[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

- **Do not** include any other keys or prose outside these two tagged blocks.
- **Do not** report stylistic or citation issues.
- Be as precise as possible about **where** (section ID or figure/table) and **what** the scientific flaw is.
- Each description for the errors must be rooted in a scientific rationale explaining why it is 'wrong' (not how it could be improved).

Begin your analysis now.

[User]
Multi-antenna towards inband shift keying is a new continuous phase frequency shift keying that is particularly suited for multi-antenna communications when the link budget is critical. It combines the constant envelope of frequency modulation with low-rate repetition coding in order to make transmit beamforming dispensable.

Although it is a frequency modulation, its transmit signal shows close to rectangular spectral shape. Similar to GSM's Gaussian minimum shift keying, it can be well approximated by linear modulation, when combined with differential precoding. This allows for easy coherent demodulation by means of a windowed fast Fourier transform.

Communication in the millimeter and terahertz bands are characterized by critical link budgets. The standard solution to this problem is to utilize transmit beamforming which concentrates the signal energy around the receiving unit. However, this comes with lots of unfavorable issues: 1) Beam alignment delays the set-up of up radio links [1]. 2) Directional transmission jeopardizes many forms of network coding. 3) The concentration of signal power raises concerns about electromagnetic compatibility [2]. 4) The narrow angular spread reduces multipath propagation [3] and, thus, diversity. This list is not exhaustive. It only serves the purpose to motivate the search for an alternative to transmit beamforming.

Transmit beamforming is a spatial repetition code. On each antenna, the same data is transmitted. If there are N transmit antennas, the code rate is 1/N. Coherent superposition in the vicinity of the receiver improves the signal-to-noise ratio (SNR) by a factor of N. In time domain, data symbols are conveniently separated by orthogonality via pulse shaping.

Information theory does not care, if we exchange space and time on an additive white Gaussian noise (AWGN) channel. It just means to give the variables in the mathematical model of the channel another physical meaning. However, the implementation may be more beneficial, if we apply the repetition code in time rather than in space.

Let all data be repeated T times. The rate of this temporal repetition code is 1/T. Coherent superposition within the receiver improves the SNR by a factor of T. In spatial domain, data symbols can be conveniently separated by orthogonality via frequency multiplex, if each subcarrier is transmitted over a different antenna.

A multi-carrier multi-antenna system with temporal repetition coding that maps every subcarrier one to one onto the antenna elements is straightforward to implement with standard linear modulations. However, this leaves much room for improvement. Linear modulation with pulse shaping also comes with one quite unfavorable issue: The amplitude varies in continuous time. The signal crest requires the amplifiers to be backed-off to avoid non-linear signal distortion and out-of-band radiation [4]. A significant amount of energy is converted into heat rather than being radiated. Amplifiers which can cope with amplitude variations are also more expensive than their counterparts that can only cope with phase changes.

Nevertheless, linear modulation has displaced their nonlinear counterparts like continuous phase modulation and frequency shift keying over the evolution of wireless standard from 1G to 5G. This was, as frequency shift keying comes with its own severe issues: 1) The spectrum is poorly confined. 2) The amplitude domain is not utilized for data transmission. Both effects result in poor spectral efficiency.

In a multi-carrier system with low-rate repetition coding, the negative issues of frequency shift keying, if properly designed, vanish, while the crest problems of linear modulation remain. This makes continuous-phase frequency shift keying (CPFSK) a promising candidate for communication in millimeter and terahertz bands.

The purpose of this paper is to propose one example of such a proper multi-carrier design of CPFSK, which is termed multi-antenna towards inband shift keying (MA-TISK), in the sequel.

For low-rate repetition coding, it is not an issue to leave the amplitude dimension of the signal unused and utilize the phase only. After all, a repetition code only utilizes one out of many dimensions in signal-space. On the AWGN channel, restricting to phase modulation has a very similar effect as a repetition code of rate one half, if the SNR is high. For low SNR, i.e. for critical link budget, the effect vanishes.

In order to exonerate CPFSK, at least for critical link budgets, it is sufficient to overcome its poor spectral confinement. A method to achieve that is presented in the following sections. Section II develops the main idea at the example of the frequency correction burst of the Global System for Mobile Communications (GSM). Section III generalizes the idea to arbitrary phase modulation and frequency pulses. Section IV provides numerical results, before Section V summarizes the conclusions and promises.

## II. REPETITION CODED GMSK

Data rate \(R\) is fundamentally related to SNR. For the complex-valued AWGN channel, the relation is well-investigated and given by
\[
R = \log_2(1 + \mathrm{SNR})
\]
If the link budget is critical, SNR is low, that means SNR ≪ 1. In that case, the logarithm in (1) is well approximated by a linear function. We have
\[
R \approx \frac{\mathrm{SNR}}{\ln 2}
\]
In that regime, you can hardly do anything better than concatenating standard methods of forward-error correction coding designed for moderate rates with repetition coding in order to get an overall very low rate.

Let us consider repetition coding modulated with Gaussian minimum shift keying (GMSK) as used in GSM. Consider a repetition code of rate \(1/T\) with two codewords, the all-one and the all-zero word. If we feed the all-zero codeword into the frequency modulator, the phase is repeatedly shifted upwards. This is equivalent to an increase of frequency. This effect is well-known and even utilized in GSM. Fig. 1 shows the short-term spectrum of GSM during the frequency correction burst which contains 148 consecutive 0-bits [5]. This temporary sine tone at the shifted frequency is used in GSM for initial frequency synchronization and coarse time slot synchronization.

For the all-one codeword, the short-term spectrum looks very similar. The frequency shift is not towards lower frequencies, but also upwards. This is as GSM uses differential precoding. Thus, a sequence of 148 1-bits is first converted into a sequence of at least 147 0-symbols before it is fed into the modulator. For a duration of at least 142 symbols¹, the instantaneous frequency is identical. However, the waveform is phase-shifted by \(\pi\).

The repetition code composed of the all-zero and the all-one codeword shapes the spectrum of GMSK. The modified spectrum has a raised carrier frequency with reduced sidelobes. During phase changes the sidelobes are predominantly left of the raised carrier frequency. The spectrum is asymmetric.

An asymmetric spectrum is well-suited for a multi-carrier system. If the rightmost subcarrier had a spectrum as shown in Fig. 1, the strong sidelobes to the left would interfere with lower subcarriers. Only the weak sidelobes to the right would contribute to out-of-band radiation. While this is good news for the top subcarrier, it is very bad at the lowest subcarrier frequency. There, the strong sidelobes would be out-of-band.

The spectrum in Fig. 1 can be flipped. For that purpose, we need to change the repetition code from \(\mathcal{R}_u = \{0000 \ldots, 1111 \ldots \}\) to \(\mathcal{R}_l = \{1010 \ldots, 0101 \ldots \}\). After differential precoding, this leads to an all-one sequence of at least 147 symbols. As MSK maps a logical 0 to a phase shift of \(\phi = \frac{\pi}{2}\) and a logical 1 to a phase shift of \(\phi = -\frac{\pi}{2}\), it results in a lowering of the instantaneous frequency.

The roadmap towards reduced out-of-band radiation should be clear now: We use the code \(\mathcal{R}_u\) for all subcarriers in the upper sideband (right of the center of the band) and the code \(\mathcal{R}_l\) for all subcarriers in the lower sideband (left of the center of the band). Though one minor issue has remained: The frequencies of the

¹GMSK modulation contains memory of up to 5 symbols.

Instead of changing the code, we could alternatively change the mapping. Let us use the code \( R_u \) for all subcarriers. However, for the subcarriers in the upper sideband, we use the mapping
\[
\phi(s) =  \begin{cases}
\frac{\pi}{2} & \text{for } s = 0 \\
-\frac{\pi}{2} & \text{for } s = 1
\end{cases} \tag{3}
\]
For subcarriers in the lower sideband, we use the mapping \( -\phi(s) \). Clearly, this gives the same result, as the two codes just differ in the sign of the symbols fed into the modulator.

We can correct the frequency shift by an offset to the phase shifts, as instantaneous frequency is the derivative of the phase. This means, we use the mappings
\[
\phi_u(s) = \phi(s) - \frac{\pi}{2} = \begin{cases}
0 & \text{for } s = 0 \\
-\pi & \text{for } s = 1
\end{cases}
\]
\[
\phi_l(s) = \frac{\pi}{2} - \phi(s) = \begin{cases}
0 & \text{for } s = 0 \\
\pi & \text{for } s = 1
\end{cases} \tag{4}
\]
for the upper and the lower sideband, respectively. Here, it is important to note that a phase shift by \(\pi\) and by \(-\pi\) are not identical in CPFSK. A positive phase shift means that the instantaneous frequency is temporarily increased, while a negative phase shift means that it is temporarily reduced. After the decay of the frequency pulse, the result does not differ, of course. If the code rate \( \frac{1}{T} \) is small, the receiver will hardly notice the difference, either, but the radio spectrum is strongly affected.

GSM is standardized with minimum shift keying as modulation format. This was based on the thought that minimum shift keying is the most efficient frequency modulation, which is correct [6]. However, GMSK is combined with differential precoding. Thus, it is a differential frequency modulation. In that case, minimum shift keying is not proven optimal. For the time-bandwidth product of BT = 0.3 in GSM, it can be well-approximated by linear modulation [7]. In this case, it is certainly not optimal.

