Discrete Hilbert Space, the Born Rule, and Quantum Gravity

This is a new paper which I only recently found the time to write up, although I have been thinking about the ideas off and on for some time.

It extends ideas first discussed in two papers with A. Zee and R. Buniy: Is Hilbert space discrete? and Discreteness and the origin of probability in quantum mechanics.

Slides from a related talk at Caltech IQIM.

The new paper connects discrete Hilbert space to specific models of quantum gravity, such as simplicial or lattice quantum gravity.

Discrete Hilbert Space, the Born Rule, and Quantum Gravity
https://arxiv.org/abs/2007.12938

Quantum gravitational effects suggest a minimal length, or spacetime interval, of order the Planck length. This in turn suggests that Hilbert space itself may be discrete rather than continuous. One implication is that quantum states with norm below some very small threshold do not exist. The exclusion of what Everett referred to as maverick branches is necessary for the emergence of the Born Rule in no collapse quantum mechanics. We discuss this in the context of quantum gravity, showing that discrete models (such as simplicial or lattice quantum gravity) indeed suggest a discrete Hilbert space with minimum norm. These considerations are related to the ultimate level of fine-graining found in decoherent histories (of spacetime geometry plus matter fields) produced by quantum gravity.

From the Discussion:

No collapse (or many worlds) versions of quantum mechanics are often characterized as extravagant, because of the many branches of the wavefunction. However it is also extravagant to postulate that spacetime or Hilbert space are infinitely continuous. Continuous Hilbert space requires that for any two choices of orientation of a qubit spin (see Figure 1), no matter how close together, there are an infinite number of physically distinct states between them, with intermediate orientation. Instead, there may only be a finite (but very large) number of distinct orientations allowed, suggesting a minimum norm in Hilbert space. No experiment can probe absolute continuity, and indeed there seem to be fundamental limits on such experiments, arising from quantum gravity itself.

We illustrated a direct connection between discrete spacetime (the simplex length a) and discrete Hilbert space (minimum non-zero distance in Hilbert space produced by time evolution), in a specific class of quantum gravity models based on Feynman path integrals. It may be the case that maximally fine-grained decoherent histories generated within quantum gravity have discrete geometries and exist in a discrete Hilbert space. Consequently histories with sufficiently small norm are never generated, thereby solving Everett's problem with maverick branches. In the remaining branches, deviations from Born Rule probabilities are almost entirely hidden from semi-classical observers. ...

See also

The Quantum Simulation Hypothesis: Do we live in a quantum multiverse simulation?

Feynman and Everett

Gork revisited

Foundation for Individual Rights in Education: "chilling academic freedom"

From the FOUNDATION FOR INDIVIDUAL RIGHTS IN EDUCATION: "FIRE’s mission is to defend and sustain the individual rights of students and faculty members at America’s colleges and universities. These rights include freedom of speech, freedom of association, due process, legal equality, religious liberty, and sanctity of conscience — the essential qualities of liberty."

Peter Bonilla (FIRE): ... it’s worth visiting the case of Stephen Hsu at Michigan State University. His case is more sobering, because the campaign succeeded in forcing his resignation as MSU’s Vice President for Research and Graduate Studies. (Hsu, a theoretical physicist, remains on MSU’s tenured faculty.) Hsu was brought down in significant part by the pressure on MSU generated by a Twitter thread and accompanying petition by MSU’s Graduate Employees Union, which cobbled together a string of decontextualized, condensed remarks on genetics and intelligence to brand Hsu “a vocal scientific racist and eugenicist.”

Hsu rebutted the claims in a post on his personal blog, accusing the petitioners of acting in bad faith. He had a point. Apart from the fact that Twitter is a woefully inadequate forum to debate intricate matters of science (or even to accurately characterize them), some of the claims against Hsu are dishonest on their face. To highlight one example, the GEU seized on the fact that a study supported in part through his office’s funding (Hsu oversaw research expenditures of roughly $700 million) suggested that there was no widespread racial bias in police shootings. GEU characterized this by saying that “Hsu’s office appears to have directed funding to research downplaying racism in bias in police shootings,” implying that this was his very goal. This is a striking charge to make against a scientist’s integrity without presenting any evidence; it’s also simply not how scientific inquiry works.

