Prompt:
What does the evidence say about necessary conditions and constraints?

EVIDENCE:
- R L TheSpin(10)chart(42.12)alsopresentspuzzles.Inanysupergeometricalgebrawithevendimensions,the spinormetricflipsallbits.Thenumberofbitsishalfthenumberofdimensions.InDirac,thenumberofbits, 2, is even, so the spinor metric preserves chirality.
- As pointed out in equation (42.97), Spin(10) chirality happens to
- The Spin(10) handedness of a spinor is the sign of the spinor under the action of the Spin(10) chiral operator κ , while chirality under Lorentz transformations is the sign of the spinor under the action of the Dirac chirality operator traditionally denoted γ .
- The most prominent striking feature is that the Spin(10) handedness coincides with the handedness, or chirality (R or L), of the spinor under Lorentz transformations.

R L TheSpin(10)chart(42.12)alsopresentspuzzles.Inanysupergeometricalgebrawithevendimensions,the spinormetricflipsallbits.Thenumberofbitsishalfthenumberofdimensions.InDirac,thenumberofbits, 2, is even, so the spinor metric preserves chirality.

As pointed out in equation (42.97), Spin(10) chirality happens to

The Spin(10) handedness of a spinor is the sign of the spinor under the action of the Spin(10) chiral operator κ , while chirality under Lorentz transformations is the sign of the spinor under the action of the Dirac chirality operator traditionally denoted γ .

The most prominent striking feature is that the Spin(10) handedness coincides with the handedness, or chirality (R or L), of the spinor under Lorentz transformations.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
What does the evidence say about necessary conditions and constraints?

EVIDENCE:
- Bylate2011,DHS DeputySecretaryJaneHollLute,UnderSecretaryforIntelligenceandAnalysisPhyllisGreen,andothers decidednottocontractBAASSwithinDHS.
- Thosemetarethe“keepersofthe secrets,”whotreatBellandMoverrudelyandtold“no,andhellno”whenasked.
- MoverandBellbecamefearfulofmovingforward.
- (PUBLICDOMAIN) -26March2011—UfologistChuckWadepresentsinformationatthe14th AnnualAztecUFOSymposiumallegedlyfromdifferentmilitarysourcesrelatedtoUAPcrashrecoveries.

Locate highest-signal evidence sentences from chunks for the question.

Check consistency, units, conservation constraints, and whether evidence supports claims.

Compose bounded answer and (if applicable) a proposal/hypothesis clearly labeled.

Generate falsification tests and identify unknowns or missing evidence.

Merge outputs, remove speculation, keep only evidence-backed facts.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Design a retrieval plan to answer a missing detail without guessing.

EVIDENCE:
- Yet this difference is smaller than small, and barely noticeable in everyday life.
- Although the belief that women are more talkative than men has been popular for decades, psychiatrist Louann Brizendine lent it new credence in her bestselling book, The Female Brain(2006).
- (1) Do women talk much more than men?
- As a rough benchmark, a Cohen’s d of .2 is considered small, .5 medium, and .8 or bigger large.

Yet this difference is smaller than small, and barely noticeable in everyday life.

Although the belief that women are more talkative than men has been popular for decades, psychiatrist Louann Brizendine lent it new credence in her bestselling book, The Female Brain(2006).

As a rough benchmark, a Cohen’s d of .2 is considered small, .5 medium, and .8 or bigger large.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
What does the evidence say about necessary conditions and constraints?

EVIDENCE:
- Following Newton, we denote differentiation dl withrespecttol byadot, aswasalreadydoneinchapterI.1.(Allvectorsinthischapter have3componentsandwillbelabeledwithanarrow.)Sincewesaidthatl isthelength, namelydl2=dX (cid:3) .dX (cid:3) ,wehavet(cid:3).t(cid:3)=1.Differentiating,weobtaint(cid:3).t ̇(cid:3)=0.Thus,theunit vectorp(cid:3)definedby t ̇(cid:3)=κp(cid:3) (1)
- The unit tangent vector is t(cid:3)≡X (cid:3) ̇ ≡ dX (cid:3) .
- “Classical”differentialgeometry IfeelthatitwouldbegoodforthosereadersseeingRiemanniangeometryforthefirsttime toworkthroughsome“classical”differentialgeometry1dealingwithcurvesandsurfaces, “real”stuffthatyoucouldactuallyseeand“holdinyourhands.”Throughoutthischapter, wewillbelivingingoodold3-dimensionalEuclideanspace.Iamgoingtotellyouhowthe greatslikeFrenetandGaussthoughtaboutcurvesandsurfaces.Noneofthefancytangent bundletalkforus;wewilljustdoit.Action,nottalk!
- Curves ConsideracurveC givenbyX (cid:3) (l)parametrizedbythelengthl alongthecurve.Inother words,westartfromanarbitrarypointOonthecurve,andpaceoffadistanceequaltol (cid:3) alongthecurve.OurlocationisthenspecifiedbythevectorX(l).