Consider a quaternary minimum shift keying [8] with Gaussian pulse filtering, i.e. a generalization of GMSK where everything stays as it is, except for having quaternary phase-shift keying (QPSK) symbols instead of its binary counterpart. Instead of the phase shifts ±π/2, we have phase shifts from the set {±π/4, ±3π/4} and apply Gray mapping. Both quadrature components can be demodulated independently from each other [8]. For the QPSK example above, the maximal frequency deviation is 3π/4 instead of π/2. This results in an at most 37% wider (slightly above the shoulder of the first sidelobe) spectrum as shown in Fig. 2. However, it doubles data rate. This is a boost in spectral efficiency of at least 46% that was not realized. An alternative way to utilize the inherent redundancy of binary GMSK is reported in [9].

In our multi-antenna concept, we are only concerned with the spectra for repetition codes. As Fig. 2 illustrates, the spectrum of quaternary GMSK is hardly worse than the spectrum of its binary counterpart. Note that only the spectrum of the upper sideband relative to the main lobe influences the out-of-band radiation. The spectrum of the lower sideband only contributes to inter-carrier interference.

In contrast to binary GMSK discussed in Section II, a modification of the phase mapping is not just a frequency shift and/or mirroring, when a repetition code is used. For quaternary phase constellations, it makes a difference, whether the repeated phase shift, i.e. the one that the symbol s = 0 is mapped to, is at the edge of the constellation, i.e. has minimum or maximum phase shift, or it is an inner phase shift between these

extremes. As noted before, a phase shift by a multiple of $2\pi$ is not irrelevant for the spectrum of CPFSK. The four different spectra are shown in Fig. 3. There are two different shapes with their respective mirrored spectra. The repeated phase shifts determine the instantaneous frequency. This can be observed in Fig. 4.

For a multi-carrier system, spectra as in Fig. 3 are quite useful. Each could represent one subcarrier. The overall spectrum of all tones is still better confined as its counterpart without repetition coding in Fig. 2. However, the carrier spacing in Fig. 3 is quite wasteful. Many more tones could be packed in-between these four ones. This issue is addressed in the next section.

Differential precoding converts any constellation of phases into a constellation of phase shifts. Any regular PSK constellation with $M$ points is mapped onto the set of phase shifts

\[
\mathcal{P} = \left\{0, \frac{2\pi}{M}, \ldots, \frac{2\pi(M - 1)}{M}\right\}.
\]

While a symbol is repeated, no phase shift occurs and the instantaneous frequency keeps constant.

## A. Antenna/Subcarrier-Dependent Constellation

We can optimize the spectral shape without influencing the performance, e.g., error and information rate, of a digital communication system. In discrete time, a phase shift by $\phi$ and by $\phi + 2\pi$ cannot be distinguished. In continuous time, however, the output signals of a CPFSK modulator differ. For the latter case, the instantaneous frequency is larger than for the former. This influences the spectral shape.

Consider the general set of PSK phase shifts
\[
\mathcal{P}(\omega) = \left\{0, \frac{2\pi}{M} - 2\pi\omega_1, \ldots, \frac{2\pi(M - 1)}{M} - 2\pi\omega_{M-1}\right\},
\]
for some binary vector $\omega = [\omega_1, \ldots, \omega_{M-1}] \in \{0, 1\}^{M-1}$. For each subcarrier, we can find the optimal vector $\omega$, such that the spectrum of the aggregated signal of all subcarriers is most confined.

For the leftmost subcarrier the optimal choice is the all-zero vector, if the number of subcarriers is large. For the rightmost subcarrier, it is the all one-vector. This is, as we prefer the sidelobes to appear towards the center of the channel band.

For other subcarriers, the optimization can be tedious and requires a clear way to measure spectral confinement. This task is left open for future work. In this work, we opt for a heuristic, but reasonable approach: We chose $\omega$ for each subcarrier in such a way that the maximal deviation of the instantaneous frequency from the center of band is as small as possible. We call this method, as well as its generalizations with more sophisticated optimization of the vector $\omega$, multi-antenna towards inband shift keying.

## B. Pulse Waveforms

In GSM, a Gaussian filtered rectangular frequency pulse shape is used. It is well optimized with respect to the trade-off between spectral confinement and limiting intersymbol interference caused by lack of temporal orthogonality. In MA-TISK, the particular choice of the frequency pulse shape is much less critical.

The repetition code ensures that the shape of the pulse hardly causes intersymbol interference. Due to the repetition code, the transmit signal is similar to a direct-sequence spread-spectrum signal with a row of the discrete Fourier transform matrix as spreading sequence. The frequency pulse shape corresponds to the chip pulse in the spread-spectrum signal. Even chip pulses that are not shift-orthogonal hardly create intersymbol interference. They may interfere with few subsequent chips. But they hardly span long enough in time to cause relevant intersymbol interference. The same considerations apply to the frequency pulses in MA-TISK.

The design of frequency pulses can overwhelmingly focus on spectral confinement. As argued above, intersymbol interference is not a critical issue due to the repetition coding. For GMSK pulses, we can thus go for lower time-bandwidth products than in GSM. It also makes sense to consider totally different pulses like the squared sine wave pulse used in [10]. Optimization of the frequency pulse shape is left open for future work.

## C. Subcarrier Separation

Ideally, the subcarriers are spaced in such a way that they can be easily demodulated via fast Fourier transform (FFT) [11] while keeping the inter-carrier interference at a minimum.

For $N$ subcarriers, the rate $R$ of the repetition code should be slightly smaller than $1/N$. If we repeat for $T > N$ times, the frequency pulse has time to decay such that the instantaneous frequency is constant for $N$ symbol periods. In that case, we can sample the received signal, window it for $N$ symbol periods and perfectly

separate the N subcarriers via FFT. In practice, some crosstalk may remain. The frequency pulse may not have fully decayed or channel dispersion may jeopardize the orthogonality. These effects can be counteracted by slightly reducing the rate R, as demonstrated in Section IV.

The system may even work with severe crosstalk. In [12], [13] a system was proposed that uses non-orthogonal spreading sequences with constant envelope instead of subcarriers. With the help of multiuser detection [14], [15], it is successfully utilized in [16] for satellite communications with a tight spectral mask that asks for a stopband attenuation of −70 dB.

We show an example to demonstrate the quantitative potential of MA-TISK. We compare the resulting spectrum to various single-carrier methods that are state-of-the-art.

We target the spectral mask of 5G downlink at 100 MHz channel bandwidth. We choose a transmit array with N = 64 antennas/subcarriers and QPSK modulation at a symbol rate of 1.25 Msymbols/s on each subcarrier resulting in a total data rate of 160 Mbit/s. The corresponding symbol period is 800 ns. We take a Gaussian filtered rectangular pulse like in the GSM standard [7] but with time-bandwidth product of 0.1 and scale it in time to fit the new data rate. This leads to a pulse duration of 91.4 ns.

In order to have the GMSK pulse largely decayed before the FFT window starts, we set T = 70. This results in a subcarrier spacing of 1.37 MHz. This is, by far, wide enough to be insensitive to Doppler-induced frequency shifts.

The resulting instantaneous frequencies are shown in Fig. 5 for 1.25 symbol periods and all 64 subcarriers. The instantaneous frequencies change over time, but they always keep between the lowest and highest subcarrier frequency. Changes are predominantly directed towards the center of the frequency band.

For most of the symbol period, MA-TISK is linear phase modulation. The instantaneous frequency is constant. The phase of each subcarrier corresponds to the PSK symbol to be transmitted. Only during phase transitions, when the instantaneous frequency changes, the modulated signal relates to the PSK symbol in a nonlinear way. For many antennas, this period is much shorter than the symbol period and can be dismissed in a similar way as the cyclic prefix in orthogonal frequency division multiplexing.

The more antennas are utilized, the larger the symbol period, as the overall bandwidth is divided into more subbands. This makes MA-TISK the more robust against channel dispersion the more antennas are used.

The power spectral density (PSD) of the transmit signal is shown in Fig. 6. While the aggregate signal is wideband, the transmit signals at any individual antenna element is very narrowband. This simplifies the design of the RF-circuitry. In fact, narrowband RF-circuitry would even help to suppress sidelobes and reduce inter-carrier interference. Furthermore, the asymmetry of the PSD at the leftmost and rightmost subcarriers is evident. Even the spectrum of the centered carrier is not symmetric, as it performs phase shifts by –π, but not by +π. The overall spectrum shows significantly less sidelobes than the spectra of the individual subcarriers. This is a direct result of the in-band direction of the frequency deviations.

When compared to single-carrier GMSK, the spectrum of MA-TISK is far better confined and competitive with linear single-carrier PSK. The latter has slight advantages when it comes to high stopband attenuation, but its spectrum deviates more from the ideal rectangular shape in the utilized spectral range.

After demodulation by means of sampling and FFT, the signal-to-interference ratio (SIR) is found as 34.4 dB. No Gaussian noise was added. Interference is solely due to crosstalk of subcarriers and intersymbol interference. The SIR heavily depends on the value of T, cf. Table I, and the duration of the frequency pulse. For T = 64 = N,

the frequency pulse is fully included in the FFT. This results in significant inter-carrier interference. Though, in combination with strong forward error-correction coding, an SIR of 8 dB is more than sufficient for reliable communication with quaternary modulation. For increasing values of \(T\), the window of the FFT begins to span only that range of the symbol periods where the instantaneous frequency is constant. Thus, the subcarriers preserve orthogonality.

The temporal combining gain is 18.1 dB. Note that the temporal combining gain can slightly exceed 18 dB as \(T = 70 > N = 64\). Setting \(T > N\) is spectrally wasteful. However, it reduces inter-carrier interference, cf. Table I. Note that linear single-carrier modulation is also spectrally wasteful due to the roll-off factor.

Transmit beamforming is a spatial repetition code and can be replaced by a temporal repetition code without reducing the data rate or the link budget. This is conveniently implemented via a multi-carrier system where each subcarrier is mapped onto one transmit antenna.

