text:
Coronavirus - Things You CANNOT Say About It
Sat, 28 Mar 2020 18:17 UTC
© Corbett Report
Pssst. You.
Are you interested in talking about . . . things? You know, the kind of things that we're not allowed to talk about anymore? You know, since the . . . uhhh . . . "The Event"?
You are? Great. I mean, you might have noticed things are getting a bit hairy out there. As in, you're likely to get your head bitten off for daring to suggest that things may not be totally ok with the "new normal."
It seems all these new social norms and cultural taboos that have arisen in the past few weeks have also created a raft of new thoughtcrimes: Things that must not be spoken for fear of being expelled from polite society . . . or worse.
That's why it's so vitally important for us to speak out about our concerns before these socially-policed thoughtcrimes become literal crimes. As I'm sure you know, if these new social norms are not confronted, voicing dissent will soon become impossible.
So, allow me to voice some thoughtcrimes of my own. But be forewarned: I assure you that you will find at least some of my ideas to be offensive. You will disagree with them strongly. You will become irate.
The real question is: What are you going to do to those voicing opinions you disagree with? Engage in dialogue with them? Or demand that agents of the state scrub their speech from the internet and lock them in a cage for their thoughtcrime?
Well, either way, I've already committed thoughtcrime numerous times in recent weeks, I might as well share them with you. Are you ready? Let's go.
1 - We have met the enemy . . . and it is our neighbors
People imagine that when the boots-on-the-ground tyranny arrives, it will be enforced by the police or the military. Newsflash: the boots-on-the-ground tyranny is here, and it is being enforced by your neighbors, Joe Sixpack and Jane Soccermom.
Need proof? How about all the new "snitch lines" that are opening up in city after city and state after state all around the globe to help good citizens tattle on neighbors who aren't practicing proper social distancing?
That's right. It's not just guys yelling out their windows in Brooklyn anymore. Now whenever you see someone within two meters of someone else it is your duty as a loyal citizen of the Brave New World Order to actively report them to the authorities so that they can be dealt with by Big Brother. Rest assured, a score card is being compiled for each jurisdiction, and the powers-that-shouldn't-be are keeping a list of who's being naughty or nice (Good job, Minnesota!).
Still, while we can all unequivocally and universally agree 100% with the idea that anyone who physically approaches another human being in this Year of our Virus 2020 deserves to be charged with manslaughter for their heinous act, maybe, just mayyyyyybe — and I'm just spitballing, so forgive me if this seems brash — we're heading into dangerous territory here. You know, what with the social distancing Stasi becoming the enforcers of our new police state nightmare and all. Call me crazy.
2 - Doctors are the new soldiers.
When 9/11 happened, there was a marked and notable intensification in the propaganda glorifying the American military. Not that such propaganda didn't exist before, but it was nothing like what we've seen since "the day that changed everything." Yes, the hero worship of veterans is one of the hallmarks of the Age of Terror that 9/11 ushered in .
So if this plandemic is the new 9/11, what's the new hero worship? Well, it should be obvious by now: Doctors are the new soldiers. Now we must dutifully show our appreciation for the brave medical workers on the front lines of this new war . . . or face yet more social castigation.
You may have noticed the interesting phenomenon making its way around the world. I call it "The Totally Spontaneous Balcony Applause Phenomenon." Yes, completely out of the blue, all the people under lockdown have decided to show their appreciation for the valiant doctors and nurses in this heroic struggle by going to their balcony at a pre-appointed time and applauding. And no, this totally spontaneous phenomenon is not just occurring in one or two countries. Or three or four countries. But in seemingly every country around the globe.
Just like that. Just out of the blue. Must be something in the zeitgeist, I guess.
Now you'll forgive me for being out of the loop, but as you know the corona madness has not quite made its way to Japan yet. (But, precisely as I predicted, the very same day that the Tokyo 2020 Games were postponed the Tokyo Governor suddenly became gravely concerned about her city, and they are now going to "have to" lockdown Tokyo unless the poor plebs behave.) So I don't know exactly how people decide on the right time to go to their balcony to applaud. Is it done by vote? What if I'm a few minutes late? Will people think I'm clapping for something else? What exactly is the etiquette here?
Here's my thoughtcrime: I find these displays creepy and off-putting. I find the glorification of doctors and nurses unsettling. Not because I think they are all quacks. Not because I think they are all evil. Not because I am not grateful for the work that (some) doctors do (some of the time). Not because I don't recognize the enormous stress that these doctors and nurses are under right now. But because this socially engineered adoration is going to be used to push an agenda exactly like the glorification of veterans was used to push the militarism agenda of the post-9/11 years.
This time, we are being asked to glorify doctors and nurses because these are the same trusted experts whose authority cannot be questioned who are going to be giving you the vaccine. You know, The Vaccine. The one that will bring an end to the then 18-month long psychological siege that we are being placed under.
What?? You still question the vaccines? You still dare to defy the authority of these brave doctors and nurses who risked their lives for us? You can't say that, you disgusting conspiracy mongering throughtcriminal, you!
Be honest, you know that this push is coming. And they are getting the public to sign on with all these "spontaneous" balcony applause sessions. So perhaps you'll forgive me for not joining in.
3 - I do not trust a single one of the numbers being reported about this outbreak.
I am still baffled by the attention that otherwise sane human beings are given to the latest reported numbers from this or that health agency about the scourge of Covid-19. People are throwing around CFRs and R0s like they've been studying epidemiology their whole lives. In truth, they're just regurgitating whatever they saw on CNN or were told in the latest Governor Cuomo press conference.
So what do we make of the baffling discrepancy in death rates from Covid-19 between different countries? Why is Italy's death rate from the disease a staggering 10% while China's is more like 4%? And what does that mean for the 70% of humanity that "experts" warn will be infected by this virus?
And while we're at it, why don't we ask some equally meaningful questions, like: What color is the Easter Bunny? How many angels can dance on the head of a pin? And just how tasty is the cheese that the moon is made of, anyway?
As I demonstrated weeks ago, methods for diagnosing this disease vary so widely from country to country that making comparisons between countries isn't even like comparing apples and oranges. It's like comparing apples and aardvarks. And diagnosing a particular type of viral infection via CT scan? How can we possibly trust the infection numbers that are being generated by such methods?
All of that would make the calculation of mortality rates for this disease problematic enough. But, to make matters worse, we don't even have an accurate tally of the number of people who have died from Covid-19. Take the infamous Italian example, for instance. We're told that the staggering death rates in Italy (roughly 10% if we go by the official numbers at press time) are a sign of just how deadly this new virus can be.
. . . But there's some problems with those numbers. As Prof Walter Ricciardi — scientific adviser to Italy's minister of health — recently revealed, "The way in which we code deaths in our country is very generous in the sense that all the people who die in hospitals with the coronavirus are deemed to be dying of the coronavirus."
So how many of the people who are reported as "Covid-19 deaths" in Italy actually had coronavirus listed as their cause of death? Just 12 per cent. What's more, according to the Italian government's own report, half of those who died had three or more other diseases at the time of the death. Nearly 80 per cent had at least two other diseases that they were fighting when they died. Only 1.7 per cent of those who died had no other disease.
But why listen to James Corbett, conspiracy theorist, or those silly Italian government health advisors on this matter? Well, I'm not alone in this suspicion of the official numbers. It turns out the "Our World in Data" research group that has been attempting to keep track of the coronavirus numbers has stopped using the World Health Organization's data because "we found many errors in the data published by the WHO when we went through all the daily Situation Reports."
And John Ioannides — who Corbett Report listeners will remember launched the replication crisis in science with his landmark 2005 paper on "Why Most Published Research Findings Are False" — has recently come out questioning whether the current Covid-19 response is "A fiasco in the making." As Ioannides observes:
"The data collected so far on how many people are infected and how the epidemic is evolving are utterly unreliable. Given the limited testing to date, some deaths and probably the vast majority of infections due to SARS-CoV-2 are being missed. We don't know if we are failing to capture infections by a factor of three or 300. Three months after the outbreak emerged, most countries, including the U.S., lack the ability to test a large number of people and no countries have reliable data on the prevalence of the virus in a representative random sample of the general population."
After this current madness passes, people will view the public's blind acceptance of these practices in the same way that we look at the public's blind acceptance of bloodletting and other methods of medical chicanery from times past.
4 - The death of a 91 year-old is a family tragedy, not an event of international concern
OK, so you still insist on taking these phony baloney numbers seriously? Then let's take another looks at that Italian report on those dying with (not of) Covid-19.
The report tells us that the median age of those who have been pronounced dead with (not of) Covid-19 is 78. To put that number in perspective, the average life expectancy in Italy is 82.8.
That means those who are dying with (not of) the disease are within years of reaching the average life expectancy (and, let's not forget, they are also suffering in the vast majority of cases from at least two other diseases). I venture to say that a similar panic could be raised about just about any viral disease in circulation if it was being reported in the same way as this coronavirus is being reported.
Since we're committing thoughtcrimes here, let's be blunt: "Elderly Patient With Multiple Complications Dies After Contracting Respiratory Illness" is NOT a news story. It's a daily fact of life.
But in fact, it is a news story. I have been keeping tabs on how the Canadian MSM have been covering the pandemic panic and saw a segment on one of the national news broadcasts about a woman whose 91-year old mother died in a nursing home. It was implied that this 91-year old woman's life was tragically cut short by the coronavirus and, to make matters worse, her daughter was unable to hold a funeral or service for her mother because Canada is currently under lockdown. I don't know if I have lost touch with reality or everyone else has, but let me reiterate: This is NOT a news story.
Don't get me wrong: Any such death is doubtless a tragedy for the family involved. My heart genuinely goes out to all those who lose their relatives in such circumstances. But this is not something that we upend our entire civilization over. We do not stop all productive human activity on the planet, collapse the economy, send millions upon millions of people to the unemployment line, institute lockdowns, and begin talking about mandatory vaccinations, internal passports and other abrogations of essential human freedoms on such a basis.
In fact, if I were to be dying at the age of 78 due to some viral respiratory illness along with my other 78-year old cohorts, I can guarantee that I would be outraged that the powers-that-shouldn't-be were using my death to upend the liberties that I had spent my life attempting to defend. It is disgusting.
"But what about the young people who die of the disease?" you ask. Fair enough. Again, according to the official reports (which, let me remind you, should not be trusted), there are people under the age of 78 who are dying from the disease as well, albeit in much smaller numbers. And, according to the "models" from the "experts" (who, let's remember, are right about everything), there could be hundreds of thousands more deaths before this pandemic runs its course.
Well, that brings me to my ultimate thoughtcrime:
5 - The idea that disease and death are unnatural or avoidable is anti-human
Sometimes they die of car accidents. Sometimes they die of work-related mishaps. Sometimes they die of old age. Sometimes they die under extremely questionable circumstances while trying to shed light on information that is uncomfortable for the deep state. And, yes, sometimes they die of respiratory illnesses during viral pandemics.
I'll go one step further: Our mortality makes us who we are. Humans are blessed and cursed with a knowledge of our own fate. No one makes it out of this life alive. And so the question of what we do with our lives becomes paramount.
But more and more, death is being removed from life. Our elderly are shipped off to nursing homes to whither away so that we don't have to face aging. The funerary industry is neat and anti-septic. Death has become an abstraction. Something that happens somewhere out there, to other people. Not to us, though, surely.
But this entire pandemic madness seems to be predicated on the notion that disease and death are somehow avoidable. That we have conquered such things. Or, at least, that no new disease could ever possibly arise (bioengineered or not) to upset our perfect balance with nature. I mean, yes, many people die of the flu every year, but that doesn't count. That's not new.
This is not to say that we shouldn't work to cure diseases and improve our health. Quite the contrary. It's just that this current bout of hysteria seems almost anti-human; as if we should be able to transcend our mortal humanity.
CJ Hopkins, in his characteristically humorous way, points out the absurdity of this "War on Death" in his latest article:
"We can't let these Russian dissension sowers, neo-Nazi accelerationists, and coronavirus-sympathizers confuse us. They want to convince us that Death is, yes, scary, and sad, but inevitable, and natural. How utterly heartless and insane is that?! No, we need to close our minds to that nonsense. People are dying! This is not normal! Death is our enemy! We have to defeat it! We need to hunt down and neutralize Death! Root it out if its hidey hole and hang it like we did with Saddam!"
I don't know why the idea that death is a part of life should be controversial. But, given that even a respected blogger like Craig Murray can be largely lambasted by his own audience for daring to post similar musings, I suppose that it is. I don't know anymore. Perhaps I'm off my rocker.
All I know is that the room to express dissent on these topics is fast disappearing. It's time for those of us who can tolerate thoughtcrime to circle the wagons. The Thought Police are closing in.
So maybe you disagree with me. Maybe you're offended by what I say. Maybe you have your own thoughtcrimes that you're afraid to express. But if we don't engage in dialogue about these ideas now, what are the chances that this information will be easier to share in the future?
So what's your thoughtcrime? Share it with the community in the comments below.
Speedz · 2020-03-29T19:50:19Z
I love this article. I have felt this way about every point.
You worded it in ways I never even thought, but especially point 2. I felt uneasy about this but could not quite put my finger on why.
But 5 and 5....."Dead on" pun intended.
Speedz 4 and 5 oops (Please can we have comment edit already?
Lana25 · 2020-04-03T17:01:58Z
Speedz Hey there ! Looking for some fun to get into? Me too! Let's get to know each other on a much more personal level [Link]
Michael L · 2020-03-29T20:02:16Z
It makes me think that we are living in "Bird Box" world. It would seem that those who having been "patriotically" watching the MSM have gone insane and are on a path to some sort of suicide. Those who did not belly up to the bar have been sparred, somewhat. From the insanity, yes, from the zombie whirling dervish patriot? Nope.
My thought crimeS? Just this morning would fill more words than the Encyclopedia Britannica.
Joyly · 2020-03-29T23:31:00Z
My thoughts: this crises is the culmination of a mind control exercise to grab control of the masses, destroy their financial future, to destroy their social and recreational life, to restrict their life choices and personal freedoms, all for the benefit and safety of the elite robbers. The elites lose nothing; governments are already conditioned to bail them out. Militarized police are already in place to enforce martial law. Technology is already distributed to provide total surveillance. Now they are turning neighbour against neighbour. There are already security employees in place in our grocery stores (in Australia). The health system was already at capacity. The schools have been shut. Even churches are closed. What else can they take away? By the tone of this article, our right to think. But we must not give in and accept any of this. By enacting all this, governments have shown they do not think for themselves, they are but puppets for the elite; they too are being manipulated.
In this chaos there is always a possibility for change , but where are the leaders we need to do this?
Joyly As far as I've read or am aware, ye are the only SOTTite who's ever got a confirmed postive of the test.
To the extent that any here are scared of this (such is hard to avoid, even for we SOTTfolk), I'd bet that most folks who've read your posts are unaware of your success in fighting it. I think you point it out and perhaps linking back to whatever article (or two: three) you said that
at would help SOTTfolk awaken others, and also be less afraid (though we ain't the panicky types.) Also, I'm confident it would give your words validly due gravitas.
But that's just me talking too much.
ReRan · 2020-04-01T13:03:59Z
Joyly Spot on, Joyly. I am so much more frightened by how this is playing with our freedoms, than I am of any disease on earth.
binra · 2020-04-01T19:14:10Z
Rowan Cocoan The test is meaningless.
What do you think it means?
That you are infected with a deadly disease?
It is being given meanings it does not merit.
Read up on the Swiss Doctor
And not it has updates each day at the bottom.
You can also read nomorefakenews.com
and off-guardian
People are fooled over and over again by the same ruses. If you were a scientist or a detective/investigate researcher/reporter - what questions would you need answered to make any sense of the statistics and the parameters? Or would you simply take everything at face value as if mistakes don't happen and people do not misrepresent or distort the facts?
If the test were meaningful as a diagnostic tool, you could well find the population is effectively already exhibiting immunity - ie CV is present and asymptomatic.
You could run a test where such trials were blinded and the testers required to QUANTIFY CV results and predict those with enough viral matter to indicate a likely cause of sickness - and then see if they score.
