diff a8edaab..ab2b37b artin_parabolic_intersections_v2.txt
diff --git a/artin_parabolic_intersections_v2.txt b/artin_parabolic_intersections_v2.txt
index 67d4b4d..881fcf0 100644
--- a/artin_parabolic_intersections_v2.txt
+++ b/artin_parabolic_intersections_v2.txt
@@ -645,3 +645,482 @@ word p and some PROPER subset J of Y.
That closes SUBCASE A of the stabilizer theorem.
-- Sections 16-19 in the next chunk --
+
+================================================================
+16. PROOF OF THE MAIN THEOREM
+================================================================
+
+We are now ready to prove the main theorem. Recall:
+
+ THEOREM. For finite Coxeter matrix M on S, subsets X, Y of S,
+ and elements g, h of A, there are Z ⊆ S and k in A with
+
+ g A_X g^{-1} ∩ h A_Y h^{-1} = k A_Z k^{-1}.
+
+Additional input needed: the "initial case".
+
+PROPOSITION 16.1 (Initial stabilizer). Fix X ⊆ S, and let f be
+the special coloring vertex of the construction. Using the family
+{E_L(1)}_L produced by the algebra Z, we have
+
+ H(S, 1) = A_X.
+
+PROOF. Direction (⊇). Every generator s of X commutes with the
+auxiliary vertex f, so B_s preserves each P_{f, L}; therefore every
+s in X lies in A_X ⊆ H(S, 1).
+
+Direction (⊆). Consider the vertex v_0 = (empty, {1}) in P(S, 1).
+Its family has only one nonzero layer, the level-0 unit vector at f.
+This is a MINIMUM-HEIGHT vertex (any competitor with a nonzero higher
+block is strictly later, and at level 0 the coordinate sum is
+minimal). Apply Proposition 12.5 with this minimum. Its visible
+set is T_+ = empty. Its invisible set K is X (colours not adjacent
+to f). So D = X and z = 1, giving
+
+ H(S, 1) ⊆ A_X. QED
+
+PROOF OF THE MAIN THEOREM.
+
+ STEP 1. Set x = h^{-1} g in A (so g = h x^{-1}). Then
+
+ g A_X g^{-1} ∩ h A_Y h^{-1}
+ = h ( x A_X x^{-1} ∩ A_Y ) h^{-1}.
+
+ (Both sides are equal after multiplying the middle expression by
+ h on left and h^{-1} on right.)
+
+ STEP 2. CLAIM: A_Y ∩ x A_X x^{-1} = H(Y, x).
+
+ Proof of claim. For a in A_Y:
+ a ∈ x A_X x^{-1}
+ iff x^{-1} a x ∈ A_X
+ iff (by Prop 16.1) B_{x^{-1} a x} preserves E_L(1) for every L
+ iff B_a preserves E_L(x) = B_x P_{f, L} for every L
+ iff a ∈ H(Y, x).
+
+ So the intersection of x A_X x^{-1} with A_Y is exactly H(Y, x).
+
+ STEP 3. By the stabilizer theorem (14.1), H(Y, x) = c^{-1} A_Z c
+ for some Z ⊆ Y and c in A_Y. Combining with Step 2:
+
+ x A_X x^{-1} ∩ A_Y = c^{-1} A_Z c.
+
+ STEP 4. Plug into Step 1:
+
+ g A_X g^{-1} ∩ h A_Y h^{-1}
+ = h (c^{-1} A_Z c) h^{-1}
+ = (h c^{-1}) A_Z (h c^{-1})^{-1}
+ = k A_Z k^{-1} with k = h c^{-1}.
+
+ This is what the theorem claims. QED.
+
+================================================================
+17. CONSEQUENCES
+================================================================
+
+COROLLARY A (arbitrary intersections, parabolic closure).
+
+ For every family F of parabolics of A, there is a subfamily
+ F_0 ⊆ F with |F_0| ≤ |S| and
+
+ (intersection over all P in F) = (intersection over all P in F_0).
+
+ In particular, arbitrary intersections of parabolics are themselves
+ parabolics. And every subset E ⊆ A lies in a unique smallest
+ parabolic subgroup PC(E), called the PARABOLIC CLOSURE of E.
+ Under inclusion, the parabolics form a complete lattice (meet =
+ intersection, join = parabolic closure of union).
