mock: the square's edges ring, and the one probe that wants a step asks for it

The square was the mock's only ideal step, which is the one shape a real producer
does not put on the wire. It is now the truncated Fourier series of the step --
`--square-harmonics N` odd harmonics, default 30 -- so each edge overshoots by the
Gibbs constant (about 9% of the 2 V swing, y = +/-1.179) and then rings, and
`--square-harmonics 0` is the ideal +/-1 step as before.

Odd harmonics only, so the series keeps the ideal wave's edge instants and its
polarity: the level-0 trigger, the glitch's inverted pulse and the buffer search
land where they did. Re-run in pairs (`--square-harmonics 0` against the default),
search, condition legs 0/1, trigrenum, disarm, trig trace, measures and events all
read exactly what they read before, with only `t0` moving. The reading that moved
is the noise probe's group-delay leg, whose `want` is the analytic `tau*ln2` of a
step: a band-limited edge reads `delay=0.002770 ratio=1.255`, the same figure on
every run. That recipe now asks for the ideal step, so its documented number stays
exact; probe-readings says why.

The series is vectorised: as a per-sample Python sum it cost ~15 sin() calls a
sample, which at the fast mock's 1.3 MS/s made the producer the bottleneck and
halved the rate the app saw (the noise probe's span-dependent counts read 22
against 48). Vectorised it adds ~0.5 ms to a 2000-sample packet, so the fast mock
runs at about nine tenths of its declared pace and those counts read within a few
percent.

docs/claude/build-and-run.md

@@ -106,6 +106,19 @@ belongs with `todo.md` item 3, alongside the relative-time toggle.
 
 `tools/mock_udpstreamer.py` has waveform-injection options used by the probes:
 
+- `--square-harmonics N` shapes `square`: `N` odd harmonics of the ideal step's Fourier
+  series, so each edge overshoots by the Gibbs constant (about 9 % of the 2 V swing at the
+  default 30, i.e. `y = ±1.179`) and then rings — a band-limited edge, which is what a
+  crossing count or a pulse-width test has to survive. The series is deterministic and the
+  odd harmonics keep the ideal wave's edge instants and its polarity, so the level-0 trigger,
+  the search and the glitch's inverted pulse still read where they did on the step; the
+  *extremes* are what move. `0` (or negative) is the ideal ±1 step, which is what a probe
+  asserting an exact extreme or an exact width asks for — the noise probe's group-delay leg is
+  the one that does, because its `want` is the analytic `tau*ln2` of a step
+  (see [verification.md](verification.md)). It costs the producer about 0.5 ms per
+  2000-sample packet on top of the ideal branch, so at the fast mock's 1.3 MS/s the mock
+  delivers at roughly nine tenths of its declared pace — a reading that scales with the
+  retained span reads that much lower, and nothing else does.
 - `--glitch-at S` / `--glitch-width W` invert the `square` signal for `W` seconds at the
   start of every square period from stream second `S` onwards — a runt pulse of a known
   width, which is what a `<` pulse-width trigger and a glitch search are checked against.

docs/claude/probe-readings.md

@@ -1027,6 +1027,12 @@ must **not** also be set (two arms on the same port race). Run with
 `--rate 2000 --batch 64 --freq 5 --glitch-at 0.25 --glitch-width 0.004` and
 `UDPSCOPE_EXIT_AFTER_MS=12000`.
 