A multi-carrier system with repetition coding can be implemented via differentially precoded continuous phase frequency shift keying while avoiding the spectral issues that come with frequency modulation in single-carrier systems. MA-TISK, a promising method of such implementation, is proposed.

MA-TISK utilizes particular sets of phase shifts that depend on the position of the subcarrier within the channel band. This tilts the spectrum of each subcarrier towards the center of the band and reduces sidelobes towards adjacent channels.

Although MA-TISK is a non-linear modulation, it can be demodulated like a linear one thanks to its combination with repetition coding and differential precoding. Intersymbol interference due to the memory of the continuous phase modulation is negligible. Cost-efficient demodulation by means of fast Fourier transform is possible.

The transition phase may introduce some minor overhead. However, the overheads of competing systems, e.g., roll-off factors or blank subcarriers, are avoided.

Thanks to its constant envelope, MA-TISK can be efficiently implemented by voltage controlled oscillators. Power amplifiers need neither be linear nor broadband and can operate without back-off at high efficiency.

MA-TISK does not require beam alignment. Radio links can be set up fast without serious overhead. This makes MA-TISK suitable for the implementation of internet of things (IoT)-applications in the millimeter-wave and terahertz bands.

MA-TISK is the more sensitive to multipath propagation the fewer antennas are deployed. This makes MA-TISK the more competitive the higher the carrier frequency.

The author would like to thank Martin Vossiek, Christian Carlowitz, and Leonhard Hahn for helpful discussions on millimeter wave circuitry. Further thanks go to Georg Fischer, Wolfgang Gerstacker, Bernhard Niemann, and Hans Rosenberger for fruitful discussions on RF amplifiers, frequency domain equalization, 6G standardization, and proof reading the manuscript, respectively.

[3] Gregory D. Durgin and Theodore S. Rappaport. Theory of multipath shape factors for small-scale fading wireless channels. IEEE Transactions on Antennas and Propagation, 48(5):682–693, May 2000.

[4] Harald Enzinger, Karl Freiberger, and Christian Vogel. Competitive linearity for envelope tracking: Dual-band crest factor reduction and 2D-vector switched digital predistortion. IEEE Microwave Magazine, 54(11):40–47, November 2018.

[5] Georg Frank and Wolfgang Koch. Initial synchronization of a mobile station in the D-net. PKI Technical Journal, (1):43–49, 1990.

[6] John G. Proakis. Digital Communications. McGraw-Hill, New York, 3rd edition, 1995.

[7] Michel Mouly and Marie-Bernadette Pautet. The GSM system for mobile communications, 1992.

[8] Nimal Ekanayake. Quaternary minimum shift keying. In Proc. of IEEE Global Telecommunications Conference (Globecom), pages 909–914, Miami, FL, November/December 1982.

[9] Raimund Meyer, Wolfgang H. Gerstacker, Robert Schober, and Johannes B. Huber. A single antenna interference cancellation algorithm for increased GSM capacity. IEEE Transactions on Wireless Communications, 5(7):1616–1621, July 2006.

[10] Ralf Müller. Transmitter and method for transmitting a data signal over multiple antennas, 2024. European Patent Application No. 24157872.3.

[11] James W. Cooley and John W. Tukey. An algorithm for the machine calculation of complex fourier series. Math. Comput., 19:297–301, 1965.

[12] Ralf R. Müller and Alexander Lampe. Spectral efficiency of random CDMA with constant envelope modulation. International Journal of Electronics and Communications, 65:701–706, 2011.

[13] Ralf R. Müller. On random CDMA with constant envelope. In Proc. of IEEE International Symposium on Information Theory (ISIT), St. Petersburg, Russia, August 2011.

[14] Ralf R. Müller. Multiuser detection for continuous phase CDMA. In Proc. of IEEE Topical Conference on Antennas and Propagation for Wireless Communications, Turin, Italy, September 2011.

[15] G. Gallinaro, N. Alagha, R. Müller, and S. Titomanlio. Quasi constant envelope CDMA for VHF maritime communications via satellite. In Ka and Broadband Communications Conference, Trieste, Italy, October 2017.

[16] ITU-R M.2092-1. Technical characteristics for a VHF data exchange system in the VHF maritime mobile band. Technical report, International Telecommunication Union, February 2022.


[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

- **Do not** include any other keys or prose outside these two tagged blocks.
- **Do not** report stylistic or citation issues.
- Be as precise as possible about **where** (section ID or figure/table) and **what** the scientific flaw is.
- Each description for the errors must be rooted in a scientific rationale explaining why it is 'wrong' (not how it could be improved).

Begin your analysis now.

[User]
ABSTRACT. We describe complex conjugation on the primitive middle-dimensional algebraic de Rham cohomology of a smooth projective hypersurface defined over a number field that admits a real embedding. We use Griffiths’ description of the cohomology in terms of a Jacobian ring. The resulting description is algebraic up to transcendental factors explicitly given by certain periods.

JEEHOON PARK, JUNYEONG PARK, AND PHILSANG YOO

Let V be a smooth projective variety over a number field k. For each embedding σ : k ↪ C, we consider the associated complex manifold V_σ := (V ×_{k,σ} C)(C) and its Betti (singular) cohomology

H^i_B(V_σ, Q) = H^i((V ×_{k,σ} C)(C), Q).

After tensoring with C, we have the Hodge decomposition

P : H^i_B(V_σ, Q) ⊗_Q C ≅ ⨁_{p+q=i} H^{p,q}

where H^{p,q} stands for the Dolbeault cohomology group of V_σ. Here and elsewhere, we always identify Betti cohomology and analytic de Rham cohomology after tensoring with C via the de Rham theorem.

Now we note that if σ is a real embedding, then the complex conjugation c ∈ Gal(C/R) acts on V_σ which induces an involution ι_σ on H^i_B(V_σ, Q). On the other hand, clearly 1 ⊗ c is also an involution on H^i_B(V_σ, Q) ⊗_Q C. It is well-known that both involutions ι_σ ⊗ 1 and 1 ⊗ c switch the Hodge components H^{p,q} and H^{q,p} under P. In particular, 1 ⊗ c corresponds to the complex conjugation ˉ• on ⨁_{p+q=i} H^{p,q}, which makes (H^i_B(V_σ, Q) ⊗_Q C)^{1⊗c} ≅ H^i_B(V_σ, Q) ⊗_Q R ≅ H^i_B(V_σ, R) into a pure R-Hodge structure of weight i.

Example 1.1. Let E be an elliptic curve defined over Q with the canonical embedding σ : Q ↪ R ⊂ C. Then one can think of an element of H^1_B(E_σ, Q) ≅ H^1(C/Z^2, Q) as of the form Qdx + Qdy, where z = x + iy is a coordinate for C. Because σ is a real embedding, the map ι_σ satisfies ι_σ(dx) = dx and ι_σ(dy) = –dy. For the Hodge decomposition, we have

H^1_B(E_σ, Q) ⊗ C ≅ H^{1,0} ⊕ H^{0,1} ≅ Cdz ⊕ Cdˉz,

where dz = dx + idy and dˉz = dx – idy. Then both ι_σ ⊗ 1 and 1 ⊗ c clearly swap H^{1,0} and H^{0,1}, because (ι_σ ⊗ 1)(dx ± idy) = dx ∓ idy and (1 ⊗ c)(dx ± idy) = dx ∓ idy.

We briefly explain the structure of the paper. We review Griffiths’ theory of primitive middle-dimensional cohomology of a smooth projective hypersurface in Subsection 2.1 and state the main result for odd dimensional hypersurfaces in Subsection 2.2. In Subsection 2.3, we provide a proof of Theorem 2.3. We give a statement for even dimensional hypersurfaces and its proof in Subsection 2.4. In Subsection 2.5 we deduce a corollary for an explicit description of real primitive cohomology in terms of the Jacobian ring.

The goal of this article is to explicitly describe \( c_{dR} \) on the primitive part of the de Rham cohomology of \( V \), when \( V \) is a smooth projective hypersurface over \( k \), using the Jacobian description of the primitive de Rham cohomology ([4] and [5]) and the result of Carlson and Griffiths ([1]).

Acknowledgement. Jeehoon Park was supported by the National Research Foundation of Korea (NRF-2021R1A2C1006696) and the National Research Foundation of Korea grant (NRF-2020R1A5A1016126) funded by the Korea government (MSIT). Junyeong Park was supported by Samsung Science and Technology Foundation under Project Number SSTF-BA2001-02. Philsang Yoo was supported by the New Faculty Startup Fund from Seoul National University, the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2022R1F1A107114212), and the LAMP Program of the National Research Foundation of Korea (NRF) grant funded by the Ministry of Education (No. RS-2023-00301976).

where \( \Omega^{\bullet}_{V/k} \) stands for the algebraic de Rham complex of \( V \). This is a finite-dimensional \( k \)-vector space with a decreasing filtration defined by

Between the two cohomology groups, the algebraic de Rham theorem provides an isomorphism

On the other hand, we consider the algebraic de Rham cohomology

\[
H^i_{dR}(V) := \mathbb{H}^i(V, \Omega^{\bullet}_{V/k})
\]

\[
F^k H^i_{dR}(V) = \mathbb{H}^i(V, \Omega^{\geq k}_{V/k})
\]

\[
I_{\sigma}: H^i_{dR}(V) \otimes_{k,\sigma} \mathbb{C} \xrightarrow{\sim} H^i_B(V_{\sigma}, \mathbb{Q}) \otimes_{\mathbb{Q}} \mathbb{C}
\]

such that \( F^k H^i_{dR}(V) \otimes_{k,\sigma} \mathbb{C} \) on the de Rham side corresponds to \( \bigoplus_{p \geq k} H^{p,q} \) on the Betti side. If \( \sigma: k \to \mathbb{C} \) is a real embedding, then the conjugation \( 1 \otimes c \) on the de Rham side corresponds to \( \iota_{\sigma} \otimes c \) on the Betti side so that we have an isomorphism \( H^i_{dR}(V) \otimes_{k,\sigma} \mathbb{R} \cong (H^i_B(V_{\sigma}, \mathbb{Q}) \otimes_{\mathbb{Q}} \mathbb{C})^{\iota_{\sigma} \otimes c} \) (see [2, Proposition 1.4]).

