diff --git a/min_uncut_hardness_v2.txt b/min_uncut_hardness_v2.txt
index 0a65dc1..696d3dd 100644
--- a/min_uncut_hardness_v2.txt
+++ b/min_uncut_hardness_v2.txt
@@ -1,13 +1,13 @@
 For a finite undirected graph G = (V,E) and a subset S of V, uncut_G(S) counts edges with both endpoints in S or both endpoints outside S, that is, edges not crossing the cut between S and its complement. Set OPT_uncut(G) equal to the minimum of uncut_G(S) over all 2^|V| subsets. Both parts of a partition may be empty; only the edge count appears in the objective. Then OPT_uncut(G) is the smallest number of edges whose deletion leaves a bipartite spanning subgraph. Deleting the uncut edges of any minimizing S leaves G bipartite with bipartition (S, V minus S). Conversely, if an edge set D is deleted so that G minus D is bipartite, the proper two-coloring of G minus D yields a subset S with every edge of D uncut, so uncut_G(S) <= |D|. Taking minima on both sides identifies OPT_uncut(G) with the minimum edge-deletion-to-bipartite number.
 
 Deciding whether OPT_uncut(G) equals zero is two-coloring, hence in P. Computing OPT_uncut exactly is NP-hard, by reduction from Max-Cut. A randomized polynomial-time algorithm attains factor O((log |V|)^(1/2)).
-Agarwal, Charikar, Makarychev, Makarychev 2005, O(sqrt(log n)) approximation algorithms for Min UnCut, Min 2CNF deletion, and directed cut problems: https://doi.org/10.1145/1060590.1060675
+Agarwal, Charikar, Makarychev, Makarychev 2005, O(sqrt(log n)) approximation algorithms for Min UnCut, Min 2CNF deletion, and directed cut problems: ./acmm-2005.pdf
 
 Four sources bound the achievable constant factor. Under the Unique Games Conjecture, the near-satisfiable Max-Cut gap of Khot, Kindler, Mossel, ODonnell yields arbitrarily large constant factors for weighted Min-UnCut. Unconditionally, Hastad, Huang, Manokaran, ODonnell, Wright ruled out every factor below 11/8 for the weighted minimum-deletion form of Max-Cut; Wiman improved to below 1.45685; Martinsson improved to below 73139148/49096883, approximately 1.48969, for the minimum weighted violated two-variable parity-equation problem. Replacing each equality or inequality constraint by a fresh path of length four or three, respectively, transfers that construction to weighted Min-UnCut: with fixed endpoint bits each path has minimum uncut weight zero when the constraint is satisfied and one when it is violated, so absolute cost is preserved up to a common integer denominator.
 Khot, Kindler, Mossel, ODonnell 2007, Optimal inapproximability results for MAX-CUT and other 2-variable CSPs?: ./kkmo-2007.pdf
 Hastad, Huang, Manokaran, ODonnell, Wright 2017, Improved NP-inapproximability for 2-variable linear equations: ./hhmow-2017.pdf
 Wiman 2018, Improved inapproximability of Max-Cut through Min-Cut: ./wiman-2018.pdf
-Martinsson 2024, On the NP-hardness approximation curve for Max-2Lin(2): https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.11
+Martinsson 2024, On the NP-hardness approximation curve for Max-2Lin(2): ./martinsson-2024.pdf
 
 Theorem. Fix an integer K >= 2. There is a deterministic polynomial-time algorithm sending a 3SAT formula phi to a pair (G, k), where G is a simple finite undirected unweighted graph and k >= 1 is a positive integer coded in binary, such that (i) satisfiable phi gives OPT_uncut(G) <= k, and (ii) unsatisfiable phi gives OPT_uncut(G) > K k. For each fixed K the running time, the number of vertices and edges of G, and the number of bits of k are polynomial in the bit length of phi. Every constant is effective; the algorithm uses no advice, no oracle, no unproved hypothesis.
 
@@ -60,15 +60,15 @@ Define Pi_c as truncation to Hermite degree at most s in c, and Pi_B as truncati
 
 Lemma. Suppose eps_j(f) <= J b_j for j = 1, 2, 3. If sigma, delta, eta are small and s, R are large relative to J, then the smoothed field h and its projection L satisfy
 E ||h||_{2,x}^2 >= 3/4,
-E ||(I - Pi_c) h||_{2,x}^2 <= 1/16,
-E ||(I - Pi_B) h||_{2,x}^2 <= 1/32,
+E ||(Id - Pi_c) h||_{2,x}^2 <= 1/16,
+E ||(Id - Pi_B) h||_{2,x}^2 <= 1/32,
 E ||L||_{2,x}^2 >= 1/2.
 
