Wear as the stabilizer: tape_loop.h and discreet.h

Every regenerating loop in this library before these files made the same promise the same way: the loop is strictly contractive because feedback is capped below one (delay.h's k_fb_max = 0.99, the comb bank's calibrated ring time). tape_loop.h and discreet.h exist to make the opposite promise — regeneration at exactly 1.0, bounded anyway — and this appendix is the derivation of why that is allowed.

A shared header, by the house rule

The family needed the same four pieces twice (discreet.h and airport.h are both tape machines), and the reuse rule sorted them cleanly. Classes with state went into a shared header the way swing_vca.h was created for the drum family: tape::reel, tape::wow_flutter, tape::wear, and a tape::ramp that is a cited copy of delay.h's anti-zipper unit. Few-line expressions stayed copies-with-citation, as ever: the Hermite polynomial inside reel is the same read as delay.h, line for line, and says so; the saturator is not copied at all but included — vca::swing_shape, the shared swing-type stage, with the reason on the include line.

reel: one wrap, two topologies

A reel is position-addressed circular storage whose reads and writes wrap modulo a settable loop length, not the buffer size. That one decision lets the same class serve both kernels. discreet.h runs it as a delay line: loop length equals capacity, an integer write head advances forever (wrapped into range each sample — a bare long head would overflow LLP64's 32-bit long in half a day of audio), and the play head trails it by the loop span. airport.h runs it as a true loop: length set per piece, one free-running head, positions handed in raw because the reel does all modular arithmetic itself. A length change is deliberately a splice — content kept, positions re-wrapped — because that is what cutting tape does.

wow_flutter: periodic on purpose

The transport error is two sines — slow-deep wow, fast-shallow flutter — returning a read-position offset in samples, phases zeroed at prepare(). The periodic term is the dominant one in the tape-echo literature (Arnardóttir, Abel, Smith, AES 2008), but the deeper reason the stochastic term is a documented non-goal is testability: the wow promise is pinned by predicting peak pitch deviation in closed form (depth · 2π · rate, so 2 ms at 0.5 Hz ⇒ ±10.9 cents) and measuring it with the YIN oracle — 10.9 measured — and that oracle test only exists because two renders are bit-identical. Determinism was a design force here, not an afterthought.

wear: the boundedness argument

One pass of generation loss is three stages in fixed order: an exact one-pole darkening lowpass (1 − e^(−2πf_c/sr), the grm_comb.h map), the shared saturator swing_shape(v, d) = tanh(d·v)/d, and the normalized DC blocker. Each carries one clause of the proof:

• tanh is bounded, so for any drive d > 0 the wear output can never exceed 1/d — whatever the loop has accumulated. That is BIBO stability at regen 1.0, unconditionally, from the saturator alone.
• The DC blocker (pole 0.999, peak gain normalized to exactly 1 — the normalization grm_comb.h earned the hard way, chasing a +0.2 dB/s swell) kills the one frequency the lowpass would happily sustain forever with an offset attached.
• The lowpass is strictly contractive above its corner and asymptotically transparent below it — which is not a leak in the proof but the musical contract: at drive 0 and regen 1.0 the sub-corner band sustains indefinitely, cleanly. The header calls this the Frippertronics contract and states it rather than hiding it.

So where delay.h proves stability by gain, this family proves it by shape: each pass survives because it is degraded. The pinned test drives regen 1.0 for ten seconds of ring and asserts non-growth — never decay, because decay would betray the contract just as surely as growth.

The doppler decision

discreet::machine gives loop_seconds an ordinary ramp and does nothing else, because nothing else is needed: moving a fractional read head is tape-speed doppler. A 0.5 → 0.75 s glide over half a second reads back an octave down mid-move (measured: 220 Hz, then re-lock within five cents) with no discontinuity, since position is continuous even where its slope is not. The rejected alternative — crossfading between two taps — would have hidden the machine, and hiding the machine is the one thing this kernel is for. The wow offset is clamped so the read can never cross the record head; at absurd depths on short loops the transport flattens against the clamp rather than wrapping, which the header files under honest limits.

A finding: the arithmetic agreed

The per-pass wear transfer is fully analytic — regen · |H_lp| · |H_dc| on the unit circle — so the notebook measured it the direct way: a two-tone burst (300 Hz under the corner, 6 kHz over it) recirculated at drive 0, each generation's tones read by Goertzel. Measured per-pass ratios: 0.292 and 0.890. Predicted: 0.292 and 0.890. Three decimals of agreement between a rendering kernel and a formula derived independently in the test is the cheapest kind of confidence this library knows how to buy, and both the test (with 15% and 5% tolerance bands it never needs) and the executed notebook carry the measurement.

The engineering ledger

The suite leans on four instruments. Analytic transfers wherever the path is linear (the per-pass darkening scenario asserts against the exact formula, both tones, both directions — highs die faster and lows barely fade, so the test cannot pass vacuously). Two-window RMS for long-run claims, inherited from the comb bank's swell story: regen 1.0 rings ten seconds and the late window may not exceed the early one. The YIN oracle for anything with a pitch: wow depth in cents against the closed form, the doppler glide and its re-lock. And bitwise assertions where the law is exact: mix endpoints, the first echo returning as literally the recorded impulse, two wow renders identical to the bit. The DC-step scenario checks the blocker's actual job — a held offset at regen 1.0 does not accumulate and the tail's mean returns below 0.02 — rather than a decay the contract never promised.

Checkpoint

One shared header, four blocks: a reel that wraps at the loop, a transport that is two deterministic sines, a wear stage whose tanh bound is the stability proof, and a cited copy of the house ramp. discreet.h composes them into the two-machine loop where regeneration legally reaches 1.0, loop moves are doppler because read heads are physical, and every claim is carried twice — discreet.ipynb executed, discreet_test.cpp pinned.