Bass has
memory.

A frequency curve does not tell the whole story.

A familiar equalizer plot tells us how strongly a fixed filter responds to steady sinusoids. That is useful, but music is not a single infinitely long sinusoid. Attacks, stops, phase relationships, and parameter changes can combine differently in time.

A recursive filter carries information from one sample to the next. In the transposed direct-form implementation used in our model, a second-order section has two stored numbers. They are usually called states. They are enough to make a filter continue responding after a short input has ended.

The key term in the first line is z₁[n]. Today's output depends on both today's input and yesterday's accumulated memory. Turning down the new input does not instantly remove the contribution already stored in the filter.

The two reported quantities answer different questions. The largest impulse sample describes one particular input: a single pulse. The sum of the response magnitudes describes how large a finite response can become when separate bounded input samples line up with the signs of the filter's memory.

For a fixed, stable, scalar linear filter with impulse response h, the exact peak-to-peak gain is the sum of |h[j]| over the entire response. A finite prefix gives a lower benchmark for that gain. To turn a finite computation into an upper guarantee, the uncomputed tail must also be bounded. That distinction becomes central later.

Nothing here says that ordinary music constantly realizes the mathematical worst case. The point of the worst case is that it makes no promise about genre, phase, duration, or whether the input happens to be friendly.

The unit of analysis is the whole machine.

The Bass Plus subsystem studied here contains 18 physical biquad sections and 36 stored state values. It is a stereo graph, with filters on the left and right paths, a middle/side transformation, middle-only processing, and further filtering after recombination. Ten coefficient groups are shared across those physical instances.

This is the original enhancement equalizer before its downstream protection. It is not the complete application signal chain. In particular, the separate Studio effects chain, nonlinear processing, and the operating system's audio path are outside the certificate discussed here.

Why not compute one maximum for every section and multiply them? That is a valid route to a conservative upper estimate in suitable settings, but it can be painfully loose. Each local maximum may require a different input waveform, a different phase relationship, or a different parameter history. Their product can describe a sequence of incompatible worst cases.

Whole-cascade analysis preserves the common input and the actual routing before taking absolute values. It can retain cancellation between pathways and coupling between filters. It also forces us to account for memory already present when a control moves.

Stable filters at isolated settings do not, by themselves, prove stability or a useful peak bound under changing coefficients. The actual transitions matter. Bass Plus smooths controls and interpolates normalized digital coefficients within processed chunks. We model that implemented digital interpolation, rather than substituting a filter reconstructed from interpolated physical settings.

Those two operations generally differ. A proof about the wrong interpolation rule would be a proof about different software, even if both versions sounded plausible in a quick test.

A guarantee has to survive an unfriendly signal.

Our target is an inequality with a clear meaning:

Here U bounds the magnitude of every stereo input sample, G is a universal amplification factor for the specified EQ model, and ε covers a separately bounded effect of tiny state resets. The guarantee must hold for every allowed input and every admitted history of settings, including initialization and switching. It must not depend on a test playlist.

Three distinct objects are easy to confuse: a large peak observed in a numerical simulation; a lower bound showing that some peak cannot be ruled out; and an upper bound ruling out everything larger. Only the last one can justify a universal headroom policy.

The gap we have not closed.

The conservative whole-source upper coefficient we currently have is 604,027,300, with a reset allowance smaller than 0.002. It is mathematically useful because it proves boundedness across the admitted domain. It is operationally unhelpful because it would imply an absurd amount of attenuation if used directly as a fixed gain budget.

At the other end, permitted extreme controls produce a source-reachable steady-state benchmark above 436 times before protection. The reset-free limiting low-frequency gain is approximately 52.8 dB. That does not mean the final application emits a signal 436 times full scale: downstream limiting and nonlinear protection are outside this isolated analysis.

This lower result still tells us something practical. A small universal allowance cannot cover the entire permitted extreme-boost range merely because a more clever proof is found. Some amplification is real; some of the huge upper estimate is mathematical conservatism. Our job is to separate the two.

