diff --git a/symmetric_tensor_categories_fiber_v2.txt b/symmetric_tensor_categories_fiber_v2.txt
index 2058a0b..550a9db 100644
--- a/symmetric_tensor_categories_fiber_v2.txt
+++ b/symmetric_tensor_categories_fiber_v2.txt
@@ -86,7 +86,7 @@ such that:
  every object has finite length;
  there are finitely many isomorphism classes of simple
         objects;
- the unit object mathbf 1bf1 has End(mathbf 1)bf1) = k;
+ the unit object 1bf1 has End(1)1) = k;
  there are enough projective objects.
 
 
@@ -134,7 +134,7 @@ in Benson--Etingof (2019).
 
 
 Intuition for Ver_p: it is the ``semisimplification'' of
-Rep_k(Z/p):bbZ/p): one kills every morphism f with tr(gf)=0
+Rep_k(Z/p):Z/p): one kills every morphism f with tr(gf)=0
 for all g; the simple objects correspond to Jordan blocks of
 lengths 1,...,p-1.  For p=2 this collapses to
 Vec_k; for p=3 it gives supervector spaces.  Each higher level
@@ -174,7 +174,7 @@ Serre closure of that subcategory.  The essential ingredients are:
 [label=Step arabic*.,ref=step:arabic*]
  Restricted tilting modulo a Steinberg module.
         For every finite-dimensional module D over the
-        elementary abelian group E_r = (Z/p)^r,bbZ/p)^r, the
+        elementary abelian group E_r = (Z/p)^r,Z/p)^r, the
         tensor product St_{r-1} ⊗  D of a fixed Steinberg
         module with D is a restriction of an SL_2-tilting
         module.  This is the Coulembier--Flake conjecture; we prove
@@ -230,11 +230,11 @@ A = End_{C}{op}.
 
 
 Proof. 
-Simple unit is the Deligne--Milne argument.  If Usubsetmathbf 1bf1
+Simple unit is the Deligne--Milne argument.  If Usubset1bf1
 is a proper subobject with cokernel Vne0, the epimorphism
-Uto V ⊗  U coming from mathbf 1bf1twoheadrightarrow V has
+Uto V ⊗  U coming from 1bf1twoheadrightarrow V has
 zero composite with the subobject V ⊗  Utwoheadrightarrow V.
-So V ⊗  U=0.  Put W=ker(mathbf 1bf1to U^*).  Then W
+So V ⊗  U=0.  Put W=ker(1bf1to U^*).  Then W
 sits inside a short exact sequence with U, V, and tensoring by
 W gives W~= V.  The endomorphism ring of the unit has no
 idempotents, so V=0.
@@ -251,7 +251,7 @@ presentation.
 
 
 We fix such a P and such an A once and for all.  In particular,
-Ptwoheadrightarrowmathbf 1,bf1, since the unit is one of the
+Ptwoheadrightarrow1,1, since the unit is one of the
 simples.
 
 ## Deligne products
@@ -267,7 +267,7 @@ the two factors (here we use k algebraically closed).
 
 [combining exact tensor functors]
 
-Let A_1,...,calA_1,...,A_s,Z be finite
+Let A_1,...,A_1,...,A_s,Z be finite
 symmetric tensor categories and let F_i:A_i ZcalZ
 be exact k-linear strong symmetric monoidal functors.  Then
 [
@@ -296,7 +296,7 @@ For finite tensor categories the Grothendieck ring K_0(C)
 is finitely freely generated by simple classes.  Symmetry makes it a
 commutative ring, and the Frobenius--Perron theorem identifies a
 unique positive normalised real character
-d = FPdim : K_0(C) RbbR with d(mathbf 1)=1bf1)=1
+d = FPdim : K_0(C) RbbR with d(1)=1bf1)=1
 and d multiplicative on classes.  Key facts, all in Etingof--
 Gelaki--Nikshych--Ostrik (2015):
 
@@ -346,7 +346,7 @@ weight a.
 