Following Newton, we denote differentiation dl withrespecttol byadot, aswasalreadydoneinchapterI.1.(Allvectorsinthischapter have3componentsandwillbelabeledwithanarrow.)Sincewesaidthatl isthelength, namelydl2=dX (cid:3) .dX (cid:3) ,wehavet(cid:3).t(cid:3)=1.Differentiating,weobtaint(cid:3).t ̇(cid:3)=0.Thus,theunit vectorp(cid:3)definedby t ̇(cid:3)=κp(cid:3) (1)

The unit tangent vector is t(cid:3)≡X (cid:3) ̇ ≡ dX (cid:3) .

Curves ConsideracurveC givenbyX (cid:3) (l)parametrizedbythelengthl alongthecurve.Inother words,westartfromanarbitrarypointOonthecurve,andpaceoffadistanceequaltol (cid:3) alongthecurve.OurlocationisthenspecifiedbythevectorX(l).

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
What does the evidence say about necessary conditions and constraints?

EVIDENCE:
- People walk past me on the sidewalk without stopping.
- After a while, I manage to sit up.
- They finally rip my backpack away, and as I bleed on the pavement, take off laughing and running down the street.
- I black out for a moment, and when I come to, I can feel their hands running up and down my body, searching—I assume—for a wallet that isn’t there.

People walk past me on the sidewalk without stopping.

They finally rip my backpack away, and as I bleed on the pavement, take off laughing and running down the street.

I black out for a moment, and when I come to, I can feel their hands running up and down my body, searching—I assume—for a wallet that isn’t there.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Summarize the most important claims and how to verify them.

EVIDENCE:
- By denying the evidence for climate change, and pulling out of the Paris Agreement, Donald Trump will cause avoidable environmental damage to our beautiful planet, endangering the natural world, for us and our children."[247] Hawking further stated that this could lead Earth "to become like Venus, with a temperature of two hundred and fifty degrees, and raining sulphuric acid".[248] Hawking was also a supporter of a universal basic income.[249] He was critical of the Israeli government's position on the Israeli–Palestinian conflict and boycotted an academic conference in Israel over the occupation of the Palestinian territories, stating that Israel's policy "is likely to lead to disaster".[250] Personal life Marriages Hawking met his future wife, Jane Wilde, at a party in 1962.
- He said, "Climate change is one of the great dangers we face, and it's one we can prevent if we act now.
- In October 1964, the couple became engaged to marry, aware of the potential challenges that lay ahead due to Hawking's shortened life expectancy and physical limitations.[128][251] Hawking later recalled: "Before I got motor neuron disease, I was bored with life.
- But he was also critical of Labour leader Jeremy Corbyn, expressing scepticism over whether the party could win a general election under him.[246] Hawking feared Donald Trump's policies on global warming could endanger the planet and make global warming irreversible.

By denying the evidence for climate change, and pulling out of the Paris Agreement, Donald Trump will cause avoidable environmental damage to our beautiful planet, endangering the natural world, for us and our children."[247] Hawking further stated that this could lead Earth "to become like Venus, with a temperature of two hundred and fifty degrees, and raining sulphuric acid".[248] Hawking was also a supporter of a universal basic income.[249] He was critical of the Israeli government's position on the Israeli–Palestinian conflict and boycotted an academic conference in Israel over the occupation of the Palestinian territories, stating that Israel's policy "is likely to lead to disaster".[250] Personal life Marriages Hawking met his future wife, Jane Wilde, at a party in 1962.

He said, "Climate change is one of the great dangers we face, and it's one we can prevent if we act now.

In October 1964, the couple became engaged to marry, aware of the potential challenges that lay ahead due to Hawking's shortened life expectancy and physical limitations.[128][251] Hawking later recalled: "Before I got motor neuron disease, I was bored with life.

But he was also critical of Labour leader Jeremy Corbyn, expressing scepticism over whether the party could win a general election under him.[246] Hawking feared Donald Trump's policies on global warming could endanger the planet and make global warming irreversible.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Summarize the most important claims and how to verify them.