You could also run other tests for other viruses in CV 'positive' csses and find many of these in many of the population without clinical symptoms. This has been done. There are.
Which 'disease' do they die of? Toss a coin. Consult the goat's entrails.
Susceptibility to more adverse effects is likely a combination of treatments and existing conditions - including existing medications of poor diet (such as low or poor fats and low cholesterol as a potential example). There is also the issue of withheld treatments that do not harm - or cost much - and often make a significant difference in positive outcomes - IV VtC and similar.
Toxicity is not given its due under the more profitable and powerful leverage of virology. Once one has caught the mind virus one cant see anything else.
The end result is attacking our immune system - which taken out of medic-talk means - attacking our life support, our consciousness, our capacity to recognise what we are doing so as to change it.
The fear given power by disproportionate reaction is very dangerous as a whole raft of changes, social, political, medical and bio-technical.
But fear also opens the opportunity to see more than its reaction would otherwise allow. I see a choice, but I am not smug or pat about it. These are challenging times.
binra Am aware. (E.g., 'test' doesn't differentiate between 'old' CV and current, but all results are classified as Plannedemic 2020 (C2019) positive - helps 'em play with their stats/fool the proles.
I dare say (tho am skeptical even here) that one should be far more scared upon having a positive test for rabies than any of these propaganda supporting 'tests.' (Sic.)
Re J, I'd bet that she's listed in Austraila as 'positive for C19' as needed to promote they hysteria narrative.
Rovica · 2020-03-30T02:18:14Z
DOWN WITH ROCKERFELLER
Ahem... sorry I don't know what came over me.
People get sick. People die. Life goes on. (Not trying to dismiss the real pain of losing a loved one, just pointing out that it is a part of life.) Those who believe in truth should not fear death anyway... but there is a severe lack of truth in the world so it is not surprising.
gail · 2020-04-02T20:27:10Z
ABSOLUTELY..... What people should be inquiring about is, why is this FLU being touted as the end all be all to life, when in fact, it has one of the lowest rates of fatalities. What really is behind the FEAR being instilled.
Rovica "Ahem... sorry I don't know what came over me."
I'd say it's the truth overriding supposed - but improper - societal perceptions of what it's 'polite to say.' (But that SOB had no aversion to state that he wished/hoped to destroy the US Constitution, so I believe his proper description is a Nazi Pro One World Order AmeriKant Traitor to America.
RC, the truth is I did not have a test; I did not qualify. Testing was limited to those who have recently returned from overseas, or been in direct contact with a person who tested positive,
My son travelled to the US in Dec, Jan, and Feb for work. I looked after his kids in feb so his wife could travel with him, and on his return I got sick. He had mild symptoms (cough) and on this last flight home he was sitting near Aussie politician Peter Dutton who was tested positive. So I am pretty certain that my son was infected (or a carrier) and absolutely certain that I was infected. At 76 my age and my predisposition to bronchial conditions put me in a high risk zone.
But I am a very self responsible person ( was not always the case - spent my life expecting my husbands to take care of everything, all 3 of them) so I took myself home, self isolated and treated myself with plenty of hydration (good water), fresh fruit and salads, vitaminC and bi-carb soda and olive leaf extract taken 5 times a day. So I started to get better after a week, and am now just rebuilding my physical strength.
My experience was moderately severe, not mild, but I have experienced far worse. So based on my experience, I would say do not worry about it, just take good care of yourself. It was a wake up call for me not to just let it run it's course; I have done that in the past and been down and out for up to 4 weeks. Also fresh air, sunshine, and light exercise is crucial. And get your supplies of powdered vitaminC straight away. If I lived in the US I would have used elderberries to make a tea I used ginger, lime and honey.
Most importantly for me was to check in on my physical health, my immune strength which is only as good as your gut health, exercise, sun etc. and educate myself. Hope this helps someone.
Joyly Whatever the specifics of your situation, I am near certain that some agency added you as one (or maybe three?) person who was 'considered' as infected. (Also, the test cannot discriminate between C2019 and C2002! Despite this, all who the tests show as 'positive' (they don't have a test for the differences) are listed as positive for C19.)
Thus, your story is one that the wise will learn from.
You are my Sunshine!
parzival · 2020-04-01T06:45:49Z
What's Up With The Italian Mortality Rate? - Questions For Corbett: [Link]
Max has some interesting sections of videos with hospitals empty and testing centers empty... hmmm... [Link]
More good news! Citizen reporters go & do what the media won't! [Link]
Gauranga · 2020-04-01T07:36:43Z
Excellent article. From the beginning I smelled the rat, that something much bigger is going on here than a virus outbreak. Similar numbers of people die every flu season from a very similar virus.
Bill Gates, the WHO and pharmaceutical company are all part of this whole new world order movement that is driving this crime. And there is evidence for this:
3010cu · 2020-04-01T11:24:40Z
Good Morning y'all I agree with every statement the author made.It irks me to no end that I have neighbors who believe this BS.They
can't accept the fact that our so called betters are attempting to pull a fast one on us.
I am an old retired farmer nearing my use by date,I died once in 2008,so I know first hand death is an intragal part of life.
Reecus · 2020-04-01T12:27:02Z
We need to focus on solutions and figure a way out of this tyranny.
One major obstacle is the legal way 'they' have jurisdiction over us the people in each country.
It seems to occur soon after our natural birth when we are registered with the state.
We are presented then with our new trademarked name in full capitalization including the family name, we presume this to be our 'name' for the rest of our lives, just look at your bank account, passport, drivers license, titles to property.....
Change your legal standing with reference to your trademarked 'name' and you will lose all state privileges but you will gain back your natural rights under Common Law.
Everything seems to be hidden in plain sight, even the 'key to the house' celebration when we turn twenty one, that was a legal opportunity to escape their matrix but we did not grasp the circumstances.
Maybe now is the time to work together on this dilemma that seems insurmountable but is worth the effort to be free men & woman, especially for the children.
katesisco · 2020-04-01T13:01:26Z
I'm responsible now also after spending 25 years coddled by my husband; but the neighborhood cop gang eliminated any future easy by taking the retirement home he built, and kidnapping the grandchildren for which they expected the full ransom from a lawsuit I was to be manipuled into filing. Year after year of slights, slurs, slings and arrows failed to make me sue so the expected pay out for the kidnappers never happened. I miss him, the family closeness i'll never have, my community I earned my home and respect in, all gone.
The community that baits will rob their neighbor, robbing your neighbor chums the sharks.
Bezel Bub · 2020-04-01T13:35:29Z
Neither the Trump administration nor Congress has the legal authority to shut down American life without due process . Business closures, restrictions on assembly and movement, and quarantines are not constitutionally permissible under some magic "emergency" doctrine.
Bezel Bub I agree with your sentiment BB but so long as they 'own' our trademarked name (see my post above), and we presume our identity is linked to this grammatically incorrect usage of the English language (see Chicago Manual of Styles & the Oxford Manual of Styles), the deception will continue. We have to break this legal bondage before we have lawful access to our Common Law rights & the Constitution. Let all of us here brainstorm this, we have the capacity to undo this deception.
Agron · 2020-04-01T13:52:29Z
Reecus It can be done only with ample number of honest lawyers who will represent the collective peoples. Or with force. No other way...
But I think the many of our rullin "elites" are getting evicted and we will see a new system being put in place. Lets hope it wont be anything like they are doing in China tho!
Reecus The people have more power than they know...we *are* the power....but only if we stand together, and therein lies the problem. TPTB have done a masterful job of creating division in this country and I honestly do not see how we get to meaningful change without a lot of suffering and drama...
Bezel Bub I say the same thing. The Change Comes From Within and we all have the power to defend our moral integrity. Ofcourse bigger and richer countries will have far greater obstacles in this regard. But once this starts globaly it will be hard not to take notice and follow good examples. Remember that theres only 8500 of them vs 7-8 billion of us. And even the ones who sold out to the 1% can decide they will now do the opposite...
I suggest optimistic outlook in any situation, even if "life threatening".
Agron "ample number of honest lawyers"
While we have too many attorneys, speaking from experience, I hope you aren't seeking more than 1% of them.
ample number of honest lawyers
LOL..good luck with that...I heard they may have spotted one in the Galapagos Islands...
Bezel Bub muahah! varan lawyers
give those poor morally upstanding-swimming in cash guys some slack... they are just using the system for what its worth.
Btw. there are a few hones ones.. you just have to have a pendulum to find them ( and a helluva lot of luck!)
Bezel Bub Didn't see that you grabbed it too.
Agron The easiest way to find the honest ones is hold out garlic and a cross. Those ~0.5% who don't flee are the good one . . (occasionally oneS).
Rowan Cocoan I knew a guy that looked JUST like me who had to find 35 000 euro on that quest. 5th!!!!! lawyer was available by the Grace Of God! ANd tons of luck... Mountains that have a renome like Everest or even bigger
Agron I've met two folks who looked like me enough that they / I reacted and commented at the same time.
However, I also met my doppelganger, too. I was in Reno, and everyone at the Walmart REFUSED to believe that I was not the slot machine repairman. After about the 4th time being asked "Why aren't you fixing the slotmachines?" I gave up and started pretending I was him.
~ 20 minutes later, I had to show my ID to the cashier who still didn't believe me.
So, I walked into the parking lot. . . and there was the handsomest guy I'd ever seen! True story and it gets crazier from there. See the AB.
Rowan Cocoan One hairdreser once assured me that I was a model for KEUNE hair products and how th f didnt I say it before and bragged about it, how humble of me . If she knew I was into this she would have made me a part of her business hahahahah... she was adamant even when I said it cant be me.
LindaMay · 2020-04-02T16:06:37Z
It is said our hair are Spirit antenae
zman · 2020-04-01T14:29:59Z
Perhaps the sheeple will now realize (not) just what the Patriot Act and the Homeland Security scams were all about. They provide the vehicles to neutralize laws and to advance agendas...without any consensus of proof, just rumor and unsubstantiated innuendo (as in the past) will do. The fact that most Americans, not to mention the rest of the world, is being hoodwinked with funny numbers and the usual MSM misinformation shows that the majority still have their blinders on, unable to see the oncoming vehicle off to the side. Much as the OPCW was corrupted for 'the plan', so are all the state controlled public health resources. The danger to TPTB is that once people realize the scope of the propaganda war, the possibility of seeing through 9/11 becomes a real threat. I expect a full frontal attack on anyone voicing disagreement or questioning of facts in regards Covid19. Logic, facts and a true assessment of the situation is the wrench in the monkey works for them. Threats, innuendo, character assassination and ignoring are their tools and we can expect a volley of all in their pursuit of chaos and control. As PompAss said, this is a live fire exercise. He wasn't kidding.
zman that the majority still have their blinders on, TRUE!
AmandaMay · 2020-04-01T15:29:43Z
IMO this is a global psyop, and the virus is being used as cover for the central bankers (owners of the Federal Reserve, the hidden oligarchy that prints money out of thin air) to bring in the new digital financial system and Agenda ID2020.
*** And if you think this is a "conspiracy theory" look at this: [Link]
Greg Mannarino warning about central bankers NWO, digital, cashless society [Link]
"Look around you. A world in lock down. Cities, states, nations shutting down. Markets in free-fall. Moreover, NOTHING you are seeing is by accident! Something I have been telling you for YEARS. Understand.. A deliberate global meltdown is occurring by design. A second Great Depression is being THRUST upon us, yes thrust. Many, many people are going do die. For how many years my EXACT words have been "It Will Be Biblical." This is all being done to bring about a NEW financial system.. and methods of control over the remaining population beyond your wildest nightmares are going to occur... what we are seeing now is JUST THE BEGINNING.... GM
- IMO this is a DELIBERATE meltdown of the world economic and financial systems. The New America is now the epicenter of a new order which will sweep the globe, quickly. I also believe it is by design that The Fed. is flooding the world with dollars now in order to usher in a one world ALL digital currency... GM" (Please listen to GM on y/t, while the MSM and US govt have been terrorizing the American people with fear porn, GM has been warning about what the bankers have been up to and how we the people are being ROBBED!!! )
The virus will be used as a cover story for the central bankers bringing in their new world digital currency (cashless society). The MSM is already floating stories about "dirty money and coins" and that they could have the so-called virus on them.
Bill Gates' Quantum Dot Digital Tattoo Implant to Track COVID-19 Vaccine Compliance [Link]
The Coronavirus COVID-19 Pandemic: The Real Danger is "Agenda ID2020" -Peter Koenig (former World Bank economist) What is the infamous ID2020? It is an alliance of public-private partners, including UN agencies and civil society. It's an electronic ID program that uses generalized vaccination as a platform for digital identity. [Link] (please read entire article)
"Another hypothesis, at this point only a hypothesis, but a realistic one, is that along with the vaccination – if not with this one, then possibly with a later one, a nano-chip may be injected, unknown to the person being vaccinated. The chip may be remotely charged with all your personal data, including bank accounts – digital money. Yes, digital money that's what "they" are aiming at, so you really have no control any more over your health and other intimate data, but also over your earnings and spending. Your money could be blocked, or taken away – as a 'sanction' for misbehavior, for swimming against the stream. You may become a mere slave of the masters. Comparatively, feudalism may appear like a walk in the park ."
Lux · 2020-04-01T16:25:35Z
AmandaMay What is GM?
Lux " Greg Mannarino ". (AT first glance I guessed "Genetically Modified?" - wrong.)
Oh, sorry "GM"= Greg Mannarino--sometimes he leaves comments on his y/t and that's how he signs off "GM"
The MSM and US govt are LYING about the hospitals overflowing with patients, bodies piling up and long lines at testing centers!! Lots of citizen journalists are going to their local hospital and finding that they are ghost towns! I'm compiling a master list of citizen videos (updated daily) showing the empty hospitals here:
Heroic Citizens Going to Local Hospitals and Fact-Checking the LYING MSM [Link]
Nothing is going on, quiet, no ambulances racing in, empty, nothing happening…It's all BS!!!! The MSM is giving the impression that hospitals are war zone, bodies are piling up, we are in the medical crisis of our lives, yet when these people visit their local hospitals, NOTHING is happening, they are ghost towns!! Tons of ambulances just sitting their parked, EMTs hanging out and playing with their phones, etc, --The MSM and US govt are LYING!!!
***Please share the link, email it out, and wake up the sheep!! The bankers are moving ahead quickly with their agenda!! and their agenda is one world govt, one world currency, and Agenda ID2020!!!
Twitter full of EMPTY HOSPITAL VIDEOS!!!!!! People are WAKING UP!!!! Looks like people are waking up--twitter full of people posting EMPTY HOSPITAL VIDEOS!!!!!!
Twitter links posted here: [Link]
***Please share the link, email it out, and wake up the sheep!!
jenny4784 · 2020-04-01T17:38:05Z
I watched a couple of these, admittedly not all. The footage certainly calls into question the media description of a war zone, BUT there's a couple things I cant square this with:
1 - none of the people were allowed inside. They are not allowing visitors, so yes the parking lots and lobbies WOULD be empty. That doesnt mean the beds- or the ICU - are empty.
2 - two people that I personally know are nurses (one definitely in Boston, the other not sure, could be NH), describing on Facebook that they are working around the clock. They are taking pretty extreme de-contimination measures when they come home. They are re-iterating messages about ventilators and dying patients. (They're both "friends" from highschool, 20 years ago for me, so although I know them I cant really message them and start asking questions about what they're actually seeing, as much as I might like to). This leads me back to point 1 that just because the exterior is a ghost town doesnt mean it's not terrible in the ICU etc.
With that being said, I find these videos incredibly interesting. It has me contemplating accidents that might get me into the ER at Mass Gen...
Definitely makes me question the hysteria of some governors though.
Oh, actually a number of them were inside. This one goes inside:
Pandemic Hunting! Hospital or Ghost Town? [Link]
Royal Colombian, New Westminster, Canada: Dude goes to 2nd floor, empty Covid unit in nearly deserted hospital.
**** Please watch this one!!! It talks about fake virus deaths, with family members saying it was NOT the virus and that it was other health issues!!!
REALIST NEWS - More empty testing facilities and fake deaths [Link]
Okay, this one is worth going through, empty testing center, *** at 2 min, news article about dead 69- yr old retired Milwaukee police office, with family member saying the media and govt are LYING and that his/her father did not die of the virus and had other health issues. At 2:47, a family member saying grandmother did NOT die of the virus, and that it was Alzheimers, a case of aspiration pneumonia that worsened. At 3:26 Gene Della Salla surprised FL announced he's dead from virus when still alive!!! "reports of my death have been exaggerated." At 4:03, Miranda Matilda saying dead family member was on hospice at nursing home and already dying (it was NOT the virus)
Also---more on how they are playing with the numbers
Comment from [Link]