A natural question is if one can describe complex conjugation in terms of \emph{algebraic} de Rham cohomology. In view of this, let us introduce an involution \( c_{dR} \) on \( H^i_{dR}(V) \otimes_{k,\sigma} \mathbb{C} \) corresponding to \( 1 \otimes c \) under \( I_{\sigma} \) (and hence \( \overline{\phantom{x}} \) under \( \mathcal{P} \circ I_{\sigma} \)). Our main question is

Is there a computable description of \( c_{dR} \)?

For each embedding \( \sigma: k \hookrightarrow \mathbb{C} \), which we assume to factor through \( \mathbb{R} \) for the main result of the paper, we consider the complex manifold \( X = (X_G \times_{k,\sigma} \mathbb{C})(\mathbb{C}) \) and its middle-dimensional primitive singular cohomology \( H^{n-1}_{\mathrm{prim}}(X, \mathbb{Q}) \). Then \( H^{n-1}_{\mathrm{prim}}(X, \mathbb{Q}) \) is a pure \( \mathbb{Q} \)-Hodge structure of weight \( n-1 \), as we have the Hodge decomposition

\[
(2.1) \qquad \mathcal{P}: H^{n-1}_{\mathrm{prim}}(X, \mathbb{Q}) \otimes_{\mathbb{Q}} \mathbb{C} \xrightarrow{\sim} H_0 := \bigoplus_{p+q = n-1} H^{p,q}_0
\]

where \( H_0^{p,q} \) is the primitive part of \( H^{p,q} \). We also consider the primitive de Rham cohomology \( H^{n-1}_{dR, \mathrm{prim}}(X_G) \) of \( X_G \) over \( k \). The standard comparison isomorphism

\[
(2.2) \qquad I_{\sigma}: H^{n-1}_{dR, \mathrm{prim}}(X_G) \otimes_{k,\sigma} \mathbb{C} \xrightarrow{\sim} H^{n-1}_{\mathrm{prim}}(X, \mathbb{Q}) \otimes_{\mathbb{Q}} \mathbb{C}
\]

exists for primitive parts (in fact, \( H^{2m+1}_{\mathrm{prim}} = H^{2m+1} \) for \( n = 2m+2 \) and \( H^{2m}_{\mathrm{prim}} \neq H^{2m} \) for \( n = 2m+1 \)).

Now we give a brief review of Griffiths’ theory ([4]) and its refined version over \( k \) ([5]). We introduce a new variable \( y \) and consider the polynomial ring \( A \) and the polynomial \( S(y,x) \in A \) as follows:

\[
A := k[y, x], \quad S(y, x) := yG(x).
\]

We also introduce two new gradings, which we call charge $ch$ and weight $wt$:

$$
ch(y) := -e, \quad ch(x_i) := \deg(x_i) = 1,\quad wt(y)=1,\quad wt(x_i)=0
$$

for $i=0, \cdots, n$. For each $m \in \mathbb{Z}$, we denote the $ch\ m$ (respectively, $wt\ m$) homogeneous component by $A_m$ (respectively, $A_{(m)}$).

For each $i = 0, \cdots, n$, let us set $G_i(x) := \frac{\partial G(x)}{\partial x_i}$. Note that the smoothness of $X_G$ implies that $G_0, \cdots, G_n$ is a regular sequence. Let $Jac(S)$ be the Jacobian ideal in $A$, i.e., the homogeneous ideal of $A$ generated by $y G_0 = \frac{\partial S}{\partial x_0}, \cdots, y G_n = \frac{\partial S}{\partial x_n}$, and $G = \frac{\partial S}{\partial y}$. Define the background charge as follows:

$$
c_X := e-(n+1).
$$

Under these notations, P. Griffiths proved the following theorem.

## Theorem 2.1. [4] There is an isomorphism

$$
\phi_C : \frac{A_{c_X} \otimes_{k,\sigma} \mathbb{C}}{Jac(S) \cap (A_{c_X} \otimes_{k,\sigma} \mathbb{C})} \stackrel{\sim}{\longrightarrow} H_0 = \bigoplus_{p+q=n-1} H_0^{p,q}
$$

which sends the $wt\ r$ homogeneous part $A_{c_X,(r)} \otimes_{k,\sigma} \mathbb{C}$ (for $r \leq n-1$)$^1$ to $H_0^{n-1-r, r}$.

Let us briefly describe $\phi_C$. Consider the relative long exact sequence for the pair $(\mathbb{P}^n, \mathbb{P}^n \setminus X)$
$$
\cdots \longrightarrow H_{n-1}(\mathbb{P}^n, \mathbb{P}^n \setminus X) \longrightarrow H_n(\mathbb{P}^n \setminus X) \longrightarrow H_n(\mathbb{P}^n) \longrightarrow H_n(\mathbb{P}^n, \mathbb{P}^n \setminus X) \longrightarrow \cdots
$$

As we have an isomorphism $H_{n-1}(\mathbb{P}^n, \mathbb{P}^n \setminus X) \cong H_{n-1}(X)$ using a tubular neighborhood of $X$ in $\mathbb{P}^n$, it induces a tubular neighborhood map
$$
T: H_{n-1}(X, \mathbb{C}) \to H_n(\mathbb{P}^n \setminus X, \mathbb{C})
$$

On the other hand, the residue map
$$
res: H^n(\mathbb{P}^n \setminus X, \mathbb{C}) \to H^{n-1}(X, \mathbb{C})
$$

is locally defined by sending $\omega = \frac{dG}{G} \wedge \omega_1 + \omega_2$, where $\omega_1, \omega_2$ are holomorphic, to $\omega_1$, so it can be shown to respect the Hodge filtration and induces

$$
res: F^p H^n(\mathbb{P}^n \setminus X, \mathbb{C}) \to F^{p-1} H^{n-1}(X, \mathbb{C}).
$$

These two maps form the following commutative diagram

\[
\begin{array}{ccc}
H_{n-1}(X, \mathbb{C}) & \xrightarrow{\otimes} & H^{n-1}(X, \mathbb{C}) \\
\downarrow T && \uparrow res \\
H_n(\mathbb{P}^n \setminus X, \mathbb{C}) & \xrightarrow{\otimes} & H^n(\mathbb{P}^n \setminus X, \mathbb{C}) \\
&& \searrow res \\
&& \mathbb{C}
\end{array}
\]

in that

$$
\int_{[\gamma]} res([\omega]) = \frac{1}{2\pi\sqrt{-1}} \int_{T([\gamma])} [\omega]
$$

holds for any $[\gamma] \in H_{n-1}(X, \mathbb{C})$ and $[\omega] \in H^n(\mathbb{P}^n \setminus X, \mathbb{C})$.

Let

$$
\Omega_x = \sum_{i=0}^n (-1)^i x_i dx_0 \wedge \cdots \wedge \widehat{dx_i} \wedge \cdots \wedge dx_n
$$

Then the Griffiths map $\phi_C$ is given by

$$
\phi_C([y^i f(x)]) = res \left( \left[ \frac{(-1)^i i! f(x) \Omega_x}{G(x)^{i+1}} \right] \right)
$$

where $\left[ \frac{(-1)^i i! f(x) \Omega_x}{G(x)^{i+1}} \right]$ represents the cohomology class in $H^n(\mathbb{P}^n \setminus X, \mathbb{C})$.

Griffiths' theorem provides us an explicitly computable description of $H^{n-1}_{prim}(X, \mathbb{C})$ and its Hodge decomposition $\bigoplus_{p+q=n-1} H_0^{p,q}$. In [5], the authors provided a refined version of $\phi_C$, that is, a $k$-linear isomorphism

$$
\phi : \frac{A_{c_X}}{Jac(S) \cap A_{c_X}} \stackrel{\sim}{\longrightarrow} H^{n-1}_{dR, prim}(X_G)
$$

----

$^1$ If $r \geq n$, then the $wt\ r$ homogeneous part belongs to the Jacobian ideal, i.e., $A_{c_X,(r)} \subset Jac(S) \cap A_{c_X,(r)}$.

which induces the isomorphism \(\varphi_{\mathbb{C}}\) after tensoring with \(\mathbb{C}\) and composing with \(\mathcal{P} \circ I_\sigma\). Thus our question regarding \(c_{\mathrm{dR}}\) can be reformulated as follows:

For a given homogeneous polynomial \(U(y, x) \in A_c^X\), can we find a polynomial \(V(y, x) \in A_c^X \otimes_{k, \sigma} \mathbb{C}\) such that the cohomology class \((\mathcal{P} \circ I_\sigma \circ \varphi)([V(y, x)])\) is same as the complex conjugation \((\mathcal{P} \circ I_\sigma \circ \varphi)([U(y, x)])\) in \(H_0\)?

## 2.2. The main result.

Let \(\mu\) be the \(\mathbb{C}\)-dimension of \(H^{n-1}_{\mathrm{prim}}(X, \mathbb{C})\). We fix an index set \(I\) of cardinality \(\mu\) and decompose

\[ I = I_0 \sqcup \cdots \sqcup I_{n-1} \]

such that the cardinality \(|I_q|\) of \(I_q\) is the \(\mathbb{C}\)-dimension of \(H_0^{n-q-1, q}\). It is well-known (see, for example, [3, 5.3]) that the diffeomorphism type of such a hypersurface \(X\) is determined by the degree \(e\) and the dimension \(n-1\), and indeed that

\[
\mu = \frac{e-1}{e} \left( (e-1)^n + (-1)^{n+1} \right)
\]

Note that if \(e = 1\), then indeed \(H^{n-1}_{\mathrm{prim}}(X, \mathbb{C}) = 0\) holds and everything vacuously works out. Henceforth, we assume \(e > 1\) and accordingly \(\mu > 0\).

