diff --git a/spec/chapters/memw.typ b/spec/chapters/memw.typ
index b6a140c96..239db58cc 100644
--- a/spec/chapters/memw.typ
+++ b/spec/chapters/memw.typ
@@ -100,7 +100,7 @@ provide these below.
 
 #render_constraint_table(alignedchip, config, groups: "assumptions")
 
-#render_constraint_table(alignedchip, config)
+#render_constraint_table(alignedchip, config, groups: ("consistency", "memory", "output"))
 
 == Padding
 The table can be padded to the next power of two with the following value assignments:
diff --git a/spec/chapters/signatures.typ b/spec/chapters/signatures.typ
index 969cfaaa7..80d491285 100644
--- a/spec/chapters/signatures.typ
+++ b/spec/chapters/signatures.typ
@@ -45,6 +45,7 @@
 
 #let interactions = signatures.signatures.filter(s => s.kind == "interaction")
 The following lists signatures of the #interactions.len() interactions in this VM.
+#show figure: set block(breakable: true)
 #figure(table(
     columns: (1fr, auto),
     inset: 7pt,
diff --git a/spec/chapters/verifier.typ b/spec/chapters/verifier.typ
index e85903478..0447bd308 100644
--- a/spec/chapters/verifier.typ
+++ b/spec/chapters/verifier.typ
@@ -1,4 +1,21 @@
+#import "../meta.typ": todo
 // TODO:
 // - sigma protocol, layout the various steps
 // - Fiat Shamir transformation into non-interactive protocol.
 // 
+
+= Auxiliary verification steps
+#todo[
+  *Verify that the number of tables used by the prover in an epoch/proof is 
+  lower than some bound.*
+  Without this restriction, a prover can construct an interaction multiplicity 
+  of $-1$ by repeating a table row a large number of times.
+  While this strategy this would result in a large increase of 
+  proof size, the excess can be purged during recursion.
+  This could e.g. be used to bypass range checks.
+
+  TODO: determine the right bound. The `IS_WH` chip with its multiplicity of type
+  `B20` induces an upper bound: it requires $approx 2^43$ rows 
+  (or $approx 2^23$ tables) to achieve a multiplicity sum of $-1$.
+  This can then be used to forge an `IS_HALF` check.
+]
\ No newline at end of file
diff --git a/spec/meta.typ b/spec/meta.typ
index 9e7967c41..107d97e02 100644
--- a/spec/meta.typ
+++ b/spec/meta.typ
@@ -66,7 +66,7 @@
   )
 )
 
-#let todo(background: white, foreground: black, name: none, body) = block(fill: background, outset: 0.4em, radius: 20%, stroke: black)[
+#let todo(background: white, foreground: black, name: none, body) = block(fill: background, outset: 0.4em, radius: .5em, stroke: black)[
   #set text(fill: foreground)
   *TODO #if name != none { [(#name)] }*: #body
 ]
diff --git a/spec/src/branch.toml b/spec/src/branch.toml
index 45e6d5618..dd2ea8d93 100644
--- a/spec/src/branch.toml
+++ b/spec/src/branch.toml
@@ -78,7 +78,7 @@ def = {idx = "i", polys = [
 
 [[variables.multiplicity]]
 name = "μ"
-type = "Bit"
+type = "Byte"
 desc = ""
 pad = 0
 
@@ -109,6 +109,11 @@ input = ["JALR"]
 [[constraint_groups]]
 name = "all"
 
+[[constraints.all]]
+kind = "template"
+tag = "IS_BYTE"
+input = ["μ"]
+
 [[constraints.all]]
 kind = "template"
 tag = "ADD"
@@ -127,7 +132,6 @@ cond = "JALR"
 kind = "template"
 tag = "IS_BYTE"
 input = [["idx", "next_pc_low", 1]]
-cond = "μ"
 
 [[constraints.all]]
 kind = "interaction"
diff --git a/spec/src/bytewise.toml b/spec/src/bytewise.toml
index b315a92b3..8e35f7579 100644
--- a/spec/src/bytewise.toml
+++ b/spec/src/bytewise.toml
@@ -26,13 +26,18 @@ pad = 0
 