Adr- I have an update. The husband of one of my mom's friends who died of heart failure on Wednesday, he was 75 and already dying, didn't actually die of Covid-19. They had given him a test that had returned a positive result. Even though he was already dying of heart failure, they marked his cause of death as Covid-19 and added him to the state total. Since none of his close contacts tested positive, they retested his body for Covid. The test came back negative. Cause of death is now listed as heart failure. The state has not reduced the Covid death stat by one as of right now. That wouldn't look good. Nor would a story of doctors being so very wrong. I wonder how many deaths have followed that same pattern?
Below is what I've been thinking they have been doing to increase the numbers of deaths from corona (claiming that elderly who died of old age died of corona) (Yes, I'm saying they are LYING)
From https://www.zerohedge.com/geopolitical/hubei-residents-riot-after-quarantine-lifted-police-beaten-their-own-shield-cop-cars)
Geocentrist- Was speaking with a guy yesterday. He was telling me his wife knew two people who each had one of their grandparents pass away recently. Both had been "on the edge" and infirmed for quite a while. Covid-19 was automatically listed as their cause of death, in fact, he said the family of one of them actually called them out on listing it as the cause of death, since they had basically no contact with anyone, apparently it caused a bit of furor. Sounds like they are creating numbers
More screwing with the numbers, calling everything the scary virus ...
adr- So the morbidly obese spouse of someone my wife works with has been home from the hospital for four days now... She went to the hospital early this month for breathing problems and they couldn't find anything wrong with her. After they couldn't diagnose her, they decided to diagnose her with Covid-19. This was because she had direct contact with someone who was at the Biogen conference in Boston. She was put in isolation at the hospital. So people have spent the past three weeks thinking she had Covid. On the call this afternoon, she said that when she was actually tested for Covid, the test came back negative, but she is glad to be recovering and back at home. So she was basically in the hospital for complications of just being morbidly obese. Another Covid story bites the dust.
sh*t they put billions of folks on lockdown for a total scam...read and weep
worldwide # of deaths from fake corona Jan 2020 thru March 25, 2002 as compared to # of deaths from other causes Jan 2020 thru March 25, 2020 [Link]
Candace Owens@RealCandaceO The number one killer in America is Heart disease. 1,002 people a day. Did you know that if you die from heart disease right now, and they determine you to be an asymptomatic carrier of Covid-19 in your post-Mortem, they legally add your death to the #Coronavirus death toll? [Link]
Mike@mnkybrd Replying to @RealCandaceO I work in medical examiners office. Suicides and murders are being classified as covid deaths if they carry the anti body. It's a numbers game, and a sham. [Link]
From comment to Jason Goodman's y/t "Is the U.S. Petrodollar the Next Fatality of Coronavirus?" [Link]
Jeremy Gebhart Please listen: My friend's friend is an army nurse with 20 years experience currently working in New York who said NY is changing ALL deaths to Coronavirus regardless of the real causes of death . She has personally handled 3 cases having no relationship to Coronavirus that NY changed to be caused by COVID-19. Anyone smell corruption? This needs investigated immediately
Oh my GOD I am so scared...almost overnight the Chinese Plague has infected my cactus...its like something out of Invasion of the Body Snatchers....normally this might be a harbinger of spring, but I just KNOW it has something to do with the Kung Flu...I'm so scared, someone please hold me[Link]
Bezel Bub What's the car? A 1950s Ford/Lincoln? (Just a guess.)
Rowan Cocoan I forget...either a 1959 or 1960 Lincoln Continental...with the sliding power rear window..
Bezel Bub Wow! I guessed it!
Got yer back BB
Excellent & far more correct descriptive term - and well analyzed article - of what our (sic) governments have done:
It's Not a Recession—It's a Government-Imposed Shutdown of the Private Sector.
Also, excellent analysis showing that this is but a variant of 'Centralized Economic Planning'.
Is COVID-19 Even Real? [Link]
marckris · 2020-04-02T07:27:37Z
Here we are again. More control, less freedom, more censhorship, etc. etc. etc. etc. etc. Welcome to the brave sick world. What a total crap. But hey, some of us have known this was coming since at least 30 years ago.
marckris Id rather give my life away than to be chipped and to let them take away our "freedoms". Or Ill be in the woods and mountains!
Lemuel Gulliver · 2020-04-02T13:02:51Z
Dear SOTTies,
You will find this futuristic scenario interesting.... Here is a link to a 2010 Rockefeller Foundation report: [Link]
Here is the text beginning on Page 18. Judge for yourself if this was mere speculation, or a plan for our future, yours and mine:
LOCK STEP: Scenario Narratives
A world of tighter top-down government control and more authoritarian leadership, with limited innovation and growing citizen pushback.
In 2012, the pandemic that the world had been anticipating for years finally hit. Unlike 2009's H1N1, this new influenza strain — originating from wild geese — was extremely virulent and deadly. Even the most pandemic-prepared nations were quickly overwhelmed when the virus streaked around the world, infecting nearly 20 percent of the global population and killing 8 million in just seven months, the majority of them healthy young adults. The pandemic also had a deadly effect on economies: international mobility of both people and goods screeched to a halt, debilitating industries like tourism and breaking global supply chains.
Even locally, normally bustling shops and office buildings sat empty for months, devoid of both employees and customers.The pandemic blanketed the planet — though disproportionate numbers died in Africa, Southeast Asia, and Central America, where the virus spread like wildfire in the absence of official containment protocols. But even in developed countries, containment was a challenge. The United States's initial policy of "strongly discouraging" citizens from flying proved deadly in its leniency, accelerating the spread of the virus not just within the U.S. but across borders.
However, a few countries did fare better — China in particular. The Chinese government's quick imposition and enforcement of mandatory quarantine for all citizens, as well as its instant and near-hermetic sealing off of all borders, saved millions of lives, stopping the spread of the virus far earlier than in other countries and enabling a swifter post-pandemic recovery.
China's government was not the only one that took extreme measures to protect its citizens from risk and exposure. During the pandemic, national leaders around the world flexed their authority and imposed airtight rules and restrictions, from the mandatory wearing of face masks to body-temperature checks at the entries to communal spaces like train stations and supermarkets. Even after the pandemic faded, this more authoritarian control and oversight of citizens and their activities stuck and even intensified. In order to protect themselves from the spread of increasingly global problems — from pandemics and transnational terrorism to environmental crises and rising poverty — leaders around the world took a firmer grip on power.
At first, the notion of a more controlled world gained wide acceptance and approval. Citizens willingly gave up some of their sovereignty — and their privacy — to more paternalistic states in exchange for greater safety and stability. Citizens were more tolerant, and even eager, for top-down direction and oversight, and national leaders had more latitude to impose order in the ways they saw fit. In developed countries, this heightened oversight took many forms: biometric IDs for all citizens, for example, and tighter regulation of key industries whose stability was deemed vital to national interests. In many developed countries, enforced cooperation with a suite of new regulations and agreements slowly but steadily restored both order and, importantly, economic growth.
Across the developing world, however, the story was different — and much more variable. Top-down authority took different forms in different countries, hinging largely on the capacity, caliber, and intentions of their leaders. In countries with strong and thoughtful leaders, citizens' overall economic status and quality of life increased. In India, for example, air quality drastically improved after 2016, when the government outlawed high-emitting vehicles. In Ghana, the introduction of ambitious government programs to improve basic infrastructure and ensure the availability of clean water for all her people led to a sharp decline in water-borne diseases. But more authoritarian leadership worked less well — and in some cases tragically — in countries run by irresponsible elites who used their increased power to pursue their own interests at the expense of their citizens.
Meanwhile, in the developed world, the presence of so many top-down rules and norms greatly inhibited entrepreneurial activity. Scientists and innovators were often told by governments what research lines to pursue and were guided mostly toward projects that would make money (e.g., market-driven product development) or were "sure bets" (e.g., fundamental research), leaving more risky or innovative research areas largely untapped. Well-off countries and monopolistic companies with big research and development budgets still made significant advances, but the IP behind their breakthroughs remained locked behind strict national or corporate protection. Russia and India imposed stringent domestic standards for supervising and certifying encryption-related products and their suppliers — a category that in reality meant all IT innovations. The U.S. and EU struck back with retaliatory national standards, throwing a wrench in the development and diffusion of technology globally.
By 2025, people seemed to be growing weary of so much top-down control and letting leaders and authorities make choices for them. Wherever national interests clashed with individual interests, there was conflict. Sporadic pushback became increasingly organized and coordinated, as disaffected youth and people who had seen their status and opportunities slip away — largely in developing countries — incited civil unrest. In 2026, protestors in Nigeria brought down the government, fed up with the entrenched cronyism and corruption. Even those who liked the greater stability and predictability of this world began to grow uncomfortable and constrained by so many tight rules and by the strictness of national boundaries. The feeling lingered that sooner or later, something would inevitably upset the neat order that the world's governments had worked so hard to establish.
Lemuel Gulliver I think Rockeffelers need a divine intervention among their ranks. ASAP! Thank You God
Agron From Zeus, perhaps. He can pretend they're IO and f' 'em; and THEN pretend they're Phaeton and blow 'em away with a thunderbolt.
Rowan Cocoan Thats an entertaining thought lolz. I think much simpler than that. Some other parasite will stab them in their backs and go down with them. Or one of their family will turn on them. Or some crap they did in the past will come back and bite em in the spine. I will say GOOD RIDDANCE when it happens
Agron God's Laws will see to it that those responsible for sin will ultimately pay the compensation due. If I seek revenge on another, now that is my sin and I will have to pay the compensation due. Revenge degrades our soul condition, I want to grow in love. Love is the only solution.
God's Laws respond to desire, if our desires are loving we get massive, immediate positive results. So desire is the key. My role as I see it is to develop strong loving desires for a future government that is fair and equitable to all, and protects individual loving rights, makes best uses of all resources for the good of all and the environment, cooperates with other countries with similar goals, and supports our life with best health practices, broad and deep education, encourages research and development that is life sustaining, and provides all necessary infrastructure.
That might seem utopian, and it is only possible if we collectively have strong positive desires, and that is not what our desires are today. We have to make a shift first and then encourage others to. This is the first time I have put this in words, so thank you for the inspiration.
Joyly You may be right, but some maintain that sometimes, Karma needs a little help.
Yeah. Grounded in compassion! And wee need to re-connect with Mother Nature and all its creatures BIGTIME! Glad you see this course of action too! That is the only valid one in my view.
Joyly Have you heard of the teachings of Dr.Grigori Grabovoi? If not I recommend some research
Grounded in compassion Joyly n A!! ... and NOT in senti-mentality!
If one has no compassion for the balanced health of one's own BOD... well, we're fucked.
Agron But what about the other gender's input? Dr, Grabovirl?
LindaMay Now, ain't no sunshine.
Free speech is what I will be saying
Dwoods44 · 2020-04-02T15:27:15Z
Hello James. I highly respect your work over the years.
You should ask why people are not getting it. You should begin by reviewing the research of the neuroscientist, Tali Shorat. Her research demonstrates that people make choices like voting or reading based on BELIEFS not Evidence. Then there is the fact that Americans are programmed and not taught to Think, i.e., to examine facts, evaluate, analyze, connect, see the bigger picture, etc. How are people programmed? Framing is one(in advertising, psychology, propaganda). Also Neuro linguistic programming like the words "conspiracy theorist" which leads people to think crazy and not bother to research.
The good thing is that beliefs can be changed. Also other research has shown that if you can show people that they are being manipulated, they will work to be not manipulated.
Dwoods44 What they have been manipulated since way back is perception/belief. Look at us, we still believe all non-sense from thousands of years ago. Even today, we make all sorts of non-sense and see how much of us believe in it. People don't want proof, they want stories. Pathetic for some of us, excitement for others.
marckris Your phrasing reminded me of* Twain's thoroughly appropriate Mysterious Stranger, free at: [Link]
"Oh, it's true. I know your race. It is made up of sheep. It is governed by minorities, seldom or never by majorities. It suppresses its feelings and its beliefs and follows the handful that makes the most noise. Sometimes the noisy handful is right, sometimes wrong; but no matter, the crowd follows it. The vast majority of the race, whether savage or civilized, are secretly kind-hearted and shrink from inflicting pain, but in the presence of the aggressive and pitiless minority they don't dare to assert themselves. Think of it! One kind-hearted creature spies upon another, and sees to it that he loyally helps in iniquities which revolt both of them. Speaking as an expert, I know that ninety-nine out of a hundred of your race were strongly against the killing of witches when that foolishness was first agitated by a handful of pious lunatics in the long ago. And I know that even to-day, after ages of transmitted prejudice and silly teaching, only one person in twenty puts any real heart into the harrying of a witch. And yet apparently everybody hates witches and wants them killed. Some day a handful will rise up on the other side and make the most noise--perhaps even a single daring man with a big voice and a determined front will do it--and in a week all the sheep will wheel and follow him, and witch-hunting will come to a sudden end.
*(again, sorry, SOTTfolks for again posting excerpt but there are few better descriptions of the proles' greatest fears.)
Rowan Cocoan I'm not familiar with this work of Twain's, but very appropo.
ReRan It's truly a classic and an eye opener re how people over time still show the same scared flaws. Twain was obviously VERY irritated with humanity by the 20th century.
Free download. READ IT! [Link]
Rowan Cocoan Thanks!
Hey, SOTTfolk: Text or TWEET this to everyone:
If you think that this C19 situation has been overhyped like 9/11/2001, and is infringing on your freedom, HONK YOUR CAR HORN, or make big noise, at these times* 9:11 a.m., 9:11p.m, and 8:19 p.m. (= 20:19 =~ CV'2019') Every Day until this inane lockdown is lifted!) *(As convenient.)
(I know what I'll be doing at 8:11a and 9:11p tonight, and at 9:11a tomorrow, until these illegal lockdowns are lifted!)
6:12 p.m., EST, 4/2/2020.
SHUT DOWN THE LOCKDOWN!If you think that this C19 situation has been overhyped like 9/11/2001, and is infringing on your freedom, HONK YOUR CAR HORN, or make big noise, at these times* 9:11 a.m., 9:11p.m, and 8:19 p.m. (= 20:19 =~ CV'2019') Every Day until this inane lockdown is lifted!
Rowan Cocoan I love your intention to take action to be heard. Personally I am only out in the middle of the day.
Joyly If stuck at home, I intend to stick the guitar amp out the window and play some funeral dirge.
Rowan Cocoan Play the Vulcan funeral dirge from Star Trek Voyager lol
drdon1972 · 2020-04-03T01:08:53Z
Out of all the issues I find disturbing about the plandemic (and there are many), one that jumps out to me is - and this pertains to Point #3 in the article - the huge discrepancies in numbers among reports, all of which favor the stoking of dismay and fear in individuals' minds. For example, on the site below, which is maintained by Johns Hopkins, there appear to be no recoveries reported anywhere in the U.S., including in King County, Washington, where the virus was first reported over two months ago. Yet, in a county not far from where I live, the latest report is that there have been 62 confirmed cases, but of these, 20 have already recovered!
Thank God for sites like SOTT to bring a little (or a lot of) sanity to the banquet of madness we're served daily.
Here is the link to the Johns Hopkins site:
What is important is to first lay the foundation. There is a deep state that is outside of political parties that has implemented secrecy, surveillance, mass manipulation throughout my lifetime, and probably before. To understand the flow you need to bring scale to thinking, by looking at decades, not the current week or month. The world is a giant chessboard of politics, wars, science, education, vaccines, global warming, pandemics where they "play" god. The perpetrators are not the NLP based "elite"; they are the "robber barons" of our time (we need to use words that are based in reality).
The key elements of the pandemic are testing and analysis and data classification. The testing has not been uniform over nations. The US refused the WHO test, which DELAYED our testing. The testing has given false positives and one scientist says the PCR test is generic not specific for coronavirus. The key to testing is when it was started and how accurate it is. The analysis of death rate is false because it does not directly attribute a positive test to the virus, and the dying have underlying disease.
Because of the testing delay, we DON'T KNOW when covid-19 was introduced in the US. One Taiwanese virologist concluded it came from the US, and another Czech scientist indicates it was engineered. This would mean that deaths due to COVID-19 were incorrectly attributed to flu or pulmonary fibrosis late last year. Now with the introduction of FULL SCALE testing of the US population we will see big numbers because it has been in the US for many months. This is why the MSM is claiming FEAR, FEAR because those behind the scenes know when it was introduced. They can then appear to be right and continue to blame Trump. David
How many people have a healthy body? How many people invest an hour or 2 per day to contribute to their own HEALTH vs contaminate one's existence?
Only healthy bodies inspire healthy minds.
So so SO much repetitive irrelevant needless nonsense talk around n around n around to nowhere FAST. ⚰
Coronavirus - Things You CANNOT Say About It ...
5G "things" you mean?