The claims against Hsu alleged no concrete misconduct or malfeasance in carrying out his administrative duties; rather, they amalgamated a disparate set of remarks, attributed the least charitable set of motives to his making them, and stated that someone holding those purported beliefs was per se unfit to hold his position. Unfortunately, MSU went for it. On June 19, MSU president Samuel Stanley demanded, and received, Hsu’s resignation as VP.

Spillover effects on academic freedom

The president of MSU’s Graduate Employees Union cheerfully expressed the opinion that the Hsu episode should have no chilling effect on academic freedom, because the GEU only sought to have Hsu removed from his administrative position and not from the tenured faculty. To be sure, there are significant differences between the two propositions, and administrators are generally considered at-will employees who can be fired any time. (Administrator cases are also outside the scope of FIRE’s mission of defending student and faculty rights.). But from someone who’d just helped mount a successful campaign that worked in part by declaring a range of opinions on certain issues effectively outside the bounds of legitimate academic inquiry, this assertion is laughable.

Faculty will most certainly take note of the actions colleges take against administrators, note the implications they may have for academic freedom, and adjust accordingly. Universities may not be so easily able to rid themselves of tenured faculty members, but several years of faculty cases suggest that universities can be quite willing to override faculty governance for the purposes of pursuing discipline (including termination and loss of tenure) against faculty for speech demonstrably protected by their academic freedom. ...

Media coverage:

A Twitter Mob Takes Down an Administrator at Michigan State (Wall Street Journal June 25)

Scholar forced to resign over study that found police shootings not biased against blacks (The College Fix)

On Steve Hsu and the Campaign to Thwart Free Inquiry (Quillette)

Michigan State University VP of Research Ousted (Reason Magazine, Eugene Volokh, UCLA)

Research isn’t advocacy (NY Post Editorial Board)

Podcast interview on Tom Woods show (July 2)

College professor forced to resign for citing study that found police shootings not biased against blacks (Law Enforcement Today, July 5)

"Racist" College Researcher Ousted After Sharing Study Showing No Racial Bias In Police Shootings (ZeroHedge, July 6)

Twitter mob: College researcher forced to resign after study finding no racial bias in police shootings (Reclaim the Net, July 8)

Horowitz: Asian-American researcher fired from Michigan State administration for advancing facts about police shootings (The Blaze, July 8)

I Cited Their Study, So They Disavowed It: If scientists retract research that challenges reigning orthodoxies, politics will drive scholarship (Wall Street Journal July 8)

Conservative author cites research on police shootings and race. Researchers ask for its retraction in response (The College Fix, July 8)

Academics Seek to Retract Study Disproving Racist Police Shootings After Conservative Cites It (Hans Bader, CNSNews, July 9)

The Ideological Corruption of Science (theoretical physicist Lawrence Krauss in the Wall Street Journal, July 12)

Foundation for Individual Rights in Education: "chilling academic freedom" (Peter Bonilla, July 22)

Genetic architecture of complex traits and disease risk predictors

The published version of the paper below is now available here:
www.nature.com/articles/s41598-020-68881-8

Genetic architecture of complex traits and disease risk predictors

Soke Yuen Yong, Timothy G. Raben, Louis Lello & Stephen D. H. Hsu

Genomic prediction of complex human traits (e.g., height, cognitive ability, bone density) and disease risks (e.g., breast cancer, diabetes, heart disease, atrial fibrillation) has advanced considerably in recent years. Using data from the UK Biobank, predictors have been constructed using penalized algorithms that favor sparsity: i.e., which use as few genetic variants as possible. We analyze the specific genetic variants (SNPs) utilized in these predictors, which can vary from dozens to as many as thirty thousand. We find that the fraction of SNPs in or near genic regions varies widely by phenotype. For the majority of disease conditions studied, a large amount of the variance is accounted for by SNPs outside of coding regions. The state of these SNPs cannot be determined from exome-sequencing data. This suggests that exome data alone will miss much of the heritability for these traits—i.e., existing PRS cannot be computed from exome data alone. We also study the fraction of SNPs and of variance that is in common between pairs of predictors. The DNA regions used in disease risk predictors so far constructed seem to be largely disjoint (with a few interesting exceptions), suggesting that individual genetic disease risks are largely uncorrelated. It seems possible in theory for an individual to be a low-risk outlier in all conditions simultaneously.