“Exact-real” also has a specific meaning. Source constants are retained as specified, while arithmetic is analyzed mathematically. A theorem in that model is not a complete verification of a shipping binary, its floating-point rounding, coefficient library, compiler, or operating system behavior. Bridging that gap is a separate engineering task.

Give the recent past a precise shape.
Put a fence around the rest.

An infinite-memory filter appears to ask for an infinite computation. One general route is to divide the past into two pieces: a finite recent window that we evaluate as a complete input/output operator, and an older tail bounded by uniform decay.

For a fixed realized coefficient trace, write the current output as a sum of recent kernel rows times past stereo inputs, plus a contribution from the state at the cut. The exact finite-window peak for independently bounded input components is the sum of the absolute values of those complete kernel rows.

The order matters: form the kernel of the whole graph, then take magnitudes. Taking magnitudes section by section first loses the cancellations we are trying to preserve.

This finite-window reduction is a general solution strategy, not yet a practical numerical answer for Bass Plus. The difficult remaining step is to bound the recent operator over all valid source histories, not just a grid of sample settings.

Initial memory matters too. An observation just after startup cannot be treated as though it followed an infinitely long warm-up. We keep the initial coefficient chord and the reachable startup state separate. Tiny coordinate resets also need their own argument: zeroing one coordinate does not necessarily decrease a quadratic energy with cross terms.

A bound that ignores those details may be impressively small and still fail at exactly the transition where a product needs protection. This is why we keep scope, initialization, and reset allowances attached to the mathematics instead of relegating them to an implementation footnote.

The model invented an impossible filter.

To make a proof computable, it is common to enclose complicated coefficient formulas in a simpler set. An enclosure is safe when it contains every actual possibility. But it may also contain impossible possibilities. Too many of those can make the resulting upper bound useless.

We found a concrete instance of that problem. The bass shelf and bass bell in Bass Plus share one smoothed frequency control. A relaxed model allowed them to behave as though their frequency axes could be chosen independently. It could pair a high-frequency shelf endpoint with a much lower-frequency bell endpoint, despite their common source control.

In a precisely specified diagnostic version of this expanded class, an exactly checked finite impulse-response prefix exceeds 1085.392. That rules out a certificate with G≤1024 for that diagnostic class. It is not a claim that the real application can realize this peak. The counterexample diagnoses the enclosure, not the software. The check retains the analysis model's rational fixed-frequency relation; it does not establish the same obstruction for a compiled floating-point optimization matrix.

The repair begins with the source. Write the shelf's physical tangent as t=tan(πF/Fs), and the actual bell tangent as r·t. The same F appears in both. Across every permitted mode and every sample rate from 8 to 384 kHz, we derived a global enclosure 0.84≤r≤17/18. This is a bound on the tangent ratio, not an assumption that tangent is linear.

We then express seven simultaneous coefficient fields using one shared positive denominator. Positive rational Bernstein weights enclose the entire physical parameter rectangle with 300 controls and at most 600 rational vertices. Crucially, the same weights apply to both filters. Because the implementation uses the same digital interpolation phase across the coefficient pair, the joint enclosure also survives the chords between physical endpoints. Adding the joint identity-coefficient point covers the initial identity-to-target chord; the remaining startup-state proof is still a separate obligation.

This construction excludes the artificial coefficient combination above. It is a source-specific mathematical improvement, not a completed new peak bound. Integrating it into a useful whole-cascade certificate is still work in progress. The convex-hull principle itself is established mathematics. [3]

“Solved” is not the same as proved.

Our other route uses joint storage functions and integral quadratic constraints, or IQCs. Instead of multiplying local peak bounds, it searches for a common inequality relating the state of the entire cascade, its input cost, its output, and source-valid coefficient relationships. Peak-to-peak IQC analysis is an established field. [2]

For one developed formulation, the 36 physical state values are accompanied by 22 analysis memories that retain parameter and product history. The model has 58 analysis states, 100 primitive product ports, and a 160-dimensional free-variable representation. Those extra analysis variables do not add audible filters to Bass Plus. The numerical searches discussed here used a 48 kHz diagnostic model; they were not a completed proof across all sample rates.