 [[variables.multiplicity]]
 name = "μ"
-type = "BaseField"
+type = "Byte"
 desc = ""
 pad = 0
 
 [[constraint_groups]]
 name = "all"
 
+[[constraints.all]]
+kind = "template"
+tag = "IS_BYTE"
+input = ["μ"]
+
 [[constraints.all]]
 kind = "interaction"
 tag = "BYTE_ALU"
diff --git a/spec/src/dvrm.toml b/spec/src/dvrm.toml
index 8467bea96..5b539ea92 100644
--- a/spec/src/dvrm.toml
+++ b/spec/src/dvrm.toml
@@ -171,13 +171,13 @@ def = ["+", "μ_q", "μ_r"]
 
 [[variables.multiplicity]]
 name = "μ_q"
-type = "BaseField"
+type = "Byte"
 desc = ""
 pad = 0
 
 [[variables.multiplicity]]
 name = "μ_r"
-type = "BaseField"
+type = "Byte"
 desc = ""
 pad = 0
 
@@ -187,6 +187,16 @@ pad = 0
 [[constraint_groups]]
 name = "range"
 
+[[constraints.range]]
+kind = "template"
+tag = "IS_BYTE"
+input = ["μ_q"]
+
+[[constraints.range]]
+kind = "template"
+tag = "IS_BYTE"
+input = ["μ_r"]
+
 [[constraints.range]]
 kind = "interaction"
 tag = "IS_HALF"
diff --git a/spec/src/eq.toml b/spec/src/eq.toml
index a06bb1b16..d2cfdee44 100644
--- a/spec/src/eq.toml
+++ b/spec/src/eq.toml
@@ -38,7 +38,7 @@ pad = 0
 
 [[variables.multiplicity]]
 name = "μ"
-type = "BaseField"
+type = "Bit"
 desc = ""
 pad = 0
 
@@ -55,6 +55,11 @@ iter = ["i", 0, 1]
 [[constraint_groups]]
 name = "all"
 
+[[constraints.all]]
+kind = "template"
+tag = "IS_BIT"
+input = ["μ"]
+
 [[constraints.all]]
 kind = "interaction"
 tag = "IS_HALF"
diff --git a/spec/src/fext_fma.toml b/spec/src/fext_fma.toml
index 1adb99eb0..83d16e10e 100644
--- a/spec/src/fext_fma.toml
+++ b/spec/src/fext_fma.toml
@@ -73,6 +73,11 @@ pad = 0
 [[constraint_groups]]
 name = "all"
 
+[[constraints.all]]
+kind = "template"
+tag = "IS_BIT"
+input = ["μ"]
+
 [[constraints.all]]
 kind = "template"
 tag = "REG"
diff --git a/spec/src/fext_load.toml b/spec/src/fext_load.toml
index 5b00c44a3..7e792155b 100644
--- a/spec/src/fext_load.toml
+++ b/spec/src/fext_load.toml
@@ -28,6 +28,11 @@ pad = 0
 [[constraint_groups]]
 name = "all"
 
+[[constraints.all]]
+kind = "template"
+tag = "IS_BIT"
+input = ["μ"]
+
 [[constraints.all]]
 kind = "template"
 tag = "REG"
diff --git a/spec/src/fext_zero.toml b/spec/src/fext_zero.toml
index c4f29ad53..10cf41cea 100644
--- a/spec/src/fext_zero.toml
+++ b/spec/src/fext_zero.toml
@@ -22,6 +22,11 @@ pad = 0
 [[constraint_groups]]
 name = "all"
 
+[[constraints.all]]
+kind = "template"
+tag = "IS_BIT"
+input = ["μ"]
+
 [[constraints.all]]
 kind = "template"
 tag = "REG"
diff --git a/spec/src/is_wh.toml b/spec/src/is_wh.toml
index 51426791f..73a38cd87 100644
--- a/spec/src/is_wh.toml
+++ b/spec/src/is_wh.toml
@@ -8,13 +8,19 @@ pad = 0
 