EVIDENCE:
- In late 2006, astronomers reported estimates of the Prograde bound orbit around a black hole rotating spin rates of black holes in The Astrophysical Journal.
- Relation with gamma ray bursts The formation of a rotating black hole by a collapsar is thought to be observed as the emission of gamma ray bursts.
- or collision of a collection of compact objects, stars, or gas with a total non-zero angular momentum.
- For this reason the term "astrophysical" black hole is usually reserved for the Kerr black hole.[5] While from an infalling observer's perspective the plunge into a rotating black hole occurs in a finite proper time and with very high rapidity (left), from Formation the perspective of a coordinate observer at infinity they slow down, approaching zero velocity at the horizon relative to a stationary probe on site while Rotating black holes are formed in the gravitational being whirled around forever by the black hole's collapse of a massive spinning star or from the collapse frame-dragging effect (right).

In late 2006, astronomers reported estimates of the Prograde bound orbit around a black hole rotating spin rates of black holes in The Astrophysical Journal.

Relation with gamma ray bursts The formation of a rotating black hole by a collapsar is thought to be observed as the emission of gamma ray bursts.

or collision of a collection of compact objects, stars, or gas with a total non-zero angular momentum.

For this reason the term "astrophysical" black hole is usually reserved for the Kerr black hole.[5] While from an infalling observer's perspective the plunge into a rotating black hole occurs in a finite proper time and with very high rapidity (left), from Formation the perspective of a coordinate observer at infinity they slow down, approaching zero velocity at the horizon relative to a stationary probe on site while Rotating black holes are formed in the gravitational being whirled around forever by the black hole's collapse of a massive spinning star or from the collapse frame-dragging effect (right).

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Which parts are established vs speculative, and what evidence supports each?

EVIDENCE:
- This is stated by the black hole uniqueness theorem: "black holes have
- Since no object can overtake a light pulse, all interior matter is imprisoned as well.
- The best-known examples are black holes: if mass is compressed into a sufficiently compact region of space (as specified in the hoop conjecture, the relevant length scale is the Schwarzschild radius, given by the equation [156]), no light from inside can escape to the outside.
- Using global geometry, later studies have revealed more general properties of black holes.

This is stated by the black hole uniqueness theorem: "black holes have

Since no object can overtake a light pulse, all interior matter is imprisoned as well.

The best-known examples are black holes: if mass is compressed into a sufficiently compact region of space (as specified in the hoop conjecture, the relevant length scale is the Schwarzschild radius, given by the equation [156]), no light from inside can escape to the outside.

Using global geometry, later studies have revealed more general properties of black holes.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Design a retrieval plan to answer a missing detail without guessing.

EVIDENCE:
- Despite this high density, however, the nuclei, taking up only a millionth of a billionth of the volume of intact atoms, can still move about freely, just as atoms or molecules do in ordinary gases.
- They can even collide with each other.
- Degenerate matter, despite its high density, therefore acts as a gas and has properties characteristic of a gas-a "nuclear gas," if you will.
- Within the electronic fluid, nuclei can move freely and can approach each other more closely, as closely as chance dictates.

Despite this high density, however, the nuclei, taking up only a millionth of a billionth of the volume of intact atoms, can still move about freely, just as atoms or molecules do in ordinary gases.

Degenerate matter, despite its high density, therefore acts as a gas and has properties characteristic of a gas-a "nuclear gas," if you will.

Within the electronic fluid, nuclei can move freely and can approach each other more closely, as closely as chance dictates.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
What does the evidence say about necessary conditions and constraints?

EVIDENCE:
- Galileo recognized that in projectile motion, the Earth's gravity affects vertical but not horizontal motion.[110] However, Galileo's idea Galileo Galilei (1564–1642) of inertia was not exactly the one that would be codified into Newton's first law.
- This idea was corrected by Isaac Beeckman, Descartes, and Pierre Gassendi, who recognized
- Galileo thought that a body moving a long distance inertially would follow the curve of the Earth.
- As long as (384–322 BCE) its impetus was sustained, the body would continue to move.[104]: 47 In the following centuries, versions of impetus theory were advanced by individuals including Nur ad-Din al-Bitruji, Avicenna, Abu'l-Barakāt al-Baghdādī, John Buridan, and Albert of Saxony.

Galileo recognized that in projectile motion, the Earth's gravity affects vertical but not horizontal motion.[110] However, Galileo's idea Galileo Galilei (1564–1642) of inertia was not exactly the one that would be codified into Newton's first law.