+
+PROOF. Start with Q = A_S (the full group), with an empty list of
+selected parabolics. While Q is not contained in some P ∈ F,
+replace Q by Q ∩ P and add P to the list.
+
+ - By the main theorem, Q ∩ P is a parabolic. Its defining subset
+ is a proper subset of Q's current one, since Q was not contained
+ in P.
+ - Proper here means: rank strictly decreases.
+ - Rank decreases by at least 1 each step, and rank ≤ |S|.
+ - So the loop terminates in at most |S| steps.
+
+The resulting Q is contained in every member of F, and equals the
+intersection of the selected (≤ |S|) parabolics. Hence intersection
+over F = intersection over F_0. The lattice structure follows at
+once. QED
+
+COROLLARY B (word and parabolic membership problems).
+
+ There is an algorithm which, given (M, X, u, v) where M is a
+ finite Coxeter matrix, X ⊆ S, and u, v are words in the generators
+ and their inverses, decides whether the element represented by v
+ lies in u A_X u^{-1}.
+
+ In particular, the WORD PROBLEM is decidable for every finitely
+ generated Artin group.
+
+PROOF SKETCH. Let w = u^{-1} v u. Build the algebra Z of Section 8
+explicitly over the number field K = Q(exp(2 pi i / N)), where N
+is the least common multiple of 2m over all finite m ≥ 4 (over Q if
+no such m). Then:
+
+ - Deciding "w ∈ A_X" ⇔ "B_w P_{f, L_0} is homotopy-equivalent to
+ P_{f, L_0} for one fixed label L_0" (Prop 16.1 plus the fact
+ that the family is preserved in a compatible way).
+
+ - Compute B_w P_{f, L_0} explicitly as a bounded complex of graded
+ projective Z-modules. This uses finite tensor products and
+ Gaussian elimination over the finite-dimensional algebra Z.
+
+ - Cancel invertible scalar differential entries (standard
+ "minimal model" reduction). Terminates because it removes two
+ summands per step.
+
+ - The result is minimal and hence unique up to isomorphism. Test
+ whether it has exactly one generator: P_{f, L_0} in placements
+ (0, 0). Yes: w ∈ A_X. No: w ∉ A_X.
+
+ For X = empty, this solves the word problem: w = 1 iff w ∈ A_empty
+ iff P_{f, 1} is preserved. QED
+
+No complexity bound is asserted. The algorithm is finite-time but
+its running time can be huge; that is not the point.
+
+COROLLARY C (non-clique dynamics).
+
+ Suppose M is irreducible (not a disjoint union of two Coxeter
+ matrices with all cross labels = 2), and m(s, t) = infinity for
+ some s ≠ t. Then:
+
+ (i) A is acylindrically hyperbolic.
+
+ (ii) Every proper parabolic subgroup P of A admits a conjugate
+ g P g^{-1} with P ∩ g P g^{-1} = {1}.
+
+Part (i) is due to a separate 2025-2026 line of work (Kato-Oguni
+/ Charney-Martin-Morris-Wright), not the main theorem. Part (ii)
+uses the main theorem: the intersection P ∩ g P g^{-1} is a
+parabolic (main theorem), it is finite (weak malnormality of proper
+parabolics, from the same second source), and any nontrivial
+parabolic contains a conjugate of a generator s, which has infinite
+order (exponent-sum homomorphism A → Z sends s to 1). So a finite
+parabolic must be {1}.
+
+COROLLARY D (complete parabolics, FA', lcH-slender).
+
+ A parabolic is COMPLETE if it is a conjugate of A_T for a subset T
+ with no INFINITE label (labels 2, 3, 4, ... allowed, sphericity NOT
+ required). Then:
+
+ (i) Every subgroup of A with property FA' (fixed-point property
+ for actions on trees without inversions) lies in a complete
+ parabolic.
+
+ (ii) Every abstract homomorphism from an almost-connected locally
+ compact Hausdorff group into A has image in a complete
+ parabolic.
+
+ (iii) If every complete standard parabolic of A is lcH-slender,
+ then A is lcH-slender.
+
+PROOF: all three parts reduce to "intersections of parabolics are
+parabolic" plus the Moller-Paris-Varghese (2024) transfer results. QED
+
+================================================================
+18. WORKED EXAMPLES
+================================================================
+
+We check the theorem by hand in five small cases. The whole point is
+to see the theorem in action where the calculation is short enough
+to verify line by line.
