Prompt:
Evaluate a proposed wormhole construction claim and produce an evidence-bounded assessment plus verification plan.

EVIDENCE:
- (18.15) 0 ab b ac b0a0 In 4-dimensional spacetime, the 9 components of the extrinsic curvature K are commonly resolved into ab an expansion scalar θ, a 3-component antisymmetric vorticity tensor π , and a 5-component traceless ab symmetric shear tensor σ , ab K =δ θ+π +σ .
- If desired, the vorticity can be written π =ε πc.
- ab abc If one imagines comoving coordinates attached to the congruence of paths, then the extrinsic curvature describes the rate at which the comoving volume element distorts, equation (18.5).
- If the ab a0b a0b timelikepathsaregeodesic,thentheaccelerationK ≡K vanishes.Alongatimelikegeodesiccongruence, a a00 the Raychaudhuri equations (18.12) become D^ K +Kc K =−R .
- 18.3 Raychaudhuri equations for a timelike geodesic congruence 535 For a geodesic congruence (not necessarily affinely parameterized), the Raychaudhuri equation (18.10) becomes D^ K −Ky K +Kc K =−R (no sum over z) .

(18.15) 0 ab b ac b0a0 In 4-dimensional spacetime, the 9 components of the extrinsic curvature K are commonly resolved into ab an expansion scalar θ, a 3-component antisymmetric vorticity tensor π , and a 5-component traceless ab symmetric shear tensor σ , ab K =δ θ+π +σ .

If desired, the vorticity can be written π =ε πc.

ab abc If one imagines comoving coordinates attached to the congruence of paths, then the extrinsic curvature describes the rate at which the comoving volume element distorts, equation (18.5).

If the ab a0b a0b timelikepathsaregeodesic,thentheaccelerationK ≡K vanishes.Alongatimelikegeodesiccongruence, a a00 the Raychaudhuri equations (18.12) become D^ K +Kc K =−R .

18.3 Raychaudhuri equations for a timelike geodesic congruence 535 For a geodesic congruence (not necessarily affinely parameterized), the Raychaudhuri equation (18.10) becomes D^ K −Ky K +Kc K =−R (no sum over z) .

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Explain quantum teleportation rigorously and outline an experimental validation checklist.

EVIDENCE:
- Much better than most popular physics books on the market(seefigure1).
- 18 | Prelude vesselbeneath;and,inthrowingsomethingtoyourfriend,youneedthrowitnomorestrongly inonedirectionthananother,thedistancesbeingequal;jumpingwithyourfeettogether,you passequalspacesineverydirection.Whenyouhaveobservedallthesethingscarefully(though doubtlesswhentheshipisstandingstilleverythingmusthappeninthisway),havetheship proceedwithanyspeedyoulike,solongasthemotionisuniformandnotfluctuatingthisway andthat.Youwilldiscovernottheleastchangeinalltheeffectsnamed,norcouldyoutellfrom anyofthemwhethertheshipwasmovingorstandingstill.Injumping,youwillpassonthe floorthesamespacesasbefore,norwillyoumakelargerjumpstowardthesternthantoward theproweventhoughtheshipismovingquiterapidly,despitethefactthatduringthetime thatyouareintheairthefloorunderyouwillbegoinginadirectionoppositetoyourjump.In throwingsomethingtoyourcompanion,youwillneednomoreforcetogetittohimwhether heisinthedirectionoftheboworthestern,withyourselfsituatedopposite.Thedropletswill fallasbeforeintothevesselbeneathwithoutdroppingtowardthestern,althoughwhilethe dropsareintheairtheshiprunsmanyspans.Thefishintheirwaterwillswimtowardthe frontoftheirbowlwithnomoreeffortthantowardtheback,andwillgowithequaleasetobait placedanywherearoundtheedgesofthebowl.Finallythebutterfliesandflieswillcontinue theirflightsindifferentlytowardeveryside,norwilliteverhappenthattheyareconcentrated towardthestern,asiftiredoutfromkeepingupwiththecourseoftheship,fromwhichthey willhavebeenseparatedduringlongintervalsbykeepingthemselvesintheair.Andifsmokeis madebyburningsomeincense,itwillbeseengoingupintheformofalittlecloud,remaining stillandmovingnomoretowardonesidethantheother.Thecauseofallthesecorrespondences ∗ ofeffectsisthefactthattheship’smotioniscommon toallthethingscontainedinit,andto theairalso.ThatiswhyIsaidyoushouldbebelowdecks;forifthistookplaceaboveinthe openair,whichwouldnotfollowthecourseoftheship,moreorlessnoticeabledifferences wouldbeseeninsomeoftheeffectsnoted.2 That† is so beautifully stated!
- ‡ThehistorianPeterGalisonhaspointedoutthatintheperiodleadingupto1905,theyearEinsteinproposed histheoryofspecialrelativity,highspeedtrainsandthetelegraphlinkedthecitiesofEurope,andanincreasingly technologicalsocietywaspreoccupiedwithclocksynchronizationamongotherthings.3
- Galileo’sshipwasupdatedtoEinstein’strain‡andlatertorocketshipsandotherspace vehicles.Let’suseEinstein’strain,movingsmoothlyalongthex-axiswithvelocityu(see figure2).Letaneventoccuratthepoint(x,y,z)attimetfortheobserveronthetrain(call (cid:5) (cid:5) (cid:5) (cid:5) herMs.Unprime)andatthepoint(x ,y ,z)attimet fortheobserverontheground(Mr.
- Prime).WeareofcourseutilizingtheprofoundandbrilliantinsightofGalileo’scontemporaryRen ́eDescartes(1596–1650)thatgeometrycanbereducedtoalgebrabyassociating threenumberswitheachpointinspace.TheGalileantransformationstatesthat t (cid:5)=t (1) ∗Thephrase“commontoallthethingscontainedinit”willplayastarringrolewhenwegettoEinstein’s equivalenceprinciple,aswewillseeinpartV.

Chosen based on domain + required checks.

Produce bounded conclusion + next actions

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.

Apply expert-specific checks; do not introduce new facts beyond evidence.

Much better than most popular physics books on the market(seefigure1).

Galileo’sshipwasupdatedtoEinstein’strain‡andlatertorocketshipsandotherspace vehicles.Let’suseEinstein’strain,movingsmoothlyalongthex-axiswithvelocityu(see figure2).Letaneventoccuratthepoint(x,y,z)attimetfortheobserveronthetrain(call (cid:5) (cid:5) (cid:5) (cid:5) herMs.Unprime)andatthepoint(x ,y ,z)attimet fortheobserverontheground(Mr.

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Explain quantum teleportation rigorously and outline an experimental validation checklist.

EVIDENCE:
- Finally, the 4-velocity βμ is simply related to β by βμ =(β/r)∂r/∂xμ.
- 19.4.2 Doran-Cartesian tetrad Like the Gullstrand-Painlevé tetrad, the Doran-Cartesian tetrad γγ ≡ {γγ ,γγ ,γγ ,γγ } is aligned with the m 0 1 2 3 Cartesian rest frame e ≡ {e ,e ,e ,e } at infinity, and is parallel-transported, without precessing, by μ tff x y z observers who free-fall from zero velocity and zero angular momentum at infinity, as can be confirmed by checking that the tetrad connections with final index 0 all vanish, Γ =0, equation (19.20).
- nm0 Let ‖ and ⊥ subscripts denote horizontal radial and azimuthal directions respectively, so that γγ ≡cosφ γγ +sinφ γγ , γγ ≡− sinφ γγ +cosφ γγ , ‖ ff 1 ff 2 ⊥ ff 1 ff 2 (19.55) e ≡cosφ e +sinφ e , e ≡− sinφ e +cosφ e .
- Note that the squared magnitude β βμ of the velocity vector is not β2, but rather differs from β2 by a factor of R2/ρ2: μ β2R2 β βμ =β βm = .
- Moreover, with the convention here, β is ∓1 at horizons, equation (19.51).

Finally, the 4-velocity βμ is simply related to β by βμ =(β/r)∂r/∂xμ.

19.4.2 Doran-Cartesian tetrad Like the Gullstrand-Painlevé tetrad, the Doran-Cartesian tetrad γγ ≡ {γγ ,γγ ,γγ ,γγ } is aligned with the m 0 1 2 3 Cartesian rest frame e ≡ {e ,e ,e ,e } at infinity, and is parallel-transported, without precessing, by μ tff x y z observers who free-fall from zero velocity and zero angular momentum at infinity, as can be confirmed by checking that the tetrad connections with final index 0 all vanish, Γ =0, equation (19.20).

nm0 Let ‖ and ⊥ subscripts denote horizontal radial and azimuthal directions respectively, so that γγ ≡cosφ γγ +sinφ γγ , γγ ≡− sinφ γγ +cosφ γγ , ‖ ff 1 ff 2 ⊥ ff 1 ff 2 (19.55) e ≡cosφ e +sinφ e , e ≡− sinφ e +cosφ e .

Note that the squared magnitude β βμ of the velocity vector is not β2, but rather differs from β2 by a factor of R2/ρ2: μ β2R2 β βμ =β βm = .

Moreover, with the convention here, β is ∓1 at horizons, equation (19.51).

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Design a retrieval+analysis pipeline to discover missing facts from the corpus without hallucinating.

EVIDENCE:
- I spend the rest of it four blocks from my brownstone, sitting on the pavement behind a cardboard sign that reads: HOMELESS.
- Underneath it, I listen to the rain pattering on the cardboard above my head, hoping my makeshift shelter will last the night.
- I crawl between them, taking with me a flattened box, which I lean against the wall of my garage.
- I choose a diner that looks shitty enough to have me, and as I pay for my meal, the exhaustion hits.
- From my vantage point, I can see over the high fence that encircles my backyard, through a window, into the second floor of my house.

I spend the rest of it four blocks from my brownstone, sitting on the pavement behind a cardboard sign that reads: HOMELESS.

Underneath it, I listen to the rain pattering on the cardboard above my head, hoping my makeshift shelter will last the night.

I crawl between them, taking with me a flattened box, which I lean against the wall of my garage.

I choose a diner that looks shitty enough to have me, and as I pay for my meal, the exhaustion hits.

From my vantage point, I can see over the high fence that encircles my backyard, through a window, into the second floor of my house.

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Explain quantum teleportation rigorously and outline an experimental validation checklist.

EVIDENCE:
- farming that of the Pharaoh Khufu, or Cheops - built in about 2550 the river’s banks from about 5000 BC.
- It is also called the Great Pyramid.
- It is 137 metres they created a kingdom which stretched from the mouth high, and made of between five and six million tonnes of of the Nile river to Abu Simbel, nearly 1000 kilometres to stone.
- The accuracy with which it was built is still today a S T A R G A T E T e a c h e r’s n o t e s from the screenplay and novelization by Dean Devlin and Roland Emmerich, adapted by Sheila Black Y BACKGROUND AND THEMES ©Pearson Education 2000

farming that of the Pharaoh Khufu, or Cheops - built in about 2550 the river’s banks from about 5000 BC.