+**That recipe needs no `--square-harmonics`**: the mock's `square` rings by default (see
+[build-and-run.md](build-and-run.md)), and the odd-harmonic series passes through zero at the ideal edge's
+own crossings, so the glitch's inverted 0.004 s pulse, the hit count, `agree` and the pulse-width leg's
+`measured=0.200000` all read what they read on the step. Only a *level* the ripple reaches and the step does
+not — the extremes, `y = ±1.179` — would need `--square-harmonics 0`, and no probe here asserts one.
+
 Measured output (port 8419):
 
 ```
@@ -1854,7 +1860,14 @@ nothing itself but *is* rebuilt under when a graph is installed at its instant -
 - **`trigger delay:`** -- the reject's group delay, measured on **`square`**, which carries no impulses: a
   one-pole turns a -1 -> +1 step into a zero crossing at exactly `tau*ln2`, so `delay` must read
   `want = 0.002206 s` and **`ratio` between 0.7 and 1.4**. A `ratio` near 0 is a pass-through; a large one
-  means `a` is far too small, i.e. `dt` never arrived at the detector.
+  means `a` is far too small, i.e. `dt` never arrived at the detector. **This is the leg the mock's
+  `--square-harmonics 0` is for**: `want` is a *step* response, and the square rings by default, which moves
+  the crossing to `delay=0.002770 ratio=1.255` — inside the band, but no longer the figure beside it, and the
+  same on every run, so it is the shape and not the run. See the recipe in
+  [verification.md](verification.md). The band/hf legs need no such flag: they count the crossings of `sine`,
+  which the ripple does not touch at all — their counts read a few percent lower at this recipe's 1.3 MS/s
+  only because the extra harmonics cost the *producer*, which
+  [verification.md](verification.md) notes.
 - `state` is `2` (Done) or `4` (Capped) on every leg. `3` (Truncated) on more than one leg means the pump loop
   is being overtaken -- nothing to fix in the feature, but say so.
 

docs/claude/verification.md

@@ -631,13 +631,24 @@ same run).
 ```bash
 cd /home/martino/Projects/tools/udpscope
 python3 tools/mock_udpstreamer.py --port 8108 --rate 1300000 --batch 1000 --duration 20 \
-  --spike-every 0.005 --spike-amp 0.5 --spike-width 2 > /tmp/noise-mock.log 2>&1 &
+  --spike-every 0.005 --spike-amp 0.5 --spike-width 2 --square-harmonics 0 \
+  > /tmp/noise-mock.log 2>&1 &
 sleep 1.5
 xvfb-run -a env UDPSCOPE_EXIT_AFTER_MS=18000 UDPSCOPE_ENABLE_ALL_MS=1000 \
   UDPSCOPE_PROBE_NOISE_MS=4000 \
   ./build/udpscope --server 127.0.0.1 --port 8108 2>&1 | grep -E 'noise-probe|CRITICAL'
 ```
 
+`--square-harmonics 0` is the one flag in that command that is not about the sine, and it is what keeps the
+probe's last reading exact: the group-delay leg measures a one-pole's *step* response on `square`, and the
+mock's square rings by default (see [build-and-run.md](build-and-run.md)). A band-limited edge moves that
+crossing off `tau*ln2` — measured `delay=0.002770 ratio=1.255` against the ideal step's `0.002206 / 1.000`,
+and the same figure on every run, so it is a property of the shape and not of the run. The leg's assertion is
+the `0.7..1.4` band, which the ripple keeps, but the ideal step is what keeps the *number* the line prints
+beside `want` as well. The other legs measure event counts on the sine, which the ripple does not touch; the
+counts do read a few percent lower under the ripple at this rate, for the producer-cost reason
+[build-and-run.md](build-and-run.md) gives.
+
 `UDPSCOPE_PROBE_NOISE_MS` is **not the time its readings are taken**. It times the probe's first stage, which
 installs a math graph through `apply_math_channels()` -- and that is a *structural rebuild* that replaces the
 buffer and drops its history. Every reading the probe takes is over retained history, so the rebuild has to

tools/mock_udpstreamer.py

@@ -8,8 +8,10 @@ once and DATA packets to whoever connected. Publish mode is Accumulate, so each
 packet carries `--batch` consecutive samples per (scalar) signal -- the shape the
 viewer's per-sample path expects.
 
-Signals: `time` (the stream's own clock, seconds), `sine`, `square`, `ramp`.
+Signals: `time` (the stream's own clock, seconds), `sine`, `square`, `ramp`,