For simplicity of presentation, we first assume \(n = 2m+2\) (\(m \geq 0\)) is even. In this case, the intersection pairing \(H_{2m+1}(X, \mathbb{Z}) \otimes H_{2m+1}(X, \mathbb{Z}) \to \mathbb{Z}\) is skew-symmetric and non-degenerate. Thus \(\mu\) is an even number \(2g\) for some positive integer \(g\). We choose a canonical basis \(\{\gamma_1, \cdots, \gamma_{2g}\}\) of \(H_{2m+1}(X, \mathbb{Z})\) (in the sense of [4, page 478]), that is, a \(\mathbb{Z}\)-basis of \(H_{2m+1}(X, \mathbb{Z})\) such that the intersection numbers \(\gamma_i \cdot \gamma_j\) give an integral skew-symmetric \(2g \times 2g\) matrix

\[
Q := \begin{pmatrix}
0 & \mathrm{id}_g \\
-\mathrm{id}_g & 0
\end{pmatrix},\quad \mathrm{id}_g = \text{the } g \times g \text{ identity matrix}
\]

On the other hand, given \(I = I_0 \sqcup \cdots \sqcup I_{2m+1}\), Hodge symmetry implies that the cardinality of \(I_0 \sqcup \cdots \sqcup I_m\) is \(g\). Then by (2.5) we can choose \(g\) elements of the form \(y^i f_{j_i}(x) \in A_c^X\), where \(i = 0, \cdots, m\) and \(j_i = 1, \cdots, |I_i|\), such that they are \(k\)-linearly independent modulo \(\mathrm{Jac}(S) \cap A_c^X\). Note that one has to have

\[
\deg f_{j_i} = c_X + i e = e - (n+1) + i e = e(i+1) - (n+1)
\]

We write these basis elements as

\[
\{U_1, U_2, \cdots, U_g\} := \{y^i f_{j_i} \mid i = 0, \cdots, m,\, j_i = 1, \cdots, |I_i|\}
\]

Having so chosen bases for

\[
\begin{array}{ccc}
H_{2m+1}(X, \mathbb{Z}) & \text{and} & \bigoplus\limits_{i=0}^{m} A_c^X,(i) / (\mathrm{Jac}(S)\cap A_c^X,(i))
\end{array}
\]

we form the \(g \times 2g\) period matrix

\[
P_X = \begin{pmatrix}
\int_{\gamma_1} \phi_{\mathbb{C}}(U_1) & \cdots & \int_{\gamma_{2g}} \phi_{\mathbb{C}}(U_1) \\
\vdots & \ddots & \vdots \\
\int_{\gamma_1} \phi_{\mathbb{C}}(U_g) & \cdots & \int_{\gamma_{2g}} \phi_{\mathbb{C}}(U_g)
\end{pmatrix}
\]

Definition 2.2. For \(n = 2m+2\) and each \(k = 1, 2, \cdots, g = \mu/2\), let

\[
U_{k,\gamma} = \sum_{i,j=1,\cdots,2g} \int_{\gamma_i} \phi_{\mathbb{C}}(U_k) \cdot (\gamma_i \cdot \gamma_j) \cdot \overline{\int_{\gamma_j} \phi_{\mathbb{C}}(U_k)} \in \mathbb{C}
\]

be the \(k\)-th diagonal entry of the following \(g \times g\) matrix

\[
P_X \cdot Q \cdot \overline{P_X}^T
\]

where \(\overline{P_X}^T\) is the transpose of the complex conjugation of the matrix \(P_X\).

To state the main result, we introduce

\[
(2.8) \qquad D = D(x) = \det\left(\frac{\partial^2 G(x)}{\partial x_i \partial x_j}\right),\quad i, j = 0, \cdots, n
\]

**Corrected OCR text:**

---
**Example 2.4** (elliptic curves). Let \( G(x) = G(x_0, x_1, x_2) \in \mathbb{Q}[x_0, x_1, x_2] \) be a homogeneous polynomial of degree 3 which defines a elliptic curve in \( \mathbb{P}^2_{\mathbb{Q}} \), that is, \( n = 2 \) and \( e = 3 \). We fix a canonical \( \mathbb{Z} \)-basis \( \{\gamma_1, \gamma_2\} \) of the homology group \( H_1(X, \mathbb{Z}) \). On the other hand, we have \( I = I_0 \sqcup I_1 \) with \( |I_0| = |I_1| = 1 \). Then for any homogeneous polynomial \( u(x_0, x_1, x_2) \) of degree 3 which does not belong to \( \mathrm{Jac}(S) \), the set \( \{[1], [y u(x_0, x_1, x_2)]\} \) is a \( \mathbb{Q} \)-basis of \( \mathbb{Q}[y, x_0, x_1, x_2] / \mathrm{Jac}(S) \). Note that \( \mathbb{D}(x) = \mathbb{D}(x_0, x_1, x_2) = \det\left(\frac{\partial^2 G(x)}{\partial x_i \partial x_j} \right) \) is homogeneous of degree 3, which by definition yields \( r_{\mathbb{D}(x)} = 1 \) and hence does not belong to \( \mathrm{Jac}(S) \). Therefore Theorem 2.3 says that

---

Note that the same formula would realize the "complex conjugation" of \( y^{n-1-i} \tilde{f}_{j_i} \) to be \( y^i f_{j_i} \) as it should be; the asymmetry in \( b_{i,j} \) cancels out the skew-symmetricity of \( Q \).

---

**Theorem 2.3.** Let \( n = 2m + 2 \) and let \( U_k = y^i f_{j_i}(x) \in \{ U_1, \ldots, U_g \} \) in (2.6). Let us choose any polynomial \( y^{n-1-i} g_{j_i}(x) \in A_{cX} \) such that \( f_{j_i}(x) \cdot g_{j_i}(x) \) does not belong to the Jacobian ideal \( \mathrm{Jac}(G) \). If we define a homogeneous polynomial \( \tilde{f}_{j_i}(x) \in (k \otimes_{k, \sigma} \mathbb{C})[x] \) of degree \( e(n-i)-(n+1) \) by

---

\[
\mathbb{C}(w, w') := \int_X w \wedge w',\quad w, w' \in \bigoplus_{p+q=n-1} H_0^{p, q}
\]

---

\[
\mu := \frac{\int_{\gamma_1} \omega \cdot \int_{\gamma_2} \omega - \int_{\gamma_2} \omega \cdot \int_{\gamma_1} \omega}{24\pi\sqrt{-1}} \in \mathbb{C}, \quad \omega = \varphi_{\mathbb{C}}([1]).
\]

---

where \( [\bullet] \) is the equivalence class of \( \bullet \) modulo \( \mathrm{Jac}(S) \cap (A_{cX} \otimes_{k, \sigma} \mathbb{C}) \) and \( \overline{\bullet} \) is the complex conjugation \( 1 \otimes c \) on \( \bigoplus_{p+q=n-1} H_0^{p,q} \) satisfying \( H_0^{p,q} = H_0^{q,p} \).

---

Then \( \mathbb{D} \in k[x] \) is a homogeneous polynomial of degree \( (n+1)(e-2) \), because \( X_G \) is assumed to be smooth. For a homogeneous polynomial \( h(x) \) of degree \( (n+1)(e-2) \), we define \( r_h \) to be the unique number in \( k \) (which exists by [6, (12.6) (ii)]) such that

---

Note that unless \( w \in H_0^{p,q} \) and \( w' \in H_0^{p',q'} \) satisfy both \( p + p' = n-1 \) and \( q+q' = n-1 \), the value \( \mathbb{C}(w,w') \) is zero. We also define a bilinear pairing

---

\[
\mathcal{R}([y^i f(x)], [y^j g(x)]) =
\begin{cases}
b_{i,j} (2\pi\sqrt{-1})^{n-1} e(e-1)^n r_{f(x)g(x)} & \text{if } i+j = n-1 \\
0 & \text{if } i+j \ne n-1
\end{cases}
\]

---

\[
\mathcal{R} : \frac{A_{cX} \otimes_{k, \sigma} \mathbb{C}}{\mathrm{Jac}(S) \cap (A_{cX} \otimes_{k,\sigma} \mathbb{C})} \times \frac{A_{cX} \otimes_{k, \sigma} \mathbb{C}}{\mathrm{Jac}(S) \cap (A_{cX} \otimes_{k, \sigma} \mathbb{C})} \to \mathbb{C}
\]

---

where \( b_{i,j} \) is given in (2.11) and \( r_h \) is defined in (2.9).

---

\[
\mathbb{C}: \bigoplus_{p+q=n-1} H_0^{p,q} \times \bigoplus_{p+q=n-1} H_0^{p,q} \to \mathbb{C}
\]

---

\[
(2.10)\qquad
\tilde{f}_{j_i}(x) := \frac{b_{i, n-1-i}\ U_{k,\gamma}}{r_{f_{j_i}, g_{j_i}}\,(2\pi\sqrt{-1})^{n-1} e(e-1)^n} \cdot g_{j_i}(x)
\]

---

\[
\varphi_\mathbb{C}([y^i f_{j_i}(x)]) = \varphi_\mathbb{C}([y^{n-1-i} \tilde{f}_{j_i}(x)]) \quad \text{in}\ H_0 = \bigoplus_{p+q=n-1} H_0^{p,q}
\]

---

\[
(2.9)\qquad
h(x) \equiv r_h \cdot \mathbb{D}(x) \bmod \mathrm{Jac}(G) = \langle G_0, \ldots, G_n \rangle
\]

---

\[
(2.11)\qquad
b_{i,j} = (-1)^{\frac{i(i+1)}{2} + \frac{j(j+1)}{2} + j^2}
\]

---

in \( k[x] \). Note that \( h(x) \) belongs to the Jacobian ideal \( \mathrm{Jac}(G) \) if and only if \( r_h = 0 \).