 [[variables.multiplicity]]
 name = "μ"
-type = "BaseField"
+type = "B20"
 desc = ""
 pad = 0
 
 [[constraint_groups]]
 name = "all"
 
+[[constraints.all]]
+kind = "interaction"
+tag = "IS_B20"
+input = ["μ"]
+multiplicity = 1
+
 [[constraints.all]]
 kind = "interaction"
 tag = "IS_HALF"
diff --git a/spec/src/is_whh.toml b/spec/src/is_whh.toml
index f00392ddb..8f46b72be 100644
--- a/spec/src/is_whh.toml
+++ b/spec/src/is_whh.toml
@@ -3,7 +3,7 @@ code = "WHH"
 
 [[variables.condition]]
 name = "cond"
-type = "BaseField"
+type = "Bit"
 desc = "the multiplicity with which to apply the interactions"
 
 [[variables.input]]
diff --git a/spec/src/keccak.toml b/spec/src/keccak.toml
index 41d52b2a8..4eba17ffc 100644
--- a/spec/src/keccak.toml
+++ b/spec/src/keccak.toml
@@ -40,6 +40,11 @@ pad = 0
 [[constraint_groups]]
 name = "output"
 
+[[constraints.output]]
+kind = "template"
+tag = "IS_BIT"
+input = ["μ"]
+
 [[constraints.output]]
 kind = "interaction"
 tag = "ECALL"
diff --git a/spec/src/keccak_round.toml b/spec/src/keccak_round.toml
index 8e11dda20..67a2cf9e3 100644
--- a/spec/src/keccak_round.toml
+++ b/spec/src/keccak_round.toml
@@ -168,6 +168,11 @@ desc = ""
 [[constraint_groups]]
 name = "io"
 
+[[constraints.io]]
+kind = "template"
+tag = "IS_BIT"
+input = ["μ"]
+
 [[constraints.io]]
 kind = "interaction"
 tag = "KECCAK"
@@ -227,7 +232,6 @@ kind = "template"
 tag = "IS_BYTE"
 input = [["idx", ["idx", "Cxz_left", "x"], "z"]]
 iters = [["x", 0, 4], ["z", 0, 7]]
-cond = "μ"
 
 [[constraints.theta]]
 kind = "template"
@@ -275,7 +279,6 @@ kind = "template"
 tag = "IS_BYTE"
 input = [["idx", ["idx", ["idx", "rot_left", "x"], "y"], "z"]]
 iters = [["x", 0, 4], ["y", 0, 4], ["z", 0, 7]]
-cond = "μ"
 ref = "keccak:c:range_rot_left"
 
 [[constraints.rho]]
@@ -283,7 +286,6 @@ kind = "template"
 tag = "IS_BYTE"
 input = [["idx", ["idx", ["idx", "rot_right", "x"], "y"], "z"]]
 iters = [["x", 0, 4], ["y", 0, 4], ["z", 0, 7]]
-cond = "μ"
 ref = "keccak:c:range_rot_right"
 
 [[constraint_groups]]
diff --git a/spec/src/load.toml b/spec/src/load.toml
index 3d3363f34..1e2e9cc07 100644
--- a/spec/src/load.toml
+++ b/spec/src/load.toml
@@ -83,6 +83,11 @@ desc = "`IS_WORD[timestamp]`"
 [[constraint_groups]]
 name = "all"
 
+[[constraints.all]]
+kind = "template"
+tag = "IS_BIT"
+input = ["μ"]
+
 [[constraints.all]]
 kind = "template"
 tag = "IS_BIT"
diff --git a/spec/src/lt.toml b/spec/src/lt.toml
index 57661333d..ace32e0e2 100644
--- a/spec/src/lt.toml
+++ b/spec/src/lt.toml
@@ -84,7 +84,7 @@ def = ["idx", "carry", 1]
 
 [[variables.multiplicity]]
 name = "μ"
-type = "Bit"
+type = "Byte"
 desc = ""
 pad = 0
 
@@ -101,6 +101,11 @@ ref = "lt:a:range_rhs"
 name = "range"
 desc = "Range-check the inputs"
 