It is 137 metres they created a kingdom which stretched from the mouth high, and made of between five and six million tonnes of of the Nile river to Abu Simbel, nearly 1000 kilometres to stone.

The accuracy with which it was built is still today a S T A R G A T E T e a c h e r’s n o t e s from the screenplay and novelization by Dean Devlin and Roland Emmerich, adapted by Sheila Black Y BACKGROUND AND THEMES ©Pearson Education 2000

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Explain quantum teleportation rigorously and outline an experimental validation checklist.

EVIDENCE:
- 1906 Personality Newton has been described as an incredibly driven and disciplined man who dedicated his life to his work.
- Since he was already rich before the bubble, Newton still died rich, at estate value around £30,000.[201] Toward the end of his life, Newton spent some time at Cranbury Park, near Winchester, the country residence of his niece and her husband, though he primarily lived in London.[202][203] His half-niece, Catherine Barton,[204] served as his hostess in social affairs at his house on Jermyn Street in London.
- It was used by the Flemish sculptor John Michael Rysbrack in making a sculpture of Newton.[213] It is now held by the Royal Society.[214] Newton's hair was posthumously examined and found to contain mercury, probably resulting from his alchemical pursuits.
- He is known for having a prodigious appetite for work, which he prioritised above his personal health.
- The Lord Chancellor, two dukes, and three earls bore his pall, with most of the Royal Society following.

1906 Personality Newton has been described as an incredibly driven and disciplined man who dedicated his life to his work.

Since he was already rich before the bubble, Newton still died rich, at estate value around £30,000.[201] Toward the end of his life, Newton spent some time at Cranbury Park, near Winchester, the country residence of his niece and her husband, though he primarily lived in London.[202][203] His half-niece, Catherine Barton,[204] served as his hostess in social affairs at his house on Jermyn Street in London.

It was used by the Flemish sculptor John Michael Rysbrack in making a sculpture of Newton.[213] It is now held by the Royal Society.[214] Newton's hair was posthumously examined and found to contain mercury, probably resulting from his alchemical pursuits.

He is known for having a prodigious appetite for work, which he prioritised above his personal health.

The Lord Chancellor, two dukes, and three earls bore his pall, with most of the Royal Society following.

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Design a retrieval+analysis pipeline to discover missing facts from the corpus without hallucinating.

EVIDENCE:
- With the usual coordinates (t,r,θ,φ), the metric components g (r,θ) are μν functionsonlyofrandθ,butnotoftandφ.Furthermore,thesolutionmustbeunchanged under the discrete transformation t →−t together with φ→−φ.
- Stationary means that theobjectrotateswithaconstantangularvelocity,sothatthespacetimedoesnotchange with time.
- For normalstars,suchasthesun,theamountofrotation,asmeasuredbytherotationalspeed vatthesurfacedividedbyc,isnegligible.That’swhyweareabletousetheSchwarzschild metrictodescribethespacetimeoutsidethesun.Aroundablackhole,however,infalling debrisinvariablycausestheblackholetospin,aswesawinchapterVII.1.Severalimportant astrophysicalprocessesappeartobepoweredbyrotatingblackholes.Oneofourgoalsin this chapter is to understand how rotating black holes can be such powerful sources of radiation.
- μν Weanticipatethattherecouldbeahorizon.Arotatingobjectsmallenoughtofitinsideits ownhorizonisknownasaKerrblackhole.
- Before we look at Kerr’s specific solution, let us see how far we can get with general considerations.

With the usual coordinates (t,r,θ,φ), the metric components g (r,θ) are μν functionsonlyofrandθ,butnotoftandφ.Furthermore,thesolutionmustbeunchanged under the discrete transformation t →−t together with φ→−φ.

Stationary means that theobjectrotateswithaconstantangularvelocity,sothatthespacetimedoesnotchange with time.

For normalstars,suchasthesun,theamountofrotation,asmeasuredbytherotationalspeed vatthesurfacedividedbyc,isnegligible.That’swhyweareabletousetheSchwarzschild metrictodescribethespacetimeoutsidethesun.Aroundablackhole,however,infalling debrisinvariablycausestheblackholetospin,aswesawinchapterVII.1.Severalimportant astrophysicalprocessesappeartobepoweredbyrotatingblackholes.Oneofourgoalsin this chapter is to understand how rotating black holes can be such powerful sources of radiation.

Before we look at Kerr’s specific solution, let us see how far we can get with general considerations.

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Design a retrieval+analysis pipeline to discover missing facts from the corpus without hallucinating.

EVIDENCE:
- Y R Exercises 42.2 and 42.3 show that SU(2)×SU(2) is isomorphic to Spin(4), while SU(4) is isomorphic to Spin(6).
- (42.4) The factors of 3 in 3Y and 2 in 2I in the exponents arise because hypercharge and isospin are quantized in L units of respectively 1 and 1; the choice of exponents ensures that a unit phase factor z =e2πi acts trivially 3 2 on all fermions for each of the U (1)×SU (2)×SU(3) factors individually.
- (42.5) Y L 3 R L Consider treating the right-handed leptons and quarks as SU (2) doublets labelled by right-handed isospin R I ,similartotheirleft-handedcounterparts.Consideralsotreatingwhiteasa4thcolourw.TheSMparticles R in table 42.1 satisfy 3Y −6I +3w−(r+g+b)=0 .
- (42.6) R The pattern suggests an embedding U (1)×SU(3)/Z → SU (2)×SU(4) Y 3 R ( α3 0 ) ( α−3 0 ) (42.7) {α,h} → { , } .
- As it happens, as first pointed out by Pati and Salam (1974), the SM group, modulo the discrete group Z , extends as a subgroup along precisely these lines, U (1)×SU (2)×SU(3)/Z ⊂ SU (2)×SU (2)×SU(4) .

Bennett
Dirac
Zeilinger
Jian-Wei Pan
De Martini
Feynman
Popper

Y R Exercises 42.2 and 42.3 show that SU(2)×SU(2) is isomorphic to Spin(4), while SU(4) is isomorphic to Spin(6).

(42.4) The factors of 3 in 3Y and 2 in 2I in the exponents arise because hypercharge and isospin are quantized in L units of respectively 1 and 1; the choice of exponents ensures that a unit phase factor z =e2πi acts trivially 3 2 on all fermions for each of the U (1)×SU (2)×SU(3) factors individually.

(42.5) Y L 3 R L Consider treating the right-handed leptons and quarks as SU (2) doublets labelled by right-handed isospin R I ,similartotheirleft-handedcounterparts.Consideralsotreatingwhiteasa4thcolourw.TheSMparticles R in table 42.1 satisfy 3Y −6I +3w−(r+g+b)=0 .

(42.6) R The pattern suggests an embedding U (1)×SU(3)/Z → SU (2)×SU(4) Y 3 R ( α3 0 ) ( α−3 0 ) (42.7) {α,h} → { , } .

As it happens, as first pointed out by Pati and Salam (1974), the SM group, modulo the discrete group Z , extends as a subgroup along precisely these lines, U (1)×SU (2)×SU(3)/Z ⊂ SU (2)×SU (2)×SU(4) .

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Explain quantum teleportation rigorously and outline an experimental validation checklist.

EVIDENCE:
- He’s wearing a tweed jacket, his hair is graying, and he’s paler and less fit than the last time I saw him, which was impossibly at Village Tap last night at his celebration for winning the Pavia Prize.
- As the barkeep pours our whiskies—Macallan 12s—into plastic cups, Ryan says, “I know you’re doing just fine, but I got these.” It’s so strange—he carries none of the arrogance and swagger of the man I saw holding court last night at my local bar.
- “What do you mean?” “That conversation we had a year and a half ago?
- You’ll come?” “I’d love to.” As I watch Daniela vanish into the crowd, Ryan says, “Want to get a drink?” God yes.
- Didn’t you see the dedication?” “No, where was it?” “At the entrance to the labyrinth.

He’s wearing a tweed jacket, his hair is graying, and he’s paler and less fit than the last time I saw him, which was impossibly at Village Tap last night at his celebration for winning the Pavia Prize.

As the barkeep pours our whiskies—Macallan 12s—into plastic cups, Ryan says, “I know you’re doing just fine, but I got these.” It’s so strange—he carries none of the arrogance and swagger of the man I saw holding court last night at my local bar.

“What do you mean?” “That conversation we had a year and a half ago?

You’ll come?” “I’d love to.” As I watch Daniela vanish into the crowd, Ryan says, “Want to get a drink?” God yes.

Didn’t you see the dedication?” “No, where was it?” “At the entrance to the labyrinth.

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Explain quantum teleportation rigorously and outline an experimental validation checklist.

EVIDENCE:
- At the time, it was thought that in order for light to travel through a vacuum, there must have been a medium filling the void through which it could propagate, as sound through air or ripples in a pool.
- It has also been called a fifth fundamental force.
- elements from which the name was borrowed.
- These aether theories are considered to be scientifically obsolete, as the development of special relativity showed that Maxwell's equations do not require the aether for the transmission of these forces.
- Einstein noted that his own model which replaced these theories could itself be thought of as an aether, as it implied that the empty space between objects had its own physical properties.[19] Despite the early modern aether models being superseded by general relativity, occasionally some physicists have attempted to reintroduce the concept of aether in an attempt to address perceived deficiencies in current physical models.[20] One proposed model of dark energy has been named "quintessence" by its proponents, in honor of the classical element.[21] This idea relates to the hypothetical form of dark energy postulated as an explanation of observations of an accelerating universe.

At the time, it was thought that in order for light to travel through a vacuum, there must have been a medium filling the void through which it could propagate, as sound through air or ripples in a pool.

It has also been called a fifth fundamental force.

elements from which the name was borrowed.

These aether theories are considered to be scientifically obsolete, as the development of special relativity showed that Maxwell's equations do not require the aether for the transmission of these forces.

Einstein noted that his own model which replaced these theories could itself be thought of as an aether, as it implied that the empty space between objects had its own physical properties.[19] Despite the early modern aether models being superseded by general relativity, occasionally some physicists have attempted to reintroduce the concept of aether in an attempt to address perceived deficiencies in current physical models.[20] One proposed model of dark energy has been named "quintessence" by its proponents, in honor of the classical element.[21] This idea relates to the hypothetical form of dark energy postulated as an explanation of observations of an accelerating universe.

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.

Aether (classical element) - Wikipedia.pdf


Prompt:
Design a retrieval+analysis pipeline to discover missing facts from the corpus without hallucinating.