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text:
\section{Introduction}
\noindent
Formal randomness notions for infinite bit
sequences can be studied via algorithmic tests. A hierarchy of such
notions has been introduced. See e.g.\ Downey
and Hirschfeldt
\cite{Downey.Hirschfeldt:book}
or Nies \cite[Ch.\ 3]{Nies:book} for
definitions and basic properties, and also Li and Vit\'anyi
\cite{Li.Vitanyi:book}. Criteria for good randomness
notions include robustness under certain computable operations on the
bit sequences. In the simplest case, such an operation is a computable
permutation of the bits. For a permutation $S$ of $\mathbb{N}$ and an
infinite bit sequence $Z$, identified with a subset of $\mathbb{N}$, by $Z
\circ S$ we denote the sequence $Y$ such that $Y(n) = Z(S(n))$.
(Note that, when viewed as a subset of $\mathbb{N}$, $Z \circ S$ equals $S^{-1}(Z)$.)
We say that a class $\mathcal C$ of bit sequences is \emph{closed
under all members of a class $\mathcal G$ of permutations} if $Z
\in \mathcal C$ implies $Z \circ S \in \mathcal C$ for each $S \in
\mathcal G$.

A central notion of randomness was introduced by
Martin-L\"of~\cite{Martin-Lof:66}. A Martin-L\"of test is a uniformly
$\Sigma^0_1$ sequence $\langle G_m
\rangle\sN m$ such that the uniform measure of $G_m$ is at most $2^{-m}$.
$Z$ fails such a test if $Z \in \bigcap_m G_m$; otherwise $Z$ passes
the test. $Z$ is Martin-L\"of random if it passes each such test.
Clearly this randomness notion is closed under computable
permutations $S$: if $Z \circ S$ fails a Martin-L\"of-test $\langle G_m
\rangle\sN m$, then $Z$ fails the test $\langle S^{-1}(G_m)\rangle\sN m$. The
weaker notion of Schnorr randomness \cite{Schnorr:75}, where one also
requires that the measure of $G_m$ is a computable real uniformly in
$m$, is closed under computable permutations
by a similar argument. Recursive randomness \cite{Schnorr:75} (see
e.g.\ \cite[Ch.\ 7]{Nies:book} as a recent reference)
is defined via failure of all computable betting strategies (martingales),
rather than by a variant of Martin-L\"of's test notion.
Nonetheless, by a more involved argument, implicit in \cite[Section
4.1]{Buhrman.etal:00}, it is closed under
computable permutations. Also see Nies~\cite[Thm.\ 7.6.24]{Nies:book}
and Kjos-Hanssen, Nguyen and Rute~\cite{Kjos-Hanssen.Nguyen.Rute:14}.

