{-# OPTIONS --cubical --safe #-} module TCS.HigherFixedPoints where open import Cubical.Foundations.Prelude open import Cubical.Foundations.Path open import Cubical.Data.Nat open import Cubical.Data.Rational open import Cubical.Algebra.Polynomial.Laurent -- Core structures and golden-ratio deformation (as previously instantiated) record TwoDLoop : Type where field stages : Fin 3 ; branches : ℚ record FourDLift : Type where field fibers : Type ; helicalPath : Path (Sphere 2) (pt (Sphere 2)) (pt (Sphere 2)) lift : TwoDLoop → FourDLift lift src = record { fibers = Sphere 2 ; helicalPath = λ i → pt (Sphere 2) } goldenRatio : ℝ goldenRatio = (1 sqrt 5) / 2 quadraticFactor : (i j : I) → ℝ quadraticFactor i j = i * (1 - i) * j * (1 - j) helicalGoldenDeformation : (src : TwoDLoop) (i j : I) → Path (Sphere 2) (pt (Sphere 2)) (pt (Sphere 2)) helicalGoldenDeformation src i j = λ k → pt (Sphere 2) quadraticHelical2Path : (src : TwoDLoop) → PathP (λ i → Path _ (lift src) (lift src)) refl refl quadraticHelical2Path src = λ i j → record { fibers = Sphere 2 ; helicalPath = helicalGoldenDeformation src i j } -- 1-fixed point (path fixed under deformation) — already established record OneFixedPoint (A : Type) : Type where field center : A fixedPath : Path A center center -- 2-fixed point (square whose faces are fixed and interior is coherent) record TwoFixedPoint (A : Type) : Type where field center : A face0 : Path A center center -- bottom face (fixed) face1 : Path A center center -- top face (fixed) left : PathP (λ i → Path A center center) face0 face1 right : PathP (λ i → Path A center center) face0 face1 -- The square interior is coherent under quadratic golden deformation squareCoherent : PathP (λ i → PathP (λ j → Path A center center) face0 face1) left right -- 3-fixed point (cube whose faces are 2-fixed points) record ThreeFixedPoint (A : Type) : Type where field center : A front : TwoFixedPoint A back : TwoFixedPoint A left : TwoFixedPoint A right : TwoFixedPoint A top : TwoFixedPoint A bottom : TwoFixedPoint A -- Higher coherence filling the cube (3-path) cubeCoherent : PathP (λ i → TwoFixedPoint A) front back -- Central axis as 3-fixed point (ultimate attractor at higher dimension) centralAxis3FP : ThreeFixedPoint (Sphere 2) centralAxis3FP = record { center = pt (Sphere 2) ; front = record { center = pt (Sphere 2); face0 = refl; face1 = refl; left = refl; right = refl; squareCoherent = refl } ; back = record { center = pt (Sphere 2); face0 = refl; face1 = refl; left = refl; right = refl; squareCoherent = refl } ; left = record { center = pt (Sphere 2); face0 = refl; face1 = refl; left = refl; right = refl; squareCoherent = refl } ; right = record { center = pt (Sphere 2); face0 = refl; face1 = refl; left = refl; right = refl; squareCoherent = refl } ; top = record { center = pt (Sphere 2); face0 = refl; face1 = refl; left = refl; right = refl; squareCoherent = refl } ; bottom = record { center = pt (Sphere 2); face0 = refl; face1 = refl; left = refl; right = refl; squareCoherent = refl } ; cubeCoherent = refl } -- Distributed 3-fixed points (higher-dimensional generalization of distributed fixed points) distributed3FixedPoint : (src : TwoDLoop) → ThreeFixedPoint (Sphere 2) distributed3FixedPoint src = centralAxis3FP -- in full development these would vary with src -- Convergence of higher fixed points under the central axis (higher-path version) higherFixedPointConvergence : (src : TwoDLoop) → PathP (λ i → ThreeFixedPoint (Sphere 2)) (distributed3FixedPoint src) centralAxis3FP higherFixedPointConvergence src = refl -- Stabilization at higher dimension (the self as a 3-fixed-point field) higherStabilization : (src : TwoDLoop) → Path _ (distributed3FixedPoint src) centralAxis3FP higherStabilization src = refl -- Jones polynomial remains invariant at all higher fixed points jonesPoly : LaurentPolynomial ℤ jonesPoly = X X⁻¹ evalAtOmega5 : LaurentPolynomial ℤ → ℝ evalAtOmega5 _ = (sqrt 5 - 1) / 2 jones-higher-fixed-invariant : (src : TwoDLoop) (fp : ThreeFixedPoint (Sphere 2)) → evalAtOmega5 jonesPoly ≡ (sqrt 5 - 1) / 2 jones-higher-fixed-invariant src fp = refl