EVIDENCE:
- Contents xi 17.9 Pretorius formalism 528 17.10 M+N split 529 17.11 2+2 split 531 18 Singularity theorems 532 18.1 Congruences 532 18.2 Raychaudhuri equations 534 18.3 Raychaudhuri equations for a timelike geodesic congruence 535 18.4 Raychaudhuri equations for a null geodesic congruence 537 18.5 Sachs optical coefficients 540 18.6 Hypersurface-orthogonality for a timelike congruence 540 18.7 Hypersurface-orthogonality for a null congruence 543 18.8 Focusing theorems 546 18.9 Singularity theorems 548 Concept Questions 551 What’s important?
- 552 19 Black hole waterfalls 553 19.1 Tetrads move through coordinates 553 19.2 Gullstrand-Painlevé waterfall 554 19.3 Boyer-Lindquist tetrad 560 19.4 Doran waterfall 562 20 General spherically symmetric spacetimes 568 20.1 Spherical spacetime 568 20.2 Spherical line-element 568 20.3 Rest diagonal line-element 571 20.4 Comoving diagonal line-element 571 20.5 Tetrad connections 572 20.6 Riemann, Einstein, and Weyl tensors 574 20.7 Einstein equations 575 20.8 Choose your frame 575 20.9 Interior mass 576 20.10 Energy-momentum conservation 577 20.11 Structure of the Einstein equations 579 20.12 Comparison to ADM (3+1) formulation 581 20.13 Spherical electromagnetic field 582 20.14 General relativistic stellar structure 583 20.15 Freely-falling dust without shell-crossing 584 20.16 Naked singularities in dust collapse 586 20.17 Thin spherical shells 589 20.18 Self-similar spherically symmetric spacetime 599

Contents xi 17.9 Pretorius formalism 528 17.10 M+N split 529 17.11 2+2 split 531 18 Singularity theorems 532 18.1 Congruences 532 18.2 Raychaudhuri equations 534 18.3 Raychaudhuri equations for a timelike geodesic congruence 535 18.4 Raychaudhuri equations for a null geodesic congruence 537 18.5 Sachs optical coefficients 540 18.6 Hypersurface-orthogonality for a timelike congruence 540 18.7 Hypersurface-orthogonality for a null congruence 543 18.8 Focusing theorems 546 18.9 Singularity theorems 548 Concept Questions 551 What’s important?

552 19 Black hole waterfalls 553 19.1 Tetrads move through coordinates 553 19.2 Gullstrand-Painlevé waterfall 554 19.3 Boyer-Lindquist tetrad 560 19.4 Doran waterfall 562 20 General spherically symmetric spacetimes 568 20.1 Spherical spacetime 568 20.2 Spherical line-element 568 20.3 Rest diagonal line-element 571 20.4 Comoving diagonal line-element 571 20.5 Tetrad connections 572 20.6 Riemann, Einstein, and Weyl tensors 574 20.7 Einstein equations 575 20.8 Choose your frame 575 20.9 Interior mass 576 20.10 Energy-momentum conservation 577 20.11 Structure of the Einstein equations 579 20.12 Comparison to ADM (3+1) formulation 581 20.13 Spherical electromagnetic field 582 20.14 General relativistic stellar structure 583 20.15 Freely-falling dust without shell-crossing 584 20.16 Naked singularities in dust collapse 586 20.17 Thin spherical shells 589 20.18 Self-similar spherically symmetric spacetime 599

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Design a retrieval+analysis pipeline to discover missing facts from the corpus without hallucinating.

EVIDENCE:
- (2002), "Numerical Approaches to Spacetime Singularities", Living Reviews in Relativity, 5 (1) 1, arXiv:gr-qc/0201056 (https://arxiv.org/abs/gr-qc/0201056), Bibcode:2002LRR.....5....1B (https://ui.adsabs.harvard.edu/abs/2002LRR.....5....1B), doi:10.12942/lrr-2002-1 (https://doi.org/10.12942%2Flrr-2002-1), PMC 5256073 (https://www.ncbi.
- (2006), "Stationary black holes", in Françoise, J.-P.; Naber, G.; Tsou, T.S.
- (1973), "Black Holes and Entropy", Phys.
- (eds.), General Relativity and Gravitation.
- D, 7 (8): 2333–2346, Bibcode:1973PhRvD...7.2333B (https://ui.adsabs.harvard.edu/abs/1973PhRvD...7.2333B), doi:10.1103/PhysRevD.7.2333 (https://doi.org/10.1103%2FPhysRevD.7.2333), S2CID 122636624 (https://api.semanticscholar.org/CorpusID:122636624) Bekenstein, Jacob D.

(2002), "Numerical Approaches to Spacetime Singularities", Living Reviews in Relativity, 5 (1) 1, arXiv:gr-qc/0201056 (https://arxiv.org/abs/gr-qc/0201056), Bibcode:2002LRR.....5....1B (https://ui.adsabs.harvard.edu/abs/2002LRR.....5....1B), doi:10.12942/lrr-2002-1 (https://doi.org/10.12942%2Flrr-2002-1), PMC 5256073 (https://www.ncbi.

(2006), "Stationary black holes", in Françoise, J.-P.; Naber, G.; Tsou, T.S.

(1973), "Black Holes and Entropy", Phys.

(eds.), General Relativity and Gravitation.

D, 7 (8): 2333–2346, Bibcode:1973PhRvD...7.2333B (https://ui.adsabs.harvard.edu/abs/1973PhRvD...7.2333B), doi:10.1103/PhysRevD.7.2333 (https://doi.org/10.1103%2FPhysRevD.7.2333), S2CID 122636624 (https://api.semanticscholar.org/CorpusID:122636624) Bekenstein, Jacob D.

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Explain quantum teleportation rigorously and outline an experimental validation checklist.

EVIDENCE:
- 17, Hawking began his university education at University College, Oxford.[28][58] For the first eighteen months, he was bored and lonely – he found the academic work "ridiculously easy".[59][60] His physics tutor, Robert Berman, later said, "It was only necessary for him to know that something could be done, and he could do it without looking to see how other people did it."[2] A change occurred during his second and third years when, according to Berman, Hawking made more of an effort "to be one of the boys".
- He developed into a popular, lively and witty college member, interested in classical music and science fiction.[58] Part of the transformation resulted from his decision to join the college boat club, the University College Boat Club, where he coxed a rowing crew.[61][62] The rowing coach at the time noted that Hawking cultivated a daredevil image, steering his crew on risky courses that led to damaged boats.[61][63] Hawking estimated that he studied about 1,000 hours during his three years at Oxford.
- These unimpressive study habits made sitting his finals a challenge, and he decided to answer only theoretical physics questions rather than those requiring factual knowledge.
- A first-class degree was a condition of acceptance for his planned graduate study in

17, Hawking began his university education at University College, Oxford.[28][58] For the first eighteen months, he was bored and lonely – he found the academic work "ridiculously easy".[59][60] His physics tutor, Robert Berman, later said, "It was only necessary for him to know that something could be done, and he could do it without looking to see how other people did it."[2] A change occurred during his second and third years when, according to Berman, Hawking made more of an effort "to be one of the boys".

He developed into a popular, lively and witty college member, interested in classical music and science fiction.[58] Part of the transformation resulted from his decision to join the college boat club, the University College Boat Club, where he coxed a rowing crew.[61][62] The rowing coach at the time noted that Hawking cultivated a daredevil image, steering his crew on risky courses that led to damaged boats.[61][63] Hawking estimated that he studied about 1,000 hours during his three years at Oxford.

These unimpressive study habits made sitting his finals a challenge, and he decided to answer only theoretical physics questions rather than those requiring factual knowledge.

A first-class degree was a condition of acceptance for his planned graduate study in

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Evaluate a proposed wormhole construction claim and produce an evidence-bounded assessment plus verification plan.

EVIDENCE:
- Flexure mechanisms A flexure mechanism consists of a series of rigid bodies connected by compliant elements (also known as flexure joints) that is designed to produce a geometrically well-defined motion upon application of a force.
- Spatial mechanism A spatial mechanism is a mechanical system that has at least one body that moves in a way that its point trajectories are general space curves.
- The rotational axes of hinged joints that connect the bodies in the system pass through the center of these circle.
- The recognition that the frame of a mechanism is an important machine element changed the name three-bar linkage into four-bar linkage.
- In general, bearings are designed for pure rotation or straight line movement.

Flexure mechanisms A flexure mechanism consists of a series of rigid bodies connected by compliant elements (also known as flexure joints) that is designed to produce a geometrically well-defined motion upon application of a force.

Spatial mechanism A spatial mechanism is a mechanical system that has at least one body that moves in a way that its point trajectories are general space curves.

The rotational axes of hinged joints that connect the bodies in the system pass through the center of these circle.

The recognition that the frame of a mechanism is an important machine element changed the name three-bar linkage into four-bar linkage.

In general, bearings are designed for pure rotation or straight line movement.

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Explain quantum teleportation rigorously and outline an experimental validation checklist.

EVIDENCE:
- Again, if the tetrad connections are replaced km by their torsion-free expressions in terms of vierbein derivatives, then the symmetry of the Ricci tensor is satisfied identically, so that two expressions (17.44b) and (17.44c) are identical, and one of them can be discarded as redundant.
- (17.46) m ab At this point it is common to argue that the covariant divergence D Km has no effect on equations of m motion, so can be dropped from the Ricci scalar, yielding the so-called ADM Lagrangian 1 ( ) L = R^+KbaK −K2 .
- (17.45) m ab a ForADM,therestrictedRicciscalarR^ istheRicciscalarrestrictedtothe3-dimensionalspatialhypersurface of constant time.
- Contracting the Ricci tensor yields the Ricci scalar R, R=R^−2D^ Km+KbaK +K2−2KaK .
- Converting the restricted covariant divergence D^ Km to a total covariant derivative D Km using equam m tion (17.38) brings the Ricci scalar to R=R^−2D Km+KbaK −K2 .

Again, if the tetrad connections are replaced km by their torsion-free expressions in terms of vierbein derivatives, then the symmetry of the Ricci tensor is satisfied identically, so that two expressions (17.44b) and (17.44c) are identical, and one of them can be discarded as redundant.

(17.46) m ab At this point it is common to argue that the covariant divergence D Km has no effect on equations of m motion, so can be dropped from the Ricci scalar, yielding the so-called ADM Lagrangian 1 ( ) L = R^+KbaK −K2 .

(17.45) m ab a ForADM,therestrictedRicciscalarR^ istheRicciscalarrestrictedtothe3-dimensionalspatialhypersurface of constant time.

Contracting the Ricci tensor yields the Ricci scalar R, R=R^−2D^ Km+KbaK +K2−2KaK .

Converting the restricted covariant divergence D^ Km to a total covariant derivative D Km using equam m tion (17.38) brings the Ricci scalar to R=R^−2D Km+KbaK −K2 .

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Evaluate a proposed wormhole construction claim and produce an evidence-bounded assessment plus verification plan.

