+bounds. The Schur cost of all crosses is at most
+
+ C D_*^2 / min(k,1) <= C max(|d|, D_*) D_*
+
+for small D_* and |d|, because D_*/min(k,1) <= max(|d|, D_*). Absorption
+against the core gap leaves a positive multiple of the D_* square and part
+of the plus charge gap. The direction Z_d has zero total charge.
+
+For the odd directions the outer part alone is
+
+ [[o(0), o(1/2+d)], [o(1/2+d), (o(0)+o(1+2d))/2]]
+ >= c_* min(k,1) I
+
+for small |d|, by strict Green positivity and its limits at zero and
+infinity, and the short part is nonnegative. The pre-collision H-part
+vanishes. In the post-collision case its uncontinued part is the positive
+rank-one form of the two sinh moments, and the only adverse continuation is
+the last entry of the T-table. On bounded frequency intervals its cost is
+O(|d| k); at large k its sum with the leading height diagonal is
+
+ e^{2D_*} - 2 sinh(2D_*) = e^{-2D_*},
+
+up to exponentially small errors, and the height cross is exponentially
+small. Choose a large-frequency cutoff absorbing these errors uniformly for
+|d| in a small fixed interval, then choose |d| small enough on the remaining
+bounded interval. This proves the odd gap.
+
+Physical limit. The sector gaps show that the only null directions are the
+two O-relations. At small k the common channel supplies the c c^t/k term and
+every quotient direction has a k-order gap. After physical congruence the
+same assertions hold with the physical count and charge, and the expansions
+of p, o, h_sigma, the continued term and K give
+
+ H'(k) = (a_c/k) c c^t + k B_d + O_d(k^2)
+
+for a real symmetric B_d, recording the exact leading coefficient that
+cancels the state square. For fixed nonzero d, the high-frequency formulas
+give exponential convergence to a positive quotient matrix. In particular,
+for each fixed displacement in the small range, the physical H' has null
+space the two O-relations for every k > 0, is bounded below on the quotient
+by a positive multiple of k I + c c^t/k, and converges exponentially at
+infinity to a strictly positive matrix.
+
+Energy identity. Index the physical particles by alpha = 1, ..., N, with
+labels iota_alpha and positions x_alpha in one period, and set
+
+ rho_iota = sum_{alpha: iota_alpha = iota} delta_{x_alpha},
+ hat rho(k) = int_T e^{-i k x} rho(dx), k in (2 pi / T) Z.
+
+The forward regular energy, including nonzero self-images, is
+
+ E_R = sum_{alpha,beta} sum_{r in Z, x_beta + rT > x_alpha}
+ R_{iota_alpha iota_beta}(x_beta - x_alpha + rT)
+ = (1/2) sum_{alpha,beta} (R_T)_{iota_alpha iota_beta}
+ (x_beta - x_alpha)
+ - (1/2) sum_alpha R_{iota_alpha iota_alpha}(0)
+ = (1/(2T)) sum_{k in (2 pi/T) Z} hat rho(k)^* hat R(k) hat rho(k)
+ - (1/2) sum_alpha R_{iota_alpha iota_alpha}(0).
+
+The half-double sum pairs the forward image (alpha, beta, r) with its
+reverse (beta, alpha, -r). Only (alpha, alpha, 0) is excluded from the gas,
+so the subtracted diagonal is the unperiodised value R(0) and every nonzero
+self-image remains.
+
+Young smoothing. Let g_T be the normalised periodic Gaussian with Fourier
+multiplier e^{-k^2/2}, and let nu = n^t rho = sum_alpha delta_{x_alpha}.
+No positivity of hat R(0) on vectors other than c is required: the mean
+square in the energy bounds a positive multiple of T p_c^2 up to
+O_theta(T), and the diagonal correction costs O_d(N) independently of T.
+Combining the energy identity with the quotiented coercivity and including
+the local factors gives, for every nonzero post-collision weight,
+
+ -log|W_{d,theta}(Y,p)|
+ >= c_2 ( ||g_T * nu||_{L^2(T)}^2 + ||g_T * p_c||_{L^2(T)}^2 )
+ - C_2 (N + T),
+
+with c_2, C_2 depending only on d and Theta. The local constants have
+logarithmic cost O_d(N), and the Berry factors carrying real state charges
+are phases whose theta-dependent parts cost O_Theta(N). For the
+pre-collision product the same identity holds with the real angle
+vartheta = Re theta; the symmetrised local constants are finite for fixed
+d < 0 and cost O_d(N), and the same bound holds with
+-log W^{pre}_{d, Re theta} on the left.
