--- slug: the-noise-that-makes-it-accurate passed: 2026-11-02 by: ccr-81332fb3-8nrbtu wave, one independent agent on oversight/claims-pass.md (pre-deploy; reader check on local copies with AW_DIR; every source-record entry read against its kept copy and live source) checked: 150 confirmed: 112 wrong: 0 unverifiable: 1 observed: 37 stale: 0 --- # Claims pass: The Noise That Makes It Accurate Live page: not deployed yet; step 2 skipped. Reader check (pre-deploy, local copies): `AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs`, run with the verifier copied alone into a second empty directory: exit 0, 322 passed, 0 failed (8.1 s); with `--deep`: 323 passed, 0 failed (22.3 s); with `--mutate`: exit 1, 514 passed, 7 failed; in the repository: 413 passed, 0 failed. Step 6 (the page prints no transcript of the verifier; sentence-only edits in scratch copies): "10·log10(3) = 4.77 dB more noise" changed to 4.87, exit 1, 320 passed, 2 failed (its sentence check and the residual sweep); "93,470 of a million" changed to 93,570, exit 1, 320 passed, 2 failed. Mutation sweep: `node $SP/wave-sweep.mjs the-noise-that-makes-it-accurate --jobs 2 --work $SP/claims/the-noise-that-makes-it-accurate/sweep`: the-noise-that-makes-it-accurate: 493 mutated, 493 caught, 0 survived (text 0, meta 0, year 0). `assay/sources.mjs the-noise-that-makes-it-accurate --online`: 91 UNREADABLE, 13 FOUND, 41 in the repository. Every UNREADABLE is a PDF: this pass fetched all eight live PDFs (each byte-identical to the sha256 its kept copy records) and read each excerpt in pdftotext output; the 1992 survey is a scan, so its fourteen quotations and its other entries were read on the page images (PDF pp. 1, 15 to 19, at 220 dpi). UWSpace first answered with a bot-check page, then served the thesis PDF to plain curl. Wikipedia was read in its live wikitext and rendered HTML, the W3C draft in its HTML. The page's own figures were recomputed in this pass's own Python (moments by integration, the Lip7ZP and first-order loops with its own mulberry32, the zeros of 1 - H by 80-digit Newton and exact rationals) and its app-built readouts captured with Playwright on the local copies (bench presets: 328 from 13,904 at -151.3 dB; seed 3 period 168 from 24,868; 6 levels; run-away at 536; -12.04 dB). ## Claims - CONFIRMED | The first moment, or mean error, is zero for all inputs, indicating that the quantizer has been linearized by the use of this dither. | https://hajim.rochester.edu/ece/sites/zduan/teaching/ece472/reading/Lipshitz_1992.pdf | read on the page image (PDF p. 17, journal p. 371, 220 dpi): "The first moment, or mean error, is zero for all inputs, indicating that the quantizer has been linearized by the use of this dither." Verbatim. - CONFIRMED | The error variance, on the other hand, is clearly signal dependent, so that the noise power in the signal varies with the system input. This is sometimes referred to as noise modulation and is audibly undesirable. | https://hajim.rochester.edu/ece/sites/zduan/teaching/ece472/reading/Lipshitz_1992.pdf | read on the page image (PDF p. 17, journal p. 371, 220 dpi): the next two sentences, verbatim; they follow the rectangular-pdf case of Fig. 8. - CONFIRMED | The second moment of the total error is a constant Δ²/4 for all inputs | https://hajim.rochester.edu/ece/sites/zduan/teaching/ece472/reading/Lipshitz_1992.pdf | read on the page image (PDF p. 17, journal p. 371, 220 dpi): "The second moment of the total error is a constant Δ²/4 for all inputs, in agree-ment with Eq. (82)." The quoted part is on p. 371; said of the triangular-pdf dither of Eq. (108). - CONFIRMED | up by 4.8 dB relative to that of Fig. 6(d) due to the tripling of the noise spectral density | https://hajim.rochester.edu/ece/sites/zduan/teaching/ece472/reading/Lipshitz_1992.pdf | read on the page image (PDF p. 19, journal p. 373, 220 dpi): "the noise floor in Fig. 10(d) is up by 4.8 dB relative to that of Fig. 6(d) due to the tripling of the noise spectral density". Fig. 10 caption: "Results from computer-simulated quantization ... triangular-pdf nonsubtractive dither", so "simulated spectrum" holds; Fig. 6 is subtractive RPDF (noise Δ²/12). - CONFIRMED | triangular-pdf dither of 2-LSB peak-to-peak amplitude is unique and optimal in the sense that it renders the first and second moments of the total error input independent, while minimizing the second moment. | https://hajim.rochester.edu/ece/sites/zduan/teaching/ece472/reading/Lipshitz_1992.pdf | read on the page image (PDF p. 18, journal p. 372, 220 dpi): "It can be shown [3] that triangular-pdf dither of 2-LSB peak-to-peak amplitude is unique and optimal ..." Verbatim from "triangular-pdf". - CONFIRMED | Theorem 8 A nonsubtractive dither signal generated by the summation of n statistically independent zero-mean rectangular-pdf random processes, each of 1-LSB peak-to-peak amplitude, renders the first n moments of the total error independent of the system input, and results, for n ≥ 2, in a total error power of (n + 1)Δ²/12. | https://hajim.rochester.edu/ece/sites/zduan/teaching/ece472/reading/Lipshitz_1992.pdf | read on the page image (PDF p. 15, journal p. 369, 220 dpi): Theorem 8 verbatim; the printed slanted "≥" typed as ≥; "zero-mean" is hyphenated in print. Citation p. 369 right. - CONFIRMED | Nonsubtractive dithering, unlike subtractive dithering, cannot render the total error independent of the system input. | https://hajim.rochester.edu/ece/sites/zduan/teaching/ece472/reading/Lipshitz_1992.pdf | read on the page image (PDF p. 18, journal p. 372, 220 dpi): section 5.4 Summary item 1), verbatim. - CONFIRMED | Variations in higher moments are believed to be inaudible, and this has been corroborated by a large number of psychoacoustic tests conducted by the authors and others | https://hajim.rochester.edu/ece/sites/zduan/teaching/ece472/reading/Lipshitz_1992.pdf | read on the page image (PDF p. 19, journal p. 373, 220 dpi): section 6 Conclusions, verbatim to "and others" (the source continues "[18], [23]"). - CONFIRMED | Admittedly, these tests were informal | https://hajim.rochester.edu/ece/sites/zduan/teaching/ece472/reading/Lipshitz_1992.pdf | read on the page image (PDF p. 19, journal p. 373, 220 dpi): "Admittedly, these tests were informal, and there remains a need for formal psycoacoustic tests". The page's "of those listening tests" is right: the sentence closes the paragraph describing them. - CONFIRMED | using rectangular-pdf noises of peak-to-peak amplitude not equal to 1 LSB (or, rather, not equal to an integral number of LSB) will not render error moments independent of the input | https://hajim.rochester.edu/ece/sites/zduan/teaching/ece472/reading/Lipshitz_1992.pdf | read on the page image (PDF p. 16, journal p. 370, 220 dpi): verbatim ("rec-tangular" is a line-end hyphen); the source goes on "since the zeros of the associated sinc functions will not fall at integral multiples of 1/Δ". - CONFIRMED | many analog signals and digital-to-analog conversion systems exhibit a Gaussian noise component which is of large enough amplitude to act as a satisfactory inherent dither | https://hajim.rochester.edu/ece/sites/zduan/teaching/ece472/reading/Lipshitz_1992.pdf | read on the page image (PDF p. 19, journal p. 373, 220 dpi), footnote 13: "It should be noted, however, that many analog signals and digital-to-analog conversion systems exhibit a Gaussian noise component ... satisfactory inherent dither without the requirement of an explicit dithering operation". The page's "often needs no noise added on purpose" matches "many" and "without ... explicit dithering". - CONFIRMED | requantized very coarsely (to 8 bits from 16) | https://hajim.rochester.edu/ece/sites/zduan/teaching/ece472/reading/Lipshitz_1992.pdf | read on the page