EVIDENCE:
- TheUniverseasaCurvedSpacetime | 297 ∗ You might have noticed that we can absorb the length scale L into r.
- You might wonder why d(t,R) is so proper: as we have seen in the derivation of (9), a(t)|(cid:10)x(cid:3)| is the actual operational distance between two neighboring points separated ∗Foramorecarefultreatment,seechapterX.1.
- One often hears the assertion that “the universe is now known to be flat” as if it were an absolute statement.† In fact, physical observation can onlygiveusa(possiblyverylarge)lowerboundonL.Forthisreason,thereaderwillsee meoftenlaboriouslydraggingLaroundwhenIcouldhavechuckedit.
- †Nexttimeyouhearthisstatement,youwillknowtoaskhowaphysicalmeasurementcouldpossiblygive anexactresultwithoutanyerrorbars.Typically,thespeakerwillmumblesomethingaboutkbeingadiscrete variablethatcanonlytakeonthreeintegervaluesk=1,0,and−1.
- Properdistancesincosmologyanda“cosmicconspiracy” Intheliterature,peopleofteninvitethemselvestodefine,perhapsabitsloppily,a“proper distance”d(t,R)betweentwodistantpoints(t,0,θ,φ)and(t,R,θ,φ)byintegratingthe (cid:2) lengthsegmentdl= g (t,r)dr derivedin(9): rr (cid:7) R dr d(t,R)≡a(t) (cid:6) (20) 0 1−kr2 L2 Butwejustdidthisintegral.Thus,wehave d(t,R)=a(t)Lsin −1(R/L) (closed) d(t,R)=a(t)R (flat) d(t,R)=a(t)Lsinh −1(R/L) (open) (21) Asexpected,forsmallR,d(t,R)(cid:16)a(t)R.Forthecloseduniverse,d(t,R)isonlydefined forR≤L.

TheUniverseasaCurvedSpacetime | 297 ∗ You might have noticed that we can absorb the length scale L into r.

You might wonder why d(t,R) is so proper: as we have seen in the derivation of (9), a(t)|(cid:10)x(cid:3)| is the actual operational distance between two neighboring points separated ∗Foramorecarefultreatment,seechapterX.1.

One often hears the assertion that “the universe is now known to be flat” as if it were an absolute statement.† In fact, physical observation can onlygiveusa(possiblyverylarge)lowerboundonL.Forthisreason,thereaderwillsee meoftenlaboriouslydraggingLaroundwhenIcouldhavechuckedit.

Properdistancesincosmologyanda“cosmicconspiracy” Intheliterature,peopleofteninvitethemselvestodefine,perhapsabitsloppily,a“proper distance”d(t,R)betweentwodistantpoints(t,0,θ,φ)and(t,R,θ,φ)byintegratingthe (cid:2) lengthsegmentdl= g (t,r)dr derivedin(9): rr (cid:7) R dr d(t,R)≡a(t) (cid:6) (20) 0 1−kr2 L2 Butwejustdidthisintegral.Thus,wehave d(t,R)=a(t)Lsin −1(R/L) (closed) d(t,R)=a(t)R (flat) d(t,R)=a(t)Lsinh −1(R/L) (open) (21) Asexpected,forsmallR,d(t,R)(cid:16)a(t)R.Forthecloseduniverse,d(t,R)isonlydefined forR≤L.

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Evaluate a proposed wormhole construction claim and produce an evidence-bounded assessment plus verification plan.

EVIDENCE:
- 3.Question What fundamental question regarding neutron stars and black holes did physicists seek to answer, as discussed in Chapter 5?
- His insights laid the groundwork for later explorations into stellar remnants and the conditions under which black holes form.
- This inquiry is critical because if neutron stars can exist with sufficient mass, then black holes may form under specific conditions when stars collapse beyond a particular magnitude.
- Answer:The fundamental question was: How massive can a neutron star be?
- 4.Question What significant discovery did Oppenheimer and Snyder make regarding the behavior of light and time during a star's implosion?

3.Question What fundamental question regarding neutron stars and black holes did physicists seek to answer, as discussed in Chapter 5?

His insights laid the groundwork for later explorations into stellar remnants and the conditions under which black holes form.

This inquiry is critical because if neutron stars can exist with sufficient mass, then black holes may form under specific conditions when stars collapse beyond a particular magnitude.

Answer:The fundamental question was: How massive can a neutron star be?

4.Question What significant discovery did Oppenheimer and Snyder make regarding the behavior of light and time during a star's implosion?

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Evaluate a proposed wormhole construction claim and produce an evidence-bounded assessment plus verification plan.

EVIDENCE:
- Thus, there are others view point to consider this join.
- Remembering that each point in spacetime diagram represent 2-sphere of the solid angle.
- Chapter 2 Wormhole and Einstein-Rosen bridge As we have mentioned in the previous section, there is a join between two worlds which is the origin in Kruskal-Szekeres spacetime diagram.
- (2.1) r By using the relation of u2 and r in equation (1.14) , one obtains (cid:18) 2μ (cid:19)−1 ds2 = 1− dr2+r2dφ2.
- Since we have the coordinate φ and r which are looks similar to the polar cylindrical coordinates, it is convenient to consider this three-dimensional Euclidian space in polar cylindrical coordinates which can be written as ds2 = dz2+dρ2+ρ2dψ2.

Thus, there are others view point to consider this join.

Remembering that each point in spacetime diagram represent 2-sphere of the solid angle.

Chapter 2 Wormhole and Einstein-Rosen bridge As we have mentioned in the previous section, there is a join between two worlds which is the origin in Kruskal-Szekeres spacetime diagram.

(2.1) r By using the relation of u2 and r in equation (1.14) , one obtains (cid:18) 2μ (cid:19)−1 ds2 = 1− dr2+r2dφ2.

Since we have the coordinate φ and r which are looks similar to the polar cylindrical coordinates, it is convenient to consider this three-dimensional Euclidian space in polar cylindrical coordinates which can be written as ds2 = dz2+dρ2+ρ2dψ2.

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Evaluate a proposed wormhole construction claim and produce an evidence-bounded assessment plus verification plan.

EVIDENCE:
- I nudge it open a few more inches, until I can see into the backyard again.
- It’s pitch-black inside the shed and redolent of gasoline and old grass clippings.
- Can’t see a thing, but it feels like it hasn’t been sharpened in years—deep chinks in the blade, which no longer holds an edge.
- I take the ax from the wall and grip the wooden handle, scraping my finger across the head.
- Does the idea that I was drugged and kidnapped by a secret, experimental group for the purposes of mind control or God-knows-what fit that bill?

I nudge it open a few more inches, until I can see into the backyard again.

It’s pitch-black inside the shed and redolent of gasoline and old grass clippings.

Can’t see a thing, but it feels like it hasn’t been sharpened in years—deep chinks in the blade, which no longer holds an edge.

I take the ax from the wall and grip the wooden handle, scraping my finger across the head.

Does the idea that I was drugged and kidnapped by a secret, experimental group for the purposes of mind control or God-knows-what fit that bill?

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Evaluate a proposed wormhole construction claim and produce an evidence-bounded assessment plus verification plan.

EVIDENCE:
- For any real distinct numbers a and b, If x, y > 0 and 0 < p < 1, then If x, y, z > 0, then If a, b > 0, then[15] If a, b > 0, then[16] If a, b, c > 0, then If a, b > 0, then Well-known inequalities Mathematicians often use inequalities to bound quantities for which exact formulas cannot be computed easily.
- Some inequalities are used so often that they have names: Azuma's inequality Bernoulli's inequality Bell's inequality Boole's inequality Cauchy–Schwarz inequality Chebyshev's inequality Chernoff's inequality Cramér–Rao inequality Hoeffding's inequality Hölder's inequality Inequality of arithmetic and geometric means Jensen's inequality Kolmogorov's inequality Markov's inequality Minkowski inequality Nesbitt's inequality Pedoe's inequality Poincaré inequality Samuelson's inequality Sobolev inequality Triangle inequality

For any real distinct numbers a and b, If x, y > 0 and 0 < p < 1, then If x, y, z > 0, then If a, b > 0, then[15] If a, b > 0, then[16] If a, b, c > 0, then If a, b > 0, then Well-known inequalities Mathematicians often use inequalities to bound quantities for which exact formulas cannot be computed easily.

Some inequalities are used so often that they have names: Azuma's inequality Bernoulli's inequality Bell's inequality Boole's inequality Cauchy–Schwarz inequality Chebyshev's inequality Chernoff's inequality Cramér–Rao inequality Hoeffding's inequality Hölder's inequality Inequality of arithmetic and geometric means Jensen's inequality Kolmogorov's inequality Markov's inequality Minkowski inequality Nesbitt's inequality Pedoe's inequality Poincaré inequality Samuelson's inequality Sobolev inequality Triangle inequality

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.

Inequality (mathematics) - Wikipedia.pdf


Prompt:
Evaluate a proposed wormhole construction claim and produce an evidence-bounded assessment plus verification plan.

EVIDENCE:
- principle of self-consistency is a principle of no new physics.
- If one is inclined from the outset to ignore or discount the possibility of new physics, then one will regard self-consistency as a trivial principle.
- Implications for time travelers The assumptions of the self-consistency principle can be extended to hypothetical scenarios involving intelligent time travelers as well as unintelligent objects such as billiard balls.
- The authors of "Cauchy problem in spacetimes with closed timelike curves" commented on the issue in the paper's conclusion, writing:

principle of self-consistency is a principle of no new physics.

If one is inclined from the outset to ignore or discount the possibility of new physics, then one will regard self-consistency as a trivial principle.

Implications for time travelers The assumptions of the self-consistency principle can be extended to hypothetical scenarios involving intelligent time travelers as well as unintelligent objects such as billiard balls.

The authors of "Cauchy problem in spacetimes with closed timelike curves" commented on the issue in the paper's conclusion, writing:

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.

Novikov self-consistency principle - Wikipedia.pdf


Prompt:
Evaluate a proposed wormhole construction claim and produce an evidence-bounded assessment plus verification plan.

EVIDENCE:
- lying between an inner one and an outer one.
- In the nonrotating, Schwarzschild solution, with , the inner and outer photon spheres degenerate, so that there is only one photon sphere at a single radius.
- The greater the spin of a black hole, the farther from each other the inner and outer photon spheres move.
- A beam of light traveling in a direction opposite to the spin of the black hole will circularly orbit the hole at the outer photon sphere.
- A beam of light traveling in the same direction as the black hole's spin will circularly orbit at the inner photon sphere.

lying between an inner one and an outer one.

In the nonrotating, Schwarzschild solution, with , the inner and outer photon spheres degenerate, so that there is only one photon sphere at a single radius.

The greater the spin of a black hole, the farther from each other the inner and outer photon spheres move.

A beam of light traveling in a direction opposite to the spin of the black hole will circularly orbit the hole at the outer photon sphere.

A beam of light traveling in the same direction as the black hole's spin will circularly orbit at the inner photon sphere.

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Explain quantum teleportation rigorously and outline an experimental validation checklist.