+[[constraints.range]]
+kind = "template"
+tag = "IS_BYTE"
+input = ["μ"]
+
 [[constraints.range]]
 kind = "interaction"
 tag = "IS_HALF"
diff --git a/spec/src/mul.toml b/spec/src/mul.toml
index e1837020a..ebfa4ee91 100644
--- a/spec/src/mul.toml
+++ b/spec/src/mul.toml
@@ -111,13 +111,13 @@ def = ["+", "μ_lo", "μ_hi"]
 
 [[variables.multiplicity]]
 name = "μ_lo"
-type = "BaseField"
+type = "Byte"
 desc = ""
 pad = 0
 
 [[variables.multiplicity]]
 name = "μ_hi"
-type = "BaseField"
+type = "Byte"
 desc = ""
 pad = 0
 
@@ -126,6 +126,16 @@ pad = 0
 [[constraint_groups]]
 name = "def"
 
+[[constraints.def]]
+kind = "template"
+tag = "IS_BYTE"
+input = ["μ_lo"]
+
+[[constraints.def]]
+kind = "template"
+tag = "IS_BYTE"
+input = ["μ_hi"]
+
 [[constraints.def]]
 kind = "template"
 tag = "IS_BIT"
diff --git a/spec/src/reg_read.toml b/spec/src/reg_read.toml
index 709f123a4..6a57ecbcc 100644
--- a/spec/src/reg_read.toml
+++ b/spec/src/reg_read.toml
@@ -2,7 +2,7 @@ name = "REG"
 
 [[variables.condition]]
 name = "cond"
-type = "BaseField"
+type = "Bit"
 desc = "the multiplicity with which to apply the interaction"
 
 [[variables.input]]
diff --git a/spec/src/reg_write.toml b/spec/src/reg_write.toml
index 411adf24c..a9d037b61 100644
--- a/spec/src/reg_write.toml
+++ b/spec/src/reg_write.toml
@@ -2,7 +2,7 @@ name = "REGW"
 
 [[variables.condition]]
 name = "cond"
-type = "BaseField"
+type = "Bit"
 desc = "the multiplicity with which to apply the interaction"
 
 [[variables.input]]
diff --git a/spec/src/rotxor.toml b/spec/src/rotxor.toml
index 9f0e398b4..73bb1cabf 100644
--- a/spec/src/rotxor.toml
+++ b/spec/src/rotxor.toml
@@ -99,7 +99,7 @@ pad = 0
 
 [[variables.multiplicity]]
 name = "μ"
-type = "BaseField"
+type = "Bit"
 desc = ""
 pad = 0
 
@@ -179,6 +179,11 @@ iter = ["i", 0, 3]
 [[constraint_groups]]
 name = "output"
 
+[[constraints.output]]
+kind = "template"
+tag = "IS_BIT"
+input = ["μ"]
+
 [[constraints.output]]
 kind = "interaction"
 tag = "ROTXOR"
diff --git a/spec/src/sha256round.toml b/spec/src/sha256round.toml
index 48c944835..a887b383b 100644
--- a/spec/src/sha256round.toml
+++ b/spec/src/sha256round.toml
@@ -276,6 +276,11 @@ cond = "μ"
 [[constraint_groups]]
 name = "output"
 
+[[constraints.output]]
+kind = "template"
+tag = "IS_BIT"
+input = ["μ"]
+
 [[constraints.output]]
 kind = "interaction"
 tag = "SHA256ROUND"
diff --git a/spec/src/shift.toml b/spec/src/shift.toml
index 81d42f9f3..771f006f5 100644
--- a/spec/src/shift.toml
+++ b/spec/src/shift.toml
@@ -182,6 +182,16 @@ tag = "IS_BIT"
 input = ["word_instr"]
 ref = "shift:c:word_instr"
 
+[[constraints.input]]
+kind = "template"
+tag = "IS_BIT"
+input = ["μ"]
+
+[[constraints.input]]
+kind = "template"
+tag = "IS_BIT"
+input = ["zbs"]
+
 [[constraint_groups]]
 name = "left_flag"