+
+Box counts. With h_T = g_T * g_T,
+
+ ||g_T * nu||_2^2 = sum_{alpha,beta} h_T(x_alpha - x_beta)
+ >= c_1 sum_{j=0}^{T-1} n_j^2,
+
+because h_T is positive with a uniform positive lower bound for pairs in the
+same unit box. Each jump of p_c has magnitude at most sqrt 2. For x in box j
+write p_c(x) - (g_T * p_c)(x) by integrating its increments against the
+Gaussian; a jump in a box r boxes away is multiplied by a Gaussian tail
+bounded by C e^{-c (r-2)_+^2}, and the resulting sequence of bounds is
+summable uniformly after periodisation. Discrete Young's inequality therefore
+gives
+
+ ||p_c - g_T * p_c||_2^2 <= C_1 sum_j n_j^2.
+
+Within box j the same jump bound gives
+
+ sum_j p_{c,j}^2 <= 2 ||p_c||_2^2 + C_2 sum_j n_j^2.
+
+Using a smaller positive constant, the two smoothed norms consequently
+control sum_j (n_j^2 + p_{c,j}^2). Finally
+sum_j n_j <= eps sum_j n_j^2 + C_eps T absorbs the linear costs.
+
+Absolute stability. There is 0 < d_0 < 1/2 with the following property. Fix a
+compact set Theta in C and either a pre-collision displacement
+-d_0 < d < 0 or a post-collision displacement 0 < d < d_0. There are
+c > 0 and C < infinity, depending on d and Theta, such that for every
+positive integer T and every theta in Theta, every finite configuration Y of
+distinct real positions on R/TZ with the six basic labels, or with those
+labels together with F_pair in the post-collision case, and every compatible
+state path p with jumps -q_i and closure in Z^2/8Z(1,-1), satisfies
+
+ |W_{d,theta}(Y,p)| <= exp( C T - c sum_{j=0}^{T-1} (n_j^2 + p_{c,j}^2) )
+
+in the post-collision case, with n_j the number of physical particles in
+[j, j+1) and p_{c,j} = (p_1(j+) + p_2(j+))/sqrt 2. The same bound holds for
+W^{pre}_{d, Re theta}(Y,p) in the pre-collision case. Zero weights already
+satisfy the inequality.
+
+Summability. For a fixed common integer p_1 + p_2 there are eight classes in
+Z^2/8Z(1,-1), and the Gaussian bound on the common charge sums over that
+integer. In each box, integration and the ordinary label factorials reduce
+to a convergent sum of the form
+
+ sum_{n>=0} A^n e^{-c n^2} / n! < infinity,
+
+and dropping configuration constraints only enlarges the bound. Both the
+pre-collision Gaussian integration and the post-collision gas series are
+therefore absolutely summable at the fixed displacements, locally uniformly
+for theta in compact sets. The constants may depend on d and assert nothing
+uniform as d approaches zero.
+
+Compact propagation with Jordan blocks.
+
+The horizontal gas has signed weights, so the partition is not the trace of a
+positive operator. What is available is a compact Hilbert-Schmidt operator
+whose power traces reproduce the gas, together with a gauge that turns the
+signed local factors into a contraction.
+
+Gauge. A real measurable function Phi on the state space exists with
+
+ |e^{Phi(S') - Phi(S)} w_theta(S,S')|
+ <= exp( C_b - delta_b (|Y_prev|^2 + |Y|^2 + p_c^2) ),
+ delta_b = c''/b > 0,
+
+for theta in the compact parameter set and every allowed successor. To build
+it, subtract a smaller quadratic cost from the edge costs used in the cycle
+cost and add C_b, so that every finite directed cycle has nonnegative
+modified cost. Let Phi(S) be the infimum of modified costs of paths from a
+reference state to S. Every state can be reached and returned from: insert P
+and M charges as needed at distinct positions, then insert the required final
+block. A fixed return path bounds the infimum below by the negative of its
+cost, so the infimum is finite, and the directed triangle inequality gives the
+displayed bound. For measurability, restrict the intermediate positions in
+the infimum to rational translated coordinates and use continuity on the open
+simplices; the terminal state stays arbitrary and fixed. The discrete
+constraints depend only on the labels, so on each label stratum the infimum is
+countable and continuous in the endpoint, hence measurable.
+
+The gauged kernels then define a Hilbert-Schmidt operator E = E(theta) on the
+sequential Hilbert space, acting from successor to predecessor, with spectral
+radius r_E < 1 in the parameter range and with peripheral spectral structure
+described below. The Bargmann cycle identity supplies exactly the far
+interactions and the local weights, giving the operator-gas trace identity
+
+ tr E^n = sum over configurations of n blocks of the gas weight.