+
+------------------------------------------------------------------
+EXAMPLE 1: Z^2 (abelian, "everything commutes")
+
+ S = {a, b}, m(a, b) = 2. A = Z^2. Every parabolic is standard
+ (conjugation does nothing in an abelian group): {0}, Z a, Z b,
+ Z^2.
+
+ Intersections:
+ Z a ∩ Z b = {0} = Z^0 OK
+ Z a ∩ Z^2 = Z a = Z^a OK
+ Z^2 ∩ Z^2 = Z^2 OK
+ {0} ∩ anything = {0} OK
+
+ Verdict: trivially true; holds for any abelian group.
+
+------------------------------------------------------------------
+EXAMPLE 2: F_2 (free group, right-angled)
+
+ S = {a, b}, m(a, b) = infinity. A = F_2, pars are {1}, <a>, <b>,
+ F_2, and conjugates. Conjugates of <a>: the groups <waw^{-1}> for
+ w in F_2 (all infinite cyclic).
+
+ By the classical theory of free groups, an intersection of two
+ infinite cyclic subgroups of a free group is cyclic (possibly
+ trivial). Everything is parabolic.
+
+ Verdict: trivially true by hand.
+
+------------------------------------------------------------------
+EXAMPLE 3: Braid_3 = <a, b | aba = bab>
+
+ S = {a, b}, m(a, b) = 3. A = Braid_3 (braid group on 3 strands).
+ Standard parabolics: {1}, <a>, <b>, Braid_3.
+
+ Take P = <a> and Q = <b a b^{-1}>. Both are infinite cyclic.
+
+ Question: is P ∩ Q trivial?
+
+ If x ∈ P ∩ Q is nontrivial, then a^k = x = (bab^{-1})^j for some
+ nonzero integers k, j. So a and bab^{-1} are both nontrivial
+ powers of the same element x, hence they COMMUTE.
+
+ But: take the natural quotient Braid_3 → S_3 sending a → (12) and
+ b → (23). Then bab^{-1} → (23)(12)(23) = (13). In S_3,
+ (12) · (13) = (132), (13) · (12) = (123),
+ which are different. So (12) and (13) do NOT commute in S_3, so
+ a and bab^{-1} do NOT commute in Braid_3. Contradiction.
+
+ Hence P ∩ Q = {1} = A_empty. Parabolic.
+
+ Verdict: hand-checkable, correct.
+
+ (Note: by the Cumplido-Gebhardt-Gonzalez-Meneses-Wiest 2019
+ theorem, ALL intersections of parabolics in spherical Braid_3 are
+ parabolic. Our example is a special case.)
+
+------------------------------------------------------------------
+EXAMPLE 4: right-angled Artin on the path a - b - c
+
+ S = {a, b, c}, m(a, b) = m(b, c) = 2, m(a, c) = infinity.
+ Presentation: A = <a, b, c | ab = ba, bc = cb>. That is: a and b
+ commute, b and c commute, a and c do not interact.
+
+ Structure: A = Z^2 *_<b> Z^2, an amalgamated free product of the
+ two Z^2 factors <a, b> and <b, c>, merged along their common
+ subgroup <b> = Z.
+
+ This is nonspherical: the associated Coxeter group is
+ W = (Z/2 × Z/2) *_<b> (Z/2 × Z/2),
+ which is infinite.
+
+ Take P = <a, b> = Z^2 and Q = <b, c> = Z^2. By the amalgam
+ structure, an element of both P and Q must lie in the amalgam
+ <b>:
+
+ P ∩ Q = <b> = A_b.
+
+ Parabolic (rank 1). GOOD
+
+ Now a conjugate version. Note that b commutes with c, so
+ c b c^{-1} = b in this group. Take
+
+ P = <a, b>, Q = c <a, b> c^{-1} = <c a c^{-1}, b> = <u, b>,
+
+ where u = c a c^{-1} is a conjugate of a.
+
+ Both P and Q contain <b>. Suppose a^k b^l = u^m b^n with k, m
+ both nonzero. Then a^k b^{l-n} = u^m = c a^m c^{-1}. The
+ left side is in the first amalgam factor Z^2 = <a, b>; the right
+ side is (c) · a^m · (c^{-1}), a normal-form expression that
+ begins and ends with elements of the second factor outside the
+ amalgam <b>. In reduced normal form, such an expression cannot
+ equal an element of the first factor (unless m = 0, reducing it
+ to 1). So m = 0, whence a^k b^{l-n} = 1 in Z^2, so k = 0 and
+ l = n. Therefore P ∩ Q = <b>.