There are a lot of detailed results in the paper, but two main points should be emphasized:

1. Much of the genetic risk identified in polygenic predictors is outside genic (protein coding) regions, and not accessible through exome sequencing.

2. The DNA regions used in disease risk predictors so far constructed seem to be largely disjoint, suggesting that most genetic disease risks are largely uncorrelated. It seems possible in theory for an individual to be a low-risk outlier in all conditions simultaneously.

The space of genetic variation is high dimensional, and extends far beyond individual (protein coding) genes. Intuitions about strong pleiotropy are likely wrong -- they were developed before we knew anything about real genetic architectures. There seem to be many causal variants that can be independently modified.


The bioRxiv (preprint) version of the paper was discussed in these earlier posts:

Live Long and Prosper: Genetic Architecture of Complex Traits and Disease Risk Predictors

Pleiotropy: Myths and Reality

From the Peiotropy post above:

1. Regions of DNA correlated to different disease risks are largely disjoint.

2. It is plausible that causal genetic variants lie in these regions. For example, the predictor SNPs themselves could be causal, or they could tag (be highly correlated in state with) nearby causal variants.

3. Hypothetically, one could edit these causal variants independently, making the beneficiary simultaneously low risk for many conditions. The number of standard deviations of effect size in the polygenic score for each disease that can be modified independently (i.e., without affecting other disease risks or traits) is large and can be directly estimated from our results.

As the figure below (source) makes clear, a few SD change (e.g., ~5 SD, from 99th percentile to 1st percentile) in polygenic score for a given disease risk can lead to a 10x or possibly 100x decrease in absolute probability in having the condition. Our results suggest that the amount of variance available for engineering is much greater than this.

Some orders of magnitude:

1E07 or ~10M common SNP differences between two individuals.

1E04 or ~10k SNPs (on average; could be much fewer) control most of the common variance for a typical complex trait.

So, in principle, there could be 1E03 or ~1k entirely independent complex traits with zero pleiotropy between them. These might include dozens of common disease risks, ~100 cosmetic traits, including facial and body morphology parameters, dozens of psychometric variables, including personality traits, etc. Clearly, individual differences are well accommodated by a ~1k dimensional phenotype space embedded in a ~10M dimensional space of genetic variants.

Of course, it is an unrealistic idealization for the traits to be entirely independent in genetic architecture. We expect that some genetic variants affect more than one trait. But our results suggest that a significant part of the genetic variance of each trait can be modified (e.g., via editing) independently of the other traits. This is simply a consequence of high dimensionality.

The Ideological Corruption of Science (Lawrence Krauss in the Wall Street Journal)

Theoretical Physicist Lawrence Krauss writes in the Wall Street Journal.

WSJ: In the 1980s, when I was a young professor of physics and astronomy at Yale, deconstructionism was in vogue in the English Department. We in the science departments would scoff at the lack of objective intellectual standards in the humanities, epitomized by a movement that argued against the existence of objective truth itself, arguing that all such claims to knowledge were tainted by ideological biases due to race, sex or economic dominance.

It could never happen in the hard sciences, except perhaps under dictatorships, such as the Nazi condemnation of “Jewish” science, or the Stalinist campaign against genetics led by Trofim Lysenko, in which literally thousands of mainstream geneticists were dismissed in the effort to suppress any opposition to the prevailing political view of the state.

Or so we thought. In recent years, and especially since the police killing of George Floyd in Minneapolis, academic science leaders have adopted wholesale the language of dominance and oppression previously restricted to “cultural studies” journals to guide their disciplines, to censor dissenting views, to remove faculty from leadership positions if their research is claimed by opponents to support systemic oppression.

... At Michigan State University, one group used the strike to organize and coordinate a protest campaign against the vice president for research, physicist Stephen Hsu, whose crimes included doing research on computational genomics to study how human genetics might be related to cognitive ability—something that to the protesters smacked of eugenics. He was also accused of supporting psychology research at MSU on the statistics of police shootings that didn’t clearly support claims of racial bias. Within a week, the university president forced Mr. Hsu to resign.

