diff --git a/CLAUDE.md b/CLAUDE.md
index 8b74a74..180fc76 100644
--- a/CLAUDE.md
+++ b/CLAUDE.md
@@ -456,6 +456,11 @@ as a grouped list does. Filling a page in is editing its `.md`; no Python follow
   long for its hanging lane steps out to the gutter on a line of its own
   (`render_hanging(whole=…, gutter=…)`), since a terminal opens a link, and a reader
   copies one, only while it sits on one line. Only a URL wider than the screen folds.
+  **A link is what `core/links.py` finds**, and nothing guesses beside it: a scheme
+  (`scheme://…`), or a host written without one where nothing else could be meant —
+  starting `www.`, or ending in a TLD IANA has delegated (`assets/tlds.txt`, the registry
+  verbatim; refreshing it is downloading it again). So `meshterm.net` is a link and
+  `file.txt` is not. A scheme-less link opens, and encodes, as `https://…`.
 - Dialogs anchor slightly above true centre, sized for their populated state.
 - Radio traffic: single transmissions or a user-chosen sample count with cooldown pacing
   — never bursts.
diff --git a/meshterm/assets/tlds.txt b/meshterm/assets/tlds.txt
new file mode 100644
index 0000000..f6957b5
--- /dev/null
+++ b/meshterm/assets/tlds.txt
@@ -0,0 +1,1438 @@
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+XN--ZFR164B
+XXX
+XYZ
+YACHTS
+YAHOO
+YAMAXUN
+YANDEX
+YE
+YODOBASHI
+YOGA
+YOKOHAMA
+YOU
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+YT
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+ZA
+ZAPPOS
+ZARA
+ZERO
+ZIP
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+ZUERICH
+ZW
diff --git a/meshterm/core/links.py b/meshterm/core/links.py
new file mode 100644
index 0000000..bd27c20
--- /dev/null
+++ b/meshterm/core/links.py
@@ -0,0 +1,104 @@
+# SPDX-License-Identifier: Apache-2.0
+"""Find the links in a line of text — written with a scheme, or without one where it is plain.
+
+A link with a scheme is anything ``scheme://…``: a scheme rather than a list of them, so a
+``meshcore://`` share pasted into a chat is found as surely as a web link is. One written
+without a scheme is accepted only where nothing else could be meant — a host starting
+``www.``, or one whose last label is a top-level domain IANA has delegated
+(``meshterm.net``, ``github.com/…``). So ``file.txt`` is not a link, ``txt`` being no TLD,
+and neither is ``3.14``, ``e.g.`` or an address like ``jp@meshterm.net``: a host only
+counts where a word begins, never inside an address, a path or another host.
+
+The registry is IANA's own list, kept verbatim in ``assets/tlds.txt`` with its version
+line on top; refreshing it is downloading the file again
+(https://data.iana.org/TLD/tlds-alpha-by-domain.txt). Some delegated TLDs are also file
+extensions — ``.py``, ``.md``, ``.zip`` — and a name ending in one reads as a link here.
+That is the registry's answer rather than a guess, and the cost of it is a code offered
+for a link nobody meant, which is the cheaper way to be wrong.
+
+Every link is handed back as the URL it opens: one written without a scheme gains
+``https://``, because a phone handed a bare host in a QR code reads it as text.
+"""
+
+from __future__ import annotations
+
+import re
+from functools import cache
+from pathlib import Path
+
+#: IANA's list of delegated top-level domains, one per line, upper case, ``#`` comments.
+TLDS_FILE = Path(__file__).resolve().parent.parent / "assets" / "tlds.txt"
+
+#: One host label: letters, digits and inner hyphens, at most 63 long.
+_LABEL = r"[a-z0-9](?:[a-z0-9-]{0,61}[a-z0-9])?"
+
+#: A link with a scheme: the scheme, ``://``, and everything up to the next space.
+_SCHEMED = r"\b[a-z][a-z0-9+.-]*://[^\s<>\"]+"
+
+#: A host written without a scheme, then an optional port and path. It must start a word —
+#: not follow a letter, a digit, ``@`` (an address), ``/`` or ``:`` (a path), or a ``.``
+#: or ``-`` (the middle of a longer name) — and its last label must be all letters, the
+#: shape of every TLD, so a version number or a decimal never gets this far.
+_BARE = (
+    r"(?<![\w@/:.-])"
+    rf"(?P<host>(?:{_LABEL}\.)+(?P<tld>[a-z]{{2,63}}))(?![\w-])"
+    r"(?::\d{1,5})?"
+    r"(?:[/?#][^\s<>\"]*)?"
+)
+
+#: Either kind. The scheme comes first, so a host inside a scheme's link is consumed with
+#: it and never found a second time on its own.
+_LINK = re.compile(rf"(?P<scheme>{_SCHEMED})|(?P<bare>{_BARE})", re.IGNORECASE)
+
+#: Punctuation that follows a link in a sentence and isn't part of it.
+_TRAILING = ".,;:!?'\""
+
+#: Each closing bracket and its opener: a closer ends a link only when it has no opener
+#: inside it, so ``(see https://example.com/a_(b))`` keeps the ``)`` that belongs to it.
+_CLOSERS = {")": "(", "]": "[", "}": "{"}
+
+
+@cache
+def tlds() -> frozenset[str]:
+    """Every delegated top-level domain, lower case — read once, on the first link looked for."""
+    lines = TLDS_FILE.read_text(encoding="ascii").splitlines()
+    return frozenset(line.strip().lower() for line in lines if line.strip()[:1] not in ("", "#"))
+
+
+def _trimmed(link: str) -> str:
+    """``link`` without the sentence's punctuation after it, or a bracket it never opened."""
+    while link:
+        last = link[-1]
+        if last in _TRAILING or (
+            last in _CLOSERS and link.count(last) > link.count(_CLOSERS[last])
+        ):
+            link = link[:-1]
+        else:
+            break
+    return link
+
+
+def urls(text: str) -> list[str]:
+    """Every link in ``text`` as the URL it opens, in reading order, each only once.
+
+    Trailing punctuation is the sentence's, not the link's; a link written without a
+    scheme comes back with ``https://`` in front of it, so ``meshterm.net`` and
+    ``https://meshterm.net`` in one message are the same link, found once.
+    """
+    found: list[str] = []
+    for match in _LINK.finditer(text):
+        link = _trimmed(match.group())
+        if match.group("scheme"):
+            if link.endswith("://"):
+                continue  # a scheme and nothing after it
+        else:
+            host = match.group("host").lower()
+            if not host.startswith("www.") and match.group("tld").lower() not in tlds():
+                continue
+            link = "https://" + link
+        if link not in found:
+            found.append(link)
+    return found
+
+
+__all__ = ["TLDS_FILE", "tlds", "urls"]
diff --git a/meshterm/ui/chat.py b/meshterm/ui/chat.py
index 3f45df1..a1df401 100644
--- a/meshterm/ui/chat.py
+++ b/meshterm/ui/chat.py
@@ -24,6 +24,7 @@ from typing import TYPE_CHECKING, Any
 from rich.console import Group, RenderableType
 from rich.text import Text
 