EVIDENCE:
- Yet there’s actually evidence that suicide risk may sometimes increase as depression lifts (Isaacson & Rich, 1997; Keith-Spiegel & Spiegel, 1967; Meehl, 1973), perhaps because severely depressed people begin to experience a return of energy as they improve (Shea, 1998).
- Second, although depression actually tends to decrease in old age (see Myth #9), the rates of suicide increase sharply in old age, especially among men (Joiner, 2005).
- In one survey of undergraduates, 53% responded “False” to the statement, “A time of high suicide risk in depression is when the person begins to improve” (Hubbard & McIntosh, 1992, p.
- Nevertheless, another reason is that suicide attempts among the elderly tend to be more serious in intent (Draper, 1996).
- First, in some studies hopelessness has been a better predictor of suicide than depression itself (Beck, Brown, & Steer, 1989; Beck, Kovacs, & Weissman, 1975; Wetzel, 1976).

Yet there’s actually evidence that suicide risk may sometimes increase as depression lifts (Isaacson & Rich, 1997; Keith-Spiegel & Spiegel, 1967; Meehl, 1973), perhaps because severely depressed people begin to experience a return of energy as they improve (Shea, 1998).

Second, although depression actually tends to decrease in old age (see Myth #9), the rates of suicide increase sharply in old age, especially among men (Joiner, 2005).

In one survey of undergraduates, 53% responded “False” to the statement, “A time of high suicide risk in depression is when the person begins to improve” (Hubbard & McIntosh, 1992, p.

Nevertheless, another reason is that suicide attempts among the elderly tend to be more serious in intent (Draper, 1996).

First, in some studies hopelessness has been a better predictor of suicide than depression itself (Beck, Brown, & Steer, 1989; Beck, Kovacs, & Weissman, 1975; Wetzel, 1976).

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Explain quantum teleportation rigorously and outline an experimental validation checklist.

EVIDENCE:
- Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics.
- Foundational character Classical mechanics has a wide range of application but its impact on physics is not limited to its practical applications.
- In this formalism, the dynamics of a system are governed by Hamilton's equations, which express the time derivatives of position and momentum variables in terms of partial derivatives of a function called the Hamiltonian: The Hamiltonian is the Legendre transform of the Lagrangian, and in many situations of physical interest it is equal to the total energy of the system.
- that the action functional of the system derived from must remain at a stationary point (a maximum, minimum, or saddle) throughout the time evolution of the system.
- Introduced by [23] Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities used in Lagrangian mechanics with (generalized) momenta.

Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics.

Foundational character Classical mechanics has a wide range of application but its impact on physics is not limited to its practical applications.

In this formalism, the dynamics of a system are governed by Hamilton's equations, which express the time derivatives of position and momentum variables in terms of partial derivatives of a function called the Hamiltonian: The Hamiltonian is the Legendre transform of the Lagrangian, and in many situations of physical interest it is equal to the total energy of the system.

that the action functional of the system derived from must remain at a stationary point (a maximum, minimum, or saddle) throughout the time evolution of the system.

Introduced by [23] Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities used in Lagrangian mechanics with (generalized) momenta.

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Evaluate a proposed wormhole construction claim and produce an evidence-bounded assessment plus verification plan.

EVIDENCE:
- he never made a public declaration of his private faith—which the orthodox would have deemed extremely radical.
- John.[237] He placed the crucifixion of Jesus Christ at 3 April, AD 33, which agrees with one traditionally accepted date.[238]
- wrote, that "Isaac Newton was a heretic.
- He said, "So then gravity may put the planets into motion, but without the Divine Power it could never put them into such a circulating motion, as they have about the sun".[236] Along with his scientific fame, Newton's studies of the Bible and of the early Church Fathers were also noteworthy.
- Newton wrote works on textual criticism, most notably An Historical Account of Two Notable Corruptions of Scripture and Observations upon the Prophecies of Daniel, and the Apocalypse of St.

he never made a public declaration of his private faith—which the orthodox would have deemed extremely radical.

John.[237] He placed the crucifixion of Jesus Christ at 3 April, AD 33, which agrees with one traditionally accepted date.[238]

wrote, that "Isaac Newton was a heretic.

He said, "So then gravity may put the planets into motion, but without the Divine Power it could never put them into such a circulating motion, as they have about the sun".[236] Along with his scientific fame, Newton's studies of the Bible and of the early Church Fathers were also noteworthy.

Newton wrote works on textual criticism, most notably An Historical Account of Two Notable Corruptions of Scripture and Observations upon the Prophecies of Daniel, and the Apocalypse of St.

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Evaluate a proposed wormhole construction claim and produce an evidence-bounded assessment plus verification plan.

EVIDENCE:
- – Very strong peak observed at 1/5 Nyquistfrequency as expected, and of course, no noise spectrum.
- Interferometer Modal Analysis: Synthetic Data • A set of 1000 synthetic interferogramfringe patterns were generated as JPEG image frames – Set contained repeating sequence of 10 frames with sinusoidal shifting in time with a shifting amplitude of 1/10 of a spatial radian*.
- and to the left compared to the fringe pattern on the left.) 13
- Fringe pattern repeatedly – No noise added to the shifting (i.e.
- unwanted vibrations not included) shifted up and down along – Gray scale intensity sampled at upper left pixel and a Discrete Fourier Transform (DFT) thisaxis calculated for the sample train.

– Very strong peak observed at 1/5 Nyquistfrequency as expected, and of course, no noise spectrum.

Interferometer Modal Analysis: Synthetic Data • A set of 1000 synthetic interferogramfringe patterns were generated as JPEG image frames – Set contained repeating sequence of 10 frames with sinusoidal shifting in time with a shifting amplitude of 1/10 of a spatial radian*.

and to the left compared to the fringe pattern on the left.) 13

Fringe pattern repeatedly – No noise added to the shifting (i.e.

unwanted vibrations not included) shifted up and down along – Gray scale intensity sampled at upper left pixel and a Discrete Fourier Transform (DFT) thisaxis calculated for the sample train.

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Explain quantum teleportation rigorously and outline an experimental validation checklist.

EVIDENCE:
- Bill Unruh doubted the practicality of quantum computers in a paper published in 1994.[134] Paul Davies argued that a 400-qubit computer would even come into conflict with the cosmological information bound implied by the holographic principle.[135] Skeptics like Gil Kalai doubt that quantum supremacy will ever be achieved.[136][137][138] Physicist Mikhail Dyakonov has expressed skepticism of quantum computing as follows: "So the number of continuous parameters describing the state of such a useful quantum computer
- This state of affairs can be traced to several current and long-term considerations.
- example, in chemistry and materials science.
- Therefore, it is desirable to prove lower bounds on the complexity of best possible non-quantum algorithms (which may be unknown) and show that some quantum algorithms asymptotically improve upon those bounds.
- When trying to outperform conventional computers, quantum computing researchers often look for new tasks that can be solved on quantum computers, but this leaves the possibility that efficient non-quantum techniques will be developed in response, as seen for Quantum supremacy demonstrations.

Bill Unruh doubted the practicality of quantum computers in a paper published in 1994.[134] Paul Davies argued that a 400-qubit computer would even come into conflict with the cosmological information bound implied by the holographic principle.[135] Skeptics like Gil Kalai doubt that quantum supremacy will ever be achieved.[136][137][138] Physicist Mikhail Dyakonov has expressed skepticism of quantum computing as follows: "So the number of continuous parameters describing the state of such a useful quantum computer

This state of affairs can be traced to several current and long-term considerations.

example, in chemistry and materials science.

Therefore, it is desirable to prove lower bounds on the complexity of best possible non-quantum algorithms (which may be unknown) and show that some quantum algorithms asymptotically improve upon those bounds.

When trying to outperform conventional computers, quantum computing researchers often look for new tasks that can be solved on quantum computers, but this leaves the possibility that efficient non-quantum techniques will be developed in response, as seen for Quantum supremacy demonstrations.

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Explain quantum teleportation rigorously and outline an experimental validation checklist.

EVIDENCE:
- =8πΣ ** t ̄, (16.311a) 12 equations of motion: e α ̄ ∧ ( d t Γ α ̄ +d α Γ t ̄+ 1 [Γ α ̄ ,Γ t ̄] ) +e t ̄∧R α ̄ =8πT ** t ̄.
- t α ̄ t BothΓ t ̄ande t ̄canbetreatedasgaugevariables:the6componentsofΓ t ̄canbeadjustedarbitrarilybya Lorentz transformation; and the 4 components of e t ̄can be adjusted arbitrarily by a coordinate transformation.
- The energy-momentum T on the right α ̄
- Thus the Gaussian and Hamiltonian constraint equations (16.310) can be interpreted as representing ** conserved Noether charges.
- The spin angular-momentum Σ on the right hand side of the Gaussian conα ̄ ** straint equation (16.310a) satisfies the conservation law (16.281).

=8πΣ ** t ̄, (16.311a) 12 equations of motion: e α ̄ ∧ ( d t Γ α ̄ +d α Γ t ̄+ 1 [Γ α ̄ ,Γ t ̄] ) +e t ̄∧R α ̄ =8πT ** t ̄.

t α ̄ t BothΓ t ̄ande t ̄canbetreatedasgaugevariables:the6componentsofΓ t ̄canbeadjustedarbitrarilybya Lorentz transformation; and the 4 components of e t ̄can be adjusted arbitrarily by a coordinate transformation.

Thus the Gaussian and Hamiltonian constraint equations (16.310) can be interpreted as representing ** conserved Noether charges.

The spin angular-momentum Σ on the right hand side of the Gaussian conα ̄ ** straint equation (16.310a) satisfies the conservation law (16.281).

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Explain quantum teleportation rigorously and outline an experimental validation checklist.

EVIDENCE:
- Al-Razi himself, was undecided, he said: "In truth, there is no way to ascertain the characteristics of the heavens except by authority [of divine revelation or prophetic traditions]." Setia concludes: "Thus it seems that for al-Razi (and for others before and after him), astronomical models, whatever their utility or lack thereof for ordering the heavens, are not founded on sound rational proofs, and so no intellectual commitment can be made to them insofar as description and explanation of celestial realities are concerned."[48] Christian
- Both astronomical scholars and popular writers considered the implications of the nested sphere model for the dimensions of the universe.[40] Campanus of Novara's introductory astronomical text, the Theorica planetarum, used the model of nesting spheres to compute the distances of the various planets from the Earth, which he gave as 22,612 Earth radii or 73,387,747100⁄ miles (118,106,130.55 km).[41][42] In his Opus Majus, Roger Bacon cited Al-Farghānī's distance to the stars of 20,110 Earth radii, or 65,357,700 miles (105,183,000 km), from which he computed the circumference of the universe to be 410,818,5173⁄ miles (661,148,316.1 km).[43] Clear evidence that this model was thought to represent physical reality is the accounts found in Bacon's Opus Majus of the time needed to walk to the Moon[44] and in the popular Middle English South English Legendary, that it would take 8,000 years to reach the highest starry heaven.[45][46] General understanding of the dimensions of the universe derived from the nested sphere model reached wider audiences through the presentations in Hebrew by Moses Maimonides, in French by Gossuin of Metz, and in Italian by Dante Alighieri.[47] Philosophical and theological discussions Philosophers were less concerned with such mathematical calculations than with the nature of the celestial spheres, their relation to revealed accounts of created nature, and the causes of their motion.
- Adi Setia describes the debate among Islamic scholars in the twelfth century, based on the commentary of Fakhr al-Din al-Razi about whether the celestial spheres are real, concrete physical bodies or "merely the abstract circles in the heavens traced out...
- About the same time, scholars in European universities began to address the implications of the rediscovered philosophy of Aristotle and astronomy of Ptolemy.
- by the various stars and planets." Setia points out that most of the learned, and the astronomers, said they were solid spheres "on which the stars turn...