 [restricted-tilting theorem]
 
-Let rge 1, E_r = (Z/p)^r,bbZ/p)^r, and iota:E_rto U(k) any
+Let rge 1, E_r = (Z/p)^r,Z/p)^r, and iota:E_rto U(k) any
 injective homomorphism.  For every finite-dimensional kE_r-module
 D there is a tilting G-module T such that
 [
@@ -363,9 +363,9 @@ Conjecture 1 of Coulembier--Flake (2025), for every prime.
 We do three things:
 [label=(alph*)]
  lift the arbitrary kE_r-module D to a coherent rational
-        U-module widehatwidehat DD of bounded ``degree'' (a bound on
+        U-module DDhat of bounded ``degree'' (a bound on
         polynomial degree of the unipotent action matrix);
- use widehatwidehat DD to construct a G-equivariant vector
+ use DDhat to construct a G-equivariant vector
         bundle M on the plane A^2_k whose
         origin fibre has all simple composition factors below a
         height bound;
@@ -385,8 +385,8 @@ matrix rho_D(u(t)); this does not depend on a chosen basis.
 [bounded lift]
 
 For every finite-dimensional kE_r-module D, there is a
-finite-dimensional rational U-module widehatwidehat DD of degree < p^r
-such that widehatwidehat D|_D|_{E_r}~= D along iota.
+finite-dimensional rational U-module DDhat of degree < p^r
+such that Dhat|_D|_{E_r}~= D along iota.
 
 
 Proof. 
@@ -405,17 +405,17 @@ rational U-module of degree <q.  Its restriction is D.
 
 ## The G-equivariant bundle on the plane
 
-Now consider the orbit map pi : GtoA^2∖bbA^2∖{0},
+Now consider the orbit map pi : GtoA^2∖A^2∖{0},
 g|-> g e_1, with e_1=(1,0).  This is a right U-torsor:
 the fibres are right U-cosets and on the standard opens xne 0,
 yne 0 it has obvious rational sections.  Form the associated
 vector bundle
 [
-  E = Gtimes^U widehatwidehat D,D,
+  E = Gtimes^U Dhat,D,
    (g,d)sim(gu,u^{-1}d),
 ]
 on the punctured plane, with G acting by left multiplication.
-The fibre at e_1 is widehatwidehat DD itself.
+The fibre at e_1 is DDhat itself.
 
 [extension and weight bound]
 
@@ -428,7 +428,7 @@ M_0 = M|_0calM|_0 has highest weight < q.
 Proof. 
 Extension.  A coherent extension F exists on the
 noetherian affine plane.  Its double dual M = mathcal
-F^calF^{**} is a coherent reflexive sheaf restricting to E.
+F^F^{**} is a coherent reflexive sheaf restricting to E.
 On the regular surface A^2 every reflexive coherent sheaf
 is locally free: at a closed point the stalk has depth two, so the
 Auslander--Buchsbaum formula gives projective dimension zero, and at
@@ -439,7 +439,7 @@ Equivariance.  Over the punctured plane we have a
 G-equivariant bundle isomorphism
 pi_2^*M a^*M for the action
 a:GtimesA^2 A^2bbA^2 and projection
-pi_2:GtimesA^2 A^2.bbA^2.  Both sides are locally
+pi_2:GtimesA^2 A^2.A^2.  Both sides are locally
 free, so the Hom-sheaf is locally free and its sections extend
 across the codimension-two subset Gtimes{0}; the extensions of
 the isomorphism and its inverse remain inverse because the two
@@ -512,7 +512,7 @@ are closed under tensor products.
 
 
 Applying cref{lem:steinberg-tilting} with m=r to M_0 = mathcal
-M|_0,calM|_0, we have:
+M|_0,M|_0, we have:
 
   T := St_{r-1} ⊗  M_0
 
@@ -531,12 +531,12 @@ N|_calN|_{e_1}~= T|_U as rational U-modules.
 
 
 Proof. 
-Jet trivialisation.  Let A=k[x,y], mathfrak m=(x,y),frakm=(x,y),
+Jet trivialisation.  Let A=k[x,y], m=(x,y),m=(x,y),
 and N=Gamma(A^2,N), a possibly
 infinite-dimensional rational G-module.  Local freeness gives
 canonical G-isomorphisms
 
-  mathfrak m^jfrakm^j N/mathfrak m^frakm^{j+1}N ~=
+  m^jfrakm^j N/m^m^{j+1}N ~=
   Sym^j(V^*) ⊗ _k T.
 