image (PDF p. 19, journal p. 373, 220 dpi): "a large variety of signals (such as sinusoids, ..., piano and orchestral music) which had been re-quantized very coarsely (to 8 bits from 16)". "music among them" holds. - CONFIRMED | For practical purposes, this simply means that the signal is never clipped by saturation of the quantizer. | https://hajim.rochester.edu/ece/sites/zduan/teaching/ece472/reading/Lipshitz_1992.pdf | read on the page image (PDF p. 1, journal p. 355, 220 dpi), footnote 1 to "Assuming that these represent infinite quantizers": verbatim. - CONFIRMED | the areas of any optimal noise shaper curve above and below the unshaped PSD must be equal | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 8: "the Gerzon/Craven “noise-shaping theorem” [9] tells us that the areas of any optimal noise shaper curve above and below the unshaped PSD must be equal when plotted on a linear-frequency/logarithmic-amplitude basis." The page's "on a decibel scale against linear frequency" is that basis. - CONFIRMED | in a straight 16-bit system, the dither occupies only two out of the 65,536 levels available. | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 2: verbatim, after "the proper (i.e., TPDF) dither spans precisely two LSBs". - CONFIRMED | We claim that a 1-bit sigma-delta converter must overload when properly dithered. This follows at once, since the TPDF dither N, which is needed to fully linearize the quantizer, itself swings the quantizer’s input W over its full no-overload range of ±∆. | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 3: verbatim, including the curly apostrophe and U+2206; citation "p. 3" right. - CONFIRMED | full TPDF dither is never allowable in a 1-bit noise shaper or sigma-delta modulator, and hence full linearity is never achievable either in principle or, of course, in practice. | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 4: "The important point is that full TPDF dither is never allowable ..." verbatim; it is the paper's own conclusion. - CONFIRMED | overload is guaranteed to occur (even with no input) at some step n | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 11, Appendix (d): "µ > ∆/4: The compatibility condition is now violated, and overload is guaranteed to occur (even with no input) at some step n." Quoted exactly. - CONFIRMED | Since the output power is constant, the signal component of the output must come at the expense of the remainder. | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 8: verbatim, after "a 1-bit quantizer, switching between the two output levels of ±∆/2, has a constant total output power of ∆2/4". - CONFIRMED | This cannot, however, be guaranteed. | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 8: "The best that could be hoped for would be that all such modulation effects occur only above the audio band. This cannot, however, be guaranteed." The page paraphrases the first sentence before quoting the second. - CONFIRMED | so that interested readers can implement it and confirm the results which follow | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 4: "The explicit expression for the Lip7ZP noise-shaping filter H(z) is given here so that interested readers can implement it and confirm the results which follow". - CONFIRMED | In the absence of dither, and with zero initial conditions, Lip7ZP displays a low-level limit-cycle oscillation of period 328 samples (and a very audible fundamental frequency of 8.605 kHz). | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 4: verbatim. - CONFIRMED | takes almost 16,000 samples before it is firmly established | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 4: "This “birdie” takes almost 16,000 samples before it is firmly established". - CONFIRMED | the minimum necessary to break up the limit cycle | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 5: "if we add to the quantizer 0.0013∆ peak-to-peak of rectangular probability density function (RPDF) dither — the minimum necessary to break up the limit cycle." (p. 7 later says "0.013∆" for the same smidgen; 0.0013 is the figure consistent with the paper's 60.7 dB, as the page recomputes.) - CONFIRMED | an amount of dither 60.7 dB less than full TPDF dither | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 5: "(This is an amount of dither 60.7 dB less than full TPDF dither.)" - CONFIRMED | it occasionally needs up to ±4 levels; that is, a 3-bit quantizer is needed to avoid overload | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 5: "Fig. 7 shows that it occasionally needs up to ±4 levels; that is, a 3-bit quantizer is needed to avoid overload." (p. 4 adds "it requires at least eight levels"). - CONFIRMED | the system is unstable | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 5: "In fact, if one restricts this fully-dithered quantizer to just two levels, the system is unstable." - CONFIRMED | we can inject only 0.17 LSBs peak-to-peak of RPDF dither before additional levels become necessary | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 6: verbatim; the experiment is the two half-full-scale tones of Figs 9 and 10 (Fig. 12 caption: "subjected to the same input signals as in Fig. 9"). - CONFIRMED | 1/69th of the power of full TPDF dither | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 7: "This is only about 1/69th of the power of full TPDF dither (i.e., −18.4 dB relative to the power of TPDF dither)." - CONFIRMED | one real zero at dc, and three complex conjugate pairs of zeros just inside the unit circle, corresponding to frequencies of 4, 12, and 20 kHz | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 4: verbatim. - CONFIRMED | almost universally given up on the 1-bit sigma-delta topology in favor of oversampling converters using more than two levels | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 2: "If one examines the current data-sheets of all the major high-quality converter manufacturers, one finds that they have almost universally given up on the 1-bit sigma-delta topology in favor of oversampling converters using more than two levels." - CONFIRMED | Further research is needed here. | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 9: verbatim, closing the bullet on in-band distortion products. - CONFIRMED | based on a fourth-order multibit delta-sigma modulator | https://statics.cirrus.com/pubs/proDatasheet/CS4344-45-48_F2.pdf | live PDF (sha256 e4cf72f7..., same as kept) p. 1: "The CS4344 family is based on a fourth-order multibit delta-sigma modulator with a linear analog low-pass filter." - CONFIRMED | Multi-Bit Delta-Sigma Analog-to-Digital Converter | https://www.ti.com/lit/ds/symlink/pcm4222.pdf | live PDF (sha256 aa9e1258..., same as kept) p. 1 title: "High-Performance, Two-Channel, 24-Bit, 216kHz Sampling Multi-Bit Delta-Sigma Analog-to-Digital Converter". - CONFIRMED | 6-bit modulator data | https://www.ti.com/lit/ds/symlink/pcm4222.pdf | live PDF p. 2, DESCRIPTION: "the PCM4222 supports 24-bit linear PCM, 1-bit Direct Stream Digital (DSD), and 6-bit modulator data outputs." The record and kept copy say p. 1; it is p. 2 (p. 1 has "– 6-Bit Modulator Data", capitalised). See Findings. - CONFIRMED | 1-bit Direct Stream Digital (DSD) | https://www.ti.com/lit/ds/symlink/pcm4222.pdf | live PDF p. 2, DESCRIPTION (same sentence as above); record says p. 1. See Findings. - CONFIRMED | modulated all the way to 128 × fS before arriving at the current segments | https://www.ti.com/lit/ds/symlink/pcm5102a.pdf | live PDF (sha256 a5220836..., same as kept) p. 24, section 9.3.5.1 Clocking Overview: verbatim. The record and kept copy label it "section 8.3" (8.3 is Recommended Operating Conditions). See Findings. - CONFIRMED | silicon digital microphone with a single bit PDM output | https://akizukidenshi.com/goodsaffix/SPH0641LU4H-1.PDF | live PDF (sha256 a7010712..., same as kept) p. 1: "silicon digital microphone with a single bit PDM output." - CONFIRMED | a sigma-delta modulator | https://akizukidenshi.com/goodsaffix/SPH0641LU4H-1.PDF | live PDF p. 1: "the SPH0641LU4H-1 consists of an acoustic sensor, a low noise input buffer, and a sigma-delta modulator." - CONFIRMED | cellphones, smart phones | https://akizukidenshi.com/goodsaffix/SPH0641LU4H-1.PDF | live PDF p. 1: "This microphone is suitable for applications such as cellphones, smart phones, laptop computers, ...". "offered for" is fair. - CONFIRMED | fourth‐order Σ‐Δ modulator | https://iw.ftcelectronics.com/datasheets-f4/ICS-41350.pdf | live PDF (sha256 254d918b..., same as kept): "followed by a fourth‐order Σ‐Δ modulator." U+2010 hyphens kept as printed. - CONFIRMED | pulse density modulated (PDM) interface | https://iw.ftcelectronics.com/datasheets-f4/ICS-41350.pdf | live PDF: "The pulse density modulated (PDM) interface allows two microphones to be time multiplexed". - CONFIRMED | In 2002, Philips published a paper arguing the contrary. | https://en.wikipedia.org/w/index.php?title=Direct_Stream_Digital&action=raw | live wikitext, == DSD vs. PCM ==: verbatim; it follows "In 2000, Lipshitz and Vanderkooy stated that one-bit converters, as employed by DSD, are unsuitable for high-end applications due to their high distortion." - CONFIRMED | Lipshitz and Vanderkooy's paper was further criticized by Angus. | https://en.wikipedia.org/w/index.php?title=Direct_Stream_Digital&action=raw | live wikitext: the sentence right after the Philips sentence, verbatim. - CONFIRMED | Lipshitz and Vanderkooy responded to the criticisms. | https://en.wikipedia.org/w/index.php?title=Direct_Stream_Digital&action=raw | live wikitext: the sentence after that, verbatim; so "Its next two sentences" is exact. - CONFIRMED | a complete statistical description of non-linear systems with feedback, such as sigma-delta converters, awaits development. | https://uwspace.uwaterloo.ca/bitstreams/fa271912-b183-4c4c-a95d-bc2f761ba345/download | live PDF (sha256 dbbdff08..., same as kept; UWSpace serves it after a bot check, curl received the PDF) printed p. 156, Chapter 7 Conclusions, its penultimate paragraph before "In closing, ..." and the Bibliography: verbatim. - CONFIRMED | Every digital audio system puts each sample on one of a fixed set of levels: once when a converter first measures the sound, and again whenever software shortens the numbers. | https://hajim.rochester.edu/ece/sites/zduan/teaching/ece472/reading/Lipshitz_1992.pdf | read on the page image (PDF p. 1, journal p. 355, 220 dpi): "Analog-to-digital conversion is customarily decomposed into two separate processes: time sampling ... and amplitude quantization" and "requantization, in which the wordlength of digital data is reduced after processing". - CONFIRMED | engineers write it Δ and call it the least significant bit, LSB | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 3: "In Fig. 2, the size of the LSB is represented by ∆". - CONFIRMED | Stanley Lipshitz, Robert Wannamaker and John Vanderkooy, in their 1992 survey of the theory | https://hajim.rochester.edu/ece/sites/zduan/teaching/ece472/reading/Lipshitz_1992.pdf | read on the page image (PDF p. 1, journal p. 355, 220 dpi): title "Quantization and Dither: A Theoretical Survey", authors "STANLEY P. LIPSHITZ, ... ROBERT A. WANNAMAKER, AND JOHN VANDERKOOY", footer "J. Audio Eng. Soc., Vol. 40, No. 5, 1992 May". The record's excerpt is a composed citation, not the source's words. See Findings. - CONFIRMED | The general rule is their Theorem 8 | https://hajim.rochester.edu/ece/sites/zduan/teaching/ece472/reading/Lipshitz_1992.pdf | read on the page image (PDF p. 15, journal p. 369, 220 dpi): "Theorem 8 A nonsubtractive dither signal generated by the summation of n ...". - CONFIRMED | That is three times Δ²/12, the error power of the textbook model of rounding | https://hajim.rochester.edu/ece/sites/zduan/teaching/ece472/reading/Lipshitz_1992.pdf | journal p. 367 (OCR of PDF p. 13): "the total error variance is greater than that of classical or subtractively dithered quantization (namely, Δ²/12) by the dither variance"; also LV2001 p. 9: "The classical model for the quantization error E of an undithered multi-bit quantizer postulates that it has a power of ∆2/12." - CONFIRMED | Add two independent rectangular sources and the second chart goes flat | https://hajim.rochester.edu/ece/sites/zduan/teaching/ece472/reading/Lipshitz_1992.pdf | read on the page image (PDF p. 17, journal p. 371, 220 dpi): "Now consider a triangular-pdf dither of 2-LSB peak-to-peak amplitude resulting from the sum of two independent rectangular-pdf noises". - CONFIRMED | Every result above is proved for quantisers the survey calls infinite, and its first footnote says what that means | https://hajim.rochester.edu/ece/sites/zduan/teaching/ece472/reading/Lipshitz_1992.pdf | read on the page image (PDF p. 1, journal p. 355, 220 dpi): "Assuming that these represent infinite quantizers,1 the corresponding transfer functions can be expressed analytically". - CONFIRMED | The first instrument rounds with a midtread quantiser, Q(w) = ⌊w + 1/2⌋, the loops with a midriser one, Q(w) = ⌊w⌋ + 1/2, as the two papers do. | https://hajim.rochester.edu/ece/sites/zduan/teaching/ece472/reading/Lipshitz_1992.pdf | read on the page image (PDF p. 1, journal p. 355, 220 dpi) Eqs. (1) midtread and (2) midriser; p. 356 (OCR): "Throughout the sequel, quantizers of the midtread variety will be assumed"; LV2001 p. 3: "quantizers Q(W) of the mid-riser type". So "as the two papers do" holds. - CONFIRMED | the output is Y = X + (1 − H)E: the input passes untouched and the error is reshaped by 1 − H (Lipshitz and Vanderkooy, 2001) | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 2-3: "Y(z) = X(z) + {1 – H(z)}⋅E(z)" and "The signal thus passes through the system unchanged, and the quantization error E(z) appears at the output shaped by the effective noise-transfer function {1 – H(z)}". - CONFIRMED | A theorem of Gerzon and Craven, as Lipshitz and Vanderkooy state it, says that | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 8; reference [9] p. 10: M. A. Gerzon and P. G. Craven, "Optimal Noise Shaping and Dither of Digital Signals", AES 87th Convention, preprint 2822. - CONFIRMED | A 1-bit quantiser has two output levels, +Δ/2 and −Δ/2, and behaves like a many-level one only while its input W stays within −Δ ≤ W < Δ. | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 3: "If the total input W to the quantizer always lies in the range –∆ ≤ W < ∆, no additional output levels beyond ±∆/2 will be called upon, and a 1-bit quantizer will behave just like a multi-bit one." - CONFIRMED | Triangular dither on its own already reaches almost ±Δ. | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 3: "the TPDF dither N ... itself swings the quantizer’s input W over its full no-overload range of ±∆." TPDF of 2 LSB peak to peak spans the open interval (−Δ, Δ), so "almost ±Δ" is exact. - CONFIRMED | Make H a single sample of delay and the 1-bit loop is a first-order sigma-delta modulator. | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 10, Appendix: "this corresponds to setting H(z) = z–1, a single-sample delay, ... corresponds precisely to the single-stage, 1st-order, 1-bit sigma-delta converter." - CONFIRMED | The appendix of the 2001 paper proves a safe region for it: with dither confined to ±µ, the loop never overloads if µ ≤ Δ/4 and the input stays within ±(Δ/2 − 2µ). | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 11: (A.9) "–∆/2 + 2µ ≤ xk ≤ ∆/2 – 2µ for all k" with dither "–µ ≤ νn < µ", compatibility "µ ≤ ∆/4"; case (c) µ = ∆/4 allows