+from ..core import links
 from ..core.channels import MENTION, split_channel_sender
 from ..core.connection import ContactNotOnDeviceError
 from ..core.events import EventKind, MeshEvent
@@ -55,18 +56,6 @@ if TYPE_CHECKING:
 _DELIVERED = ("✓", "ok")
 _FAILED = ("✗", "err")
 
-#: A URL in a message body: a scheme, ``://``, and everything up to the next space. A
-#: scheme rather than a list of them, so a ``meshcore://`` share pasted into a chat gets
-#: its code as surely as a web link does.
-_URL = re.compile(r"\b[a-z][a-z0-9+.-]*://[^\s<>\"]+", re.IGNORECASE)
-
-#: Punctuation that follows a URL in a sentence and isn't part of it.
-_TRAILING = ".,;:!?'\""
-
-#: Each closing bracket and its opener: a closer ends a URL only when it has no opener
-#: inside it, so ``(see https://example.com/a_(b))`` keeps the ``)`` that belongs to it.
-_CLOSERS = {")": "(", "]": "[", "}": "{"}
-
 #: A whitespace-delimited word — the unit the wrap moves whole, so the run a URL must not be
 #: cut across is the word it sits in, any bracket or full stop beside it included.
 _WORD = re.compile(r"\S+")
@@ -87,24 +76,6 @@ def _bind(platform: Platform) -> None:
     _URL_CODES = platform.url_codes
 
 
-def _urls(body: str) -> list[str]:
-    """Every URL in ``body`` in reading order, each only once, without trailing punctuation."""
-    found: list[str] = []
-    for match in _URL.finditer(body):
-        url = match.group()
-        while url:
-            last = url[-1]
-            if last in _TRAILING or (
-                last in _CLOSERS and url.count(last) > url.count(_CLOSERS[last])
-            ):
-                url = url[:-1]
-            else:
-                break
-        if not url.endswith("://") and url not in found:
-            found.append(url)
-    return found
-
-
 class ChatScreen(Screen):
     """A live conversation: a scrolling transcript above a pinned input line.
 
@@ -609,11 +580,11 @@ class ChatScreen(Screen):
         prefix.append("❯ " if selected else "  ", style="cursor" if selected else None)
         prefix.append(f"{stamp:%H:%M}  ", style="cursor" if selected else "muted")
         body_text = self._body_text(body, message, selected=selected)
-        whole = [m.span() for m in _WORD.finditer(body_text.plain) if _URL.search(m.group())]
+        whole = [m.span() for m in _WORD.finditer(body_text.plain) if links.urls(m.group())]
         lines = render_hanging(
             prefix, body_text, width, indent=prefix.cell_len, whole=whole, gutter=_URL_GUTTER
         )
-        urls = _urls(body) if _URL_CODES else []
+        urls = links.urls(body) if _URL_CODES else []
         if urls:
             lines += render_lines(qr_strip(urls, width, indent=prefix.cell_len), width)
         return lines
@@ -836,7 +807,7 @@ class ChatScreen(Screen):
         if self._selected is None or not self._messages:
             return []
         message = self._messages[max(0, min(self._selected, len(self._messages) - 1))]
-        return _urls(self._sender_and_body(message)[1])
+        return links.urls(self._sender_and_body(message)[1])
 
     def _open_codes(self) -> None:
         """Show the picked message's links as QR codes on the share screen (^U).
diff --git a/tests/test_chat.py b/tests/test_chat.py
index bb618d6..8bfc160 100644
--- a/tests/test_chat.py
+++ b/tests/test_chat.py
@@ -2332,23 +2332,6 @@ def _decode_braille(rows: list[str], left: int, size: int) -> list[list[bool]]:
     ]
 
 
-def test_urls_are_found_whole_once_and_without_the_sentence_around_them() -> None:
-    """A URL ends where its sentence's punctuation begins, and keeps a bracket it opened."""
-    from meshterm.ui.chat import _urls
-
-    body = (
-        "map at https://meshterm.net. see (https://en.wikipedia.org/wiki/Foo_(bar)), "
-        "'https://example.org/a?b=1' and meshcore://channel/add?name=Ops&secret=ab — "
-        "again https://meshterm.net! nothing here: https:// or note:x"
-    )
-    assert _urls(body) == [
-        "https://meshterm.net",
-        "https://en.wikipedia.org/wiki/Foo_(bar)",
-        "https://example.org/a?b=1",
-        "meshcore://channel/add?name=Ops&secret=ab",
-    ]
-
-
 def test_on_the_picocalc_each_url_hangs_a_code_under_its_message_side_by_side() -> None:
     """Two links, two braille codes on the same rows, hung at the body's own indent.
 