Our main purpose is to study analogs in computational complexity
theory of this result. In order to guarantee compatibility with the
theory developed in Downey and Hirschfeldt~\cite{Downey.Hirschfeldt:book} and Nies~\cite{Nies:book} we view
  sets of numbers (i.e., infinite bit sequences), rather than sets of strings over an alphabet of size at least 2, as our principal
objects of study.
We note that work of Lutz, Mayordomo, Ambos-Spies and others,
beginning in the 1980s and surveyed in Ambos-Spies and Mayordomo
\cite{Ambos.Mayordomo:97}, studied sets of strings: martingales bet on
the strings in length-lexicographical order. Such languages can be
identified with bit sequences via this order of strings, but the time
bounds imposed on martingales are   exponentially larger when
they bet on strings.

To be able to apply the notions of resource bounded computability to
bit sequences and permutations,
we will  identify infinite bit sequences with subsets of the set $\{0\}^*$
of unary strings. Such sets are called tally languages. We view
permutations as acting on $\{0\}^*$.
A bit sequence is \emph{polynomial
time random} if no polynomial time computable bettings strategy
succeeds on the sequence. This notion was briefly introduced by Schnorr~\cite{Schnorr:75},
studied implicitly in the above-mentioned work of Lutz, Mayordomo, Ambos-Spies and others, and in more explicit
form in Yongge Wang's 1996 thesis \cite{Wang:96}.

Our leading question
is: \emph{under which polynomial time
computable permutations $S$ is polynomial time randomness closed?}
If $S^{-1}$ is not polynomially bounded, we build a
polynomial time random bit sequence $Z$ such that $Z \circ S$ is not
polynomial time random. After that we assume that $S$ satisfies
the extra condition that its inverse is polynomially bounded. Now
the closure depends on additional assumptions in complexity theory:
\begin{itemize}
\item The first result, Theorem~\ref{thm:BPP}, shows
that if $\mathsf{BPP}$
contains a superpolynomial deterministic time class, such as
$\mathsf{DTIME}(n^{\log n})$,
then polynomial time randomness is not preserved by some
permutation $S$ such that both $S$ and its inverse are in $\P$.