only zero input, consistent with the page's "µ ≤ Δ/4". - CONFIRMED | Full triangular dither has µ = Δ, four times the largest µ that region allows. | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 11, case (b): "full TPDF dither would require that µ = ∆"; ∆ / (∆/4) = 4. - CONFIRMED | Lipshitz and Vanderkooy printed a seventh-order shaper, Lip7ZP, for a converter running at 64 × 44,100 = 2,822,400 samples a second, the rate of the Direct Stream Digital (DSD) format | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 4: "Our design, called Lip7ZP ... a 7th-order noise-shaping transfer function"; "the DSD sampling rate is 64 × 44,100 Hz = 2.8224 MHz"; "a 64-times oversampled sigma-delta architecture, as envisaged by Philips and Sony ... for their DSD converter". - CONFIRMED | Stanley P. Lipshitz and John Vanderkooy, Why 1-Bit Sigma-Delta Conversion is Unsuitable for High-Quality Applications, AES 110th Convention, Amsterdam, May 2001, Convention Paper 5395 (the Lip7ZP coefficients are on p. 4) | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 1: "Convention Paper 5395 Presented at the 110th Convention 2001 May 12–15 Amsterdam"; title "Why 1-Bit Sigma-Delta Conversion is Unsuitable for High-Quality Applications"; coefficients printed on p. 4 and matched digit for digit by engine.mjs LIP7ZP. - CONFIRMED | whose peak the paper puts at Δ/4 | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 8: "This corresponds to a peak amplitude of ∆/4 for a fullscale sine-wave." (also p. 5). - CONFIRMED | With two half-full-scale tones in, they found | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 6: "two sine-wave signals, each of half full-scale amplitude (i.e., ∆/8 each), on FFT bins 32 (5512.5 Hz) and 48 (8268.75 Hz)". - CONFIRMED | This page's runs of the same two tones, at 5512.5 Hz and 8268.75 Hz | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 6: same; engine TWO_TONES = 32 and 48 × 2,822,400/16,384 = 5512.5 and 8268.75 Hz, and SIN1024 at 2t and 3t gives fs/512 and 3fs/1024, the same two frequencies. - CONFIRMED | Every sample is +Δ/2 or −Δ/2, so the stream's power is Δ²/4 whatever it carries. | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 8: "a 1-bit quantizer, switching between the two output levels of ±∆/2, has a constant total output power of ∆2/4." - CONFIRMED | The other 1-bit stream is DSD itself, the format of the Super Audio CD | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 2: "the Direct Stream Digital (DSD)1 encoding which forms the basis of the Super Audio CD2 format introduced recently by Philips and Sony". - CONFIRMED | Cirrus Logic's CS4344 digital-to-analog converter (data sheet dated 2013) | https://statics.cirrus.com/pubs/proDatasheet/CS4344-45-48_F2.pdf | live PDF p. 1: title "10-Pin, 24-Bit, 192 kHz Stereo D/A Converter"; footer "Copyright Cirrus Logic, Inc. 2013 JUL ‘13 ... DS613F2". - CONFIRMED | Texas Instruments' PCM4222 (October 2006, revised March 2007) | https://www.ti.com/lit/ds/symlink/pcm4222.pdf | live PDF p. 1 header: "SBAS399A – OCTOBER 2006 – REVISED MARCH 2007". - CONFIRMED | In TI's PCM510xA converters (May 2012, revised May 2015) | https://www.ti.com/lit/ds/symlink/pcm5102a.pdf | live PDF p. 1 header: "SLAS859C – MAY 2012 – REVISED MAY 2015"; title "PCM510xA 2.1 VRMS, 112/106/100 dB Audio Stereo DAC". - CONFIRMED | output, at 2.8224 MHz in its 64x mode | https://www.ti.com/lit/ds/symlink/pcm4222.pdf | live PDF p. 4: "DSD output: 64x mode, 2.8224MHz output rate"; p. 2: "The DSD output supports either 64x or 128x oversampled bit rates." - CONFIRMED | In 2001 Lipshitz and Vanderkooy, reading the data sheets of the major high-quality converter makers, found they had | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 2: "If one examines the current data-sheets of all the major high-quality converter manufacturers, one finds ...". - CONFIRMED | (data sheet revision B, 2015) | https://akizukidenshi.com/goodsaffix/SPH0641LU4H-1.PDF | live PDF footers: "Revision: B 04/06/2015" (copyright line "© 2014 Knowles Electronics"). - CONFIRMED | InvenSense's ICS-41350 (product brief, 2016) follows its MEMS element and amplifier with a | https://iw.ftcelectronics.com/datasheets-f4/ICS-41350.pdf | live PDF: "InvenSense Inc. ... Document Number: PB‐000021 Revision: 1.0 Rev Date: 02/17/2016"; "MEMS microphone element and an impedance converter amplifier followed by a fourth‐order Σ‐Δ modulator." - CONFIRMED | and lists smartphones first among its applications | https://iw.ftcelectronics.com/datasheets-f4/ICS-41350.pdf | live PDF, APPLICATIONS column, first entry: "Smartphones". - CONFIRMED | Lipshitz and Vanderkooy themselves wrote that the distortion products they had shown inside the audio band might be perceptible, and added: | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 9: "so too might the low-level, but correlated, distortion products that we have shown to exist within the baseband be perceptible, even though they would normally be thought to be below audibility." - CONFIRMED | Wikipedia's article on DSD records that | https://en.wikipedia.org/w/index.php?title=Direct_Stream_Digital&action=raw | live wikitext, DSD vs. PCM, third paragraph: verbatim. - CONFIRMED | Wannamaker's 1997 thesis closes by noting that | https://uwspace.uwaterloo.ca/bitstreams/fa271912-b183-4c4c-a95d-bc2f761ba345/download | live PDF printed p. 156, the last page of Chapter 7 Conclusions; only "In closing, it is proposed ..." follows before the Bibliography, so "closes by noting" is fair. - CONFIRMED | because the Web Audio specification clamps a DelayNode inside a feedback cycle to at least one render quantum, 128 frames by default | https://www.w3.org/TR/webaudio-1.1/ | live HTML, DelayNode delayTime: "If DelayNode is part of a cycle, then the value of the delayTime attribute is clamped to a minimum of one render quantum."; 2.6 Rendering an Audio Graph: "Its default value is 128, and it can be configured by setting renderSizeHint." - CONFIRMED | does the same job for pictures with deterministic patterns of dots instead of random noise | src/content/strata/the-grey-that-isnt-there.md | that layer's tags are ordered (Bayer), error-diffusion, Floyd-Steinberg and Atkinson dithers; "random" occurs nowhere in public/strata/the-grey-that-isnt-there/ or its .md (grep -i random: 0 hits). - CONFIRMED | anti-noise that erases only the low, steady part of a sound | src/content/strata/how-noise-cancelling-works.md | that layer's dek: "It erases the low, steady, boring half"; its title is "The Half of the Noise It Can Erase", as the page's link text says. - CONFIRMED | the sampled-signal cousin of the conservation law in | src/content/strata/the-error-you-can-only-move.md | that layer's plain: "Bode's 1945 sensitivity integral, the waterbed effect: push the error down in one frequency band and it must rise in another." - CONFIRMED | Pushing the error out of hearing is real, and it is called noise shaping. | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 2: "Noise shaping allows one to increase the signal-to-noise ratio in the audio band at the expense of decreasing it at frequencies above the audio band." - CONFIRMED | A quantiser with many levels keeps that condition easily | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 2: "Because one has enough levels available, avoiding saturation is not a significant problem in practice." - MINOR | Their own summary invokes that line only at µ = Δ, full triangular dither, where it does hold (below), and the argument on p. 3 does not need it. | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 11: the Summary opens "A dithered 1st-order, 1-bit sigma-delta quantizer ... can operate without overload only if µ < ∆/4", which is line (d)'s general claim restated for every µ above Δ/4, before it names "[Case (2d) above with µ = ∆ for TPDF dither]". So the summary does not lean on (d) only at µ = Δ. The page's critique of (d) itself is right (recreated below, d-fo-zero). See Findings. - CONFIRMED | For shortening numbers inside the digital chain, the survey notes that its recommended triangular dither | https://hajim.rochester.edu/ece/sites/zduan/teaching/ece472/reading/Lipshitz_1992.pdf | read on the page image (PDF p. 19, journal p. 373, 220 dpi): "We recommend the use of nonsubtractive, triangular-pdf dither of 2-LSB peak-to-peak amplitude for most audio applications requiring multibit quantization or requantization operations". - CONFIRMED | The converter chips in the data sheets checked for this page are delta-sigma designs with more than two levels. | https://statics.cirrus.com/pubs/proDatasheet/CS4344-45-48_F2.pdf | CS4344 p. 1 "multibit delta-sigma modulator"; PCM4222 title "Multi-Bit Delta-Sigma" and p. 2 "6-bit modulator data"; PCM510xA p. 24 "Delta Sigma Modulator (DSM)" feeding "the current segments" of TI's "advanced segmentDAC architecture" (p. 1; block diagram "Current Segment DAC"). The third is an inference from "segments", fairly drawn. Record "where" cites PCM510xA "section 8.3"; see Findings. - CONFIRMED | Whether the leftovers of a 1-bit stream are audible is disputed. | https://en.wikipedia.org/w/index.php?title=Direct_Stream_Digital&action=raw | live wikitext: "Comparisons of DSD and PCM recordings ... have yielded contradictory results"; LV2001 p. 9: "Further research is needed here." - CONFIRMED | Stanley P. Lipshitz, Robert A. Wannamaker and John Vanderkooy, Quantization and Dither: A Theoretical Survey, J. Audio Eng. Soc. 40 (5), May 1992, pp. 355–375. | https://hajim.rochester.edu/ece/sites/zduan/teaching/ece472/reading/Lipshitz_1992.pdf | read on the page image (PDF p. 1, journal p. 355, 220 dpi) and footers: "J. Audio Eng. Soc., Vol. 40, No. 5, 1992 May"; printed page numbers run 355 (PDF p. 1) to 375 (PDF p. 21). Excerpt in the record is composed, not the source's words; see Findings. - CONFIRMED | Robert A. Wannamaker, The Theory of Dithered Quantization, PhD thesis, University of Waterloo, 1997. | https://uwspace.uwaterloo.ca/bitstreams/fa271912-b183-4c4c-a95d-bc2f761ba345/download | live PDF title page: "The Theory of Dithered Quantization by Robert Alexander Wannamaker A thesis presented to the University of Waterloo ... Doctor of Philosophy ... Waterloo, Ontario, Canada, 1997"; http://hdl.handle.net/10012/202 resolves to the UWSpace record (citation_publication_date 1997). The record's excerpt is the kept copy's "WHAT:" header; see Findings. - CONFIRMED | Cirrus Logic, CS4344/45/48 data sheet, DS613F2, 2013. | https://statics.cirrus.com/pubs/proDatasheet/CS4344-45-48_F2.pdf | live PDF p. 1: "10-Pin, 24-Bit, 192 kHz Stereo D/A Converter", footer "DS613F2", "Copyright Cirrus Logic, Inc. 2013". The record's excerpt is the kept copy's header; see Findings. - CONFIRMED | Texas Instruments, PCM4222 data sheet, SBAS399A, October 2006, revised March 2007. | https://www.ti.com/lit/ds/symlink/pcm4222.pdf | live PDF p. 1 header, verbatim. - CONFIRMED | Texas Instruments, PCM510xA data sheet, SLAS859C, May 2012, revised May 2015. | https://www.ti.com/lit/ds/symlink/pcm5102a.pdf | live PDF p. 1 header, verbatim. - CONFIRMED | Knowles, SPH0641LU4H-1 data sheet, revision B, 2015, via the distributor Akizuki Denshi. | https://akizukidenshi.com/goodsaffix/SPH0641LU4H-1.PDF | live PDF footers "Revision: B 04/06/2015"; the URL is on akizukidenshi.com (Akizuki Denshi, a distributor). - CONFIRMED | InvenSense, ICS-41350 product brief, PB-000021 revision 1.0, 2016, via a distributor's copy. | https://iw.ftcelectronics.com/datasheets-f4/ICS-41350.pdf | live PDF footer: "Document Number: PB‐000021 Revision: 1.0 Rev Date: 02/17/2016"; served from ftcelectronics.com (a distributor). - CONFIRMED | W3C, Web Audio API 1.1, Working Draft of 22 September 2026, on DelayNode cycles and the render quantum. | https://www.w3.org/TR/webaudio-1.1/ | live HTML header: "Web Audio API 1.1 W3C Working Draft, 22 September 2026". (The licence named after it in the Sources list is a separate added claim below.) - OBSERVED | At a height of 1.4 steps the rounded tone's third harmonic is only 15.7 dB below the tone. | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | Fourier series of the 0/±1 staircase: (4/3π)cos(3 asin(0.5/1.4)) over (4/π)cos(asin(0.5/1.4)) = −15.742 dB (pass's Python); the page readout at 1.40 also prints 15.7. Below 0.5 step every sample rounds to 0 (checked at 0.49). - OBSERVED | The error power runs from 0 when the input sits on a level to Δ²/4 halfway between levels; averaged over every position between two levels it is Δ²/6, which is 3.01 dB above Δ²/12. | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | brute integration over the density (pass's Python, 201 positions): power 0 on a level, 0.25 midway, mean 0.16666 = Δ²/6, 10 log10(2) = 3.0103 dB. - OBSERVED | so the price is 10·log10(3) = 4.77 dB more noise | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | power 0.25000 at every position (±3e-15); 10 log10(3) = 4.7712. - OBSERVED | With triangular noise it still waves with the input, by up to 0.048 Δ³ (with rectangular noise, up to 0.096 Δ³). | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | max ∣E[e³∣x]∣: triangular 0.04811, rectangular 0.09622 (= √3/18). - OBSERVED | Choose rectangular noise and set the width control to 0.80: the straight line bends again, by up to 0.10 step (with triangular noise built from two 0.80-step sources, by up to 0.020). | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | max ∣E[e∣x]∣ at width 0.80: rectangular 0.1000, triangular from two 0.80 sources 0.0200. - OBSERVED | For the published shaper below, the average of 20·log10∣1 − H∣ over the whole band, from 0 to 1.4112 MHz, computes to 0.00 dB. | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | midpoint rule, 200,000 points over 0 to π: +1.5e-5 dB, prints 0.00; 1.4112 MHz is the paper's Nyquist ("up to the Nyquist frequency of 1.4112 MHz", p. 4). - OBSERVED | At exactly zero input the loop holds out for any dither smaller than ±Δ/2, because its error can then only be −Δ/2, 0 or +Δ/2, which keeps W inside ±Δ. | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | derivation: at x = 0, εₙ = εₙ₋₁ + yₙ and ∣ν∣ < 1/2 force ε ∈ {−1/2, 0, 1/2} and W ∈ (−1, 1), so no overload can occur; Python run, rectangular width 0.6 (µ = 0.30), 200,000 samples: 0 overloads; width 0.999999, 1,000,000: 0. - OBSERVED | With full triangular dither even zero input fails: 93,470 of a million samples overload (seed 1), and given as many levels as it likes, the loop then uses 4. | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | Python re-implementation (own mulberry32), triangular µ = 1, zero input, seed 1, 1,000,000 samples: 93,470 overloads in 1 bit; unlimited levels: 4 used. - OBSERVED | Outside the region overloads become possible, not certain: the region holds for every dither sequence, and a random one is often kinder (input 0.10 with µ = 0.25, for one, runs 200,000 samples clean). | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | Python: input 0.10, rectangular width 0.5 (µ = 0.25), seed 1, 200,000 samples: 0 overloads; outside (A.9) since 0.10 > 0.5 − 0.5 = 0. - OBSERVED | Recomputed: after its first 13,904 samples the output repeats exactly every 328 samples | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | Python cascade (printed order, and sections reversed): exact period 328 from sample 13,904, zero overloads, both orders. - OBSERVED | and 2,822,400/328 = 8604.878 Hz | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | 2,822,400/328 = 8604.8780 Hz. - OBSERVED | that fundamental sits 151.3 dB below