We derived an invertible steady-input coordinate shear for that complete representation. We also assemble the dissipation expression through an exact difference identity before conversion to floating point, avoiding subtraction of large nearly equal quantities. These transformations preserve the mathematical model. They change how the calculation is represented, not what audio the application processes.

The broader numerical idea is known: delta-operator formulations and careful scaling have long been used to improve discrete-time matrix inequalities. We do not claim to have invented it. [4]

The most recent transformed calculation is a useful negative result. Its solver returned an inaccurate solution status, but independent reconstruction found violations in 277 of 382 required auxiliary conditions, as well as the main inequalities. It also assigned a negative value to a quantity required to be nonnegative. We rejected it.

The solver's projected cone slack was much closer to positive semidefinite than its affine equations were to consistency. Looking only at one side of that residual would have hidden the failure. Checking retained variables against the original constraints exposed it.

This does not prove that a good certificate is impossible. It shows that this candidate is invalid. The model-enclosure counterexample in the preceding section answers a different question: why a particular target bound is impossible for an overly broad class. Keeping those diagnoses separate prevents us from treating every numerical failure as a new mathematical discovery.

What if protection could manage memory, too?

There is a second, more interventionist route. Instead of asking for a tight peak bound on an unchanged equalizer, we can design an explicitly changed processing rule whose safety follows directly from how it chooses the next input and state.

For one sample frame, the outputs and pre-reset next states are linear forms of the new input u and old state x. Scale the new input by α and all 36 old state values by a common β. Each monitored quantity becomes

Here wᵢ is the new-input contribution and oᵢ is the stored-memory contribution. Requiring every output, internal signal, and next state to stay inside its chosen limit creates a two-dimensional feasible polygon in (α,β). The point (0,0) is always feasible in the mathematical construction.

The general proof is an induction over the whole graph. Accepted outputs and next states satisfy their bounds; the next frame starts inside the bounded state set; the same argument applies again. Coordinate deadzones cannot increase a coordinate's absolute magnitude, so they preserve the raw state box used in this proof.

Using one β for the entire memory has a useful structural property. In the reset-free linear model, for a fixed realized coefficient trace and fixed scale sequence, each old whole-cascade impulse contribution is multiplied by a nonnegative factor. This preserves the internal cancellation within that contribution. It does not imply that every particular output sample becomes smaller: reducing one age of memory can remove cancellation with a different age.

There is also an important practical qualification. The actual deadzone breaks a simple linear superposition across a multi-sample block. Our block prototype therefore uses linear calculations only to propose a choice. It privately computes the actual trial, checks the resulting outputs and state, and commits exactly those checked values. A bounded-work fallback handles failed proposals.

The prototype is transparent when the nominal trial is already safe. When intervention is needed, its priorities matter. Preserving new input at the expense of memory can reshape a resonant response aggressively. Preserving memory can reduce new input more. Numerical experiments exposed strong waveform changes in the frame-by-frame version; the block approach still had boundary discontinuities. Those are quality concerns, not listening-test results.

A safety proof is not a sound-quality proof. This regulator is a research candidate, not an announcement of transparent overload protection in the application. Integration would require performance measurements, distortion analysis, transition testing, and careful listening. None of those requires an acoustics laboratory to begin, but none is replaced by the invariant-set theorem.

The results, without the victory lap.

Research is easier to assess when each result is attached to what it actually establishes. These are the main outcomes of this investigation as of the date of this article.

The most valuable recent change is a better diagnosis. We can now point to both a genuine numerical failure and a concrete modeling relaxation that admits an impossible setting combination. Those lead to different remedies. More solver iterations do not restore a source relationship that the model discarded; a tighter source enclosure does not automatically fix a numerically invalid solution.

The next mathematical target remains a useful, independently checkable upper bound for the unchanged complete cascade. We are not counting one successful frequency, one fixed preset, or a dense parameter grid as a universal result. A grid can guide the search; a full-domain argument must close it.