\item The second result, Theorem~\ref{thm:scan}, shows that
$\mathsf{PSPACE}$-ran\-dom\-ness is preserved by polynomial time permutations
with polynomially bounded inverse; so if $\P = \mathsf{PSPACE}$ then
polynomial time randomness is preserved by such permutations.
\end{itemize}
Broadly speaking, the idea for Theorem~\ref{thm:BPP}  is as follows. Choose an  $O(n^{\log n})$ time computable martingale $M$ only betting on odd positions  $1,3,5, \ldots$  that  dominates   (up to a positive factor) all polynomial time computable  martingales that only bets on odd  positions.   Use the hypothesis in order to take a language $A \in \mathsf{BPP}$    which tells at which extension  of a string of odd length $M$ does not increase.  Now let $B$ be a highly random set (albeit $B$ can be chosen in $E$). Let $Z$ be the bit sequence that copies $B(n)$ at position $2n$, and takes the value of $A$ at the string $Z(0)\ldots Z(2n)$ at position $2n+1$. Then one can verify that  $Z$ is polynomial time random. If  $\widehat Z$ is a  rearrangement of  the bits of $Z$ so that a sufficiently large block of  bits of $B$ is  interspersed between bits determined by $A$, then we can use these bits of $B$ as random bits required in a randomised polynomial time algorithm for $A$. This will show that $\widehat Z$ is not polynomial time random.

Theorem~\ref{thm:scan} closely follows
Buhrman, van Melkebeek, Regan, Sivakumar and Strauss
\cite[Section 4.1]{Buhrman.etal:00}, which introduces and studies resource-bounded betting games. It actually shows that
$\mathsf{PSPACE}$-randomness is
closed under certain polynomial time scanning functions, which, unlike
permutations, can uncover the bits of a set in an order determined by
previous bits. Each permutation in question can be seen as a scanning
function of the appropriate kind. (We note that removing the  resource bounds from  Theorem~\ref{thm:scan}
 yields a proof that recursive randomness is
closed under computable permutations, and in fact under computable scanning functions that scan each position.)
Thm.\ 5.6 in Buhrman et al.\ \cite{Buhrman.etal:00} is a related result based on the same methods developed there; however, in that result an assumption on the existence of certain pseudorandom generators is made, while our   Theorem~\ref{thm:scan} does not rest on any unproven assumptions.

We note another notion of robustness for randomness notions. One can
easily adapt all the randomness notions to an alphabet
other than $\{0,1\}$. Base invariance says that the notion is
preserved when one replaces a sequence over one alphabet by a sequence
in a different alphabet that
denotes the same real number. Brattka, Miller and Nies
\cite{Brattka.Miller.ea:16} have shown this for recursive randomness,
and Figueira and Nies \cite{Figueira.Nies:13}
have shown it for polynomial time randomness, each time relying on the
connection of randomness of a real with differentiability at the real
of certain effective functions.

Using Figueira and Nies \cite{Figueira.Nies:13},
Nies \cite{Nies:14} provides a characterisation of polynomial time
randomness for real numbers in terms of differentiability of all
polynomial time computable nondecreasing functions on the reals.

\section{Preliminaries} \label{s:prelim}
For a bound $h$, as usual $\mathsf{DTIME}(h)$ denotes the languages $A$
computable in time $O(h)$. Informally we often say that $A$ is
computable in time $h$.  As in Ambos-Spies and Mayordomo~\cite{Ambos.Mayordomo:97}, we require that martingales have rational values.
\begin{definition} \label{df:mg}
A martingale $M$ is a function from $\{0,1\}^*$ to
$\{q \in {\mathbb Q}: q > 0\}$ satisfying $M(x) = (M(x0)+M(x1))/2$
for all $x \in \{0,1\}^*$. A martingale succeeds on a set $Z$
if $\limsup_n M(Z\upharpoonright n)=\infty$. One says that a martingale
\emph{does not bet at
a position $n$} if $M(x0) = M(x1)$ for each $x \in \{0,1\}^n$.
\end{definition}

\noindent
One says that $Z$ is recursively random if no computable martingale
succeeds on $Z$.

Each polynomial in this paper will be non-constant and have
natural number coefficients. For a polynomial time version of
recursive randomness, we have
to be careful how to define polynomial time computability for a
martingale: as in \cite{Buhrman.etal:00}, a positive rational number $q$ is presented by a pair $\langle
k, n \rangle$ of
denominator and numerator (written in binary) such that $q = k/n$ in
lowest terms. A martingale $M$ is
polynomial time computable if on input $x$ one can determine $M(x)$ in
this format in polynomial time. $Z$ is polynomial time random if no
such martingale succeeds on $Z$. In a similar way one defines
exponential time randomness.

A martingale $M$ is polynomial space computable if $M(x)$ can be
computed in polynomial space (including the space needed to write
the output). $Z$ is polynomial space random if no
such martingale succeeds on $Z$.

We first show that polynomial time randomness fails to be closed under
polynomial time computable permutations $S$ that are ``dishonest'' in
the sense that $S(n)$ can be much less than $n$.

\begin{theorem} \label{prop:S}
Let $S$ be a polynomial time
permutation of $\{0\}^*$ such that for each polynomial $p$, there are
infinitely many $n$ with $p(S(n)) \le n$. There is a polynomial time
random $Z$ computable in time $2^{O(n)}$ such that $Z \circ S$ is
not polynomial time random.
\end{theorem}

\noindent
Clearly a permutation $S$ as in Proposition~\ref{prop:S} exists: Let
$(p_k)$ list the non-constant polynomials with natural coefficients in
such a way that for $u \le n$, $O(n^2)$ steps suffice to verify
whether $p_k(u) \le n$.
On input $n$ of the form $\langle k,i \rangle$, see whether $p_k(\langle k,0
\rangle) \le n$. If not let $S(n) = \langle k,i+1 \rangle$. If so and $n$ is
least such, let $S(n) = \langle k,0 \rangle$. Otherwise $S(n) = n$.

\begin{proof}[Proof of Proposition~\ref{prop:S}] Nies \cite[Section
7.4]{Nies:book} provided a construction template for recursively
random sets, going back to Schnorr's work. We adapt some
parts of this template to the resource bounded setting.

Let $\langle B_k \rangle$ be an effective listing of the polynomial time
martingales with positive rational values. We may assume that $B_k$
is computable in time $p_k(n)= k (n^k+1)$.

For each $n$, let $B_{k,n}$ be the martingale with initial capital
$1$ that does not bet until its input reaches length $n$, and then
uses the same betting factors as $B_k$. Thus, \[B_{k,n}(x) =
\frac{B_k(x)}{B_k(x \upharpoonright n)}\] for any string $x$ of length at least $n$.
Let $\widetilde p_{k,n}$ be a polynomial so that $B_{k,n}(x)$ for $|x|
\ge n$ can be computed in time $\widetilde p_{k,n}(|x|)$.

\iffalse
computed in time $q_k = 2p_k + C$.
\fi

We inductively define a sequence of numbers. Let $n_0 = 0$, and let
$n_{k+1} $ be the least $n> n_k$ such that $q_k(S(n)+1) \le n$,
where $q_k$ is a polynomial time bound for the martingale $\sum_{r\le
k} 2^{-r} B_{r, n_{r}}$ and $q_k(n) \ge n+2$.
Let $L= \sum_r 2^{-r} B_{r, n_{r}}$. Note that $L$ is a
rational-valued martingale, because on inputs of length at most
$n_{k}$, all the $B_{r, n_r}$ for $r > k$ together contribute $2^{-k}$.

Let now $Z$ be the left-most non-ascending path of $L$: $Z(m) = 0$ if
$L(Z \upharpoonright m \hat {\ } 0) \le L(Z \upharpoonright m)$, and $Z(m) = 1$ otherwise.
Since $L$ does not succeed on $Z$ and $L$ multiplicatively dominates
each $B_k$, the set $Z$ is polynomial time random.

Note that since $S \in \P$, from $n$ we can in polynomial time
recursively recover the sequence $n_0, q_0, n_1, q_1, \ldots$ and
thereby compute the maximal $k$ such that $n_k < n$. In particular we
can decide whether $n$ is of the form $n_{k+1}$ for some $k$. By
definition, for $n= n_{k+1}$ we have $q_k(S(n) +1) \le n$ and hence
$S(n) +1 < n_{k+1}$.
Since $q_k $ as a time bound is sufficient to determine $L(y)$ for
strings $y$ of length $S(n)+1$, the bit $Z \circ S(n)$ can be
computed in time polynomial in $n$. Hence $Z \circ S$ is not
polynomially random.

We can ensure such a set $Z$ is computable in time $2^{O(n)}$ by choosing the
listing $\langle B_k \rangle$ appropriately.
\end{proof}

\begin{remark} We note that methods involving the $\langle B_{k,n} \rangle$
similar to the above can be used to show that each class $\mathsf{DTIME}(h)$
with superpolynomial time constructible $h$ contains a polynomial time
random (tally) set. We have to initiate a copy $\langle B_{k,n} \rangle$ of
$B_k$ finitely many times until a length $n$ is reached such that
for $m \ge n$, $h(m)$ time is sufficient to simulate its behaviour on
strings of length $m$.
\end{remark}

\section{If $\mathsf{BPP}$ Contains a Superpolynomial Time Class Then Closure Fails}

\begin{definition} \label{def:FPC} A permutation $S$ of $\{0\}^*$ is
called  \emph{fully polynomial time computable} if both $S$ and $S^{-1}$ are
polynomial time computable.
\end{definition}

\noindent

A complexity theoretic assumption considerably  weaker than $\mathsf{BPP} =
\mathsf{EXP}$ suffices for non-closure.

\begin{theorem} \label{thm:BPP}
Suppose that $\mathsf{DTIME}(h) \subseteq \mathsf{BPP}$ for some time constructible
function $h$ that dominates all the polynomials. Then there are a
polynomial time random set $Z \in \mathsf{DTIME}(2^{3n})$
and a fully polynomial time computable permutation
$S$ such that $Z \circ S$ is not polynomial time random.
\end{theorem}

\begin{proof}
We may assume that $h(n) \le n^{\log n}$.
It is well-known that whenever a martingale
in a certain complexity class succeeds on a set $Z$ then there
is also a successful martingale in the same class betting only on even
positions,
or there is a successful martingale betting only on the odd positions.

The construction has two steps. Firstly, by standard methods discussed
at the end of Section~\ref{s:prelim}, one can build a martingale $M$ in
$\mathsf{DTIME}(h)$ which bets only on odd positions, and   dominates up to a
multiplicative constant all
polynomial time martingales betting on odd positions. Let
$$
   A = \{x \in \{0,1\}^*: x \text{ has odd length and } M(x1) < M(x0)\}.
$$
The set $A$ is in $\mathsf{DTIME}(h)$ and hence by assumption in $\mathsf{BPP}$.

