observer: genuine exponential divergence reports `converged` in BOTH channels once saturation clamps — contradicts #422's stated guarantee

[InauguralPhysicist]

Finding

#422 states, as the boundary of the value channel's known blind spot:

Exponential/geometric divergence (fixed relative step) IS caught; constant-increment or polynomial growth is not.

It is not caught. A textbook geometric runaway saturates at ±1e308 (the "finite by construction" rule), which turns an unbounded trajectory into a fixed point — and both channels then classify it as the strongest rest band.

Reproduction

v0.38.0, release build at 078e759.

define run as:
    z is 2.0
    for i in range of 20:
        z is z * z
    print of ("report=" + (report of z) + " report_value=" + (report_value of z) + " value=" + (str of z))
    print of ("  diverging? " + (str of (diverging of z)) + " converged? " + (str of (converged of z)))
    return 0
run of 0

report=converged report_value=converged value=1e+308
  diverging? 0 converged? 1

Expected: diverging, or at minimum not converged.

Mechanism

z reaches 1e308 after ~6 steps and stays pinned there for the remaining 14, so the dH window fills with zeros in both channels. On the entropy side H(1e308) ≈ 0 < h_low, so the entropy < h_low clause that normally holds converged back from information-rich fixed points is satisfied too. Every clause of the converged definition (PREDICATES.md kind 0) is legitimately met — the window really is quiet and the entropy really is low. The problem is upstream: saturation destroyed the evidence of divergence before the observer sampled it.

This is why report_value does not rescue it. #422 was closed on the reasoning that the relative-step channel catches fixed-relative-step growth; that reasoning holds only while the value can still grow. Past 1e308 the relative step is exactly 0.

Why it matters

README.md:158 sells the trajectory words for "instability alerts". This is the canonical instability, and the runtime reports maximal stability with rc=0. It is also the one failure mode where a wrong answer is most costly: a diverging solver that reports converged returns 1e308 as an answer rather than an error.

Options

1. Flag the saturation event on the observer slot and let diverging read it (cheapest; makes the clamp visible rather than silent).
2. Track magnitude alongside dH so an unbounded-growth signal exists independently of step size — this is option (a) from Observer value channel (report_value) is blind to sub-exponential divergence and sub-deadband oscillation (relative-delta normalization) #422, and would also cover the polynomial case Observer value channel (report_value) is blind to sub-exponential divergence and sub-deadband oscillation (relative-delta normalization) #422 left open.
3. Reconsider the saturation rule itself (filed separately: the associativity/detectability consequences).

Related: #422, #674, #294, #412.

[InauguralPhysicist]

Reopening: PR #889 closed the reproducer, not the defect. I wrote that fix; this is a correction to its scope, not a new finding.

The gate lands at |v| >= 1e308. The false converged starts ~305 decades earlier.

Measured, at 27a22e7

x *= 1.5   first certifies `converged` at x = 287,626      (~1e5)
x *= 2.0   first certifies `converged` at x = 8,388,608    (~1e6)

The issue text guessed the runaway would need to be large. It does not. A fixed-relative-step geometric runaway — exactly the class #422 recorded as caught — certifies converged at five decimal digits.

The boundary is a region, not a point

h_low = 0.1 against H(x) = H_b(1/(1+|x|)) puts every |x| > ~76 under the entropy floor. Holding a constant value and asking:

held value	report	converged
5	equilibrium	0
76	equilibrium	0
77	converged	1
88	converged	1
1e15	converged	1
1e307	converged	1
1e308	diverging	0

Any runaway that flattens anywhere in [77, 1e307] satisfies every clause of converged legitimately. #889 closed the single point at the top.

A linear runaway makes it concrete — this terminates, at y = 88:

y is 1.0
loop while not (converged of y):
    y is y + 1.0

Why the original fix passed its own gate

The repro and the planted fault were both derived from the same wrong frame — "saturation destroys the evidence" — so both agreed with it. Nothing in that PR was positioned to ask whether the boundary was a point or a region. The saturation gate itself is still correct and stays; it is just not sufficient, and #889's regression test blessed 1e307 in a way that reads as endorsing the region (a constant at 1e307 does converge, but the entropy channel cannot distinguish that from a runaway that flattened there).

What decides this

A convergence corpus with analytically known ground truth — 27 sequences, no implementation consulted — scores the shipped predicate at:

ENTROPY CHANNEL (what converged/report use): correct=16/27  FP=3  FN=8
VALUE CHANNEL   (report_value)             : correct=26/27  FP=0  FN=1
precision = 4/7 of firings were real
recall    = 4/12 of real convergences detected

The already-implemented value channel scores 26/27 on the same corpus. Landing that corpus as a scored gate is the immediate next step (separate PR); the routing/h_low decision it argues for touches the entropy-constant fence and is not mine to make.

Related: #862 (the H(x) ≡ H(1/x) level sets), #865 (saturation indistinguishability).

[InauguralPhysicist]

Auto-closed by #891's merge (the PR body said 'Reopens' but the linked-issue automation read the reference as a close). Reopening — #891 landed the measurement instrument; the routing fix is in flight.