observer: entropy is a function of |x| alone with level sets {±x, ±1/x}, so H(x) ≡ H(1/x) — the entropy channel cannot see sign-flip or reciprocal oscillation

[InauguralPhysicist]

Finding

entropy_of_num (src/eigenscript.c:257) is H₂(1/(1+|x|)). Two identities follow from the formula alone:

• H₂(p) = H₂(1−p), and p(1/x) = 1/(1+1/x) = x/(1+x) = 1 − p(x), so H(x) ≡ H(1/x) exactly.
• H reads fabs(num), so H(x) ≡ H(−x).

The level set of every value is therefore {x, −x, 1/x, −1/x}. Any trajectory confined to that set has dH = 0 at every step and is invisible to all six predicates and to report.

Numerically confirmed at v0.38.0 / 078e759:

H(2)     = 0.9182958340544896      H(0.5)   = 0.9182958340544896
H(1000)  = 0.011397802630112312    H(0.001) = 0.011397802630110919
H(7)     = 0.5435644431995964      H(-7)    = 0.5435644431995964

H(1000) and H(0.001) agree to ~1e-15 — three orders of magnitude below dh_zero = 0.001.

Reproduction

define a as:
    x is 5.0
    for i in range of 20:
        x is 0 - x               # ±5 forever: the simplest possible oscillator
    print of ("sign-flip: report=" + (report of x) + " oscillating=" + (str of (oscillating of x)))
    return 0
a of 0

define b as:
    y is 1000.0
    for i in range of 20:
        y is 1.0 / y             # 1000 <-> 0.001, ratio 1e6
    print of ("recip:     report=" + (report of y) + " converged=" + (str of (converged of y)))
    return 0
b of 0

sign-flip: report=equilibrium oscillating=0
recip:     report=converged converged=1

The second case is the worse one: a sequence swinging over six orders of magnitude fires the strongest band.

Distinct from #422

#422 covers the value channel's sub-deadband oscillation blindness (a flip below dh_zero is not counted). This is the entropy channel, and the mechanism is different: the steps here are enormous, not sub-deadband — they are simply invisible to a magnitude-only, non-injective measure. report_value classifies both cases correctly as oscillating, which is the point: the channel that is wrong is the one every bare predicate, report, and loop while not converged read by default.

Why it matters

README.md:150-157 advertises oscillation detection with a 5.0 / 2.0 alternation. That example works only because |5| ≠ |2| and 5 ≠ 1/2. Move it onto a level set and detection silently inverts. docs/PREDICATES.md:479 already documents the non-monotonicity (entropy peaks at |x| = 1); the non-injectivity is undocumented and has sharper consequences.

Options

1. Make the value channel the default for report and the bare predicates, demoting entropy to an opt-in diagnostic. Note this trades these failures for Observer value channel (report_value) is blind to sub-exponential divergence and sub-deadband oscillation (relative-delta normalization) #422's and observer: report_value classifies monotone CONVERGENCE as 'diverging' (rate > ~0.87) #674's, so it needs the value channel's floor raised first — not a free swap.
2. Keep entropy as the default but break the x → 1/x symmetry (e.g. sign-aware or signed-log measure).
3. Document the level sets in PREDICATES.md next to the existing |x| = 1 note, and have report cross-check the value channel before returning a rest band.

Related: #422, #294, #412, #735.

[InauguralPhysicist]

Status after #892 (predicates routed to the value channel): the H(x) ≡ H(1/x) ≡ H(−x) level sets no longer decide any numeric classification — the sign-flip and reciprocal oscillators this issue demonstrates now read oscillating from every live surface (report, the predicate words, report_value, tape/DAP/step).

What remains of this finding: the entropy signal keeps the blind spot, reachable via classify of [t, "entropy"] and for non-numeric bindings (whose values have no reciprocal structure, so the level sets are moot there in practice). PREDICATES.md now characterizes it in the saturation-rule and entropy-route sections.

Leaving open only for the question of whether the written characterization satisfies the original ask, per the CoS routing (lang-review's call, not a code change). If the docs note suffices, this can close.