spec/ECSM: fix padding bug

Author: erik-3milabs

spec/ecsm.typ

@@ -132,7 +132,8 @@ We emphasize here that @ec:c:c0_63_is_zero is required to ensure the sum evaluat
 The constraints @ec:c:c0_0 and @ec:c:c0_i, as well as the magic number $8160$ in @ec:c:range_c0 are discussed in @ecsm-limb_carry.
 #render_constraint_table(ecsm_chip, config, groups: "xG2")
 
-Next, we restrict the witness pair $(y_G, #`q1`)$:
+Next, we restrict the witness pair $(y_G, #`q1`)$.
+Note there that @ec:c:c1_0 and @ec:c:c1_i multiply `B` by `μ` to simplify the padding; there are no other side-effects to this since $#`μ` = 1$ on non-padding rows (@ec:c:mu_isbit).
 
 #render_constraint_table(ecsm_chip, config, groups: "yG")
 

spec/src/ecsm.toml

@@ -347,26 +347,27 @@ ref = "ec:c:q1_257"
 
 [[constraints.yG]]
 kind = "arith"
-constraint = "$2^8 dot #`c1`_0 = #`YG`_0 dot #`YG`_0 - #`X2`_0 dot #`XG`_0 - #`B`_0 + #`P`_0 dot #`P`_0 - #`Q1`_0 dot #`P`_0$"
+constraint = "$2^8 dot #`c1`_0 = #`YG`_0 dot #`YG`_0 - #`X2`_0 dot #`XG`_0 - #`μ` dot #`B`_0 + #`P`_0 dot #`P`_0 - #`Q1`_0 dot #`P`_0$"
 poly = ["-", 
     ["+", 
         ["*", ["idx", "YG", 0], ["idx", "YG", 0]], 
         ["*", ["idx", "P", 0], ["idx", "P", 0]], 
     ], 
     ["*", ["idx", "X2", 0], ["idx", "XG", 0]], 
-    ["idx", "B", 0], 
+    ["*", "μ", ["idx", "B", 0]], 
     ["*", ["idx", "Q1", 0], ["idx", "P", 0]], 
     ["*", ["^", 2, 8], ["idx", "c1", 0]]
 ]
+ref = "ec:c:c1_0"
 
 [[constraints.yG]]
 kind = "arith"
-constraint = "$2^8 dot #`c1`_i = #`c1`_(i-1) - #`B`_i + sum_(j=0)^i (#`YG`_j dot #`YG`_(i-j) + #`P`_j dot #`P`_(i-j) - #`X2`_j dot #`XG`_(i-j) - #`Q1`_j dot #`P`_(i-j))$"
+constraint = "$2^8 dot #`c1`_i = #`c1`_(i-1) - #`μ` dot #`B`_i + sum_(j=0)^i (#`YG`_j dot #`YG`_(i-j) + #`P`_j dot #`P`_(i-j) - #`X2`_j dot #`XG`_(i-j) - #`Q1`_j dot #`P`_(i-j))$"
 poly = ["+", 
     ["-", 
         ["idx", "c1", ["-", "i", 1]], 
         ["*", ["^", 2, 8], ["idx", "c1", "i"]],
-        ["idx", "B", "i"]
+        ["*", "μ", ["idx", "B", "i"]]
     ], 
     ["sum", ["=", "j", 0], "i", 
         ["-", 
@@ -380,6 +381,7 @@ poly = ["+",
     ]
 ]
 iter = ["i", 1, 63]
+ref = "ec:c:c1_i"
 
 [[constraints.yG]]
 kind = "arith"