---

where \( U_{k,\gamma} \) is given in Definition 2.2, \( r_{f_{j_i}, g_{j_i}} \) is a non-zero number given in (2.9) by the assumption, and

---

then we have

---

where \(\Omega^p_Y\) is a sheaf of holomorphic \(p\)-forms on \(Y\) and \(\Omega^p_{\mathbb{P}^n}(\log X)\) is a sheaf of meromorphic \(p\)-forms \(\omega\) on \(\mathbb{P}^n\) such that \(\omega\) and \(d\omega\) are regular on \(\mathbb{P}^n \setminus X\) and have at most a pole of order one along \(X\). See [6, page 444] for more details. Let \(\delta\) be the coboundary map in the Poincaré residue sequence induced from (2.12):

where \(\operatorname{Res}_0\) is the Grothendieck residue map (see [6, (12.3)] for its definition and well-definedness). This follows from [6, Remark (12.10)], where we set \(d_0 = e-1\), \(m=n\), and \(F_i = \frac{\partial G}{\partial x_i}\) in their notations. Moreover, we use the following result [6, (12.5)] on the computation of the Grothendieck residue:

The relationship between the long exact sequence of hypercohomology of (2.12) and the Gysin sequence is given by the following commutative diagram (see, for instance, [6, Proposition (11.4)]):

A main result of [1] gives an explicit Čech-type formula for the coboundary map \(\delta\) of the cup product of two cohomology classes: let \(i\) and \(j\) be non-negative integers such that \(i+j = n-1\). For homogeneous polynomials \(A(x), B(x) \in \mathbb{C}[x]\) such that \(\deg(A) = e(i+1)-(n+1)\) and \(\deg(B) = e(j+1)-(n+1)\),

where \(r_h \in \mathbb{C}\) is defined in (2.9) for a homogeneous polynomial \(h\) of degree \((n+1)(e-2)\) and otherwise set to be zero. Thus we have

Here all the vertical maps are isomorphisms, the map \(\delta\) in (2.13) can be regarded as a part of \(\tilde{\delta}\), and the lower shriek \(\eta_!\) of \(\eta\) can be constructed as the composition of three maps:

Let \(\eta : X \to \mathbb{P}^n\) be a given closed embedding. Let us consider the following exact sequence of complexes of sheaves on \(\mathbb{P}^n\):

(2.16) \[
\int_{\mathbb{P}^n} \left[\frac{h(x)\Omega_x}{\frac{\partial G}{\partial x_0} \frac{\partial G}{\partial x_1} \ldots \frac{\partial G}{\partial x_n}}\right] = (2\pi \sqrt{-1})^n (e-1)^n r_{h(x)}
\]

where \(\cdot\) is the cup product of the singular cocycles and

(2.14) \[
\delta\left(\operatorname{res}\left(\frac{A(x)\Omega_x}{G(x)^{i+1}}\right) \cdot \operatorname{res}\left(\frac{B(x)\Omega_x}{G(x)^{j+1}}\right)\right) = c_{i,j} \frac{A(x)B(x)\Omega_x}{\frac{\partial G}{\partial x_0} \frac{\partial G}{\partial x_1} \ldots \frac{\partial G}{\partial x_n}},
\]

(2.13) \[
\delta : H^{n-1}(X, \Omega_X^{n-1}) \to H^n(\mathbb{P}^n, \Omega_{\mathbb{P}^n}^n)
\]

Proof. This essentially follows from [1, Theorem 3] and results of [6].

\[
H^{k-2}(X, \mathbb{C}) \overset{\sim}{\to} H_{2n-k}(X, \mathbb{C}) \xrightarrow{\eta_*} H_{2n-k}(\mathbb{P}^n, \mathbb{C}) \overset{\sim}{\to} H^k(\mathbb{P}^n, \mathbb{C}),
\]

\[
\operatorname{Res}_0 \left\{\frac{h(x)}{\frac{\partial G}{\partial x_0} \cdots \frac{\partial G}{\partial x_n}}\right\} = (e-1)^{n+1} r_h
\]

(2.12) \[
0 \to \Omega_{\mathbb{P}^n}^\bullet \to \Omega_{\mathbb{P}^n}^\bullet(\log X) \xrightarrow{\mathrm{res}} \eta_*\Omega_X^{\bullet-1} \to 0
\]

For any \(h(x) \in \mathbb{C}[x]\), we have the following formula:

(2.15) \[
c_{i,j} = \frac{(-1)^{\frac{i(i+1)}{2} + \frac{j(j+1)}{2} + n-1 + j^2}}{i!j!} e.
\]

\[
\int_{\mathbb{P}^n}\left[\frac{h(x)\Omega_x}{\frac{\partial G}{\partial x_0} \frac{\partial G}{\partial x_1} \ldots \frac{\partial G}{\partial x_n}}\right] = \frac{(2\pi\sqrt{-1})^n}{e-1}\operatorname{Res}_0 \left\{\frac{h(x)}{\frac{\partial G}{\partial x_0} \cdots \frac{\partial G}{\partial x_n}}\right\}
\]

Lemma 2.5. The following diagram commutes

See [1, Theorem 3] for details.

Here is the corrected text from the provided images, in reading order:

---

where the first and third maps are the Poincaré duality isomorphisms. Thus when \(k = 2n\), we have
\[
\int_X \mu = \int_{\mathbb{P}^n} m(\mu)
\]
for \(\mu \in H^{2n-2}(X, \mathbb{C}) \cong H^{n-1}(X, \Omega_X^{n-1})\). Moreover, because \(\delta = 2\pi\sqrt{-1} \cdot m!\), we have

\[
\tag{2.17}
\int_X \mu = \frac{1}{2\pi\sqrt{-1}} \int_{\mathbb{P}^n} \delta(\mu).
\]

Therefore we have
\[
\int_X \operatorname{res}\left( \frac{(-1)^i i! f(x)\Omega_x}{G(x)^{i+1}} \right)
\cdot \operatorname{res}\left( \frac{(-1)^j j! g(x)\Omega_x}{G(x)^{j+1}} \right)
\]
\[
\tag{2.17}
= \frac{1}{2\pi\sqrt{-1}} \int_{\mathbb{P}^n}
\delta\left( \operatorname{res}\left( \frac{(-1)^i i! f(x)\Omega_x}{G(x)^{i+1}} \right)
\cdot \operatorname{res}\left( \frac{(-1)^j j! g(x)\Omega_x}{G(x)^{j+1}} \right)\right)
\]
\[
\tag{2.14}
= \frac{1}{2\pi\sqrt{-1}} \int_{\mathbb{P}^n}
c_{i, j} \frac{(-1)^{i+j} i! j! f(x)g(x)\Omega_x}
{\frac{\partial G}{\partial x_0} \frac{\partial G}{\partial x_1} \cdots \frac{\partial G}{\partial x_n}}
\]
\[
\tag{2.15}
= \frac{b_{i, j} \cdot e}{2\pi\sqrt{-1}}
\int_{\mathbb{P}^n} \frac{f(x)g(x)\Omega_x}
{\frac{\partial G}{\partial x_0} \frac{\partial G}{\partial x_1} \cdots \frac{\partial G}{\partial x_n}}
\]
\[
\tag{2.16}
= b_{i, j}(2\pi\sqrt{-1})^{n-1} e(e-1)^n r_{f(x)g(x)}.
\]

If we use (2.4), then the lemma follows from the above computation. \(\Box\)

Proof of Theorem 2.3. The following diagram commutes

\[
\begin{array}{ccc}
\bigoplus_{p+q=2m+1} H^{p,q}_0 & \times & \bigoplus_{p+q=2m+1} H^{p,q}_0 \\
\downarrow{\cong} & & \downarrow{\cong} \\
H_{2m+1}(X, \mathbb{C}) & \times & H_{2m+1}(X, \mathbb{C})\\
 & \searrow{\mathcal{B}} \\
 & & \mathbb{C}
\end{array}
\]

where the bottom pairing \(\mathcal{B}\) is the intersection pairing, and the isomorphism \(H_0 \stackrel{\sim}{\rightarrow} H_{2m+1}(X, \mathbb{C})\) is given by

\[
\omega \mapsto \left(\int_{\gamma_1} \omega\right) \cdot \gamma_1 + \cdots + \left(\int_{\gamma_{2g}} \omega\right) \cdot \gamma_{2g}.
\]

Therefore with respect to the complex conjugation \(1 \otimes c\), the value
\[
\mathcal{C}(\phi_{\mathbb{C}}(U_k), \phi_{\mathbb{C}}(U_k)) = \int_X \phi_{\mathbb{C}}(U_k) \wedge \overline{\phi_{\mathbb{C}}(U_k)}
\]
is the same as

\[
\mathcal{B} \left( \sum_{i=1}^{2g} \left(\int_{\gamma_i}\phi_{\mathbb{C}}(U_k)\right) \cdot \gamma_i,
\sum_{j=1}^{2g} \left(\int_{\gamma_j} \overline{\phi_{\mathbb{C}}(U_k)}\right) \cdot \gamma_j \right)
\]
which is \(U_{k,\gamma}\) in Definition 2.2. Then in view of Lemma 2.5, finding \(\phi_{\mathbb{C}}^{-1}(\phi_{\mathbb{C}}(U_k))\) for \(U_k = y^i f_{j_i}(x)\) amounts to finding \(h_{j_i}(x)\) such that \(R(y^i f_{j_i}, y^{n-1-i}h_{j_i}) = U_{k,\gamma}\). Now it is clear that \(h_{j_i} = \tilde{f}_{j_i}\) as defined in (2.10) satisfies the desired condition. \(\Box\)