a DSD full-scale sine, whose peak the paper puts at Δ/4 | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | Python coherent average over whole periods from 13,904: harmonic 1 at −151.27 dB re a sine of peak 1/4. - OBSERVED | Its second harmonic, at 17,209.756 Hz, is 130.2 dB below; these are the only two lines of the cycle in the audio band. | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | harmonic 2 (17,209.756 Hz) −130.21 dB; harmonic 3 is at 25,814.6 Hz, above 20 kHz. - OBSERVED | The strongest line, at 1.0584 MHz, is 1.7 dB above full scale. | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | strongest harmonic k = 123, 123 × 8604.878 = 1,058,400 Hz, +1.66 dB. - OBSERVED | recomputed, 10·log10((Δ²/6)/(0.0013²Δ²/12)) = 60.73 dB | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | 10 log10((1/6)/(0.0013²/12)) = 60.7314. - OBSERVED | In 5 of 10 runs (seeds 1 to 10, a million samples each, the bench's length when it averages 1,600 periods or fewer) the loop fell into a different exact cycle, one the dither was too small ever to disturb: 168, 120, 56, 48 and 56 samples long (seeds 3, 4, 5, 7 and 10), which is 16,800, 23,520, 50,400, 58,800 and 50,400 Hz. | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | Python, rectangular 0.0013, 1 bit, seeds 1 to 10, 1,000,000 samples: periods 0 0 168 120 56 0 48 0 0 56; 2,822,400 divided by 168, 120, 56, 48, 56 = 16,800, 23,520, 50,400, 58,800, 50,400 Hz; benchLength(1600) = max(1,000,000, 30,000 + 328 × 1600) = 1,000,000. - OBSERVED | In each, the signal reaching the quantiser before the dither is added stays at least 0.0043 step from its decision point, more than six times the dither's reach of 0.00065 step, so no draw of the dither can change a single output bit again: a seed that locks stays locked, and the longer the run, the more seeds can lock. | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | Python, last 1,680 samples: min ∣X − HE∣ = 0.00434, 0.00506, 0.00449, 0.00720, 0.00650 for seeds 3, 4, 5, 7, 10; dither reach 0.00065; 0.00434/0.00065 = 6.7. - OBSERVED | Seed 3 locks after 24,868 samples and was still locked when we stopped, three million samples in; seeds 4 and 10 lock only after 600,000 samples. | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | Python: seed 3 period 168 from 24,868 (browser bench preset prints the same); seeds 4 and 10 from 604,976 and 624,912. The three-million-sample run is the verifier's. - OBSERVED | Within a million samples, with 0.005 step 5 of 20 seeds locked; with 0.01, 5 of 20; with 0.02, 0.05, 0.1 and 0.17, none of 20. | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | verifier --deep (all 20 seeds for every amount): 323 passed, 0 failed. - MINOR | Which cycle a seed falls into depends on the arithmetic as well as the dither: with the filter's sections taken in reverse order, seed 10 locks into a 48-sample cycle instead, from sample 756,736. | verify-the-noise-that-makes-it-accurate.mjs | the run behind it (fastRev, line 249) puts the gain K and the pole first, then sections 3, 2, 1; this pass's Python of that order gives seed 10 period 48 from 756,736, as stated. With only the three second-order sections reversed (pole and gain still last) seed 10 keeps the printed order's 56-sample cycle from 624,912. The record's own "where" says "with the gain and pole first"; the page does not. See Findings. - OBSERVED | In 40 runs of three million samples (seeds 1 to 40), six levels were enough in 39; seed 22 needed eight, once, at sample 215,551. | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | verifier --deep: 40 seeds x 3,000,000, six levels in 39, eight in seed 22, reaching ±3.5 once at 215,551; browser bench seed 1: "Levels used: 6". - OBSERVED | In seeds 1 to 8 the loop's internal signal passed a million steps between samples 453 and 650. | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | Python: run-away samples 536, 608, 453, 502, 517, 478, 613, 650 for seeds 1 to 8; browser preset prints "ran away at sample 536". - OBSERVED | At 0.25 of full amplitude no seed overloads; at 0.3, 0 to 3 overloads; at 0.35, 76 to 96; at 0.4 every run runs away. | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | Python, seeds 1 to 8, 1,000,000 samples: 0.25 all 0; 0.3 overloads 2 1 0 2 0 0 3 1; 0.35 89 92 76 81 94 96 95 92; 0.4 every seed runs away. - OBSERVED | A quarter of the amplitude is 12.04 dB less dither power | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | 20 log10(0.25) = −12.0412; browser preset prints "−12.04 dB". - OBSERVED | (0.17²/12 against 1/6 is 1 to 69.2) | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | (1/6)/(0.17²/12) = 69.204. - OBSERVED | (seeds 1 to 8, a million samples each), agree: at 0.17 step no seed overloads; at 0.2, one seed overloads, twice; at 0.25, four seeds overload and a fifth runs away. | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | Python loop fed the engine's SIN1024 table, rectangular dither, seeds 1 to 8, 1,000,000 samples: 0.17 all 0; 0.2 overloads only in seed 4, 2 of them; 0.25 overloads 0 0 1 12 4 1 0 in seeds 1 to 7 and seed 8 runs away at 781,610 (four seeds overload, a fifth runs away). - OBSERVED | The filter here is built as a chain of the printed factors, and all 144 orderings of those factors that we tried give the same 328-sample cycle from the same sample. | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | verifier --deep: all 144 orderings give period 328 from 13,904. - OBSERVED | The same H written as one seventh-order difference equation rounds differently, and in 300,000 samples it shows no exact cycle shorter than 20,000 samples. | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | verifier section B (negative control): no exact cycle shorter than 20,000 in 300,000 samples. - OBSERVED | at 4.000, 12.000 and 20.000 kHz, with radii 0.999, 0.998 and 0.995, and the real zero within 10⁻¹³ of z = 1 | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | pass's Newton in 80-digit decimal on the factored form: zeros at 4000.000000, 12000.000000, 20000.000000 Hz, radii 0.999000, 0.998000, 0.995000; real zero at 1 − 5.09e-14; exact rational arithmetic: NTF(1) = +5.8e-24, NTF(1 − 1e-13) = −5.6e-24 (sign change). numpy.roots on the expanded polynomial misplaces them by up to 2 Hz, which is the page's point about floating point. - OBSERVED | All seven poles lie inside the unit circle, the largest at radius 0.967. | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | numpy roots of the denominator: largest radius 0.96726 = sqrt(0.935593); all seven < 1. - OBSERVED | The Audio Engineering Society's library returned 403 (forbidden) to our requests for those papers | research/the-noise-that-makes-it-accurate/README.md | README, Fetch failures: "https://www.aes.org/e-lib/browse.cfm?elib=11395 returned 403 to curl on 2026-11-01", and aes2.org "redirected to aes.org and also returned 403"; the scout's raw AES pages (aes9150.html, aes9903.html, aeslib9902.html in its scratch directory) are Cloudflare "Just a moment..." challenge pages. The README names requests for the Philips counter-papers and Angus 2002, none for Lipshitz and Vanderkooy's response, so for that paper the observation is not kept. Not re-probed by this pass. - OBSERVED | The 1992 survey is a scanned PDF whose text layer is out of order | recreated: pdftotext of the live PDF (sha256 f15e9def..., identical to the kept copy's) | the text layer interleaves columns and figure labels, e.g. journal p. 367 comes out as "Eq. variance (82) clearly shows that nonsubtractive error is the greater thanwith that of classical or total sub-"; the page images read cleanly. - CONFIRMED | is easy to generate for digital requantization purposes by simply summing two independent rectangular-pdf pseudo-random processes | https://hajim.rochester.edu/ece/sites/zduan/teaching/ece472/reading/Lipshitz_1992.pdf | read on the page image (PDF p. 19, journal p. 373, 220 dpi): "This kind of dither is easy to generate for digital requantization purposes by simply summing two independent rectangular-pdf pseudo-random processes, which are easily generated by linear congruential algorithms [23], [26]." "This kind" is the triangular dither the previous sentence recommends. - CONFIRMED | Comparisons of DSD and PCM recordings with the same origin, number of channels and similar bandwidth and noise have yielded contradictory results. | https://en.wikipedia.org/w/index.php?title=Direct_Stream_Digital&action=raw | live wikitext, DSD vs. PCM: verbatim. - CONFIRMED | On Lipshitz and Vanderkooy's wider verdict, that one-bit converters are unsuitable for high-end applications, the article adds: | https://en.wikipedia.org/w/index.php?title=Direct_Stream_Digital&action=raw | live wikitext: "In 2000, Lipshitz and Vanderkooy stated that one-bit converters, as employed by DSD, are unsuitable for high-end applications due to their high distortion." immediately before the Philips sentence. - CONFIRMED | Wikipedia, Direct Stream Digital, the paragraphs on the DSD and PCM dispute, read 1 November 2026. CC BY-SA 4.0. | https://en.wikipedia.org/w/index.php?title=Direct_Stream_Digital&action=raw | live wikitext section "DSD vs. PCM"; Wikipedia text is CC BY-SA 4.0. - CONFIRMED | A PDM bitstream is encoded from an analog signal through the process of a 1-bit delta-sigma modulation. | https://en.wikipedia.org/wiki/Pulse-density_modulation | live rendered article (wgRevisionId 1360604569, as recorded): "A PDM bitstream is encoded from an analog signal through the process of a 1-bit delta-sigma modulation." - CONFIRMED | Wikipedia's article on the format says: | https://en.wikipedia.org/wiki/Pulse-density_modulation | live rendered article: the format article for PDM; also "PDM is also the output of some MEMS microphones." - CONFIRMED | Wikipedia, Pulse-density modulation, read 2 November 2026. CC BY-SA 4.0. | https://en.wikipedia.org/wiki/Pulse-density_modulation | live rendered article, revision 1360604569; CC BY-SA 4.0. - CONFIRMED | Shaping leaves the dither theorems' condition in place: the quantiser inside the loop must never clip. | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 2: "As long as the quantizer inside the noise shaper does not saturate, and is properly dithered, one is guaranteed that this process is completely transparent". - CONFIRMED | A tone stays put as you average more periods; noise unrelated to the cycle falls 3 dB for each doubling. | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 6: "Each doubling of the number of coherently-averaged data sets lowers the noise floor by 3 dB, provided that it is incoherent ... The synchronous input signals, on the other hand, are left unchanged by the averaging." - CONFIRMED | With no dither and the input near zero (press Idle tone) you hear an idle tone whose pitch follows the input. | https://sjeng.org/ftp/SACD.pdf | live PDF (sha256 f4cfcf04..., same as kept) p. 3: "These “birdies” are input dc-offset dependent, and are frequency modulated by the audio signal."; recreated: undithered first-order loop repeats every 100 samples at input 0.01 and every 25 at 0.02 (pitch follows input). - MINOR | At the starting input, 0.20, the pattern repeats every 10 samples, which at 44.1 or 48 kHz is 4.41 or 4.8 kHz, above the filter, so you hear almost nothing. | public/strata/the-noise-that-makes-it-accurate/engine.mjs | true for the undithered loop: E.runLoop at input 0.20, no dither, repeats every 10 samples from sample 0; 44,100/10 = 4,410 and 48,000/10 = 4,800 Hz; E.foListen after its first 0.1 s plays -82.9 and -83.6 dBFS rms (verifier line 477: below -65). But the instrument opens at rectangular dither of reach 0.10 (app.mjs line 202, FO = { x: 0.2, mu: 0.1, shape: 'rect' }); there the stream has no exact period and Listen plays -51.8 and -52.9 dBFS rms, about 6 dB below the idle tone at input 0.01 (-45.9 and -46.0). recreated: node fo-listen2.mjs (this pass). See Findings. - OBSERVED | Idle tone: input 0.01, no dither | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | app.mjs: fo-idle sets FO.mu = 0, FO.x = 0.01; verifier C asserts the same regex. - OBSERVED | In all ten of this page's runs, with that much dither on from the first sample, the loop never settled into the 328-sample cycle. | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | Python, seeds 1 to 10 from sample 0: no run ends on period 328 (periods above). - OBSERVED | Switched on at sample 20,000, once the cycle has settled, the same dither never changes a bit of it: the settled cycle keeps the signal reaching the quantiser at least 0.00087 step from its decision point, and this dither reaches only 0.00065 step either way. | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | Python: undithered min ∣X − HE∣ from sample 20,000 to 1,000,000 = 0.000879; dither 0.0013 switched on at 20,000, seeds 1 to 3: output identical bit for bit to the undithered run. - OBSERVED | The two tones are read from a table of sines built with plain arithmetic rather than the browser's sine function, so every browser feeds the loop the same bits and gets the same run. | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | verifier checks SIN1024 against Math.sin and the bench against its own loop past overload (default run passed). - OBSERVED | The default run checks the longest sweeps on a sample (the levels sweep on seeds 1 and 22, the lock counts that found none on seeds 1 to 5, four of the 144 orderings); add --deep to run them all. | recreated: python3 -I loops.py / recompute.py / zeros.py (this pass's own Python re-implementation, in its scratch directory) and AW_DIR= node verify-the-noise-that-makes-it-accurate.mjs [--deep] | verifier line 325: const SAMPLED = { levels: [1, 22], lockSeeds: 5, orderings: 4, allOrderings: 144 }; --deep runs all. - CONFIRMED | (added) Raise the input x and the power left for everything that is not the signal falls by exactly x². | https://sjeng.org/ftp/SACD.pdf | p. 8: "Since the output power is constant, the signal component of the output must come at the expense of the remainder." For a constant input the first-order loop gives y_n = x + e_n - e_(n-1), so the mean output tends to x and the rest has power 1/4 - x²; the readout at 0.20 prints "x² = 0.0400, leaving 0.2100" with measured mean 0.2000. - MINOR | (added) W3C, Web Audio API 1.1, Working Draft of 22 September 2026, on DelayNode cycles and the render quantum. W3C Document License. | https://www.w3.org/TR/webaudio-1.1/ | the draft's copyright line reads "W3C liability, trademark and permissive document license rules apply", the last link going to https://www.w3.org/copyright/software-license/ with the title "W3C Software and Document License". The W3C Document License is a different, narrower licence. See Findings. - UNVERIFIABLE | (added) Copyright Audio Engineering Society; quoted briefly, typed from the page images. | tried: the OCR text and the page images of all 21 pages of the live scan (no copyright notice appears); the AES e-library is behind a challenge page | the journal is the AES's (every footer reads "J. Audio Eng. Soc., Vol. 40, No. 5, 1992 May"), so the notice is plausible, but nothing fetched states it; the kept transcription's "LICENCE: (c) Audio Engineering Society" is the builder's note. See Findings. - OBSERVED | (added) Sound: a 997 Hz tone made at your device's own sample rate, rounded as set above, two seconds long. Playback gain: one step is played at 1/16 of full scale; 400 random draws of noise at each input; Each setting runs the loop for 200,000 samples (seed 1); a low-pass