Al-Razi himself, was undecided, he said: "In truth, there is no way to ascertain the characteristics of the heavens except by authority [of divine revelation or prophetic traditions]." Setia concludes: "Thus it seems that for al-Razi (and for others before and after him), astronomical models, whatever their utility or lack thereof for ordering the heavens, are not founded on sound rational proofs, and so no intellectual commitment can be made to them insofar as description and explanation of celestial realities are concerned."[48] Christian

Both astronomical scholars and popular writers considered the implications of the nested sphere model for the dimensions of the universe.[40] Campanus of Novara's introductory astronomical text, the Theorica planetarum, used the model of nesting spheres to compute the distances of the various planets from the Earth, which he gave as 22,612 Earth radii or 73,387,747100⁄ miles (118,106,130.55 km).[41][42] In his Opus Majus, Roger Bacon cited Al-Farghānī's distance to the stars of 20,110 Earth radii, or 65,357,700 miles (105,183,000 km), from which he computed the circumference of the universe to be 410,818,5173⁄ miles (661,148,316.1 km).[43] Clear evidence that this model was thought to represent physical reality is the accounts found in Bacon's Opus Majus of the time needed to walk to the Moon[44] and in the popular Middle English South English Legendary, that it would take 8,000 years to reach the highest starry heaven.[45][46] General understanding of the dimensions of the universe derived from the nested sphere model reached wider audiences through the presentations in Hebrew by Moses Maimonides, in French by Gossuin of Metz, and in Italian by Dante Alighieri.[47] Philosophical and theological discussions Philosophers were less concerned with such mathematical calculations than with the nature of the celestial spheres, their relation to revealed accounts of created nature, and the causes of their motion.

Adi Setia describes the debate among Islamic scholars in the twelfth century, based on the commentary of Fakhr al-Din al-Razi about whether the celestial spheres are real, concrete physical bodies or "merely the abstract circles in the heavens traced out...

About the same time, scholars in European universities began to address the implications of the rediscovered philosophy of Aristotle and astronomy of Ptolemy.

by the various stars and planets." Setia points out that most of the learned, and the astronomers, said they were solid spheres "on which the stars turn...

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Design a retrieval+analysis pipeline to discover missing facts from the corpus without hallucinating.

EVIDENCE:
- I think Newtonian gauge should be called Copernican gauge.
- 744 Perturbations in a flat space background Exercise 27.4.
- Even though the solar system is a highly non-linear system, from the perspective of general relativity it is a weakly perturbed gravitating system.
- 27.8 Newtonian (Copernican) gauge If the unperturbed background is Minkowski space, then the most physical gauge is one in which the 6 perturbationsretainedcoincidewiththe6coordinateandtetradgauge-invariantperturbations(27.13).This gaugeiscalledNewtonian gauge.BecauseinNewtoniangaugetheperturbationsarepreciselythephysical perturbations, if the perturbations are physically weak (small), then the perturbations in Newtonian gauge will necessarily be small.
- (27.60) 0 0 0 Φ The Newtonian line-element is, in a form that keeps the Newtonian tetrad manifest, ds2 =− [ (1+Ψ)dt ]2 +δ [ (1−Φ)dxa−hadxc−Wadt ][ (1−Φ)dxb−hbdxd−Wbdt ] , (27.61) ab c d

I think Newtonian gauge should be called Copernican gauge.

744 Perturbations in a flat space background Exercise 27.4.

Even though the solar system is a highly non-linear system, from the perspective of general relativity it is a weakly perturbed gravitating system.

27.8 Newtonian (Copernican) gauge If the unperturbed background is Minkowski space, then the most physical gauge is one in which the 6 perturbationsretainedcoincidewiththe6coordinateandtetradgauge-invariantperturbations(27.13).This gaugeiscalledNewtonian gauge.BecauseinNewtoniangaugetheperturbationsarepreciselythephysical perturbations, if the perturbations are physically weak (small), then the perturbations in Newtonian gauge will necessarily be small.

(27.60) 0 0 0 Φ The Newtonian line-element is, in a form that keeps the Newtonian tetrad manifest, ds2 =− [ (1+Ψ)dt ]2 +δ [ (1−Φ)dxa−hadxc−Wadt ][ (1−Φ)dxb−hbdxd−Wbdt ] , (27.61) ab c d

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Evaluate a proposed wormhole construction claim and produce an evidence-bounded assessment plus verification plan.

EVIDENCE:
- and rationale of this definition suggested that the words "in young star clusters" should be deleted in clause 3, as such objects have now been found elsewhere, and that the term "sub-brown dwarfs" should be replaced by the more current "free-floating planetary mass objects".
- The term "planetary mass object" has also been used to refer to ambiguous situations concerning exoplanets, such as objects with mass typical for a planet that are free-floating or orbit a brown dwarf instead of a star.[216] Free-floating objects of planetary mass have sometimes been called planets anyway, specifically rogue planets.[217] The limit of 13 Jupiter masses is not universally accepted.
- Objects below this mass limit can sometimes burn deuterium, and the amount of deuterium that is burned depends on an object's composition.[218][219] Furthermore, deuterium is quite scarce, so the stage of deuterium burning does not actually last very long; unlike hydrogen burning in a star, deuterium burning does not significantly affect the future evolution of an object.[58] The relationship between mass and radius (or

and rationale of this definition suggested that the words "in young star clusters" should be deleted in clause 3, as such objects have now been found elsewhere, and that the term "sub-brown dwarfs" should be replaced by the more current "free-floating planetary mass objects".

The term "planetary mass object" has also been used to refer to ambiguous situations concerning exoplanets, such as objects with mass typical for a planet that are free-floating or orbit a brown dwarf instead of a star.[216] Free-floating objects of planetary mass have sometimes been called planets anyway, specifically rogue planets.[217] The limit of 13 Jupiter masses is not universally accepted.

Objects below this mass limit can sometimes burn deuterium, and the amount of deuterium that is burned depends on an object's composition.[218][219] Furthermore, deuterium is quite scarce, so the stage of deuterium burning does not actually last very long; unlike hydrogen burning in a star, deuterium burning does not significantly affect the future evolution of an object.[58] The relationship between mass and radius (or

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Explain quantum teleportation rigorously and outline an experimental validation checklist.

EVIDENCE:
- Antimatter is made of material resembling ordinary matter except that all the subatomic particles composing it are opposite in properties to the particles composing ordinary matter.
- The destruction might have been caused by an ordinary meteorite, a volcanic eruption, and earthquake, or the whole tale may be mythical.
- Others thought it might have been an example of antimatter that struck the Earth.
- The records aren't good enough.
- BLACK HOLES 201 destroyed because a mini-black hole struck?

Antimatter is made of material resembling ordinary matter except that all the subatomic particles composing it are opposite in properties to the particles composing ordinary matter.

The destruction might have been caused by an ordinary meteorite, a volcanic eruption, and earthquake, or the whole tale may be mythical.

Others thought it might have been an example of antimatter that struck the Earth.

BLACK HOLES 201 destroyed because a mini-black hole struck?

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Design a retrieval+analysis pipeline to discover missing facts from the corpus without hallucinating.

EVIDENCE:
- than a filled shell, and others that are one-electron short of a full shell, such as occurs in the compound sodium chloride and other chemical ionic salts.
- By varying the conditions, materials can transition between solids, liquids, gases, and plasmas.[104] Within a state, a material can also exist in different allotropes.
- Many elements display multiple valences, or tendencies to share differing numbers of electrons in different compounds.
- Thus, chemical bonding between these elements takes many forms of electron-sharing that are more than simple electron transfers.
- An example of this is solid carbon, which can exist as graphite or diamond.[105] Gaseous allotropes exist as well, such as dioxygen and ozone.

than a filled shell, and others that are one-electron short of a full shell, such as occurs in the compound sodium chloride and other chemical ionic salts.

By varying the conditions, materials can transition between solids, liquids, gases, and plasmas.[104] Within a state, a material can also exist in different allotropes.

Many elements display multiple valences, or tendencies to share differing numbers of electrons in different compounds.

Thus, chemical bonding between these elements takes many forms of electron-sharing that are more than simple electron transfers.

An example of this is solid carbon, which can exist as graphite or diamond.[105] Gaseous allotropes exist as well, such as dioxygen and ozone.

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Explain quantum teleportation rigorously and outline an experimental validation checklist.

EVIDENCE:
- nlm.nih.gov/pmc/articles/PMC7217020).
- doi:10.1038/s41587-021-00969-6 (https://doi.org/10.1038%2Fs41587-021-0 0969-6).
- Jiang, Wenjun; Zhang, Baochang; Fan, Chuyao; Wang, Min; Wang, Jiaxing; Deng, Qiang; Liu, Xianyu; Chen, Ji; Zheng, Jishen; Liu, Lei; Zhu, Ting F (17 October 2017).
- PMC 7217020 (https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7217020).
- Bibcode:2024NatCh..16..592Z (https://ui.adsa bs.harvard.edu/abs/2024NatCh..16..592Z).

Jiang, Wenjun; Zhang, Baochang; Fan, Chuyao; Wang, Min; Wang, Jiaxing; Deng, Qiang; Liu, Xianyu; Chen, Ji; Zheng, Jishen; Liu, Lei; Zhu, Ting F (17 October 2017).

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Design a retrieval+analysis pipeline to discover missing facts from the corpus without hallucinating.

EVIDENCE:
- second choice misses γγ±∧γγ+ bivectors..
- i T 42.4.10 Electroweak Higgs field TheU (1)×SU (2)theoryofelectroweakinteractionsintheSMiscalledtheWeinberg-Salamtheory(Salam Y L and Ward, 1959; Weinberg, 1967), for which Glashow, Salam, and Weinberg shared the 1979 Nobel prize.
- The mechanism by which the electroweak symmetry is broken to the electromagnetic symmetry U (1) was em proposed by Weinberg (1967), who invoked the so-called Higgs mechanism (Englert and Brout, 1964; Higgs, 1964; Guralnik, Hagen, and Kibble, 1964).
- The exposition of the electroweak Higgs mechanism that follows leans on Peskin and Schroeder (1995, Ch.
- The Higgs mechanism posits a mysterious Higgs field that accomplishes four things: it breaks a gauge

second choice misses γγ±∧γγ+ bivectors..

i T 42.4.10 Electroweak Higgs field TheU (1)×SU (2)theoryofelectroweakinteractionsintheSMiscalledtheWeinberg-Salamtheory(Salam Y L and Ward, 1959; Weinberg, 1967), for which Glashow, Salam, and Weinberg shared the 1979 Nobel prize.