2.4. The case of odd \(n\). For odd \(n = 2m+1\) (\(m \geq 0\)), according to [4, Proposition 7.1], for

\[
H_{2m}(X, \mathbb{Z})_0 = \ker(H_{2m}(X, \mathbb{Z}) \to H_{2m}(\mathbb{P}^n, \mathbb{Z})),
\]

the intersection pairing

\[
H_{2m}(X, \mathbb{Z})_0 \otimes H_{2m}(X, \mathbb{Z})_0 \to \mathbb{Z}
\]

is symmetric and non-degenerate, and \(\det(\gamma_\alpha \cdot \gamma_\beta) = \pm e\). We choose a standard basis (in the sense of [4, page 485]) \(\gamma_1, \dots, \gamma_\mu\) for \(H_{2m}(X, \mathbb{Z})_0\), that is, a \(\mathbb{Z}\)-basis of \(H_{2m}(X, \mathbb{Z})_0\) for which the intersection pairing matrix is of the form

\[
P = \begin{pmatrix}
P_0 & 0 \\
 & P_1 \\
 & & \ddots \\
0 & & & P_r
\end{pmatrix}, \qquad
P_1 = \begin{pmatrix}
0 & d_1 \\
d_1 & 0
\end{pmatrix}, \ldots,
P_r = \begin{pmatrix}
0 & d_r \\
d_r & 0
\end{pmatrix}
\]

where \(P_0\) is an anisotropic symmetric integral matrix, that is, \(v^T P_0 v \neq 0\) for any integral vector \(v\).

which is k-linearly independent modulo Jac(S) ∩ AcX. We also form the a × μ period matrix PX as in (2.7).

be the k-th diagonal entry of the following a × a matrix

PX · P · PX‾ᵀ

where PX‾ᵀ is the transpose of the complex conjugation of the matrix PX.

Uk,γ = ∑_{i,j=1,...,μ} ∫_{γi} φC(Uk) · (γi • γj) · ∫_{γj} φC(Uk)‾ ∈ ℂ

{U₁, U₂, ···, Uₐ} := {yⁱfⱼᵢ | i = 0, 1, ···, m−1, jᵢ = 1, ···, |Iᵢ|} ⊔ {yᵐfⱼₘ | jₘ = 1, ···, ⌊|Iₘ| + 1/2⌋}

Let Uk = yⁱfⱼᵢ(x) ∈ {U₁, U₂, ···, Uₐ} in (2.18). Then the same statement in Theorem 2.3 holds for n = 2m + 1 case by the same argument.

2.5. Pure ℝ-Hodge structure. For given n, we consider elements of the form yⁱfⱼᵢ ∈ AcX that are k-linearly independent modulo Jac(S) ∩ AcX as considered in (2.6) and (2.18). For a notational convenience, we regard jᵢ as an element of Iᵢ. Then the following corollary is immediate from Theorem 2.3.

Corollary 2.7. For n even, let

(2.19)
wⱼᵢ = (yⁱfⱼᵢ + yⁿ⁻¹⁻ⁱf̃ⱼᵢ) / 2

for i = 0, ..., (n−2)/2 and jᵢ = 1, ..., |Iᵢ|. Then we have

(φC⁻¹ ◦ P)(Hⁿ⁻¹_{prim}(X, ℝ)) = ∑_{j ∈ I₀ ⊔···⊔ I_{(n−2)/2}} ( ℝ·[wⱼ] + ℝ·[√−1 wⱼ] )

For n = 2m+1 odd, under the same notation (2.19) and

wⱼₘ = (yᵐfⱼₘ + yᵐf̃ⱼₘ) / 2

for jₘ = 1, ..., ⌊|Iₘ| + 1/2⌋, we have

(φC⁻¹ ◦ P)(Hⁿ⁻¹_{prim}(X, ℝ)) = {
  ∑_{j ∈ I₀ ⊔···⊔ Iₘ} (ℝ·[wⱼ] + ℝ·[√−1 wⱼ])    if |Iₘ| is even
  ∑_{j ∈ I₀ ⊔···⊔ Iₘ \ {α}} (ℝ·[wⱼ] + ℝ·[√−1 wⱼ]) + ℝ·[w_α]   if |Iₘ| is odd
}
where we write α = ⌊|Iₘ| + 1/2⌋ ∈ Iₘ.

Example 2.8 (elliptic curves). With the notations of Example 2.4, we have

(φC⁻¹ ◦ P)(Hⁿ⁻¹_{prim}(X, ℝ)) = ℝ [ (1 + μ·y D(x))/2 ] + √−1 ℝ [ (1 + μ·y D(x))/2 ]

Jeehoon Park: QSMS, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, South Korea
Email address: jpark.math@gmail.com

Junyeong Park: Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, UNIST-gil 50, Ulsan 44919, South Korea

Philsang Yoo: Department of Mathematical Sciences & Research Institute of Mathematics, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, South Korea


[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

- **Do not** include any other keys or prose outside these two tagged blocks.
- **Do not** report stylistic or citation issues.
- Be as precise as possible about **where** (section ID or figure/table) and **what** the scientific flaw is.
- Each description for the errors must be rooted in a scientific rationale explaining why it is 'wrong' (not how it could be improved).

Begin your analysis now.

[User]
Mihail Stoian*

We study the fundamental scheduling problem 1‖∑pjUj. Given a set of n jobs with processing times pj and deadlines dj, the problem is to select a subset of jobs such that the total processing time is maximized without violating the deadlines. In the midst of a flourishing line of research Fischer and Wennmann have recently devised the sought-after Õ(P)-time algorithm, where P = ∑pj is the total processing time of all jobs. This running time is optimal as it matches conditional lower bounds based on popular conjectures.

However, P is not the sole parameter one could parameterize the running time by. Indeed, they explicitly leave open the question of whether a running time of Õ(n + max dj) or even Õ(n + max pj) is possible. In this work, we show, somewhat surprisingly, that by a refined implementation of their original algorithm, one can obtain the asked-for Õ(n + max dj)-time algorithm.

The 1∥∑pjUj problem is a classical NP-hard problem that generalizes the well-known Subset Sum problem. Given n jobs with processing times pj and deadlines dj, the problem is to select a subset of jobs such that the total processing time is maximized under the constraint that their deadlines are observed. Its history goes back to the 55-year old algorithm by Lawler and Moore [9], whose running time reads O(nP), where P = ∑pj.

Multiple Machines. The problem also appears in the multiple-machine flavor as the so-called Pm∥∑pjUj problem: In this setting, there are m machines among which the jobs can be distributed; the task remains the same as before. For simplicity, we assume that m is a constant.

Related Work. Interestingly, the running time of Lawler and Moore's algorithm remained the state-of-the-art for quite some time: Only recently did Bringmann, Fischer, Hermelin, Shabtay and Wellnitz [2] manage to lower it to Õ(P^{7/4}) by introducing, as a by-product, a novel convolution — the skewed convolution.¹ Following their work, a line of research started to improve either the algorithmic reduction or the skewed convolution itself, resulting in a Õ(P^{7/5})-time algorithm [8, 11]. Recently, Fischer and Wennmann [5] showed that the skewed convolution can be forgotten altogether: Indeed, they provide a randomized algorithm in time O(P log P) and, alternatively, a deterministic one in time O(P log^{1+o(1)} P). In particular, their algorithm is optimal (up to lower-order factors) under the Strong Exponential Time Hypothesis or the Set Cover Hypothesis.

Similar to the Subset Sum problem, our problem also has other parameters that can be used to parametrize the running time by. Prominent examples are dmax = max dj and pmax = max pj. Prior research also introduced algorithm w.r.t. to these parameters: an Õ(min{P·D#, P+D})-time algorithm [2], where #D is the number of distinct deadlines and D is the sum of all distinct deadlines, and an Õ(n + p^3max)-time algorithm [8]. Fischer and Wennmann explicitly leave open the question of whether a time of Õ(n + dmax) or even Õ(n + pmax) is possible [5].

Our Results. We show that an Õ(n + dmax)-time algorithm does indeed exist. Perhaps surprisingly, our algorithm is a simple refined implementation of Fischer and Wennmann's. Our analysis also extends to the multi-machine setting, resulting in a Õ(n + d^mmax)-time algorithm.

Notation. We use [n] := {1, . . . , n} (note that Fischer and Wennmann's notation is 0-based [5]). The interval [i..j] denotes {i, . . . , j}, while [i..j) denotes [i, j]\{j}; analogously, define (i..j]. Furthermore, we use the standard sumset notation A+b = {a+b | a ∈ A} and A+B = {a+b | a ∈ A, b ∈ B}.

Earliest-Deadline-First. We employ the key observation due to Lawler and Moore [9] that w.l.o.g. the early, i.e., non-tardy, jobs may be scheduled in the sorted order of their deadlines. Thus, the problem reduces to computing a subset of jobs J* ⊆ [n] that maximizes ∑j∈J* pj and observes the deadlines of the jobs within J*, i.e., ∑i∈J*:i≤j pi ≤ dj for all j ∈ J.

Lawler and Moore’s Algorithm. The algorithm due to Lawler and Moore is a simple dynamic program. Namely, one recursively defines the sets S0, . . . , Sn ⊆ [0..P] as in the following:

¹: The term Õ(P) = P(log P)^{O(1)} suppresses polylogarithmic factors.