... Shortly after Mr. Hsu resigned, the authors of the psychology study asked the Proceedings of the National Academy of Science to retract their paper—not because of flaws in their statistical analysis, but because of what they called the “misuse” of their article by journalists who argued that it countered the prevailing view that police forces are racist. They later amended the retraction request to claim, conveniently, that it “had nothing to do with political considerations, ‘mob’ pressure, threats to the authors, or distaste for the political views of people citing the work approvingly.” As a cosmologist, I can say that if we retracted all the papers in cosmology that we felt were misrepresented by journalists, there would hardly be any papers left.

Actual censorship is also occurring. A distinguished chemist in Canada argued in favor of merit-based science and against hiring practices that aim at equality of outcome if they result “in discrimination against the most meritorious candidates.” For that he was censured by his university provost, his published review article on research and education in organic synthesis was removed from the journal website, and two editors involved in accepting it were suspended.

An Italian scientist at the international laboratory CERN, home to the Large Hadron Collider, had his scheduled seminar on statistical imbalances between the sexes in physics canceled and his position at the laboratory revoked because he suggested that apparent inequities might not be directly due to sexism. A group of linguistics students initiated a public petition asking that the psychologist Steven Pinker be stripped of his position as a Linguistics Society of America Fellow for such offenses as tweeting a New York Times article they disapproved of.

Whenever science has been corrupted by falling prey to ideology, scientific progress suffers. This was the case in Nazi Germany, the Soviet Union—and in the U.S. in the 19th century when racist views dominated biology, and during the McCarthy era, when prominent scientists like Robert Oppenheimer were ostracized for their political views. To stem the slide, scientific leaders, scientific societies and senior academic administrators must publicly stand up not only for free speech in science, but for quality, independent of political doctrine and divorced from the demands of political factions.

Mr. Krauss a theoretical physicist, is president of the Origins Project Foundation and author of “The Physics of Climate Change,” forthcoming in January.

Media coverage:

A Twitter Mob Takes Down an Administrator at Michigan State (Wall Street Journal June 25)

Scholar forced to resign over study that found police shootings not biased against blacks (The College Fix)

On Steve Hsu and the Campaign to Thwart Free Inquiry (Quillette)

Michigan State University VP of Research Ousted (Reason Magazine, Eugene Volokh, UCLA)

Research isn’t advocacy (NY Post Editorial Board)

Podcast interview on Tom Woods show (July 2)

College professor forced to resign for citing study that found police shootings not biased against blacks (Law Enforcement Today, July 5)

"Racist" College Researcher Ousted After Sharing Study Showing No Racial Bias In Police Shootings (ZeroHedge, July 6)

Twitter mob: College researcher forced to resign after study finding no racial bias in police shootings (Reclaim the Net, July 8)

Horowitz: Asian-American researcher fired from Michigan State administration for advancing facts about police shootings (The Blaze, July 8)

I Cited Their Study, So They Disavowed It: If scientists retract research that challenges reigning orthodoxies, politics will drive scholarship (Wall Street Journal July 8)

Conservative author cites research on police shootings and race. Researchers ask for its retraction in response (The College Fix, July 8)

Academics Seek to Retract Study Disproving Racist Police Shootings After Conservative Cites It (Hans Bader, CNSNews, July 9)

The Ideological Corruption of Science (theoretical physicist Lawrence Krauss in the Wall Street Journal, July 12)

Foundation for Individual Rights in Education: "chilling academic freedom" (Peter Bonilla, July 22)

Wall Street Journal: Moral Panic at MSU

Professor Cesario and co-authors have retracted their PNAS paper Officer characteristics and racial disparities in fatal officer-involved shootings. I know that they had a difficult time with this decision. See further below for WSJ coverage.

My understanding from communication with Cesario is that he and his co-authors stand by the data and statistical analysis used in their paper. He also stands by the remarks in this Manifold (podcast) interview: Joe Cesario on Police Decision Making and Racial Bias in Deadly Force Decisions.

See this statement about the retraction from Cesario and co-author:

... One problem with such benchmarking approaches is that debate arises about whether it is more informative to compare the number of civilians shot to overall population proportions or to proxies for violent crime proportions. Indeed, one will obtain different results depending on what one thinks is the relevant comparison group: calculating P(shot|race) for the entire population will likely show evidence of anti-Black disparity, whereas calculating P(shot|race) for the pool of civilians who have violent interactions with the police will likely show no evidence of anti-Black disparity.