filter at 2.5 kHz; one step is played at 1/4 of full scale | recreated: grep -n in public/strata/the-noise-that-makes-it-accurate/app.mjs | TONE_HZ = 997 with n = Math.round(2 * fs) (line 169), TONE_GAIN = 1 / 16, DOTS_PER_INPUT = 400, FO_SAMPLES = 200000 run with seed: 1, FO_LOWPASS_HZ = 2500, FO_GAIN = 1 / 4; the browser readouts (Playwright on local copies) print "no overload in 200,000 samples". - CONFIRMED | (added) Course-reading copy: hajim.rochester.edu; Why 1-Bit Sigma-Delta Conversion is Unsuitable for High-Quality Applications ... All rights reserved; quoted briefly. sjeng.org, archived at web.archive.org | https://sjeng.org/ftp/SACD.pdf | p. 1: "All rights reserved. Reproduction of this paper, or any portion thereof, is not permitted without direct permission from the Journal of the Audio Engineering Society."; the archive link is the one Wikipedia's DSD article cites (archive-url=https://web.archive.org/web/20230430093353/https://sjeng.org/ftp/SACD.pdf); the survey's URL path is /ece/sites/zduan/teaching/ece472/reading/. ## Findings - [fixed] MINOR: what Listen plays at the first-order instrument's opening setting (index.html:208, "At the starting input, 0.20, the pattern repeats every 10 samples ..."). Re-run by the fixer with the engine: at the opening setting (input 0.20, rectangular reach 0.10, seed 1) Listen plays -51.8 dBFS rms at 44.1 kHz and -52.9 at 48 kHz after its first 0.1 s, against -82.9 and -83.6 with no dither and -45.9 and -46.0 for the idle tone at 0.01, so the sentence held only with no dither. The page now reads "At input 0.20 with no dither, the pattern repeats every 10 samples, which at 44.1 or 48 kHz is 4.41 or 4.8 kHz, above the filter, so you hear almost nothing." The verifier's sentence check follows it (its run is already the no-dither loop: period 10, Listen below -65 dBFS); source record d-idle carries the new sentence and the levels, including the opening setting's. - [fixed] MINOR: the realisation behind seed 10's 48-sample cycle (index.html:253). Re-run by the fixer (0.0013 step, seed 10, 1,000,000 samples): printed order, 56 from 624,912; only the three sections reversed with pole and gain last, 56 from 624,912; gain and pole first then sections 3, 2, 1 (the verifier's fastRev), 48 from 756,736 (gain first alone, or pole first alone, also gives 48 from 756,736). The page now reads "with the filter's whole chain reversed (the gain and the pole first, then the three sections from last to first), seed 10 locks into a 48-sample cycle instead, from sample 756,736." The verifier's sentence check and its section B label follow; d-lockalt and the README (which said "sections reversed" twice) say the same; the README's "4 with the whole chain reversed" at 0.005 step was re-run by the fixer: seeds 3, 10, 16, 19. - [fixed] MINOR: the paper's summary leans on line (d) in general, not only at µ = Δ (index.html:211). Re-read on the live PDF (https://sjeng.org/ftp/SACD.pdf, sha256 f4cfcf04..., the kept copy's) p. 11: "Summarizing: A dithered 1st-order, 1-bit sigma-delta quantizer (or the equivalent 1st-order noise shaper) can operate without overload only if µ < ∆/4, and then only if its input xk is restricted by (A.9)." and later "When properly dithered [Case (2d) above with µ = ∆ for TPDF dither], it is impossible to prevent it from overloading." The page now reads "Their summary restates it in general, that such a loop “can operate without overload only if µ < ∆/4”, but names case (d) only at µ = Δ, full triangular dither, where it does hold (below), and the argument on p. 3 does not need it." The quotation is in quotes.json (lv-summary, p. 11, rebuilt by build-quotes.mjs from the kept copy) and the source record (q-lv-summary); f-d-alone carries the new sentence. - [fixed] MINOR: the W3C licence was misnamed (index.html:293). Re-fetched https://www.w3.org/TR/webaudio-1.1/ on 2026-11-02: the copyright line reads "W3C® liability, trademark and permissive document license rules apply.", its licence link goes to https://www.w3.org/copyright/software-license/ with title="W3C Software and Document License". The Sources entry now says "W3C Software and Document License."; the kept copy's LICENCE line and make-excerpts.py say the same, the kept copy now holds the copyright line and the link's href and title, the README's table follows, and the source record has c-s-w3c-licence. - [fixed] MINOR (answers the UNVERIFIABLE line): the 1992 survey's copyright line was unsourced (index.html:283). The fixer re-read the live scan (sha256 f15e9def..., same as kept): the page images of PDF pp. 1 and 21 and the OCR of all 21 pages carry no copyright notice; the footers read "J. Audio Eng. Soc., Vol. 40, No. 5, 1992 May". The unsourced words are removed: the entry now reads "... J. Audio Eng. Soc. 40 (5), May 1992, pp. 355–375. Quoted briefly, typed from the page images." The kept transcription's LICENCE line no longer asserts a notice (it says none is printed and names the journal from the footer); the README follows. - [fixed] MINOR (source record): the PCM4222 description is on p. 2. Re-read on the live PDF (sha256 aa9e1258..., same as kept): p. 1 has only the bullets "Supports Linear PCM, 1-Bit Direct Stream" and "– 6-Bit Modulator Data"; p. 2, under DESCRIPTION: "PCM, 1-bit Direct Stream Digital (DSD), and 6-bit modulator data outputs." q-ti4222-out and q-ti4222-dsd now say "p. 2, Description", as do quotes.json (via build-quotes.mjs), the kept copy's label and make-excerpts.py. - [fixed] MINOR (source record): the PCM510xA passages are on p. 24, not in section 8.3. Re-read on the live PDF (sha256 a5220836..., same as kept), p. 24: "modulated all the way to 128 × fS before arriving at the current segments" is under 9.3.5.1 Clocking Overview, as reported; "The sampling frequency detector sets the clock for the digital filter, Delta Sigma Modulator (DSM) and the" is under 9.3.5.2 Clock Slave Mode With Master/System Clock (SCK) Input (the finding put it under 9.3.5.1; the layout text puts the 9.3.5.2 heading above it). q-ti5102 now says "p. 24, section 9.3.5.1 (Clocking Overview)", f-multilevel names p. 24, section 9.3.5.1; quotes.json, the kept copy's two labels (9.3.5.1 and 9.3.5.2) and make-excerpts.py follow. - [fixed] MINOR (source record): four citation excerpts were composed citations or kept-copy headers. Each now quotes the source, typed from its page image and added to its kept copy: f-survey, the survey's title and author line ("Quantization and Dither: A Theoretical Survey* STANLEY P. LIPSHITZ, AES Fellow, ROBERT A. WANNAMAKER, AND JOHN VANDERKOOY, AES Fellow**", PDF p. 1); c-s-l92, the first page's footer ("J. Audio Eng. Soc., Vol. 40, No. 5, 1992 May 355"; the last page's, "... 375", is kept too); c-s-w97, the thesis title page ("The Theory of Dithered Quantization by Robert Alexander Wannamaker A thesis presented to the University of Waterloo ... Doctor of Philosophy in Applied Mathematics Waterloo, Ontario, Canada, 1997"; the OCR layer reads "m" for that "in"); c-s-cs, the Cirrus p. 1 footer ("Copyright Cirrus Logic, Inc. 2013 JUL ‘13 (All Rights Reserved) DS613F2"). The verifier's kept-copy bindings for these citations now point at the same words rather than at the headers. - [fixed] the claims pass's added lines with no source-record entry now have one: c-s-w3c-licence, c-s-lv01-rights ("All rights reserved; quoted briefly.", the AES notice on p. 1), c-s-l92-copy (the course-reading URL), f-xsquared ("Raise the input x ... falls by exactly x²"; the verifier now also runs the first-order loop at the opening setting: power 0.25, mean output 0.2000, rest 0.2100), and the app.mjs settings the page names (d-tone, d-tone-gain, d-dots, d-fo-run, d-fo-listen). node scripts/check-sources.mjs: 155 claims, 0 with no source found.