+
+ Verdict: hand-checkable, correct.
+
+------------------------------------------------------------------
+EXAMPLE 5: a nonspherical case with braid relations
+
+ S = {a, b, c}, m(a, b) = m(b, c) = 3, m(a, c) = infinity.
+ Diagram: a - b - c path, edges labelled 3, 3; no relation between
+ a and c.
+
+ Associated Coxeter group:
+ W = <a, b, c | a^2 = b^2 = c^2 = 1, (ab)^3 = 1, (bc)^3 = 1>.
+ The subgroups <a, b> and <b, c> are each S_3 (order 6); they
+ share <b> = Z/2. So
+ W = S_3 *_<b> S_3,
+ an infinite amalgamated free product. Nonspherical.
+
+ Standard parabolics:
+ {1}, <a> = Z, <b> = Z, <c> = Z,
+ <a, b> = Braid_3 (spherical),
+ <b, c> = Braid_3 (spherical),
+ <a, c> = F_2 (free, since m(a, c) = infinity),
+ <a, b, c> = A (nonspherical).
+
+ Take P = <a, b> and Q = <b, c>. Both are Braid_3-type. They
+ share <b>. Is the intersection bigger than <b>?
+
+ Suppose x ∈ <a, b> ∩ <b, c> is not in <b>. Then x is a
+ nonempty reduced word in {a, b, a^{-1}, b^{-1}} (in the first
+ subgroup) that is also a reduced word in {b, c, b^{-1}, c^{-1}}
+ (in the second). In the free product structure underlying A, an
+ element is uniquely determined by its normal form. An element
+ in the subgroup generated by a, b has normal form using only
+ a-appearances and b-appearances; an element in the subgroup
+ generated by b, c has normal form using only b-appearances and
+ c-appearances. An element in BOTH must use only b-appearances
+ (else it would have an a or c appearance, contradicting
+ membership in the other subgroup; the subgroup <a, b> really
+ contains elements with an "a" in the normal form, and similarly
+ for c). So x ∈ <b>. Therefore:
+
+ P ∩ Q = <b>.
+
+ Parabolic (rank 1). GOOD
+
+ This is the simplest honest case where the ambient group A is
+ nonspherical, both intersected parabolics are spherical, and the
+ intersection is a proper subset that still sits as a standard
+ parabolic. The main theorem says this works for ANY Artin group,
+ not just ones where such a hand check is possible.
+
+================================================================
+19. WHAT IS AND IS NOT CLAIMED
+================================================================
+
+WHAT IS CLAIMED:
+
+ - The intersection of two parabolics in any finite-rank Artin
+ group is parabolic. No assumption of sphericity. No finiteness
+ of labels.
+
+ - Arbitrary intersections of parabolics are parabolic, and at
+ most |S| members are ever needed.
+
+ - The word problem and the "membership in u A_X u^{-1}" problem
+ are decidable for every finite-rank Artin group.
+
+ - The Parabolic Intersection Conjecture implies (combined with
+ other work) acylindrical hyperbolicity of non-clique Artin
+ groups, and reduces "complete parabolic" membership results
+ (FA', lcH-slender) to the complete parabolics themselves.
+
+WHAT IS NOT CLAIMED:
+
+ - No complexity bound is supplied for the algorithm of Corollary B.
+
+ - The construction of Z is nontrivial in practice (it involves
+ fusion-labelled Temperley-Lieb-Jones categories, Jones-Wenzl
+ idempotents, and exact linear algebra over a specific number
+ field). The paper shows it is COMPUTABLE in principle, not
+ that it is EFFICIENT.
+
+ - This document does not give the full inner proofs of the
+ categorical and homological lemmas (Sections 8, 9, 11, 13 of
+ the research paper). The shape of those arguments is described
+ but not reproduced.
+
+================================================================
+20. GLOSSARY
+================================================================
+
+ARTIN GROUP (A_M): the group generated by s in S with braid
+ relations of length m(s,t). Section 3.
+
+CANDIDATE: a typed coset satisfying the two-point prescription.
+ Section 12.
+
+CAP (at level q on D): no generator above level q at any colour
+ of D. Section 11.
+
+COCHAIN / INTERNAL DEGREE: the two indices on a generator of a
+ graded Z-complex. Section 9.