It is in this context that Johnson et al. (2019) was produced. Rather than debating which pool of civilians is the correct comparison group, we tested a less broad question that could be answered with current data: Is there a relationship between the race of officers and the civilians they fatally shot? However, this does not address the larger question of how race impacts the probability of being shot by police.

Cesario et al. maintain that the results described in the second paragraph above still stand: i.e., race of officer does not affect race of civilians shot. The issue is whether the result can be used to infer something about the conditional probability P(shot|race) discussed in the first paragraph. Other papers by Cesario, and his own broad conclusions from years of research in this area, suggest that P(shot|race) does NOT show the level of bias sometimes claimed by activists or in the media.

Note:

1. The PNAS paper that has been retracted is one out of several produced by Cesario and collaborators. In the podcast show notes we linked to Is There Evidence of Racial Disparity in Police Use of Deadly Force? Analyses of Officer-Involved Fatal Shootings in 2015–2016, which appeared in the journal Social Psychological and Personality Science. Cesario tells me that the following language from that paper still stands -- it is unaffected by the PNAS retraction:

When adjusting for crime, we find no systematic evidence of anti-Black disparities in fatal shootings, fatal shootings of unarmed citizens, or fatal shootings involving misidentification of harmless objects. Multiverse analyses showed only one significant anti-Black disparity of 144 possible tests. Exposure to police given crime rate differences likely accounts for the higher per capita rate of fatal police shootings for Blacks, at least when analyzing all shootings.

2. Some of their work is based on statistical analysis of police incident data, but other work is based on simulator studies of actual police officers under stress.

The small amount of funding that Professor Cesario received from my office in 2016 was for simulator studies (specifically, to cover production costs for realistic video of police stops). The request for funding was strongly endorsed by the Associate Dean for Research in the College of Social Science at MSU, and the College of Social Science matched the contribution from my office. The project was interdisciplinary, in collaboration with researchers in our School of Criminal Justice. Professor Cesario has also received National Science Foundation grants to study this topic. This 2018 award, Understanding Race Bias in the Decision to Shoot with an Integrated Model of Decision Making, is for $620k.

Obviously this is very important research and should not be politicized.

WSJ: The Proceedings of the National Academy of Sciences is a peer-reviewed journal that claims to publish “only the highest quality scientific research.” Now, the authors of a 2019 PNAS article are disowning their research simply because I cited it.

Psychologists Joseph Cesario of Michigan State and David Johnson of the University of Maryland analyzed 917 fatal police shootings of civilians from 2015 to test whether the race of the officer or the civilian predicted fatal police shootings. Neither did. Once “race specific rates of violent crime” are taken into account, the authors found, there are no disparities among those fatally shot by the police. These findings accord with decades of research showing that civilian behavior is the greatest influence on police behavior.

In September 2019, I cited the article’s finding in congressional testimony. I also referred to it in a City Journal article, in which I noted that two Princeton political scientists, Dean Knox and Jonathan Mummolo, had challenged the study design. Messrs. Cesario and Johnson stood by their findings. Even under the study design proposed by Messrs. Knox and Mummolo, they wrote, there is again “no significant evidence of anti-black disparity in the likelihood of being fatally shot by the police.”

My June 3 Journal op-ed quoted the PNAS article’s conclusion verbatim. It set off a firestorm at Michigan State. The university’s Graduate Employees Union pressured the MSU press office to apologize for the “harm it caused” by mentioning my article in a newsletter. The union targeted physicist Steve Hsu, who had approved funding for Mr. Cesario’s research. MSU sacked Mr. Hsu from his administrative position. PNAS editorialized that Messrs. Cesario and Johnson had “poorly framed” their article—the one that got through the journal’s three levels of editorial and peer review.

Mr. Cesario told this page that Mr. Hsu’s dismissal could narrow the “kinds of topics people can talk about, or what kinds of conclusions people can come to.” Now he and Mr. Johnson have themselves jeopardized the possibility of politically neutral scholarship. On Monday they retracted their paper. They say they stand behind its conclusion and statistical approach but complain about its “misuse,” specifically mentioning my op-eds.

The authors don’t say how I misused their work. Instead, they attribute to me a position I have never taken: that the “probability of being shot by police did not differ between Black and White Americans.” To the contrary, I have, like them, stressed that racial disparities in policing reflect differences in violent crime rates. The only thing wrong with their article, and my citation of it, is that its conclusion is unacceptable in our current political climate.