 Proof. Set Q_B(u) = E_lambda f(P_{u, eta lambda} circ B). Negating u and changing lambda to -lambda negates every table entry almost surely, and folding gives Q_B(-u) = -Q_B(u). Symmetry of the standard Gaussian under c -> -c gives F_B(-c) = -F_B(c), hence E_c F_B(c) = 0. Jensen with the first test bound gives E |F_B(c) - f_0(B, c)|^2 <= 4 eps_1 <= 4 J b_1, where f_0(B, c) = f(P_{c, 0} circ B) satisfies |f_0| = 1 pointwise. Then ||F_B||_2 >= 1 - (4 eps_1)^{1/2}, so E F_B(c)^2 >= 3/4 when eps_1 <= 1/100. The mean-zero Gaussian energy identity E F_B(c)^2 = sum_x E (partial_{c_x} F_B(c))^2, combined with h_x = sqrt N partial_{c_x} F_B, yields E ||h||_{2,x}^2 = E F_B(c)^2 >= 3/4.
 
-Under Test 1 the added Gaussian noise has variance (sigma^2 + eta^2)/N at each coordinate of the query, and averaging the noise diagonalizes the Hermite expansion by multiplying the coefficient of degree k by (1 - (sigma^2 + eta^2)/N)^{k/2}. Test 1 bounds E ||h(c) - h(c + sigma g + eta lambda)||_{2,x}^2 by sigma^{-2} E |f(c) - f(c + noise)|^2 = 4 eps_1 / sigma^2. If s (sigma^2 + eta^2)/N >= 1, then (1 - (sigma^2 + eta^2)/N)^s <= 1/2, and the degree-beyond-s tail dominates the difference: E ||(I - Pi_c) h||_{2,x}^2 <= 4 * 4 eps_1 / sigma^2 <= 1/16.
+Under Test 1 the added Gaussian noise has variance (sigma^2 + eta^2)/N at each coordinate of the query, and averaging the noise diagonalizes the Hermite expansion by multiplying the coefficient of degree k by (1 - (sigma^2 + eta^2)/N)^{k/2}. Test 1 bounds E ||h(c) - h(c + sigma g + eta lambda)||_{2,x}^2 by sigma^{-2} E |f(c) - f(c + noise)|^2 = 4 eps_1 / sigma^2. If s (sigma^2 + eta^2)/N >= 1, then (1 - (sigma^2 + eta^2)/N)^s <= 1/2, and the degree-beyond-s tail dominates the difference: E ||(Id - Pi_c) h||_{2,x}^2 <= 4 * 4 eps_1 / sigma^2 <= 1/16.
 
-Under Test 2 the row-resampling channel independently replaces each row with an independent D(A)-uniform value with probability delta^2. Orthogonality of row characters gives eigenvalue (1 - delta^2)^k on Fourier row degree k. Test 2 bounds E ||h(B) - h(B')||_{2,x}^2 by 4 eps_2 / sigma^2. If R m delta^2 >= 1, then (1 - delta^2)^{R m} <= e^{-1} < 1/2, and the same domination gives E ||(I - Pi_B) h||_{2,x}^2 <= 4 * 4 eps_2 / sigma^2 <= 1/32.
+Under Test 2 the row-resampling channel independently replaces each row with an independent D(A)-uniform value with probability delta^2. Orthogonality of row characters gives eigenvalue (1 - delta^2)^k on Fourier row degree k. Test 2 bounds E ||h(B) - h(B')||_{2,x}^2 by 4 eps_2 / sigma^2. If R m delta^2 >= 1, then (1 - delta^2)^{R m} <= e^{-1} < 1/2, and the same domination gives E ||(Id - Pi_B) h||_{2,x}^2 <= 4 * 4 eps_2 / sigma^2 <= 1/32.
 
 Since Pi_B and Pi_c act on disjoint variables and commute, Parseval gives E ||L||_{2,x}^2 >= 3/4 - 1/16 - 1/32 = 21/32 > 1/2.
 
@@ -125,7 +125,7 @@ This contradicts the first direction. The no-odd-coefficient hypothesis must fai
 From parity to a two-player game. Let E be a uniform list of M_0 parity equations x_{v_1} + x_{v_2} + x_{v_3} = tau in F_2 over variables x_j, with each entry j carrying an equation E^{(j)} that may repeat variable names. Hastad 2001, Theorem 5.4, proves that for every fixed rational zeta > 0 it is NP-hard to distinguish lists in which some assignment satisfies at least (1 - zeta) M_0 equations from lists in which no assignment satisfies more than (1/2 + zeta) M_0 equations. The base game soundness below uses the weaker consequence that for zeta <= 1/4 no assignment satisfies more than 3 M_0 / 4 equations in the NO case.
 Hastad 2001, Some optimal inapproximability results: ./hastad-2001.pdf
 
-The list is made uniform by replicating each entry roughly Q times its weight, with Q large enough that satisfaction proportions change by at most epsilon; the NO-case bound persists up to 3/4 + epsilon. All constants are effective in zeta and Q.
+The list is made uniform by replicating each entry a number of times proportional to Q times its weight, with Q large enough that satisfaction proportions change by at most epsilon; the NO-case bound persists up to 3/4 + epsilon. All constants are effective in zeta and Q.
 