This idea was corrected by Isaac Beeckman, Descartes, and Pierre Gassendi, who recognized

Galileo thought that a body moving a long distance inertially would follow the curve of the Earth.

As long as (384–322 BCE) its impetus was sustained, the body would continue to move.[104]: 47 In the following centuries, versions of impetus theory were advanced by individuals including Nur ad-Din al-Bitruji, Avicenna, Abu'l-Barakāt al-Baghdādī, John Buridan, and Albert of Saxony.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.

Bennett
Dirac
Zeilinger
Jian-Wei Pan
De Martini
Feynman
Popper


Prompt:
Summarize the most important claims and how to verify them.

EVIDENCE:
- ∗Asexplainedalsointheprecedingchapter,thehomogeneityproblemimpliestherewasnotenoughtime fortheirregularitiesatthetimeofdecouplingtogrowintothestructuresweseetoday.Thatproblemwassolved bydarkmattergettingaheadstartongrowingstructures.
- InflationaryCosmology | 531 η A B C η η O P x Figure1 Causaldomainsintheearlyuniverse:EventsAand Barecausallycorrelated;similarly,eventsBandCarecausally correlated,butnoteventsAandC.
- Flatnessproblem From (VIII.2.29), we learned that (cid:24) , which measures the curvature of the universe, k evolvesaccordingto(cid:24) ̇ =H(cid:24) (2(cid:24) +(cid:24) −2(cid:24) ).Atonetime,itwasthoughtthat(cid:24) is k k r m (cid:27) (cid:27) strictlyzero,andso (cid:24) ̇ =H(cid:24) (2(cid:24) +(cid:24) ) (1) k k r m Since (2(cid:24) +(cid:24) )>0, any deviation away from (cid:24) =0 is unstable.
- As long as (cid:24) is r m k k not 0, its magnitude would grow regardless of its sign.

InflationaryCosmology | 531 η A B C η η O P x Figure1 Causaldomainsintheearlyuniverse:EventsAand Barecausallycorrelated;similarly,eventsBandCarecausally correlated,butnoteventsAandC.

Flatnessproblem From (VIII.2.29), we learned that (cid:24) , which measures the curvature of the universe, k evolvesaccordingto(cid:24) ̇ =H(cid:24) (2(cid:24) +(cid:24) −2(cid:24) ).Atonetime,itwasthoughtthat(cid:24) is k k r m (cid:27) (cid:27) strictlyzero,andso (cid:24) ̇ =H(cid:24) (2(cid:24) +(cid:24) ) (1) k k r m Since (2(cid:24) +(cid:24) )>0, any deviation away from (cid:24) =0 is unstable.

As long as (cid:24) is r m k k not 0, its magnitude would grow regardless of its sign.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Which parts are established vs speculative, and what evidence supports each?

EVIDENCE:
- doi:10.1016/S0032-0633(00)00039-8 (https://doi.org/ 10.1016%2FS0032-0633%2800%2900039-8).
- "Bayesian analysis of Enceladus's plume data to assess methanogenesis" (https://www.nature.com/articles/s41550-021-01372-6).
- doi:10.1038/s41550-021-01372-6 (https://doi.org/10.1038%2Fs41550-021-01372-6).
- ISSN 20411723 (https://search.worldcat.org/issn/2041-1723).

"Bayesian analysis of Enceladus's plume data to assess methanogenesis" (https://www.nature.com/articles/s41550-021-01372-6).

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Which parts are established vs speculative, and what evidence supports each?

EVIDENCE:
- In the chiral representation constructed in part 5, all orthonormal basis vectors γγ , and all orthonormal basis p-vectors a γγ ≡γγ ∧...∧γγ ,aretraceless(exceptfortheunitbasiselement1),unitary,andeitherHermitian a1...ap a1 ap (if[p/2]iseven,i.e.p=0,1,4,5,...)orskew-Hermitian(if[p/2]isodd,i.e.p=2,3,6,7,...)2[N/2]×2[N/2] matrices.
- 38.15 The super geometric algebra in arbitrarily many spatial dimensions 993 SinceN isodd,thischoiceofP reflectsanoddnumberofaxes,soindeedreversesparity.Theoperation P of reflecting all axes (other than the scalar axis γγ or γγ ) is rotationally invariant with respect to N N+1 rotations in N dimensions (with the scalar axis γγ or γγ fixed).
- Right- and left-handed chiral subalgebras in even dimensions.
- A basis spinor ε is a right-orleft-handeddependingonwhetherthenumberofspinflipsoftheindexa=a ...a ,relative 1 [N/2] to the all-up index ↑↑...↑, is even or odd, ( ∏ ) κ ε = (−) ε .