 The target has a good filtration.  Tensor products of Sym's of
@@ -551,20 +551,20 @@ Now T has both a Weyl and a good filtration, so for every j,
 ]
 by Cline--Parshall--Scott--van der Kallen.
 
-Starting from the identity s_1:Tto N/mathfrak mfrakm N, we lift
-successively to G-maps s_j:Tto N/mathfrak m^jfrakm^j N that reduce
-modulo mathfrak m^jfrakm^j to s_{j-1}; the obstruction is exactly the
+Starting from the identity s_1:Tto N/m N, we lift
+successively to G-maps s_j:Tto N/m^jfrakm^j N that reduce
+modulo m^jfrakm^j to s_{j-1}; the obstruction is exactly the
 Ext group just computed.  Extending scalars, s_j gives an
-A/mathfrak m^j-linearfrakm^j-linear map
-(A/mathfrak m^j)frakm^j) ⊗ _k Tto N/mathfrak m^jfrakm^j N, which is an
+A/m^j-linearfrakm^j-linear map
+(A/m^j)m^j) ⊗ _k Tto N/m^jfrakm^j N, which is an
 isomorphism by Nakayama (both sides free of the same rank and the
 map is an isomorphism on the residue field).  The inverse limit is a
 G-equivariant isomorphism
-k[[x,y]] ⊗ _k Txrightarrow{sim}widehatwidehat N.N.
+k[[x,y]] ⊗ _k Txrightarrow{sim}Nhat.N.
 
 Fibre comparison.  Restrict to the fixed line y=0, pass
 to K = k((x)): this gives a U_K-isomorphism
-phi:T_Kcong N_{xe_1}.  To cancel the parameter
+phi:T_K~= N_{xe_1}.  To cancel the parameter
 mismatch we use the torus element h = diag(x,x^{-1}):
 since hu(t)h^{-1} = u(x^2 t), the composite
 f = b_h^{-1}phirho_T(h) intertwines the original
@@ -584,7 +584,7 @@ Apply cref{prop:extension-bound} to get M with origin
 fibre M_0 whose simple factors have highest weight <q=p^r.
 By cref{lem:steinberg-tilting}, T = St_{r-1} ⊗  M_0 is
 tilting.  Apply cref{lem:formal-comparison} to
-N = St_{r-1} ⊗ _kM:calM: its origin fibre is
+N = St_{r-1} ⊗ _kM:M: its origin fibre is
 T, so
 [
   N|_calN|_{e_1} ~= T|_U.
@@ -604,7 +604,7 @@ does T.  Set T_{out} := T; the theorem follows.
 Fix rge 1, q=p^r, and an embedding
 E = E_r ↪ U(k).  We write R for restriction of
 SL_2-modules along this embedding, and W = Ver_{p^r}(k).
-Let T⊆calT⊆ Rep_k(SL_2) be the category of tilting
+Let T⊆T⊆ Rep_k(SL_2) be the category of tilting
 modules, and let I_r be the tensor ideal generated by
 St_r = L(p^r-1).
 
@@ -669,7 +669,7 @@ C is an object of C
 equipped with a commuting E-action.  Applying Θ to the
 underlying E-module and to the action maps produces an A-module
 object in W, hence an object of
-C W.calW.  This is the internal
+C W.W.  This is the internal
 extension Θ_{C} of Θ (Coulembier--Flake
 2026, Example 3.2.4 and Proposition 3.2.6; requires only
 additivity of Θ, not exactness).
@@ -832,7 +832,7 @@ proper subgroup of E_r is projective.
 