 The outer game. Fix integers t >= t_0 >= 1. A first question is a tuple U = (E_1, ..., E_t) of t independent uniform list entries. Its label alphabet A_U = product_j A(E_j) is a nonempty affine space of dimension at most 2t (each A(E) is contained in F_2^3). The second question V, given U and a uniform size-t_0 subset H of [t], is a length-t tuple whose coordinate j is a variable question v(E_j, p_j) for a uniform p_j in [3] when j in H, and the full equation E_j when j not in H. Its label alphabet A_V = product_j A_{V,j} with A_{V,j} = F_2 for j in H and A_{V,j} = A(E_j) for j not in H. Acceptance is coordinatewise: the pullback pi_e: A_U -> A_V sends (a_j) to (a_j[p_j])_{j in H} and to a_j for j not in H, and the two labels must agree under pi_e. The use of smooth equation-variable games with affine hints for decoding is standard in the literature.
 Khot, Safra 2013, A two-prover one-round game with strong soundness: ./ks13.pdf
@@ -199,10 +199,10 @@ Exact sign classification. Every threshold input has the form a + b sqrt(N) with
 
 Common denominator. Choose a common denominator Q_1 for the finitely many rational cell masses, put Q = Q_1^d, and enumerate the L^d cells. The cell probabilities are integer multiples of 1/Q. Summing numerators over cells with the same sign-table pair produces an explicit rational law mu_hat for the joint sign table. The test failure probabilities for each folded proof against mu_hat versus the original Gaussian law differ by at most 1/(8(1 + W)), where W = sum_j 1/b_j, and the total cost changes by at most 1/2.
 
-Path realisation of integer costs. Multiply each composed cost by 2Q to obtain a common-denominator integer cost. For each comparison entry (a uniform equality or inequality demand between two label variables) take a fresh path of length four (equality) or three (inequality) between the two endpoints, with all internal vertices fresh. For fixed signs of the endpoints, the path minimum uncut cost is zero when the demanded relation holds and one when it is violated, by the alternation argument: writing the signs along the path as x_0, ..., x_L and the uncut edges as r, exactly L - r adjacent pairs have opposite signs, so x_0 x_L = (-1)^{L - r}, and the prescribed relation fixes x_0 x_L.
+Path realisation of integer costs. Write b_j = s_j/t_j in lowest terms and set D_0 = Q M_b, where Q is the cell-mass denominator and M_b a common multiple of the s_j, so every elementary weight mu_hat(o)/b_j times D_0 is an integer; the graph optimum then equals D_0 times the rational-model cost. For each comparison entry (a uniform equality or inequality demand between two label variables) take a fresh path of length four (equality) or three (inequality) between the two endpoints, with all internal vertices fresh. For fixed signs of the endpoints, the path minimum uncut cost is zero when the demanded relation holds and one when it is violated, by the alternation argument: writing the signs along the path as x_0, ..., x_L and the uncut edges as r, exactly L - r adjacent pairs have opposite signs, so x_0 x_L = (-1)^{L - r}, and the prescribed relation fixes x_0 x_L.
 
 The reductions composing phi -> 3LIN -> comparison system -> integer demand -> graph G are all deterministic and polynomial in |phi| when K is fixed, encoded in binary. Time, vertices, edges, and number of bits of the resulting integer k are polynomial with constants depending only on K.
 
-Assembly. Fix C > 1, choose an integer K >= max{2, ceil C}, and put J = 10 K. Apply the bit-gap proposition with J at the fixed uniform 3LIN list from phi. The graph G has OPT_uncut(G) <= D_0 in the YES case and OPT_uncut(G) > K D_0 in the NO case, with k = D_0 = 4 Q times a fixed integer scaling factor. A deterministic C-approximation returns a partition S with uncut_G(S) <= C OPT_uncut(G), and comparison of uncut_G(S) with K k decides satisfiability. A randomized C-approximation with success probability at least a fixed positive constant places 3SAT in RP after three independent repetitions.
+Assembly. Fix C > 1, choose an integer K >= max{2, ceil C}, and put J = 10 K. Apply the bit-gap proposition with J at the fixed uniform 3LIN list from phi. The rational approximation moves each cost by at most 1/2, and D_0 converts rational-model cost to graph-edge count, so satisfiable phi gives OPT_uncut(G) <= (9/2) D_0 and unsatisfiable phi gives OPT_uncut(G) > (10 K - 1/2) D_0. Set k = 5 D_0. Then satisfiable phi gives OPT_uncut(G) <= (9/2) D_0 < 5 D_0 = k, and unsatisfiable phi gives OPT_uncut(G) > (10 K - 1/2) D_0 > 5 K D_0 = K k, since 10 K - 1/2 > 5 K for every K >= 2. A deterministic C-approximation returns a partition S with uncut_G(S) <= C OPT_uncut(G), and comparison of uncut_G(S) with K k decides satisfiability. A randomized C-approximation with success probability at least a fixed positive constant places 3SAT in RP after three independent repetitions.
 
 This completes the reduction from 3SAT to Min-UnCut at every prescribed constant factor.