In the chiral representation constructed in part 5, all orthonormal basis vectors γγ , and all orthonormal basis p-vectors a γγ ≡γγ ∧...∧γγ ,aretraceless(exceptfortheunitbasiselement1),unitary,andeitherHermitian a1...ap a1 ap (if[p/2]iseven,i.e.p=0,1,4,5,...)orskew-Hermitian(if[p/2]isodd,i.e.p=2,3,6,7,...)2[N/2]×2[N/2] matrices.

38.15 The super geometric algebra in arbitrarily many spatial dimensions 993 SinceN isodd,thischoiceofP reflectsanoddnumberofaxes,soindeedreversesparity.Theoperation P of reflecting all axes (other than the scalar axis γγ or γγ ) is rotationally invariant with respect to N N+1 rotations in N dimensions (with the scalar axis γγ or γγ fixed).

Right- and left-handed chiral subalgebras in even dimensions.

A basis spinor ε is a right-orleft-handeddependingonwhetherthenumberofspinflipsoftheindexa=a ...a ,relative 1 [N/2] to the all-up index ↑↑...↑, is even or odd, ( ∏ ) κ ε = (−) ε .

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Summarize the most important claims and how to verify them.

EVIDENCE:
- SomeProjectBlueBookcasesatthetimeremindedsomeCIAofficialsofghosts, demons,etc.
- Dukealsoclaimsearlymembers oftheCollinsElitewereaskedbysomeintheCIADirectorateofPlansin1952todetermineifUAPhad demonicorigin.
- DukeclaimsOperationOftenandCollinsElitepersonnelusedacoverstorythattheywereUS ArmypsychologistsworkingonaprojectfortheUniversityofSouthCarolinatounderstandmental traumainmilitarypersonnel;thegroup’sactualgoalwastoanalyzethenightmaresofPaulGarrattaftera near-deathexperience.
- OftenpersonnelconcludedtheNDEsuggestedsaucersmightbeharvesting humansoulsandtheyweredemonicinnature.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.


Prompt:
Design a retrieval plan to answer a missing detail without guessing.

EVIDENCE:
- The astute reader might realize that there is a third equation, the definition of l: (dx)2+(dy)2=1.
- A thorough understanding of this example will be important in mastering Einstein gravity later, so to make sure that you follow, let me spell out the steps here.
- The more important lesson here, however, is to observe how (3) simplifies with the “preferred”orna(cid:2)turalparametrization,namelytouseforλthelengthl alongthecurve, definedbydl= dx2+dy2,sothat " (cid:3) (cid:4) (cid:3) (cid:4) (cid:2) dx 2 dy 2 dx2+dy2 + = =1 dl dl dl Replaceλbylin(3).Watchthesquarerootsdisappear,sothat(3)simplifiesto d2x =0and dl2 d2y =0.WewillexploitthissimplificationruthlesslywhenwegettoEinstein’stheoryof dl2 gravity.
- TheShortestDistancebetweenTwoPoints | 125 (cid:6) (cid:8) thatE= dx 1+(dy)2.Obviously,thelengthisminimizedbysetting dy =0.Imposing dx dx theboundaryconditions,weobtainthesolutiony(x)=0.

The astute reader might realize that there is a third equation, the definition of l: (dx)2+(dy)2=1.

A thorough understanding of this example will be important in mastering Einstein gravity later, so to make sure that you follow, let me spell out the steps here.

The more important lesson here, however, is to observe how (3) simplifies with the “preferred”orna(cid:2)turalparametrization,namelytouseforλthelengthl alongthecurve, definedbydl= dx2+dy2,sothat " (cid:3) (cid:4) (cid:3) (cid:4) (cid:2) dx 2 dy 2 dx2+dy2 + = =1 dl dl dl Replaceλbylin(3).Watchthesquarerootsdisappear,sothat(3)simplifiesto d2x =0and dl2 d2y =0.WewillexploitthissimplificationruthlesslywhenwegettoEinstein’stheoryof dl2 gravity.

TheShortestDistancebetweenTwoPoints | 125 (cid:6) (cid:8) thatE= dx 1+(dy)2.Obviously,thelengthisminimizedbysetting dy =0.Imposing dx dx theboundaryconditions,weobtainthesolutiony(x)=0.

If evidence lacks a detail, retrieve more chunks rather than guessing.

List tests or derivations implied by the evidence.