 Proof. 
 Suppose all M(a) projective.  Fix d and an elementary abelian
-p-subgroup Hlemathfrak S_d.frakS_d.  Every H-orbit in
+p-subgroup HleS_d.S_d.  Every H-orbit in
 {1,...,d} is regular because H is abelian; orbitwise,
 P^{ ⊗  d} is a product of P^{ ⊗  p^{a_i}}, each
 projective over k E_{a_i} by assumption.  By
@@ -840,21 +840,21 @@ cref{lem:product-actions}, omega(P^{ ⊗  d}) is projective
 over kH for the product of orbit groups, hence over the diagonal
 copy of H (which is an injective image).  Chouinard's theorem then
 says projective over every elementary abelian p-subgroup implies
-projective over kmathfrak S_d.frakS_d.
+projective over kS_d.S_d.
 
-Now apply Ptwoheadrightarrowmathbf 1bf1 (mathfrak S_d-equivariant,frakS_d-equivariant,
+Now apply Ptwoheadrightarrow1bf1 (S_d-equivariant,S_d-equivariant,
 trivial action on the target) and exactness of omega followed by
 coinvariants:
 [
- omega(P^{ ⊗  d})_{mathfrak S_d}frakS_d twoheadrightarrow
- omega(mathbf 1)bf1)ne0.
+ omega(P^{ ⊗  d})_{S_d}S_d twoheadrightarrow
+ omega(1)1)ne0.
 ]
 By cref{lem:norm-projectivity}, the total symmetrizer
-b_d = sum_{sigmainmathfrak S_d}frakS_dsigma acts nontrivially on
+b_d = sum_{sigmainS_d}S_dsigma acts nontrivially on
 P^{ ⊗  d}.
 
 On the other hand the action map
-kmathfrak S_dfrakS_dtoEnd_{C}{ ⊗  d}) cannot be
+kfrakS_dtoEnd_{C}{ ⊗  d}) cannot be
 injective: length grows exponentially in d (bounded by
 ell(P)b^{d-1} with b=max_iell(S_i ⊗  P)) and the
 dimension of Hom(X,Y) is bounded by ell(X)ell(Y), whereas
@@ -875,7 +875,7 @@ diagonal.
 ## Rank varieties and support
 
 Let kE_r-modules be graded by the augmentation ideal
-mathfrak mfrakm = (g_i-1)_i.  The projectivised rank variety of
+m = (g_i-1)_i.  The projectivised rank variety of
 a finite-dimensional module D is
 [
  supp(D) =
@@ -900,7 +900,7 @@ F_p-linearlybbF_p-linearly independent coordinates.
 
 Proof. 
 For cin(F_p^r)^, let H=ker c and
-N = Ind_H^{E_r}k, the regular representation of E_r/Hcong
+N = Ind_H^{E_r}k, the regular representation of E_r/H~=
 Z/pbbZ/p (identified with k[z]/(z^p), generator acts by
 1+z).  The element sum_ialpha_i(g_i-1) acts on N by
 h_alpha(z) = sum_ialpha_i((1+z)^{c_i}-1), whose linear
@@ -975,7 +975,7 @@ additive polynomial f(X)=sum_{s=0}^{r-2}a_s X^{p^s} vanishes on
 the p^r-element span of the beta_i, but deg fle p^{r-2}, so
 f=0.
 
-Row space of B contains no nonzero F_p^r-vector:bbF_p^r-vector: if
+Row space of B contains no nonzero F_p^r-vector:F_p^r-vector: if
 c_i = f(beta_i)inF_p, then the polynomial f^p-f
 vanishes on the same p^r-element set and has degree
 le p^{r-1}; so f^p-f=0, forcing f=0 (since deg f<p^{r-1}
@@ -1032,7 +1032,7 @@ projective restriction to every proper subgroup of E_r.
 By cref{prop:support-selection} pick lambda so that
 R_lambda(St_{r-1}) ⊗  M is nonprojective.
 By cref{thm:restricted-tilting},
-R_lambda(St_{r-1}) ⊗  Mcong R_lambda(T) for a tilting
+R_lambda(St_{r-1}) ⊗  M~= R_lambda(T) for a tilting
 module T.  By cref{lem:theta-detection},
 [
  0 ne Θ(R_lambda(T))
@@ -1048,7 +1048,7 @@ Phi is exact.
 % =================================================================
 
 We now have an exact Phi:C CcalCboxtimesmathcal
-W,calW, q=p^r, W={p^r}(k), semilinear.
+W,W, q=p^r, W={p^r}(k), semilinear.
 Its image closes up on the first factor to the Serre closure of a
 Frobenius-exact subcategory.
 