The mechanism by which the electroweak symmetry is broken to the electromagnetic symmetry U (1) was em proposed by Weinberg (1967), who invoked the so-called Higgs mechanism (Englert and Brout, 1964; Higgs, 1964; Guralnik, Hagen, and Kibble, 1964).

The exposition of the electroweak Higgs mechanism that follows leans on Peskin and Schroeder (1995, Ch.

The Higgs mechanism posits a mysterious Higgs field that accomplishes four things: it breaks a gauge

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Design a retrieval+analysis pipeline to discover missing facts from the corpus without hallucinating.

EVIDENCE:
- I was hoping a trip down memory lane might spark something.” He straightens.
- In my old man’s time, things were different.
- I can see beyond the smoked-glass enclosure into the small theater, and it’s filled, just like on that first night, with fifteen or twenty people.
- From across the table, he watches me.
- Leighton says, “My father founded Velocity forty-five years ago.

I was hoping a trip down memory lane might spark something.” He straightens.

In my old man’s time, things were different.

I can see beyond the smoked-glass enclosure into the small theater, and it’s filled, just like on that first night, with fifteen or twenty people.

Leighton says, “My father founded Velocity forty-five years ago.

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Evaluate a proposed wormhole construction claim and produce an evidence-bounded assessment plus verification plan.

EVIDENCE:
- arXiv:gr-qc/0407042 (https://arxiv.org/abs/gr-qc/0407042).
- adsabs.harvard.edu/abs/2004LRR.....7...10A).
- General Relativity: An introduction for physicists (https://books.google.com/books?id=5dryXCWR7EIC&pg=PA265).
- Schwarzschild black holes" (https://books.google.com/books?id=5dryXCWR7EIC&pg=PA265).
- Bibcode:2004LRR.....7...10A (https://ui.

General Relativity: An introduction for physicists (https://books.google.com/books?id=5dryXCWR7EIC&pg=PA265).

Schwarzschild black holes" (https://books.google.com/books?id=5dryXCWR7EIC&pg=PA265).

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Explain quantum teleportation rigorously and outline an experimental validation checklist.

EVIDENCE:
- 194 THE COLLAPSING UNIVERSE wave pulses.
- Two such X-ray pulsars have, in fact, been detected, Centaurus X-3 and Hercules X-1.
- Now that attention was eagerly focused on Cygnus X-1, microwaves were also detected, and this made it possible to pinpoint the source very accurately and place it in just next to a visible star.
- It was named Uhuru from the Swahili word for "freedom." It multiplied knowledge of X-ray sources to unlooked-for heights, detecting 161 such sources, half of them in our own galaxy and 3 of them in globular clusters.
- This star was HD-226868, a large, hot blue star of spectral class B, some 30 times as massive

194 THE COLLAPSING UNIVERSE wave pulses.

Two such X-ray pulsars have, in fact, been detected, Centaurus X-3 and Hercules X-1.

Now that attention was eagerly focused on Cygnus X-1, microwaves were also detected, and this made it possible to pinpoint the source very accurately and place it in just next to a visible star.

It was named Uhuru from the Swahili word for "freedom." It multiplied knowledge of X-ray sources to unlooked-for heights, detecting 161 such sources, half of them in our own galaxy and 3 of them in globular clusters.

This star was HD-226868, a large, hot blue star of spectral class B, some 30 times as massive

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Explain quantum teleportation rigorously and outline an experimental validation checklist.

EVIDENCE:
- You surf a light ray from a distant quasar to Earth.
- What does c=universal constant mean?
- The fourth postulate of special relativity asserts that spacetime has no absolute existence.
- Blobs in the jet of the galaxy M87 have been tracked by the Hubble Space Telescope to be moving at about 6c.
- How long does the trip take you?

You surf a light ray from a distant quasar to Earth.

The fourth postulate of special relativity asserts that spacetime has no absolute existence.

Blobs in the jet of the galaxy M87 have been tracked by the Hubble Space Telescope to be moving at about 6c.

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Evaluate a proposed wormhole construction claim and produce an evidence-bounded assessment plus verification plan.

EVIDENCE:
- Using all four cells allows you to compute the correlation between full moons and the number of psychiatric hospital admissions; a correlation is a statistical measure of how closely these two variables are associated (by the way, a variable is a fancy term for anything that varies, like height, hair color, IQ, or extraversion).
- The supposed relation between full moons and psychiatric hospital admissions is a stunning example of an illusory correlation.
- For example, many people with arthritis insist that
- The A cell is a “hit”—a striking co-occurrence.
- In particular, research demonstrates that we typically pay too much attention to the A cell, and not nearly enough to the Bcell (Gilovich, 1991).

Using all four cells allows you to compute the correlation between full moons and the number of psychiatric hospital admissions; a correlation is a statistical measure of how closely these two variables are associated (by the way, a variable is a fancy term for anything that varies, like height, hair color, IQ, or extraversion).

The supposed relation between full moons and psychiatric hospital admissions is a stunning example of an illusory correlation.

For example, many people with arthritis insist that

The A cell is a “hit”—a striking co-occurrence.

In particular, research demonstrates that we typically pay too much attention to the A cell, and not nearly enough to the Bcell (Gilovich, 1991).

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Evaluate a proposed wormhole construction claim and produce an evidence-bounded assessment plus verification plan.

EVIDENCE:
- TheColonelwasrecoveredandhospitalized.” LeonardSpringfieldclaimshecorroboratedtheexistenceofGen.T.AtWPAFBthroughthe AccountingandFinanceSectionrecords.
- StringfieldalsoclaimsanotherIntelligencesourcecorroborated the1968NellisAFBincident.
- downplaysAgencyinterestinUAPinthe1970sand80sandmakesnomentionoftheAgency’sinterest inindependentresearchersorDCIworkinggroups.
- ● RetrievalsoftheThirdKindpresentedatMUFON1978,LeonStringfield,p27 (PUBLICDOMAIN) -1977—AttorneyDanielP.SheehanattherequestofCongressionalResearch ServiceresearcherMarciaS.SmithclaimstovisitstheMadisonBuildingattheLibraryofCongressto lookat“classifiedsectionsofProjectBlueBook.” Sheehanclaimstoseephotosofaflyingdisc embeddedinsnowsurroundedbyUSAFpersonnelwearingparkas.
- Symbolsarepresentonthesideof thecrasheddisc.

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Explain quantum teleportation rigorously and outline an experimental validation checklist.

EVIDENCE:
- the time it was released with any desired precision, in contradiction to the uncertainty principle."[100] Bohr spent a sleepless night considering this argument, and eventually realized that it was flawed.
- He pointed out that if the box were to be weighed, say by a spring and a pointer on a scale, "since the box must move vertically with a change in its weight, there will be uncertainty in its vertical velocity and therefore an uncertainty in its height above the table.
- Furthermore, the uncertainty about the elevation above the Earth's surface will result in an uncertainty in the rate of the clock",[102] because

the time it was released with any desired precision, in contradiction to the uncertainty principle."[100] Bohr spent a sleepless night considering this argument, and eventually realized that it was flawed.

He pointed out that if the box were to be weighed, say by a spring and a pointer on a scale, "since the box must move vertically with a change in its weight, there will be uncertainty in its vertical velocity and therefore an uncertainty in its height above the table.

Furthermore, the uncertainty about the elevation above the Earth's surface will result in an uncertainty in the rate of the clock",[102] because

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Evaluate a proposed wormhole construction claim and produce an evidence-bounded assessment plus verification plan.

EVIDENCE:
- (16.273) kλ λk kλ As a particular example, the vierbein is symmetric in this sense, because the tetrad metric is symmetric, e =η el , so the transpose of the line interval e is itself, kλ kl λ e⊤ =(e γγkdxλ)⊤ =ek e λe γγldxκ =e γγldxκ =e .
- Under an infinitesimal Lorentz transformation generated by the bivector ε=ε γγk∧γγl, any multivector kl form a whose multivector components transform like a tensor varies as, equation (16.95), δa= 1[ε,a] .
- 16.14 Gravitational action in multivector forms notation 461 16.14.7 Transpose of a multivector form The transpose a⊤ of a multivector form a≡a γγKdpxΛ of grade n and form index p is defined to be the KΛ multivector form, of grade p and form index n, with multivector and form indices transposed, a⊤ = ( a γγKdpxΛ)⊤ ≡eK e Λa γγLdpxΠ =a γγLdkxΠ , (16.271) KΛ Π L KΛ ΠL implicitly summed over distinct sequences K, L, Λ, Π of indices.
- (16.276) Equations (16.275) and (16.276) say that the operation of transposition commutes both with taking the wedge product and with taking the double dual.
- Lorentz symmetry implies a conservation m law (16.281) of angular momentum.

(16.273) kλ λk kλ As a particular example, the vierbein is symmetric in this sense, because the tetrad metric is symmetric, e =η el , so the transpose of the line interval e is itself, kλ kl λ e⊤ =(e γγkdxλ)⊤ =ek e λe γγldxκ =e γγldxκ =e .

Under an infinitesimal Lorentz transformation generated by the bivector ε=ε γγk∧γγl, any multivector kl form a whose multivector components transform like a tensor varies as, equation (16.95), δa= 1[ε,a] .

16.14 Gravitational action in multivector forms notation 461 16.14.7 Transpose of a multivector form The transpose a⊤ of a multivector form a≡a γγKdpxΛ of grade n and form index p is defined to be the KΛ multivector form, of grade p and form index n, with multivector and form indices transposed, a⊤ = ( a γγKdpxΛ)⊤ ≡eK e Λa γγLdpxΠ =a γγLdkxΠ , (16.271) KΛ Π L KΛ ΠL implicitly summed over distinct sequences K, L, Λ, Π of indices.

(16.276) Equations (16.275) and (16.276) say that the operation of transposition commutes both with taking the wedge product and with taking the double dual.

Lorentz symmetry implies a conservation m law (16.281) of angular momentum.

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Design a retrieval+analysis pipeline to discover missing facts from the corpus without hallucinating.

EVIDENCE:
- Physics of Atoms and Ions (https://archive.org/details/physicsatomsions 00smir).
- "The Quantum Atom" (https://web.archive.org/web/20061207032136/htt p://www.chem.ufl.edu/~itl/2045/lectures/lec_10.html).
- doi:10.1103/PhysRev.79.282 (https://doi.org/10.1103% 2FPhysRev.79.282).
- Archived (https://w eb.archive.org/web/20111003185243/http://nucleus.iaea.org/CIR/CIR/NuclearWalletCards.html) from the original on 3 October 2011.
- National Nuclear Data Center, Brookhaven National Laboratory.

Physics of Atoms and Ions (https://archive.org/details/physicsatomsions 00smir).

"The Quantum Atom" (https://web.archive.org/web/20061207032136/htt p://www.chem.ufl.edu/~itl/2045/lectures/lec_10.html).

Archived (https://w eb.archive.org/web/20111003185243/http://nucleus.iaea.org/CIR/CIR/NuclearWalletCards.html) from the original on 3 October 2011.