Secondly, let $B \subseteq \{0\}^*$ be a language on which no
martingale in $\mathsf{DTIME}(2^{4 \cdot n})$ succeeds.
Again by standard methods one can ensure that $B$ is in $\mathsf{DTIME}(2^{5 \cdot n})$.
Define a set $Z \subseteq \mathbb N$ as follows:
\iffalse
\begin{eqnarray*}
   Z(2n)   & = & B(n); \\
   Z(2n+1) & = & A( Z\upharpoonright {2n+1}).
\end{eqnarray*}
\fi
$$
   Z(2n) = B(n); \ Z(2n+1) = A(Z \upharpoonright 2n+1).
$$
\noindent We may visualise $Z$ as follows:

\medskip

 $  \begin{tabular}{|c|c|c|c|c|c|c|c|c| c|c|c|c| c}
\hline

B &  {A} &   B  &   {A} & B  &  {A} & B&   {A}  &B  &   {A} & B &  {A}
 & B & \ldots \\

\hline

\end{tabular}$
\smallskip

  {\tiny $  B(0)  \,   { A(Z\upharpoonright 1)}  \,  B(1 ) \,   {A (Z\upharpoonright 3)}
\, B(2)  \,   {A (Z \upharpoonright 5)}  \ldots$ }

\medskip
\noindent
Clearly $Z \in \mathsf{DTIME}(2^{3n})$. It is claimed that $Z$ is polynomial time random.
As the martingale $M$ only bets on odd positions, $Z$ is defined such
that $M$ never gains capital on $Z$. As $M$ is universal among the
martingales computable in polynomial time with this property, no
martingale betting on the odd positions succeeds on $Z$.

Suppose now that $L$ is a polynomial time martingale which bets on the even
positions and note that one can compute in time $O(h(n))$
from $B(0),B(1),\ldots,B(n)$ inductively the values $Z(0),Z(1),\ldots,Z(2n+1)$,
as for every $x$ of length $2n+1$ the value $A(x)$ can be computed in
time $h(n)$. Thus if $L$ succeeds on $Z$ then there is a new martingale $N$
succeeding on $B$ which satisfies that
$$
   N(B\upharpoonright n+1) = L(Z\upharpoonright 2n)
$$
and which uses that $Z(2n) = B(n)$ while the bits of $Z$ at odd positions
on which $L$ does not bet can be computed as indicated above from the
other bits. To compute $N(x)$ for $x$ of length $2n$ takes $q(n) +
\sum_{i< n} h(2i+1)$ steps for some polynomial $q$. So $N \in \mathsf{DTIME}
(n^{O(\log n)})$, which contradicts the assumption that no such
martingale computable in time $O(2^{4n})$ succeeds on $B$. This
verifies the claim.

Since $A \in \mathsf{BPP}$, there is a polynomial $p$ such that
an appropriate randomised algorithm $\mathcal R$ on input $x \in
\{0,1\}^{2n+1}$ computes $A(x)$ in time $p(n)$,
with error probability $2^{-4n-2}$, using $p(n)$ random bits.
Now consider the sequence $\widehat Z$ consisting for $n=0,1,\ldots$
of $p(n)$ bits taken from $B$ followed by the bit $Z(2n+1)$. Again we
visualise $\widehat Z$:

\medskip

$ \begin{tabular}{|c|c|c|c|c|c|c|c|c| c|c|c|c| c}
\hline

B &  {A} & B & B  & B &   {A}  &B  & B& B  &B  & B& B &  {A}  & \ldots \\

\hline

\end{tabular}$

\smallskip

   $p(0)$ \hspace{1.1cm}  $p(1)$ \hspace {3cm} $p(2)$

\medskip
\noindent Formally one can define $\widehat Z$ from $Z$ as follows:
\begin{eqnarray*}
  \mbox{ for $m< p(n)$,} \hspace{1.5cm} \mbox{ } & & \\
  \widehat Z((\sum_{k<n} p(k))+n+m) & = &  B((\sum_{k<n} p(k))+m) \ = \
       Z(2(\sum_{k<n} p(k)+m)); \\
  \widehat Z((\sum_{k\leq n} p(k))+n) & = & Z(2n+1)= A( Z\upharpoonright {2n+1}).
\end{eqnarray*}
This mapping is given by a permutation $S$ so that
$\widehat Z(r) = Z(S(r))$ for all positions $r$.
So if $r = (\sum_{k<n} p(k))+n+m$ then $S(r) = 2(\sum_{k<n} p(k))+2m$
and if $r = (\sum_{k\leq n} p(k))+n$ then $S(r) = 2n+1$, for all $m,n$
with $m < p(n)$. The permutation $S$ and its inverse satisfy that the
mappings $0^k \mapsto 0^{S(k)}$ and $0^k \mapsto 0^{S^{-1}(k)}$
on the unary strings $\{0\}^*$ are polynomial time computable, thus
the $S$ is of the form as required; to see this note that for a polynomial
$p$ also the mapping $n \mapsto \sum_{k<n} p(k)$ is a polynomial; similarly
for a function bounded by a polynomial.

Now it will be shown that $\widehat Z$ is not polynomial time random.
Note that there are $2^{2n+1}$ strings of length $2n+1$. Given a string
of $p(n)$ random bits, the probability that when using these bits the randomised
algorithm $\mathcal R$ computes $A(x)$ correctly
for all $x \in \{0,1\}^{2n+1}$ is at least $1-2^{2n+1} \cdot 2^{-4n-2}
= 1-2^{-2n-1}$.
We want to show that $B$ provides random bits that allow $\mathcal R$
to correctly compute $A$ for almost all inputs. Otherwise, we can
build a martingale $M$ computable in time $2^{10 \cdot n}$
which succeeds on $B$:
The martingale $M$ splits its
capital into bins of value $2^{-n-1}$ and for each block
of $p(n)$ bits starting at $\sum_{k < n} p(k)$, it takes the value
$2^{-n-1}$ from the corresponding bin and bets it on the strings $y$
consisting of
$p(n)$ bits that do not compute all values of $A(x)$ with $x \in
\{0,1\}^{2n+1}$
correctly using $\mathcal R$. This condition can be checked for these bits
in the time bound given as it involves running $\mathcal R$ with $y$
as the random bits
on all strings $x$ of length $2n+1$ and comparing the result with $A(x)$ for all
$2^{p(n)}$ choices of random bits $y$. After these simulations, $M$
distributes the capital from the bin evenly on those
strings of random bits which cause $\mathcal R$ to make an error.
After having processed the
bits from the block of $p(n)$ bits, the capital in this bin remains
unchanged by future
bets. The set of random strings $y$ on which the computation of some of the
$A(x)$ in $x \in \{0,1\}^{2n+1}$ is false has at most
the probability $2^{-4n-2} \cdot 2^{2n+1} = 2^{-2n-1}$. Therefore
the capital from the bin multiplies at least by $2^{n+1}$ during the block and
reaches the value $1$.

For the time bound on $M$, whenever the input has length between
$\sum_{k<n} p(k)$ and $\sum_{k \leq n} p(k)$,
the martingale computes $2^{n+1}$ values $A(x)$ for
$x \in \{0,1\}^{2n+1}$ with respect to $p(n)$ random bits taking $2^{p(n)}$
possible choices. However, for all polynomials and almost all $n$,
$p(n) \leq \sum_{k < n} p(k)$, as the degree of the sum-polynomial of
$p$ is by one above the degree of $p$ and the polynomial $p$ is
positive. Thus, for such $n$, when $n' = \sum_{k<n} p(k)$ is a lower
bound on the
length of the input to the martingale $M$ then $p(n) \leq n'$ and
$2n+1 \leq n'$ and thus the whole computations can be handled in time
$O(2^{3n'})$.

If there are infinitely many blocks in $B$ where the random bits of this
block do not compute all $A(x)$ with $x$ of the corresponding length correctly,
then this martingale succeeds, contrary to the assumption on $B$.
So, for almost all $n$, the block of $p(n)$ random bits in $\widehat Z$ before
$A(Z \upharpoonright {2n})$ permits to compute this value correctly.

Now this property will be used to
construct a polynomial time martingale $H$ which succeeds on $\widehat Z$. Let
$\tilde A(n)$ denote $A( Z\upharpoonright {2n+1})$. Given $p(n)$ random bits
from $B$ preceding $\tilde A(n)$ in $\widehat Z$, the martingale $H$ archives
these bits without
betting on them. It then bets half of its capital on the value for
$\tilde A(n)$ computed from these random bits; note that due to
$\tilde A(0),\tilde A(1),\ldots,\tilde A(n-1)$ and $B(0),B(1),\ldots,B(n)$
being coded in $\widehat Z$ in positions before that of $\tilde A(n)$, when the
bet for $\tilde A(n) = Z(2n+1)$ has to be made, one can
retrieve besides the random bits also $Z(0)Z(1)\ldots Z(2n)$ from the
history. So one can use the random bits to compute the value almost
always correctly.
Thus the martingale $H$ will only finitely often place a wrong bet and lose
some of its capital, but for almost all $\tilde A(n)$ predict the value
correctly and multiply its capital by $3/2$. Thus the martingale
succeeds. As all the operations above are polynomial time computable,
the set $\widehat Z$ is not polynomial time random.
\end{proof}

\noindent
The proof of Theorem~\ref{thm:BPP} can be adjusted to obtain a corollary.

\begin{corollary} \label{cor:BPP}
Let $A,B \subseteq \{0\}^*$. Suppose that $A$ is in $\mathsf{BPP}$
and $B$ is $\mathsf{EXP}$-random relative to $A$.
Then $A$ is polynomial time computable relative to $B$,
and in particular not polynomial time random relative to $B$.
\end{corollary}

\begin{proof}
For the ease of notation, we often write $A(n)$ in place of $A(0^n)$ and so on;
however, both $A$ and $B$ are viewed as subsets of $\{0\}^*$.

There is a polynomial time algorithm and a polynomial $p$ such that
the algorithm uses $p(n)$ random bits to compute $A(n)$
with error probability $2^{-n}$. As in the theorem above,
one can now query $B$ for getting the random bits and the places
where the queries are asked are different for $n,m$ whenever $n \neq m$.
So there is a polynomial $q$ with $q(n)+p(n) = q(n+1)$ for all $n$
and where the algorithm asks the bits of $B$ at
$q(n),q(n)+1,\ldots,q(n)+p(n)-1$ to compute $A(n)$.

If now there is an error, then an exponential time martingale relative
to $A$ can make sufficient profit, as only a slim minority of the possiblities
of the bits of $B$ from $q(n)$ to $q(n)+p(n)-1$ are realised. This
contradicts the assumption that $B$ is random relative to $A$.
Hence $A$ can be computed relative to $B$ by this algorithm with
only finitely many errors; these can then be corrected by a finite
table holding the correct values for the positions where the algorithm
makes an error.
\end{proof}

\begin{remark}
In the proof of Theorem~\ref{thm:BPP}, $Z = \tilde A \oplus B$
is polynomial time random; however, $\tilde A$ is not polynomial time
random relative to $B$, as the rearrangement with $S$ shows.
Note that van Lambalgen's Theorem
\cite{Lambalgen:90} says that in a recursion-theoretic setting,
$\tilde A \oplus B$ is random iff (a) $B$ is random and
(b) $\tilde A$ is random relative to $B$.
Thus, under the assumption that $\mathsf{BPP} = \mathsf{EXP}$, one of the directions of
the van Lambalgen Theorem does
not hold for polynomial time randomness.