@@ -1070,7 +1070,7 @@ itself.  F is not assumed exact.
 
 [mod-p identity in K_0]
 
-In K_0(C)/pK_0(calC)/pK_0(C),
+In K_0(C)/pK_0(C)/pK_0(C),
 [
   [Y]^p = sum_{i=1}^{p-1}i[F_i(Y)].
 ]
@@ -1097,7 +1097,7 @@ ring and applying to concrete objects gives the iterated form.
 
 An exact strong monoidal sigma-semilinear functor still preserves
 Frobenius--Perron dimension: it sends nonzero objects to nonzero
-objects (from mathbf 1↪bf1↪ X ⊗  X^*), so
+objects (from 1↪1↪ X ⊗  X^*), so
 composing with FP dimension gives a positive normalised character on
 K_0(C); the FP characterization forces it to be the FP
 dimension of the source.  In particular
@@ -1105,7 +1105,7 @@ dimension of the source.  In particular
  d(Phi(X)) = d(X).
 ]
 
-[first-component inequality]
+[first-component ineuality]
 
 Let SinC be simple and Aboxtimes B a composition
 factor of Phi(S).  Then d(A)le d(S).  If equality holds then
@@ -1115,7 +1115,7 @@ ordinary Frobenius functor is exact).
 
 
 Proof. 
-Inequality. d(S)=d(Phi(S))=sum d(A_k)d(B_k)ge d(A_1)
+Ineuality. d(S)=d(Phi(S))=sum d(A_k)d(B_k)ge d(A_1)
 for any factor, using dge 1 for nonzero objects.
 
 Equality. If d(A)=d(S) then the FP decomposition sum is a
@@ -1142,7 +1142,7 @@ using d(F(Y_{t-1}))ge d(Y_t) and d(ell_{i_t})ge1:
 [
  d(Y_{t-1})ge d(F(Y_{t-1}))ge d(Y_t).
 ]
-But d(Y_r)ge d(A)=d(S)=d(Y_0), so all inequalities are equalities;
+But d(Y_r)ge d(A)=d(S)=d(Y_0), so all ineualities are equalities;
 in particular d(F(S))=d(S), so SinC_{ex}.
 
 
@@ -1154,7 +1154,7 @@ Sto A whenever Aboxtimes B appears in some composition factor
 of Phi(S).
 
 Every vertex has at least one outgoing arrow.  By
-cref{lem:first-component-inequality}, dimension is nonincreasing
+cref{lem:first-component-ineuality}, dimension is nonincreasing
 along arrows.  On a directed cycle, dimension is constant, all
 vertices belong to C_{ex}, and each vertex has
 exactly one outgoing arrow.  Hence a path entering a cycle is
@@ -1200,7 +1200,7 @@ centering
 caption{A schematic path in the graph of simple objects, with an
 illustrative three-vertex cycle.  Dimension is nonincreasing along
 every arrow; arrows before the cycle may also preserve dimension.
-On a cycle, cref{lem:first-component-inequality} gives equality,
+On a cycle, cref{lem:first-component-ineuality} gives equality,
 membership in C_{ex}, and a unique outgoing
 neighbor at every vertex, so the path cannot leave the cycle.}
 
@@ -1213,9 +1213,9 @@ A^{[sigma]} have the same objects and additive morphism
 groups but scalar action twisted by sigma^{-1}.  Then
 sigma-semilinear functors out of A = k-linear
 functors out of A^{[sigma]}, and
-(A B)^calB)^{[sigma]}
+(A B)^B)^{[sigma]}
  = A^{[sigma]}boxtimesmathcal
- B^calB^{[sigma]}, so products of sigma-semilinear functors are
+ B^B^{[sigma]}, so products of sigma-semilinear functors are
 sigma-semilinear.
 
 Each Ver_{p^s}(k) is defined over F_p; scalar transport
@@ -1248,7 +1248,7 @@ Autoequivalences on other factors send simples to simples and
 don't touch the first component.
 