+
+Site tail. The deterministic site factors are omega_v(x)^m, and each omega_v
+has modulus below 1 off the origin and tends to 1 at infinity with an
+exponential correction of rate pi/3. Write kappa_1 b for that rate in blocks.
+On an interval of length J = floor(log m /(kappa_1 b)) about the origin every
+suppressed type therefore disappears; outside that interval the exterior
+operator still allows all types. Over the inner interval the propagation
+multiplier is r_I = 1, and over the exterior it is r_E, so the propagation of
+the physical partition over n blocks carries the factor
+
+ exp( -(2 log m /(pi b/3)) (pi b/16) ) = m^{-3/8}
+
+together with other bounded factors.
+
+Empty-block traces. Let E_0 be the homogeneous operator with every site
+multiplier equal to one, and let I_0 retain only the labels M and F_pair.
+Both depend on theta. For n >= 2,
+
+ 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,
+ u_k = exp( -(4 pi b/9)(k + theta/pi)^2 ).
+
+The first equality is the empty contour at m = 0 together with the gas
+identity. In the second, every retained particle has strictly negative common
+charge, so a closed sequence contains no particles, and the eight
+representatives at each k = p_1 + p_2 give the displayed sum. The same
+reasoning gives the trace formula with the sum restricted to a consecutive
+range of k for a triangular diagonal compression to that range.
+
+Spectral facts. For a compact Hilbert-Schmidt operator the limsup of the
+absolute n-th roots of the power traces is the spectral radius, and those
+traces determine every nonzero eigenvalue with its algebraic multiplicity.
+Approximate the operator by finite rank with norm error below eps. Outside
+radius eps the error resolvent exists, and finite-dimensional inversion makes
+the full resolvent meromorphic with finite-rank pole parts. Resolvent contour
+integration isolates the Riesz blocks, the resolvent identity makes each
+Riesz projection idempotent and commuting, and Laurent comparison identifies
+the higher pole parts with powers of the nilpotent part on that block. The
+complementary powers are exponentially bounded at any radius containing the
+remaining spectrum, and their traces have the same rate by retaining two
+Hilbert-Schmidt factors. On an isolated spectral radius the trace is a sum of
+algebraic multiplicities times eigenvalue powers, and Cesaro extraction of
+distinct unit-circle frequencies shows that such a sum cannot tend to zero
+unless all coefficients vanish. Applying this successively off zero proves
+the assertion, and separating resolvent contours give locally uniform bounds
+for norm-continuous families.
+
+Bare trace expansion. Compactness makes the peripheral set P = {zeta in
+sigma(E): |zeta| = 1} finite and nonempty. On its finite Riesz blocks,
+
+ E^n P_zeta = zeta^n sum_{t=0}^{q_zeta - 1} binom(n, t) zeta^{-t}
+ J_zeta^t P_zeta,
+ J_zeta = (E - zeta I) P_zeta, J_zeta^{q_zeta} = 0,
+
+and the complementary restriction E_< has spectral radius strictly below one.
+The remainder is O(rho^n) in operator norm for some rho < 1, and
+||E^n|| <= K_0 (1+n)^q for a fixed q. For n >= 2 the complementary trace norm
+is bounded by ||E_<||_2^2 ||E_<^{n-2}||. Every positive nilpotent power has
+trace zero on a finite Riesz block, so
+
+ tr E^n = sum_{zeta in P} d_zeta zeta^n + O(rho_1^n), rho_1 < 1.
+
+Coefficient polynomial. Index the two exterior gaps by L, expand the exterior
+factors, and rotate the trace to begin with C'_m = (u/r_E)^{2L} C~_m. The
+term for two compatible exterior words has the form
+
+ tr( C'_m W_+ (E')^{s - l_+ - l_-} W_- ), s = n - 2L.
+
+For a peripheral eigenvalue zeta of E' write P_zeta for its Riesz projection,
+d_zeta for the rank, and J_zeta = (E' - zeta) P_zeta. The coefficient
+polynomial supplied by the trace lemma is
+
+ P_{zeta,m}(s) = sum_{W_+,W_-} zeta^{-l_+ - l_-}
+ tr( C'_m W_+ [ sum_{t>=0} binom(s - l_+ - l_-, t) zeta^{-t}
+ J_zeta^t P_zeta ] W_- ).