The corollary also shows that one can choose, under the assumption
that $\mathsf{BPP} $ contains a superpolynomial time class, sets $A,B
\subseteq \{0\}^*$ such that $A$ is
polynomial time random,
$B$ is polynomial time random relative to $A$ and $A$ is polynomial
time computable relative
to $B$. Hence this assumption implies that $A$ is a basis for
polynomial time randomness
even though $A$ is polynomial time random itself. This contrasts with the
setting of Martin-L\"of randomness randomness in recursion theory: a basis for
Martin-L\"of randomness has to
be trivial and therefore cannot be random
\cite{Hirschfeldt.Nies.Stephan:07,Nies:05}. On the other hand, the
bases for recursive randomness include every
set below the halting problem that is not diagonally noncomputable (DNC),
but no set of PA degree \cite{Hirschfeldt.Nies.Stephan:07}. Every high set computes a recursively random set, and an incomplete  high r.e.\ set is not DNC. So a recursively random set can   be a basis for recursive randomness.
\end{remark}

\section{If $\P = \mathsf{PSPACE}$ Then Closure Holds}

\noindent
We say that $Z \subseteq \mathbb{N} $ is \emph{polynomial space random} if no
  martingale computable in polynomial space  succeeds on $Z$. In this section we
show that polynomial space randomness is closed under fully polynomial time
computable permutations in the sense of Definition \ref{def:FPC}. If
$\P = \mathsf{PSPACE}$ this closure property applies to polynomial time
randomness as well.

In fact we show a stronger closure property where the permutations
are generalised to certain non-monotonic scanning rules (adaptively
specifying an order in which bits are read). We modify the argument given
by Buhrman, van Melkebeek, Regan, Sivakumar and Strauss
\cite[Section 4.1]{Buhrman.etal:00}, which was not concerned with
polynomial space randomness, but rather was geared to the context of Lutz's
theory of resource bounded measure. As already mentioned, in that
theory, the positions a
martingale bets on
are strings in some non-unary alphabet.
Such strings can be suitably
encoded by natural numbers; however, the resource bounds change
when one converts such a martingale into one in the sense of our
Definition~\ref{df:mg}.

\begin{definition} A \emph{scanning function} is a function $V \colon
\{0,1\}^* \to \{0\}^*$ such that $V(\alpha) \neq V(\alpha \upharpoonright i)$ for
each $\alpha \in \{0,1\}^* $ and each $i < |\alpha|$. In the context of $V$,
we will call a string $\alpha$ a \emph{run of $V$}, thinking of
$\alpha$ as a sequence of answers to oracle queries. We will call
$V(\alpha \upharpoonright i)$ the \emph{$i$-th query in the run of
$V$ on $\alpha$}.

As before, subsets of $\mathbb{N}$ will be identified with languages over
the unary alphabet~$\{0 \}$. For $Z \subseteq \mathbb{N}$ let $Z \circ V
\subseteq \mathbb{N}$ be the set $Y$ such that $Y(i) = Z(V (Y \upharpoonright i))$
for each~$i$. In the following we review some key technical concepts
\cite[Section 4.1]{Buhrman.etal:00}, somewhat changing the
terminology in order to make it compatible with the one of Nies \cite[Section
7.5]{Nies:book} where non-monotonic randomness notions are studied.

For a function $g \colon \mathbb{N} \to \mathbb{N}$, one says that $V$ is
\emph{$g$-filling} if for each $n$ and each run $\alpha$ of length $g(n)$,
\[ \forall r < n \, \exists i \, V(\alpha \upharpoonright i) = r. \]
\end{definition}

\begin{definition}
A \emph{non-monotonic betting strategy} $G$ is
a pair $(V,B)$ such that $V$ is a scanning function and $B$ is a
martingale. $G$ succeeds on $Z\subseteq \{0\}^*$ if $\lim_n
B(Z \circ V\upharpoonright n) = \infty$.

One says that a non-monotonic betting strategy $G$ is computable in
polynomial space if both $V$ and $B$ are computable in polynomial
space. One says that $Z\subseteq \mathbb{N} $ is \emph{non-monotonically
polynomial space random} if no such betting strategy succeeds on $Z$.
\end{definition}

\noindent
The final concept we need is that of consistency between a run
$\alpha$ of $V$ and a string~$w$.

\begin{definition}
For bit strings $\alpha, w$, we write $\alpha \sim_V w $ if for
each $j < |\alpha|$, if the $j$-th query $x$ in the run of $V$ on
$\alpha$ is less than $|w|$, then $w(x) = \alpha(j)$.

\begin{lemma} Suppose $V$ is $g$-filling. Let $|\alpha| \ge i:=
g(|w|)$. Then $\alpha \sim_V w $ iff $\alpha \upharpoonright i \sim_V w$.
\end{lemma}

\noindent
To see this, note that any query $q$ with $q< |w|$ has to be
asked before stage $g(|w|)$ by the definition of the function $g$.
\end{definition}

\begin{theorem} \label{thm:scan}
Let $V$ be a scanning function in $\mathsf{PSPACE}$ that is $g$-filling for
a polynomial bound $g$. If $Z$ is polynomial space random, then so is
$Z \circ V$.
\end{theorem}

\begin{proof}
Suppose $Z \circ V$ is not polynomial space random. Let $G=(V,B)$
be a betting strategy in $\mathsf{PSPACE}$ that succeeds on $Z$;
thus, $B$ succeeds on $Z \circ V$.

We define a martingale $D$ in $\mathsf{PSPACE}$ that succeeds on $Z$.
We may assume that $g(n) \ge n$. For $t \ge g(|w|)$ let
\[ D(w) = 2^{|w|-t} \sum_{|\alpha| = t \ \land \ \alpha \sim_V w} B(\alpha).\]
By the claim above and since $B$ is a martingale, this definition
is independent of $t$. Note that among the runs $\alpha$ of length
$t$, a fraction of $2^{-|w|}$ satisfy that $\alpha \sim_V w$; so
$D(w)$ is simply the average value of $B(\alpha)$ over all such $\alpha$.

If we let $t = g(|w|)$, by the hypotheses that $G$ is in $\mathsf{PSPACE}$
and that $g$ is a polynomial, $D$ is in $\mathsf{PSPACE}$.

The rest of the argument somewhat simplifies the one of
\cite{Buhrman.etal:00} in the present context.

\begin{lemma}
$D$ is a martingale.
\end{lemma}

\noindent Let $w$ be a string of length $n$. If $|\alpha| = g(n+1)$ and
$\alpha \sim_V w$, then either $\alpha \sim_V w0$ or $\alpha \sim_V w1$.
Letting $u= g(n+1)$, for each $r =0,1$ we have
\begin{center}
  $D(wr) = 2^{|w|+1 -u} \sum_{|\alpha| = u \ \land \ \alpha \sim_V {wr} }
           B(\alpha)$.
\end{center}
Hence, since the definition of $D(w)$ does not depend on the choice
of $t\ge g(|w|)$,
\begin{center}
$D(w0) + D(w1) =  2^{|w|+1 -u} \sum_{|\alpha| = u \
\land \ \alpha \sim_V {w} } B(\alpha) = 2 D(w)$.
\end{center}
\iffalse
\fi

\begin{lemma} $D$ succeeds on $Z$. \end{lemma}

\noindent
We may assume that $B(x)>0$ for each $x$. The Savings Lemma (see
e.g.\ Nies \cite[7.1.14]{Nies:book})
states that each computable martingale $M$ can be turned into a
computable martingale $\widehat M$ that succeeds on the same sets, and
has the extra property that $\widehat M(\beta) \ge \widehat M(\alpha)
-2$ for each strings $\beta \supseteq \alpha$ (namely, $\widehat M$
never loses more than $2$). It is easy to see from the proof that if
$M$ is computable in polynomial space, then so is $\widehat M$. So we
may assume that $B$ has this property.

This implies that for each
$c\in \mathbb{N}$ there is a prefix $\alpha$ of $Z \circ V$ such that
\begin{center}
  $B(\beta ) \ge c$ for each string $\beta \succeq \alpha$.
\end{center}
By definition of $Z \circ V$ we have $\alpha(i) = Z(V(\alpha\upharpoonright i))$ for each
$i < |\alpha|$. Let $r = 1+ \max_{i< |\alpha|} V(\alpha\upharpoonright i)$ be $1+$
the maximum query asked in the run of $V$ on $\alpha$, and let $w = Z
\upharpoonright r$. So $g(r) \ge |\alpha|$.

If $\beta\sim_V w$ is a string such that $|\beta| =g(r)$, then $\beta
\succeq \alpha$, for $\alpha(r) \neq \beta(r)$ for some $r < |\alpha|$
would imply that $\beta \not \sim_V w$ as $w$ answers all such
queries correctly. So $B(\beta) \ge c$. Hence $D(w) \ge c$ because
$D(w)$ is the average over values $B(\beta)$ for all such $\beta$.
\end{proof}

\begin{corollary} Let $S$ be a polynomial time computable permutation
of $\{0\}^*$ such that $S^{-1}$ is polynomially bounded. If $Z$ is
polynomial space random, then so is $Z \circ S$.
\end{corollary}

\begin{proof}
The permutation $S$ can be viewed as a scanning function $V_S$ that only
looks at the length of the input: $V_S(\alpha) = S(|\alpha|)$.
By hypothesis on $S$, the scanning function $V_S$
is polynomially filling. So $Z\circ S = Z \circ V_S$ is polynomial
space random by the theorem.
\end{proof}

\noindent
The foregoing corollary can be restated in terms of randomness on
languages in the sense of \cite{Ambos.Mayordomo:97}: Let $S$ be a
exponential time computable permutation
of $\{0,1\}^*$ such that $|S^{-1}(x)| = O(|x|)$ for each string $x$.
If a language $Z$ is exponential space random, then so is $Z \circ S$.

We end with a question. Recall that $\mathsf{PP}$ denotes
probabilistic polynomial time, a subclass of $\mathsf{PSPACE}$.
If $\P= \mathsf{PP}$, is polynomial time randomness closed
under permutations $S$ of $\{0\}^*$ such that $S, S^{-1}$ are polynomial time
computable?

\medskip
\noindent
{\bf Acknowledgments.} The authors would like to thank Eric Allender, Klaus Ambos-Spies, Alexander Galicki, Elvira Majordomo,
and Wolfgang Merkle for discussions and comments.

{A.~Nies is supported
in part by the Marsden Fund of the Royal Society of New Zealand, UoA
13-184. F.~Stephan
is supported in part by the Singapore Ministry of Education Academic
Research Fund Tier 2
grant MOE2016-T2-1-019 / R146-000-234-112. Part of this work was done
while F.~Stephan
was on sabbatical leave at the University of Auckland. The work was
completed while  Nies  visited the Institute for Mathematical Sciences
at NUS during the  2017 programme ``Aspects of Computation''.}

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redpajama_set_name: RedPajamaArXiv