This retraction bodes ill for the development of knowledge. If scientists must disavow their findings because they challenge reigning orthodoxies, then those orthodoxies will prevail even when they are wrong. Political consensus will drive scholarship, and not the reverse. The consequences for the policing debate are particularly dire. Researchers will suppress any results that contravene the narrative about endemic police racism. That narrative is now producing a shocking rise in shootings in American cities. The victims, including toddlers, are almost exclusively black.

Ms. Mac Donald is a fellow at the Manhattan Institute.

Media coverage:

A Twitter Mob Takes Down an Administrator at Michigan State (Wall Street Journal June 25)

Scholar forced to resign over study that found police shootings not biased against blacks (The College Fix)

On Steve Hsu and the Campaign to Thwart Free Inquiry (Quillette)

Michigan State University VP of Research Ousted (Reason Magazine, Eugene Volokh, UCLA)

Research isn’t advocacy (NY Post Editorial Board)

Podcast interview on Tom Woods show (July 2)

College professor forced to resign for citing study that found police shootings not biased against blacks (Law Enforcement Today, July 5)

"Racist" College Researcher Ousted After Sharing Study Showing No Racial Bias In Police Shootings (ZeroHedge, July 6)

Twitter mob: College researcher forced to resign after study finding no racial bias in police shootings (Reclaim the Net, July 8)

Horowitz: Asian-American researcher fired from Michigan State administration for advancing facts about police shootings (The Blaze, July 8)

I Cited Their Study, So They Disavowed It: If scientists retract research that challenges reigning orthodoxies, politics will drive scholarship (Wall Street Journal July 8)

Conservative author cites research on police shootings and race. Researchers ask for its retraction in response (The College Fix, July 8)

Academics Seek to Retract Study Disproving Racist Police Shootings After Conservative Cites It (Hans Bader, CNSNews, July 9)

The Ideological Corruption of Science (theoretical physicist Lawrence Krauss in the Wall Street Journal, July 12)

Foundation for Individual Rights in Education: "chilling academic freedom" (Peter Bonilla, July 22)

"Preposterously Flimsy" -- podcast interview on Twitter Mob and MSU Moral Panic

"Steve Hsu is professor of theoretical physics at Michigan State University. Until recently, he was vice president for research and graduate studies. Despite over 1000 letters on his behalf including from top academics like Harvard's Steven Pinker, the administration caved to a Twitter mob and asked for Hsu's resignation. The reason is so preposterously flimsy that you just need to hear it for yourself."

Much more background here, including an article from the Wall Street Journal. (Note, there are ~2000 signatories to the letter of support, and many dozens of individual letters from scientists and professors that have been sent to the MSU President on my behalf.)

Here is one of the 150+ comments on the YouTube page:

Media coverage:

A Twitter Mob Takes Down an Administrator at Michigan State (Wall Street Journal June 25)

Scholar forced to resign over study that found police shootings not biased against blacks (The College Fix)

On Steve Hsu and the Campaign to Thwart Free Inquiry (Quillette)

Michigan State University VP of Research Ousted (Reason Magazine, Eugene Volokh, UCLA)

Research isn’t advocacy (NY Post Editorial Board)

Podcast interview on Tom Woods show (July 2)

College professor forced to resign for citing study that found police shootings not biased against blacks (Law Enforcement Today, July 5)

"Racist" College Researcher Ousted After Sharing Study Showing No Racial Bias In Police Shootings (ZeroHedge, July 6)

Twitter mob: College researcher forced to resign after study finding no racial bias in police shootings (Reclaim the Net, July 8)

Horowitz: Asian-American researcher fired from Michigan State administration for advancing facts about police shootings (The Blaze, July 8)

I Cited Their Study, So They Disavowed It: If scientists retract research that challenges reigning orthodoxies, politics will drive scholarship (Wall Street Journal July 8)

Conservative author cites research on police shootings and race. Researchers ask for its retraction in response (The College Fix, July 8)

Academics Seek to Retract Study Disproving Racist Police Shootings After Conservative Cites It (Hans Bader, CNSNews, July 9)

The Ideological Corruption of Science (theoretical physicist Lawrence Krauss in the Wall Street Journal, July 12)

Foundation for Individual Rights in Education: "chilling academic freedom" (Peter Bonilla, July 22)