+
+CONJUGATE: g P g^{-1} for some g in A. Section 4.
+
+COXETER GROUP (W_M): quotient of the Artin group by s^2 = 1.
+ Section 3.
+
+ELIGIBLE COLOUR: a colour whose actual level-q count is strictly
+ below the harmonic interpolation, or equal with a "+1" sign mask.
+ Section 12.
+
+HEIGHT: the lexicographic comparison of level blocks at a vertex
+ of P(Y, x). Section 12.
+
+H(Y, x) (STABILIZER): { a in A_Y : a preserves E_L(x) for all
+ labels L }. Section 10.
+
+INVISIBLE: colours whose harmonic interpolation and that of their
+ exterior neighbours all vanish. Section 12.
+
+LAYER (Y^d(E)): the generators at level d, with their differential.
+ Section 9.
+
+LEVEL (d = c + u): cochain degree plus internal degree. Section 9.
+
+m(s, t): Coxeter matrix entry. Section 2.
+
+PARABOLIC CLOSURE (PC(E)): smallest parabolic containing E.
+ Section 17.
+
+PARABOLIC SUBGROUP: a conjugate of a standard A_X. Section 4.
+
+POSITIVE WORD: a word in the generators (no inverses). Section 1.
+
+SPHERICAL (Coxeter matrix): its Coxeter group is finite. Section 2.
+
+STANDARD PARABOLIC (A_X): subgroup generated by X ⊆ S. Section 4.
+
+STABILIZER THEOREM: H(Y, x) is a conjugate of a standard parabolic
+ of A_Y. Section 14.
+
+TEST (e_a E): right projective at a ⊗ E; its cohomology detects
+ the layer's top and socle. Section 9.
+
+TRUNCATION TRIANGLE: 0 -> t_{<=k}E -> E -> t_{>=k+1}E -> 0.
+ Section 9.
+
+VISIBLE (T_+): complement of the invisible set in the type T.
+ Section 12.
+
+WEIGHTED FUSION: the identity Sum_L w_L [D ⊗ L : H] = w_H dim D.
+ Section 8.
+
+Z: the auxiliary finite-dimensional graded algebra. Section 8.
+
+================================================================
+21. FURTHER READING
+================================================================
+
+Coxeter and Artin groups: Bourbaki, "Lie Groups and Lie Algebras,
+Chapters 4-6" (2002). Davis, "The Geometry and Topology of Coxeter
+Groups" (2008).
+
+Standard parabolics and their intersections: van der Lek's thesis
+(1983). Paris, "Parabolic subgroups of Artin groups", J. Algebra
+157 (1997). Charney-Paris, "Convexity of parabolic subgroups in
+Artin groups", Bull. LMS 46 (2014).
+
+Prior cases of the Parabolic Intersection Conjecture:
+ Cumplido-Gebhardt-Gonzalez-Meneses-Wiest, Ann. of Math. 190 (2019)
+ - spherical.
+ Duncan-Kazachkov-Remeslennikov, J. Algebra 316 (2007) - right-angled.
+ Morris-Wright, J. Pure Appl. Algebra 225 (2021) - FC, one side.
+ Moller-Paris-Varghese, Israel J. Math (2024) - one side spherical;
+ and generalization to arbitrary intersections assuming pairwise.
+ Antolin-Foniqi, Proc. Edinburgh Math. Soc. 65 (2022) - even FC.
+ Cumplido-Martin-Vaskou, Math. Proc. Cambridge (2023) - large type.
+
+Categorical action: Khovanov-Seidel, JAMS 15 (2002). Huerfano-
+Khovanov, J. Algebra 246 (2001). Heng-Licata, "Fusion-stable
+modules over a twisted algebra" (2024).
+
+Acylindrical hyperbolicity and non-clique behaviour: Charney-Martin-
+Morris-Wright (2026). Kato-Oguni (2025).
+
+lcH-slender and complete parabolics: Moller-Paris-Varghese (2024).
+
+The full research paper: "Parabolic Intersections in Artin Groups,"
+source files included in this repository (01_intro.tex, 02_category.tex,
+03_layers.tex, 04_caps_twopoint.tex, 05_null.tex, 06_heights.tex,
+07_spherical.tex, 08_stabilizers.tex). That paper gives every lemma
+and proof in full, at the level of graduate algebra and topology.
+
+================================================================
+END
+================================================================