@@ -2530,3 +2513,24 @@ def test_ctrl_u_is_named_only_on_a_picked_message_with_a_link() -> None:
     for screen in (linked, direct, scoped):
         assert "^U QR" in screen.footer_hint
         assert len(screen.footer_hint) <= 72, screen.footer_hint
+
+
+def test_a_link_written_without_a_scheme_gets_its_code_too() -> None:
+    """``meshterm.net/map`` is as much a link as its https form, and opens as that."""
+    from meshterm.ui.qr import QrScreen
+
+    opened: list = []
+
+    class Session(_StubSession):
+        async def run_screen(self, screen):  # noqa: ANN001, ANN202
+            opened.append(screen)
+
+    async def scenario() -> None:
+        screen = _url_chat("map at meshterm.net/map, notes in file.txt", Session())
+        screen.handle("up")
+        assert "^U QR" in screen.footer_hint
+        screen.handle("url_code")
+        await asyncio.sleep(0)
+        assert isinstance(opened[0], QrScreen) and opened[0].urls == ("https://meshterm.net/map",)
+
+    asyncio.run(scenario())
diff --git a/tests/test_links.py b/tests/test_links.py
new file mode 100644
index 0000000..199ab3a
--- /dev/null
+++ b/tests/test_links.py
@@ -0,0 +1,60 @@
+# SPDX-License-Identifier: Apache-2.0
+"""Finding links in a line of text — with a scheme, and without one where it is plain."""
+
+from __future__ import annotations
+
+from meshterm.core.links import TLDS_FILE, tlds, urls
+
+
+def test_urls_are_found_whole_once_and_without_the_sentence_around_them() -> None:
+    """A URL ends where its sentence's punctuation begins, and keeps a bracket it opened."""
+    body = (
+        "map at https://meshterm.net. see (https://en.wikipedia.org/wiki/Foo_(bar)), "
+        "'https://example.org/a?b=1' and meshcore://channel/add?name=Ops&secret=ab — "
+        "again https://meshterm.net! nothing here: https:// or note:x"
+    )
+    assert urls(body) == [
+        "https://meshterm.net",
+        "https://en.wikipedia.org/wiki/Foo_(bar)",
+        "https://example.org/a?b=1",
+        "meshcore://channel/add?name=Ops&secret=ab",
+    ]
+
+
+def test_a_link_without_a_scheme_is_found_by_www_or_a_delegated_tld() -> None:
+    """``www.`` or a real TLD says *link*; each comes back as the https URL it opens."""
+    body = (
+        "map at meshterm.net/map, flasher at flasher.meshcore.co.uk. "
+        "WWW.Example.org (or www.anything.local) and github.com/jpmartineau/MeshTerm?tab=1!"
+    )
+    assert urls(body) == [
+        "https://meshterm.net/map",
+        "https://flasher.meshcore.co.uk",
+        "https://WWW.Example.org",
+        "https://www.anything.local",  # www. is enough on its own; .local is no TLD
+        "https://github.com/jpmartineau/MeshTerm?tab=1",
+    ]
+
+
+def test_a_dotted_word_that_is_no_link_stays_text() -> None:
+    """A file, a number, an abbreviation, an address, a path: no TLD, or not a word's start."""
+    body = (
+        "see file.txt and notes.pdf, pi is 3.14 and v1.2.3, e.g. this, i.e. that, "
+        "mail jp@meshterm.net, open C:/logs/meshterm.log or ./run.sh, @Alice.Smith"
+    )
+    assert urls(body) == []
+
+
+def test_the_same_link_written_two_ways_is_found_once() -> None:
+    """``meshterm.net`` and ``https://meshterm.net`` open the same page — one code, not two."""
+    assert urls("meshterm.net or https://meshterm.net") == ["https://meshterm.net"]
+    # A host inside a schemed link is part of it, never a second link of its own.
+    assert urls("https://github.com/x/meshterm.net") == ["https://github.com/x/meshterm.net"]
+
+
+def test_the_tld_registry_is_ianas_list_read_whole() -> None:
+    """The file is IANA's, version line and all; the set is every TLD in it, lower case."""
+    assert TLDS_FILE.read_text(encoding="ascii").startswith("# Version ")
+    assert {"com", "net", "org", "dev", "io", "ca", "uk"} <= tlds()
+    assert not {"txt", "pdf", "log", "local"} & tlds()
+    assert len(tlds()) > 1000