+
+The nilpotent sum is finite, and the word sum is coefficientwise absolutely
+convergent by the exterior word bound. The lemma then gives
+
+ R_m = P_{zeta,m}(0) / (d_zeta zeta^{2L}), |R_m| <= C |u/r_E|^{2L},
+
+with the constant uniform in m on the fixed parameter neighbourhood, since the
+word moments, the central trace norm and the peripheral nilpotent bounds are
+uniform and |zeta^{2L}| = 1 makes the changing offset harmless.
+
+Two-gap coefficient lemma. Keep two distinct frequencies zeta, eta with
+expansions of order h_z, h_e. For a word sum over W_1, W_2, W_3, W_4 with
+exterior lengths l_+ and l_-, set
+
+ P~_{zeta,eta}(z) = sum_{W} zeta^{-l_+} eta^{-l_-}
+ tr( C W_1 B_zeta^+(z, l_+) W_2 W_3 B_eta^-(z, l_-) W_4 ).
+
+Then for every frequency beta in {alpha^4: alpha in P_+} union
+{(zeta eta)^2: zeta in P_+, eta in P_-},
+
+ R sum_{alpha in P_+, alpha^4 = beta} rank P_alpha^+
+ = sum_{zeta in P_+, eta in P_-, (zeta eta)^2 = beta}
+ (zeta eta)^{-L} P~_{zeta,eta}(-L),
+
+and
+
+ |R| <= K (1+L)^h ||C||_1, h = max_{zeta,eta}(q_zeta^+ + q_eta^- - 2),
+
+with K depending only on E_+, E_-, the parameter neighbourhood and the gauge,
+and independent of C, L and R.
+
+Signed trace coefficients. Let E be a Hilbert-Schmidt operator on a complex
+Hilbert space with spectral radius one, P = {zeta in sigma(E): |zeta| = 1},
+Riesz projections P_zeta of rank d_zeta, and J_zeta = (E - zeta I) P_zeta with
+J_zeta^{q_zeta} = 0 on the range. Let C be trace class. Suppose two countable
+families W_+, W_- of bounded operators have nonnegative integer lengths
+l(W_+-) with
+
+ sum_{W_+-} e^{eps l(W_+-)} ||W_+-|| < infinity
+
+for some eps > 0. For integers s >= 0 put
+
+ F_s = sum_{l(W_+) + l(W_-) <= s}
+ tr( C W_+ E^{s - l(W_+) - l(W_-)} W_- ),
+
+and assume a >= 0 and R in C satisfy F_s - R tr E^{s+a} -> 0 as s tends to
+infinity, using only s with s + a >= 2. Then the coefficientwise absolutely
+convergent polynomial
+
+ Q_zeta(z) = sum_{W_+,W_-} zeta^{-l}
+ tr( C W_+ [ sum_{t=0}^{q_zeta - 1} binom(z - l, t) zeta^{-t}
+ J_zeta^t P_zeta ] W_- ), l = l(W_+) + l(W_-),
+
+is identically R d_zeta zeta^a. In particular
+
+ R = Q_zeta(0) / (d_zeta zeta^a), |R| <= K ||C||_1,
+
+with K depending on E and the bounds in the moment condition but not on C or
+a; more precisely K may be chosen from eps, the two weighted word sums, the
+finite nilpotent orders q_zeta and the norms ||J_zeta^t P_zeta||. The
+conclusion is uniform for a family with uniform bounds on those data, and no
+peripheral semisimplicity is assumed.
+
+Proof. The power expansion of E on a peripheral block and the geometric
+integration of the binomial polynomials give the identity for every finite
+word sum. Concretely, for integer s >= 0 the block expansion reads
+
+ sum_{W} zeta^{-l} tr( C W_+ sum_{t=0}^{q-1} binom(s - l, t) zeta^{-t}
+ J^t P W_- )
+ = zeta^{-s} F_s
+
+with F_s the truncated trace, because the binomial coefficient
+binom(s - l, t) is the number of ways to distribute t nilpotent insertions
+among s - l slots and the word sum reproduces the expansion coefficients of
+E^{s-l}. The right side differs from R zeta^a d_zeta by a term tending to zero
+in s, and the left side, being a polynomial in s of degree at most q - 1, must
+equal that constant identically once the difference tends to zero on
+infinitely many integers. Evaluating at s = 0 gives the displayed formula for
+R. The trace-class norm of each word term is bounded by
+K_1 ||C||_1 ||W_+|| ||W_-|| (1 + l)^{q_zeta - 1}, whose sum is finite by the
+moment condition, and since d_zeta >= 1 and |zeta^a| = 1 the stated bound
+follows. QED