 
-Let D⊆calD⊆C be the full subcategory whose
+Let D⊆D⊆C be the full subcategory whose
 simple composition factors lie in C_{ex}.  This
 is the Serre closure of C_{ex}.
 
@@ -1279,7 +1279,7 @@ e-isotypic parts, and H/HeH is finite-dimensional and finite
 projective dimension, so D is finite with enough
 projectives.
 
-Factoring through D:calD: cref{lem:first-components}.
+Factoring through D:D: cref{lem:first-components}.
 
 
 % =================================================================
@@ -1320,12 +1320,12 @@ So in all cases:
 Set n=max(2,r) and Z = Ver_{p^n}(k).  The tower
 inclusions Ver_{p^r}(k)⊆Ver_{p^n}(k)
 (Benson--Etingof--Ostrik 2023, Theorem 4.34) give a k-linear exact
-strong symmetric monoidal I:W Z.calZ.
+strong symmetric monoidal I:W Z.Z.
 
 The functors E:D ZcalZ and I:W
 Z combine, by cref{lem:combine-functors}, into
 [
- H:D W^calW^{boxtimes N}toZ,
+ H:D W^W^{boxtimes N}toZ,
  
  H(Dboxtimes W_1boxtimes...boxtimes W_N)
  = E(D) ⊗  I(W_1) ⊗ ... ⊗  I(W_N).
@@ -1340,7 +1340,7 @@ sigma).  Let J_{sigma^{-N}} be scalar transport on Z
 along b|-> b^{p^{-N}}.  Then
 [
  F := J_{sigma^{-N}}circ HcircPsi_N :
- C→calC→Z
+ C→C→Z
 ]
 is k-linear, exact, strong symmetric monoidal.
 
@@ -1355,15 +1355,15 @@ F is faithful.
 
 Proof. 
 Suppose Xne0.  The coevaluation
-mathbf 1bf1to X ⊗  X^* is nonzero (triangle identity) and by
-simplicity of mathbf 1bf1 it is injective; applying exact F shows
+1bf1to X ⊗  X^* is nonzero (triangle identity) and by
+simplicity of 1bf1 it is injective; applying exact F shows
 F(X)ne0.  For a nonzero f:Xto Y: exactness gives
 imF(f)~= F(imf), and
 imfne0, so F(f)ne0.
 
 
 Applying cref{lem:faithful} to F shows F is
-faithful.  Hence F:calF:C{p^n}(k) is the
+faithful.  Hence F:F:C{p^n}(k) is the
 required fiber functor, and cref{thm:main} follows.
 
 % =================================================================
@@ -1399,7 +1399,7 @@ pge 5 & Vec_k, sVec_k,
 
 Proof. 
 Direction ⇐: Vec_k has trivial
-End(mathbf 1)=k,bf1)=k, so any strong symmetric tensor functor out of
+End(1)=k,1)=k, so any strong symmetric tensor functor out of
 it is fully faithful (fill in the linear maps on simples); the
 supervector-space case is similar (an odd simple has a nontrivial
 sign in its double dual, which cannot match a vector space).
@@ -1413,7 +1413,7 @@ Then F corestricts to a surjective-in-the-tensor-category-sense
 functor onto I (Etingof--Gelaki--Nikshych--Ostrik 2015,
 Definition 1.8.18, Prop 1.8.19, Section 6.3).  Incompressibility
 of C forces this corestriction to be fully faithful.
-The two FP-dimension inequalities of finiteness (inclusion vs
+The two FP-dimension ineualities of finiteness (inclusion vs
 surjection) then force equality, hence an equivalence.
 Benson--Etingof--Ostrik 2023, Corollary 4.61 then restricts
 I to a full or plus sub-Verlinde at height le n.
@@ -1479,7 +1479,7 @@ Cref{lem:modules-finite-alg} makes C, together with
 V={p^n}(k), a ``pretannakian category'' in the
 sense of Coulembier--Etingof--Ostrik (2024).  The chosen F is a
 tensor functor, and their Theorem 4.2.1 produces the stated
-V-group;calV-group; conversely, forgetting G-structure gives the
+V-group;V-group; conversely, forgetting G-structure gives the
 inverse functor.  Composition is the equivalence.