Note that we follow the two-step construction of \(S_j\) proposed by Fischer and Wennmann [5], since it allows for a modular implementation. The optimal value can be then read off as max \(S_n\).

**Fischer and Wennmann’s Algorithm.** The algorithm proposed by Fischer and Wennmann [5] uses the dynamic string data structure [6, 7], recently employed in modular subset sum [1, 3, 10]. Their key observation is that one can use the same data structure to support the deletions in Eq. (3c). In particular, one has to interpret the sets \(S_j\) as indicator strings of size \(P\).

**Our Approach.** However, observe that in Eq. (3c), we anyway trim the current set \(S_j + \{0, p_j\}\) by its processing times greater than \(d_j\). Hence, if we had a mechanism to only add valid processing times, i.e., l.e.q. \(d_j\), we could interpret the sets \(S_j\) indeed as indicator strings of size \(d_{max} + 1\).^2

First, observe that, by construction, the following invariant holds: \(S_{j-1} \subseteq [0..d_{j-1}]\). The new algorithm works as follows. Instead of directly performing Eq. (3b) as is, we first split \(S_j\) into two parts w.r.t. to \(d_j - p_j\), i.e., we obtain \(L_{j-1} := S_{j-1} \cap [0..d_j - p_j]\) and \(R_{j-1} := S_{j-1} \cap (d_j - p_j..d_j]\). We will see later how \(L_{j-1}\) and \(R_{j-1}\) can be computed via the dynamic string data structure. We then compute \(\tilde{S}_j = L_{j-1} + p_j\). Finally, we need to merge \(\tilde{S}_j\) with both \(L_{j-1}\) and \(R_{j-1}\) to obtain the final \(S_j\). Better said, we are left with merging \(\tilde{S}_j\) with \(S_{j-1}\) to get \(S_j = \tilde{S}_j \cup S_{j-1}\).

In the following, we outline the implementation details.

**Leveraging Dynamic Strings.** The dynamic string data structure is an important toolbox in Fischer and Wennmann’s algorithm [5]. In the sequel, we use their presentation of the data structure and outline only the operations we will need:

---
^2The offset of “+1” is necessary to account for deadlines of value 0.

**Lemma 3.1** (Dynamic String Data Structure [6, 7]). The dynamic string data structure maintains a dynamic collection \( X \) of non-empty strings, supporting the following operations:

- **make_string(x):** Given any string \( x \in \Sigma^+ \), insert \( x \) into \( X \).
- **concat(x_1, x_2):** Given \( x_1, x_2 \in X \), insert the concatenated string \( x_1x_2 \) into \( X \).
- **split(x, i):** Given \( x \in X \) and \( i \in [0..|x|) \), insert \( x[0..i] \) and \( x(i..|x|] \) into \( X \).

Let \( s \) be the maximum of the total length of all strings and the number of executed operations. Then all operations run in randomized time \( O(\log s) \) or in deterministic time \( O(\log^{1+o(1)} s) \), except for **make_string** which takes time \( O(|x| + \log s) \) and \( O(|x| \cdot \log^{o(1)} s) \), respectively.

Let \( u \) be the string size (this will be set to \( d_{\max} + 1 \) later). We represent a set \( S \subseteq [n] \) by its corresponding indicator string \( x_S \in \{0, 1\}^u \), i.e., \( i \in S \) if and only if \( x_S[i] = 1 \).

To initialize the data structure, we build \( 0^u \) by inserting the word \( w = 0 \) and self-concatenating it \( \log u \) times. This takes time \( O(\log^2 u) \).

Then, to obtain \( L_{j-1} \), we split \( x_{S_{j-1}} \) at position \( d_j - p_j \). Afterward, we can compute \( \widetilde{S}_j \) by shifting \( x_{L_{j-1}} \) by \( p_j \) positions. Namely, we perform **concat(\( 0^{p_j}, x_{L_{j-1}} \))**, where the string \( 0^{p_j} \) can be obtained by splitting off the above pre-computed \( 0^u \) via **split(\( 0^u, p_j \))**. We finally trim the resulting string to length \( u \) via **split(\( 0^{p_j} x_{L_{j-1}}, u \))**. Merging \( \widetilde{S}_j \) with \( S_{j-1} \) is now a bit-wise OR of \( x_{\widetilde{S}_j} \) and \( x_{S_{j-1}} \), and can be implemented in the same way as by Fischer and Wennmann [5].

**Running Time Analysis.** Our running time analysis closely follows that by Fischer and Wennmann [5], with the key difference that, using our new implementation, we manage to bound the number of insertions by \( d_{\max} + 1 \) instead of \( 2P + 1 \):

**Lemma 3.2.** It holds that \( \sum_{i \in [n]} |\widetilde{S}_j \setminus S_{j-1}| \leq d_{\max} + 1 \).

**Proof.** A benefit of the new algorithm is that, compared to Ref. [5], we do not need to split w.r.t. \( d_j \) since the set \( S_j \) is, by construction, already filtered out of processing times greater than \( d_j \). Hence, for any \( x \in [0..d_{\max}] \), once \( x \in \widetilde{S}_j \setminus S_{j-1} \), then \( x \in S_j, \ldots, S_n \) and, hence, we have
\[
|\{i \in [n] \mid x \in \widetilde{S}_j \setminus S_{j-1}\}| \leq 1.
\]
With this observation, the cumulative number of insertions reads
\[
\sum_{i \in [n]} |\widetilde{S}_j \setminus S_{j-1}| = \sum_{x \in [0..d_{\max}]} |\{i \in [n] \mid x \in \widetilde{S}_j \setminus S_{j-1}\}| \leq d_{\max} + 1.
\]
\(\Box\)

Using the dynamic string data structure with string-size \( u = d_{\max} + 1 \), we obtain an \(\tilde{O}(n + d_{\max})\)-time algorithm.

**Retrieving the Schedule.** We have shown that we can compute the optimal value in time \( \tilde{O}(n + d_{\max}) \) by a refined implementation of the original algorithm. Obtaining the optimal schedule can be, indeed, done in a similar way, since the sets themselves are after each iteration the same as in Fischer and Wennmann’s algorithm. We skip the details and refer the reader to Ref. [5, Sec. 3.2].

Our refined analysis can be extended to the setting of \(m\) machines. In that context, the original dynamic program slightly changes, but the core idea remains the same. In particular, we can define similar sets \(S_0, \ldots, S_n\) which are this time subsets of \([0..P]^m\), i.e., the \(i\)-th entry of each vector corresponds to the \(i\)-th machine:

A feasible schedule of total processing time \(t\) exists if and only if there exists \(s \in S_n\) such that \(s_1 + \ldots + s_m = t\) [9]. It is now easy to see that we can apply our refined algorithm for the single-machine setting to this setting as well. The key idea is to apply our trimming strategy point-wise, for each of the \(m\) (independent) entries of the vectors.

We can similarly bound the number of insertions:

**Lemma 4.1.** It holds that \(\sum_{j \in [n]} \left| \widetilde{S}_j \setminus S_{j-1} \right| \leq (d_{\max} + 1)^m\).

**Proof.** We closely follow our proof for Lemma 3.2. Namely, we make the observation that for any \(x \in [0..d_{\max}]^m\), once \(x \in \widetilde{S}_j \setminus S_{j-1}\), then \(x \in S_j, \ldots, S_n\). Hence, we have

\[
|\{ j \in [n] \mid x \in \widetilde{S}_j \setminus S_{j-1} \}| \leq 1.
\]

Therefore, we can bound the total number of insertions by

\[
\sum_{j \in [n]} \left| \widetilde{S}_j \setminus S_{j-1} \right| = \sum_{x \in [0..d_{\max}]^m} |\{ j \in [n] \mid x \in \widetilde{S}_j \setminus S_{j-1} \}| \leq (d_{\max} + 1)^m.
\]

The rest of the algorithm, in particular the adapted multi-dimensional dynamic string data structure, remains the same as in Fischer and Wennmann [5].

We settled an open question explicitly asked by Fischer and Wennmann [5]. Somewhat surprisingly, we obtained it by slightly modifying their original algorithm, resulting in a \(\tilde{O}(n + d_{max})\)-time algorithm. A pressing open question is whether the problem can be solved in \(\tilde{O}(n + p_{max})\)-time. Indeed, it would be (again) surprising if the same algorithm could be adapted to this setting. However, such an algorithm would have a wider implication, since it would also represent a breakthrough for Subset Sum with small items. In spite of that, the recent break-through by Chen, Lian, Mao and Zhang [4], namely an \(\tilde{O}(n + \sqrt{wt})\)-time algorithm for the Subset Sum problem, shows that a similar running time for the \(1||\sum p_i U_i\) problem may not be completely out of reach.

[9] E. L. Lawler and J. M. Moore. A functional equation and its application to resource allocation and sequencing problems. Management Science, 16(1):77–84, 1969. URL: http://www.jstor.org/stable/2628367. 1, 4

[10] Krzysztof Potepa. Faster deterministic modular subset sum. In Petra Mutzel, Rasmus Pagh, and Grzegorz Herman, editors, 29th Annual European Symposium on Algorithms, ESA 2021, September 6–8, 2021, Lisbon, Portugal (Virtual Conference), volume 204 of LIPIcs, pages 76:1–76:16. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ESA.2021.76, doi:10.4230/LIPIcs.ESA.2021.76. 2

[11] Baruch Schieber and Pranav Sitaraman. Quick minimization of tardy processing time on a single machine. In Algorithms and Data Structures: 18th International Symposium, WADS 2023, Montreal, QC, Canada, July 31–August 2, 2023, Proceedings, pages 637–643, Berlin, Heidelberg, 2023. Springer-Verlag. doi:10.1007/978-3-031-38906-1_42. 1