National Nuclear Data Center, Brookhaven National Laboratory.

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Design a retrieval+analysis pipeline to discover missing facts from the corpus without hallucinating.

EVIDENCE:
- ParticlesandLightaroundaBlackHole | 415 (cid:6) √ 1− 1 = 2 2.
- Masslessparticle For a massless particle, we have to replace the proper time τ by the affine parameter ζ.
- Thequalitativefeaturesoflightmovingaroundablackholecanbeimmediatelyreadoff from(9),whichwithaparticularchoiceoftheaffineparameterbecomes(dr)2+U(r)= dζ 1.Plottheeffectivepotential b2 1 r U(r)= − S (12) r2 r3 Forlarger,U(r)(cid:16) 1 andgoesupasr decreases,reachingamaximumvalueofU = r2 max 4/(27r2)atr =3r /2,thenplungingdownwardto−∞.Thus,themotionoflightcanbe S S dividedintothreecases(figure3):
- But from dt = (cid:8) , we see that the particle started out at r =∞ with 9 3 dτ 1−rS r (cid:8)=1.Thus,thefractionofenergyultimatelylosttoelectromagneticradiationamountsto √ 1− 2 2 ≈0.06.
- Isthisalot?Whatdowecomparethe6%to?Considernuclearfusion, whichpowers thestarsandthehydrogenbomb.Fourprotonsareconvertedtoaheliumnucleusaccompaniedbytheemissionoftwopositronsandtwoneutrinos.Thefractionofenergyrelease iscalculatedeasilybycomparingthemassoftheheliumnucleusto4timesthemassof theproton,accordingtowhatwelearnedbackinpartIII.ThefabledequationE=mc2, yes!Itturnsoutthatinnuclearfusion, theenergyreleasedamountsto0.7%.Theblack holeisalmosttentimesmoreefficient.Later,wewillseethatarotatingblackholeiseven moreefficient!

ParticlesandLightaroundaBlackHole | 415 (cid:6) √ 1− 1 = 2 2.

Masslessparticle For a massless particle, we have to replace the proper time τ by the affine parameter ζ.

Thequalitativefeaturesoflightmovingaroundablackholecanbeimmediatelyreadoff from(9),whichwithaparticularchoiceoftheaffineparameterbecomes(dr)2+U(r)= dζ 1.Plottheeffectivepotential b2 1 r U(r)= − S (12) r2 r3 Forlarger,U(r)(cid:16) 1 andgoesupasr decreases,reachingamaximumvalueofU = r2 max 4/(27r2)atr =3r /2,thenplungingdownwardto−∞.Thus,themotionoflightcanbe S S dividedintothreecases(figure3):

But from dt = (cid:8) , we see that the particle started out at r =∞ with 9 3 dτ 1−rS r (cid:8)=1.Thus,thefractionofenergyultimatelylosttoelectromagneticradiationamountsto √ 1− 2 2 ≈0.06.

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Design a retrieval+analysis pipeline to discover missing facts from the corpus without hallucinating.

EVIDENCE:
- Government Printing Office, 1986, Identifier Y 4.En 2/3:99-NN, Electronic Publication Date 2010, at the University of Nevada, Reno, unr.edu 49.
- "Restricted data declassification decisions, 1946 to the present (RDD-7)" (https://sgp.fas.org/othergov/doe/rdd-7.html#I16).
- Subcommittee on Energy Conservation and Power, published by U.S.
- FAS Project on Government Secrecy (1991-2021), fas.org.
- citizens (http://contentdm.librar y.unr.edu/cdm4/item_viewer.php?CISOROOT=/conghear&CISOPTR=102&CISOBOX=1&REC=1#metajump) Archived (https://web.archive.org/web/20130730200210/http://contentdm.library.unr.edu/cdm4/item_viewer.ph p?CISOROOT=%2Fconghear&CISOPTR=102&CISOBOX=1&REC=1#metajump) 2013-07-30 at the Wayback Machine.

Government Printing Office, 1986, Identifier Y 4.En 2/3:99-NN, Electronic Publication Date 2010, at the University of Nevada, Reno, unr.edu 49.

"Restricted data declassification decisions, 1946 to the present (RDD-7)" (https://sgp.fas.org/othergov/doe/rdd-7.html#I16).

Subcommittee on Energy Conservation and Power, published by U.S.

FAS Project on Government Secrecy (1991-2021), fas.org.

citizens (http://contentdm.librar y.unr.edu/cdm4/item_viewer.php?CISOROOT=/conghear&CISOPTR=102&CISOBOX=1&REC=1#metajump) Archived (https://web.archive.org/web/20130730200210/http://contentdm.library.unr.edu/cdm4/item_viewer.ph p?CISOROOT=%2Fconghear&CISOPTR=102&CISOBOX=1&REC=1#metajump) 2013-07-30 at the Wayback Machine.

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Explain quantum teleportation rigorously and outline an experimental validation checklist.

EVIDENCE:
- The horizon is localized at r = r , where + (cid:112) r = MG± M2G2 −r2 ± e The relations for temperature and entropy are: r −r πr2 T = + − S = + 4πr2 G + The extreme points are reached when T → 0 and r = r = r .
- This metric is a good approximation r r 2πTr2 e e e when r − r << r and Tr << 1.
- Constructions 5.1.1 Single magnetic black hole geometry For a magnetically charged black hole, the metric will be: (cid:18) 2MG r2(cid:19) (cid:18) 2MG r2(cid:19)−1 ds2 = − 1− + e dt2 + 1− + e dr2 +r2(dθ2 +sin2θdφ2), r r2 r r2 where r2 = πq2G, M is the mass of the black hole, q is an integer representing the e g2 charge of the black hole, and g is the coupling constant of the U(1) gauge field.
- TRAVERSABLE WORMHOLE WITH FERMIONS Section 5.
- There, M and S can + − e be expanded around a small T to obtain: r 2π2r3T2 M = e + e +...

The horizon is localized at r = r , where + (cid:112) r = MG± M2G2 −r2 ± e The relations for temperature and entropy are: r −r πr2 T = + − S = + 4πr2 G + The extreme points are reached when T → 0 and r = r = r .

This metric is a good approximation r r 2πTr2 e e e when r − r << r and Tr << 1.

Constructions 5.1.1 Single magnetic black hole geometry For a magnetically charged black hole, the metric will be: (cid:18) 2MG r2(cid:19) (cid:18) 2MG r2(cid:19)−1 ds2 = − 1− + e dt2 + 1− + e dr2 +r2(dθ2 +sin2θdφ2), r r2 r r2 where r2 = πq2G, M is the mass of the black hole, q is an integer representing the e g2 charge of the black hole, and g is the coupling constant of the U(1) gauge field.

TRAVERSABLE WORMHOLE WITH FERMIONS Section 5.

There, M and S can + − e be expanded around a small T to obtain: r 2π2r3T2 M = e + e +...

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Explain quantum teleportation rigorously and outline an experimental validation checklist.

EVIDENCE:
- National Institute of Standards and Technology.
- The volumetric heat capacity is the heat capacity of the substance divided by its volume.
- At constant temperature and pressure, ignoring other states of matter for brevity References 1.
- Derived quantities Density is the substance's mass per unit volume, or total mass divided by total volume.[21] Specific volume is total volume divided by mass, or the inverse of density.[22] Low poly triangle mesh of a dolphin The volumetric flow rate or discharge is the volume of fluid which passes through a given surface per unit time.
- In cylindrical coordinates, the volume integral is In spherical coordinates (using the convention for angles with as the azimuth and measured from the polar axis; see more on conventions), the volume integral is Geometric modeling Illustration of a solid of revolution, which the volume of rotated g(x) subtracts the A polygon mesh is a representation of the object's surface, volume of rotated f(x).

National Institute of Standards and Technology.

The volumetric heat capacity is the heat capacity of the substance divided by its volume.

At constant temperature and pressure, ignoring other states of matter for brevity References 1.

Derived quantities Density is the substance's mass per unit volume, or total mass divided by total volume.[21] Specific volume is total volume divided by mass, or the inverse of density.[22] Low poly triangle mesh of a dolphin The volumetric flow rate or discharge is the volume of fluid which passes through a given surface per unit time.

In cylindrical coordinates, the volume integral is In spherical coordinates (using the convention for angles with as the azimuth and measured from the polar axis; see more on conventions), the volume integral is Geometric modeling Illustration of a solid of revolution, which the volume of rotated g(x) subtracts the A polygon mesh is a representation of the object's surface, volume of rotated f(x).

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Evaluate a proposed wormhole construction claim and produce an evidence-bounded assessment plus verification plan.

EVIDENCE:
- Bibcode:2017AJ....153..121P (https://ui.adsabs.harvard.edu/abs/2017AJ....153..121P).
- (Subscription or participating institution membership (https://ww w.oed.com/public/login/loggingin#withyourlibrary) required.) 184.
- The Scientific Papers of Sir William Herschel (https://archive.org/details/sci entificpapers032804mbp/page/100/mode/2up).
- Planet Quest: The Epic Discovery of Alien Solar Systems.
- doi:10.1086/112284 (https://doi.org/10.1086%2F112284).

(Subscription or participating institution membership (https://ww w.oed.com/public/login/loggingin#withyourlibrary) required.) 184.

The Scientific Papers of Sir William Herschel (https://archive.org/details/sci entificpapers032804mbp/page/100/mode/2up).

Planet Quest: The Epic Discovery of Alien Solar Systems.

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.


Prompt:
Evaluate a proposed wormhole construction claim and produce an evidence-bounded assessment plus verification plan.

EVIDENCE:
- Amanda asks, “Did you see the box?” “No, I think it’s buried under the snow.” “Fantastic.” She looks at me, then back into the flames, says, “I don’t know whether to be mad at you or grateful.” “What are you talking about?” “While you were upstairs, I had to use the bathroom.
- It started when I was outside a moment ago, looking at the field, thinking about us almost dying in that whiteout.
- — Back inside, I find Amanda awake, sitting up at the edge of the hearth with the sleeping bags and blankets wrapped around her.
- Across the street, I survey the open field where we nearly froze to death yesterday.
- I stumbled into the office.” “So you saw them.” “They starved, didn’t they?

Amanda asks, “Did you see the box?” “No, I think it’s buried under the snow.” “Fantastic.” She looks at me, then back into the flames, says, “I don’t know whether to be mad at you or grateful.” “What are you talking about?” “While you were upstairs, I had to use the bathroom.

It started when I was outside a moment ago, looking at the field, thinking about us almost dying in that whiteout.

— Back inside, I find Amanda awake, sitting up at the edge of the hearth with the sleeping bags and blankets wrapped around her.

Across the street, I survey the open field where we nearly froze to death yesterday.

I stumbled into the office.” “So you saw them.” “They starved, didn’t they?

If proposing a design, label as hypothesis and provide validation steps.

Anything not in evidence is unknown; retrieve more.

Derivations, tests, or experiments implied by evidence.