diff b1f9bc4..56bc728 switch_chain_polynomial_mixing_v2.txt
diff --git a/switch_chain_polynomial_mixing_v2.txt b/switch_chain_polynomial_mixing_v2.txt
index c2abd2c..afd24d6 100644
--- a/switch_chain_polynomial_mixing_v2.txt
+++ b/switch_chain_polynomial_mixing_v2.txt
@@ -193,50 +193,6 @@ Remarks on the shape of the bound.
2^{-n^2}, this matches the theorem's 2 n^8 up to logarithmic
factors. Nothing exotic is hiding in the constants.
---------------------------------------------------------------------------
-1.4 History: what did we already know?
---------------------------------------------------------------------------
-
-Approximate uniform generation of graphs with prescribed degrees is a
-classic algorithmic problem, with applications in statistics,
-network science, and combinatorics. The switch chain (a.k.a. the
-degree-preserving network randomization or the swap chain) was
-proposed as a practical algorithm long before anyone could analyse it.
-
-The Kannan-Tetali-Vempala 1999 paper launched the switch-chain
-program, analysed regular bipartite sequences, and raised the
-possibility of broad applicability. Cooper-Dyer-Greenhill 2007 (with
-a 2012 corrigendum) proved rapid mixing for regular sequences. Feder-
-Guetz-Mihail-Saberi (2006) handled early irregular families. Greenhill
-(2015) and Greenhill-Sfragara (2018) proved rapid mixing under certain
-degree-sparsity restrictions. Amanatidis-Kleer (2020) gave the strong-
-stability route. Jerrum-Sinclair (1990) and Jerrum-McKay-Sinclair
-(1992) introduced the notion of stability of degree sequences that
-becomes the algorithmic toolkit. Erdos-Greenhill-Mezei-Miklos-Soltesz-
-Soukup (2022) unified all "P-stable" families. Erdős-Miklós-Soukup
-(2025) and Erdős-Lippner-Nevo-Soukup (2025) developed further rapidly
-mixing families via geometric methods.
-
-The modern form of the Kannan-Tetali-Vempala conjecture predicted rapid
-mixing for EVERY realizable degree specification, in each of three
-parallel models: bipartite graphs, directed graphs, and simple
-undirected graphs. Fu-Qin-Wang (2026) proved the bipartite form.
-Theorem 1 here proves the simple undirected form. The three
-conjectures are genuinely distinct, though the techniques overlap
-heavily.
-
-Along the way, an "undirected Curveball" resampling was studied by
-Verhelst (2008), Strona et al. (2014, 2016), and Carstens-Berger-Strona
-(2018). Carstens-Berger-Strona already observed that the trades on
-disjoint vertex pairs can be DEPENDENT; the current paper is the first
-to quantify that dependence tightly enough to prove polynomial mixing
-for all graphical d.
-
-The exact sampling approach used in Section 10 is a rare-branch
-correction due to Gobel-Liu-Manurangsi-Pappik (2024), combined with
-exact tabulation of the residual law. A similar tabulation appears in
-the recent contingency-table sampler of OpenAI (2026).
-
--------------------------------------------------------------------------
SECTION 2. PAIR RESAMPLINGS AND THE OPERATOR H
--------------------------------------------------------------------------
@@ -377,40 +333,6 @@ trades" in 2018). So the engine inequality is not a triviality that
falls out of a commuting family of operators. Each family needs its
own estimate.
---------------------------------------------------------------------------
-2.4 Plan of attack, section by section
---------------------------------------------------------------------------
-
-Split the sum over ordered pairs (a, b) of vertex pairs into two
-groups:
-
- (TRIPLES) a and b share a vertex, so a union b is a vertex TRIPLE
- T. On a fixed triple T, there are three vertex pairs,
- and every ordered pair of distinct pairs has union T.
- Contribution: sum over triples T of H_T^2, where
- H_T = sum_{a in C(T,2)} h_a is the projection to a single
- triple. Sections 3-4 prove H_T^2 >= H_T.
-
- (DISJOINT) a and b do not share a vertex. Contribution:
- D = sum_{a, b disjoint} h_a h_b. Sections 5-7 prove D
- is positive semidefinite.
-
-If both pieces hold, then
-
- H^2 = sum_T H_T^2 - (n - 3) H + D
- >= sum_T H_T - (n - 3) H + 0
- = (n - 2) H - (n - 3) H = H,
-
-where the combinatorial identity H^2 = sum_T H_T^2 - (n - 3) H + D is
-a regrouping of the ordered double sum (see Section 8). This is
-exactly ENGINE, and it is the core of the paper.
-
-Sections 3-4 handle the triple terms with a variance inequality for a
-3-row binary matrix. Sections 5-7 handle the disjoint terms, first by
-identifying a rank-one correlation between the two resamplings, then
-bounding that correlation with a slice inequality. Sections 8-10 wrap
-everything up.
-
--------------------------------------------------------------------------
SECTION 3. THE THREE-ROW VARIANCE INEQUALITY
--------------------------------------------------------------------------
@@ -2173,43 +2095,4 @@ execution is impossible: computing |Omega_d| exactly (which the branch
does) is #P-hard in general.
--------------------------------------------------------------------------
-SECTION 10. FINAL REMARKS
---------------------------------------------------------------------------
-
-What we have proved. For every graphical labeled degree vector d on
-n >= 4 vertices, the simple-undirected switch chain mixes in
-total-variation distance 1/4 from any start in at most 2 n^8 steps, and
-there is an exactly uniform sampler for the same degree vector running
-in expected polynomial bit time. This resolves the simple-undirected
-form of the Kannan-Tetali-Vempala conjecture.
-
-What is missing. The bipartite and directed forms are analogous
-problems with their own techniques; Fu-Qin-Wang (2026) proved the
-bipartite form. The general-host versions (prescribed degrees in
-arbitrary hosts, weighted edges, etc.) are open in general. Bounds
-for sampling FROM a specific distribution other than uniform remain
-an active problem.
-
-The big idea you should take away. Four moves, all simple in
-hindsight:
-
- 1. Replace "one switch at a time" by "resample a whole fiber at once".
- Fibers are uniform q-subsets, so the analysis reduces to classical
- slice combinatorics.
-
- 2. Show one operator inequality, H^2 >= H, by splitting into "three
- pairs on a triple" (a three-row variance bound) and "disjoint
- pairs" (a rank-one correlation corrected by a slice inequality).
-
- 3. Compare back to the switch chain using the same fiber structure.
- The gap is 1/(24 n^2 C(n,4)).
-
- 4. Upgrade to exact uniform via a rare-branch correction.
-
-The whole paper is these four moves, developed carefully.
-If you followed this far, you know what a spectral gap is, why mixing
-time depends on it, how to compare one Markov chain to another, and how
-to correct a small bias to make sampling exactly uniform. Those are
-four of the most useful ideas in modern probability, and they are all
-here.