-Everything is float64 and unquantised.
+`noise`. Everything is float64 and unquantised. `square` is band-limited by
+default: its edges are the truncated Fourier series of a step, so each one
+overshoots and rings (`--square-harmonics 0` gives the ideal +/-1 step).
 
     ./tools/mock_udpstreamer.py --port 8000 --rate 10000 --batch 100
 """
@@ -85,6 +87,7 @@ def build_data(
     spike_every=0.0,
     spike_amp=0.0,
     spike_width=0,
+    square_harmonics=30,
 ):
     stamps = [t0 + i * dt for i in range(batch)]
     # `time_spike` models a device whose clock glitches for one packet: the stamps jump
@@ -116,9 +119,33 @@ def build_data(
         hit = into < (spike_width * dt)
         sign = np.where((period.astype(np.int64) & 1) == 0, 1.0, -1.0)
         sine = np.array(sine) + np.where(hit, sign * spike_amp, 0.0)
-    square = [
+    # A real square is band-limited: an edge is not a step but an overshoot followed by a
-        1.0 if math.sin(2.0 * math.pi * freq * 0.5 * t) >= 0.0 else -1.0 for t in stamps
+    # decaying ring. The truncated Fourier series is exactly that shape, and it is
-    ]
+    # deterministic -- unlike the interferer -- so a probe can still assert the spacing and the
+    # count of the events it finds. Only odd harmonics, so the sum keeps the ideal wave's edge
+    # instants (it passes through zero at the same crossing) and its polarity: the trigger's
+    # level-0 edges, the glitch's inverted pulse widths and the buffer search's hits all land
+    # where they did on the step. What moves is the extremes: 30 harmonics overshoot by the
+    # Gibbs constant, about 9% of the 2 V swing, so y = +/-1.179 rather than +/-1 -- which is
+    # why a probe asserting an exact extreme or an exact step width asks for
+    # `--square-harmonics 0`, the ideal +/-1 step that was the only shape here before.
+    if square_harmonics > 0:
+        # The same series as a per-sample Python sum costs ~15 sin() calls a sample, and at the
+        # fast mock's rates that makes the *producer* the bottleneck: it falls behind its own
+        # declared pace, and a reading that scales with the retained span (the noise probe's
+        # crossing counts) then moves for a reason that has nothing to do with the signal.
+        t_arr = np.asarray(stamps)
+        harmonics = np.arange(1.0, float(square_harmonics) + 1.0, 2.0)
+        square = (
+            (4.0 / math.pi)
+            * np.sin(2.0 * math.pi * harmonics[:, None] * (freq * 0.5) * t_arr[None, :])
+            / harmonics[:, None]
+        ).sum(axis=0)
+    else:
+        square = [
+            1.0 if math.sin(2.0 * math.pi * freq * 0.5 * t) >= 0.0 else -1.0
+            for t in stamps
+        ]
     ramp = [(freq * t) % 1.0 * 2.0 - 1.0 for t in stamps]
     noisy = np.sin(100 * math.pi * np.array(stamps) + 0.5) + np.random.normal(
         0, 0.5, batch
@@ -207,6 +234,19 @@ def main():
         "what the exact probe assertions (peak to peak = 2, top = 2/pi) are read against; "
         "0.1 is the figure the measurement hysteresis is sized against",
     )
+    ap.add_argument(
+        "--square-harmonics",
+        type=int,
+        default=30,
+        help="odd harmonics of `square`'s Fourier series: a real square is band-limited, so "
+        "each edge overshoots by the Gibbs constant (about 9%% of the 2 V swing at 30 "
+        "harmonics) and then rings -- the shape a crossing count, a pulse-width test or a "
+        "search has to survive. The odd harmonics keep the ideal wave's edge instants and "
+        "its polarity, so the level-0 trigger, the search and the glitch's inverted pulse "
+        "still read where they did on the step; the extremes move to +/-1.179. 0 (or "
+        "negative) = the ideal +/-1 step, which is what an assertion about an exact extreme "
+        "or an exact width asks for",
+    )
     ap.add_argument(
         "--clock-jitter",
         type=float,
@@ -457,6 +497,7 @@ def main():
             args.spike_every,
             args.spike_amp,
             args.spike_width,
+            args.square_harmonics,
         )
         burst = max(1, args.drop_burst)
         if args.drop_every > 0 and drop_left == 0 and counter % args.drop_every == 0: