diff --git a/iosMath/lib/MTMathAtomFactory.m b/iosMath/lib/MTMathAtomFactory.m
index 727d86e..f3945ed 100644
--- a/iosMath/lib/MTMathAtomFactory.m
+++ b/iosMath/lib/MTMathAtomFactory.m
@@ -98,11 +98,15 @@ + (nullable MTMathAtom *)atomForCharacter:(unichar)ch
 {
     NSString *chStr = [NSString stringWithCharacters:&ch length:1];
     if (ch < 0x21 || ch > 0x7E) {
-        // skip non ascii characters and spaces. Non-Latin text must be
-        // wrapped in \text*, \textbf{...}, etc.
+        // No atom for control characters, spaces, or non-ASCII literals. The
+        // builder decides what to do with these: whitespace is silently ignored,
+        // everything else raises MTParseErrorInvalidCharacter. Non-Latin text
+        // must be wrapped in \text*, \textbf{...}, etc.
         return nil;
     } else if (ch == '$' || ch == '%' || ch == '#' || ch == '&' || ch == '~' || ch == '\'') {
-        // These are latex control characters that have special meanings. We don't support them.
+        // LaTeX control characters with special meanings. They have no atom of
+        // their own; the builder handles them (& / ~ / ' are consumed before
+        // reaching here, while $ % # raise MTParseErrorInvalidCharacter).
         return nil;
     } else if (ch == '^' || ch == '_' || ch == '{' || ch == '}' || ch == '\\') {
         // more special characters for Latex.
diff --git a/iosMath/lib/MTMathListBuilder.h b/iosMath/lib/MTMathListBuilder.h
index 69a69e7..15245b7 100644
--- a/iosMath/lib/MTMathListBuilder.h
+++ b/iosMath/lib/MTMathListBuilder.h
@@ -89,6 +89,10 @@ typedef NS_ENUM(NSUInteger, MTParseErrors) {
     MTParseErrorInvalidLimits,
     /// The LaTeX nesting depth exceeded the safe parsing limit.
     MTParseErrorNestingTooDeep,
+    /// A character in the string is not a valid LaTeX input character in math
+    /// mode (e.g. a non-ASCII literal like π, or a special character such as
+    /// %, #, $ that has no meaning here).
+    MTParseErrorInvalidCharacter,
 };
 
 @end
diff --git a/iosMath/lib/MTMathListBuilder.m b/iosMath/lib/MTMathListBuilder.m
index e87288c..fb51fc4 100644
--- a/iosMath/lib/MTMathListBuilder.m
+++ b/iosMath/lib/MTMathListBuilder.m
@@ -358,11 +358,30 @@ - (MTMathList*)buildInternal:(BOOL) oneCharOnly stopChar:(unichar) stop
         } else if (_spacesAllowed && ch == ' ') {
             // If spaces are allowed then spaces do not need escaping with a \ before being used.
             atom = [MTMathAtomFactory atomForLatexSymbolName:@" "];
+        } else if (ch == '~') {
+            // Tilde is a non-breaking space in LaTeX; render it as an ordinary space.
+            atom = [MTMathAtomFactory atomForLatexSymbolName:@" "];
         } else {
             atom = [MTMathAtomFactory atomForCharacter:ch];
             if (!atom) {
-                // Not a recognized character
-                continue;
+                // Characters TeX silently discards: whitespace (catcode 10/5,
+                // ignored in math mode) and NUL (catcode 9). Note that other
+                // control characters are *not* spaces in TeX (form feed is \par,
+                // vertical tab is an ordinary "other" character), so they fall
+                // through to the error below, as they should.
+                if (ch == ' ' || ch == '\t' || ch == '\n' || ch == '\r' || ch == '\0') {
+                    continue;
+                }
+                // Any other unrecognized character is an error: a non-ASCII literal
+                // (e.g. π, ×, ≤) or a special character with no meaning in math mode
+                // (% is a comment, # a macro parameter, $ toggles math mode). Callers
+                // should use the corresponding LaTeX command (e.g. \pi, \%, \#).
+                // ch is a single UTF-16 code unit; we just report its value (an
+                // above-BMP character reports its leading surrogate, which is fine
+                // for an error message).
+                [self setError:MTParseErrorInvalidCharacter
+                       message:[NSString stringWithFormat:@"Unknown character U+%04X is not a valid LaTeX input character in math mode. Use the corresponding LaTeX command instead.", ch]];
+                return nil;
             }
         }
         NSAssert(atom != nil, @"Atom shouldn't be nil");
diff --git a/iosMathTests/MTMathListBuilderTest.m b/iosMathTests/MTMathListBuilderTest.m
index 4bfe8c1..24c2b95 100644
--- a/iosMathTests/MTMathListBuilderTest.m
+++ b/iosMathTests/MTMathListBuilderTest.m
@@ -69,6 +69,8 @@ - (void)tearDown
              @[ @"x \\ y", @[  @(kMTMathAtomVariable), @(kMTMathAtomOrdinary), @(kMTMathAtomVariable)], @"x\\  y"],
              // spacing
              @[ @"x \\quad y \\; z \\! q", @[  @(kMTMathAtomVariable), @(kMTMathAtomSpace), @(kMTMathAtomVariable),@(kMTMathAtomSpace), @(kMTMathAtomVariable),@(kMTMathAtomSpace), @(kMTMathAtomVariable)], @"x\\quad y\\; z\\! q"],
+             // tilde is a non-breaking space (renders as an ordinary space, same as a literal space)
+             @[ @"x~y", @[  @(kMTMathAtomVariable), @(kMTMathAtomOrdinary), @(kMTMathAtomVariable)], @"x\\  y"],
              ];
 }
 
@@ -1488,6 +1490,16 @@ - (void) testDisplayLines
               @[@"x^\\choose y", @(MTParseErrorInvalidCommand)],
               @[@"x^\\brack y",  @(MTParseErrorInvalidCommand)],
               @[@"x^\\brace y",  @(MTParseErrorInvalidCommand)],
+              // REN-5: non-ASCII literal characters should produce MTParseErrorInvalidCharacter
+              @[@"π", @(MTParseErrorInvalidCharacter)],          // π (U+03C0)
+              @[@"3 × 4", @(MTParseErrorInvalidCharacter)],      // 3 × 4
+              @[@"x ≤ y", @(MTParseErrorInvalidCharacter)],      // x ≤ y
+              @[@"x 𝑎 y", @(MTParseErrorInvalidCharacter)],      // above-BMP literal (U+1D44E, surrogate pair)
+              // Special characters with no meaning in math mode are errors (match LaTeX:
+              // % is a comment, # is a macro parameter, $ toggles math mode - none valid here).
+              @[@"a % b", @(MTParseErrorInvalidCharacter)],
+              @[@"a # b", @(MTParseErrorInvalidCharacter)],
+              @[@"a $ b", @(MTParseErrorInvalidCharacter)],
               ];
 };
 
@@ -1508,6 +1520,26 @@ - (void) testErrors
         }
 }
 
+// REN-5: characters TeX silently discards (whitespace catcode 10/5 and NUL
+// catcode 9) must continue to parse without error. Guards against the error
+// path swallowing legitimate whitespace.
+- (void) testIgnoredWhitespaceCharacters
+{
+    unichar nulChars[3] = { 'x', 0x0000, 'y' };
+    NSString* withNul = [NSString stringWithCharacters:nulChars length:3];
+    NSArray* inputs = @[ @"x\ty", @"x\ny", @"x\ry", withNul ];
+    for (NSString* str in inputs) {
+        NSError* error = nil;
+        MTMathList* list = [MTMathListBuilder buildFromString:str error:&error];
+        NSString* desc = [NSString stringWithFormat:@"whitespace input %@", str];
+        XCTAssertNotNil(list, @"%@", desc);
+        XCTAssertNil(error, @"%@", desc);
+        XCTAssertEqual(list.atoms.count, 2u, @"%@", desc);
+        XCTAssertEqual([list.atoms[0] type], kMTMathAtomVariable, @"%@", desc);
+        XCTAssertEqual([list.atoms[1] type], kMTMathAtomVariable, @"%@", desc);
+    }
+}
+
 // REN-6: \over inside an explicit-brace script group must still parse correctly.
 - (void) testOverInScriptBraces
 {