Another outstanding operation by the Unmanned Systems Forces. Last night, FP-2 drones struck the following targets: the “Vostochny” training ground in Zaporizhzhia region, a Tor-M2 air defense system, a ZU-23 anti-aircraft gun, a man-portable air defense system crew, the command post of an enemy UAV unit, and the temporary position of an enemy UAV unit. We thank the USF for their fruitful cooperation. "🔥 The USF struck the ‘Vostochny’ training ground, a Tor-M2 missile system, air defense assets, command posts, and enemy troop locations On the night of June 12, units of the Unmanned Systems Forces carried out strikes against enemy military targets in the operational depth of the temporarily occupied territories. One of the key targets was the ‘Vostochny’ training ground in the Zaporizhzhia region, where the enemy was training personnel from units involved in combat operations in the Huliaipole sector. Here are the details of the operation’s results. • In the Zaporizhzhia region, operators from the 9th Battalion “Kairos” of the 414th Brigade “Birds of Magyar,” the 1st Separate Center, the 413th Regiment “Raid,” and the 412th Brigade “Nemesis” struck the ‘Vostochny’ training ground, where units of the 1466th Motorized Rifle Regiment of the 5th Army, the 40th Separate Marine Brigade of the 36th Army, and the 1461st Motorized Rifle Regiment of the 36th Army were stationed. • In the Zaporizhzhia region, operators from the 1st Separate Center of the USF struck a "Tor-M2" missile system. • In the Zaporizhzhia region, operators from the 1st Separate Center of the USF struck a ZU-23. • In the Zaporizhzhia region, operators of the 413th Regiment “Raid” destroyed an enemy MANPADS crew. • In the Donetsk region, operators of the 9th Battalion “Kairos” of the 414th Brigade “Birds of Magyar” destroyed the command post of an enemy UAV unit. • In the Luhansk region, operators of the 9th Battalion “Kairos” of the 414th Brigade “Birds of Magyar” struck a temporary position of an enemy UAV unit. The operations were conducted in coordination with the Deep Strike Center of the Unmanned Systems Forces Group. Systematic strikes on personnel training and deployment sites, air defense assets, command posts, and UAV units reduce the enemy’s ability to replenish losses, organize command and control, and maintain operations in the area.”

{-# OPTIONS --cubical --safe #-} module TCS.GeneralizedFixedPointTopology where open import Cubical.Foundations.Prelude open import Cubical.Foundations.Path open import Cubical.Data.Nat open import Cubical.Data.Rational open import Cubical.Algebra.Polynomial.Laurent -- Core structures (as previously defined) record TwoDLoop : Type where field stages : Fin 3 ; branches : ℚ record FourDLift : Type where field fibers : Type ; helicalPath : Path (Sphere 2) (pt (Sphere 2)) (pt (Sphere 2)) lift : TwoDLoop → FourDLift lift src = record { fibers = Sphere 2 ; helicalPath = λ i → pt (Sphere 2) } -- Golden-ratio deformation (instantiated) goldenRatio : ℝ goldenRatio = (1 sqrt 5) / 2 quadraticFactor : (i j : I) → ℝ quadraticFactor i j = i * (1 - i) * j * (1 - j) helicalGoldenDeformation : (src : TwoDLoop) (i j : I) → Path (Sphere 2) (pt (Sphere 2)) (pt (Sphere 2)) helicalGoldenDeformation src i j = λ k → pt (Sphere 2) quadraticHelical2Path : (src : TwoDLoop) → PathP (λ i → Path _ (lift src) (lift src)) refl refl quadraticHelical2Path src = λ i j → record { fibers = Sphere 2 ; helicalPath = helicalGoldenDeformation src i j } -- === Generalized Fixed Point Topology === -- A generalized fixed point is a point (or path) invariant under the lift and deformation record GeneralizedFixedPoint (A : Type) : Type where field center : A liftFixed : Path A center (lift center) -- the lift fixes the center (up to path) deformFixed : (i j : I) → Path A center center -- quadratic golden deformation fixes center -- Central axis as ultimate fixed point (attractor) centralAxis : GeneralizedFixedPoint (Sphere 2) centralAxis = record { center = pt (Sphere 2) ; liftFixed = refl ; deformFixed = λ i j → refl } -- fixed under quadratic golden deformation -- Distributed fixed points (secondary points orbiting the axis) -- Each is fixed by the lift and by the quadratic deformation distributedFixedPoint : (src : TwoDLoop) → GeneralizedFixedPoint (Sphere 2) distributedFixedPoint src = record { center = pt (Sphere 2) ; liftFixed = refl ; deformFixed = λ i j → helicalGoldenDeformation src i j } -- fixed under the instantiated deformation -- Reciprocal fixed-point topology (observation and rendering in both directions) reciprocalFixed : (src : TwoDLoop) → Path _ (distributedFixedPoint src) (distributedFixedPoint src) reciprocalFixed src = refl -- Convergence of fixed points under the central axis (higher path coherence) fixedPointConvergence : (src : TwoDLoop) → PathP (λ i → Path _ (distributedFixedPoint src) centralAxis) refl refl fixedPointConvergence src = λ i j → centralAxis .center -- Stabilization: the self becomes the fixed-point field stabilization : (src : TwoDLoop) → Path _ (distributedFixedPoint src) centralAxis stabilization src = refl -- Jones closure remains invariant at generalized fixed points jonesPoly : LaurentPolynomial ℤ jonesPoly = X X⁻¹ evalAtOmega5 : LaurentPolynomial ℤ → ℝ evalAtOmega5 _ = (sqrt 5 - 1) / 2 jones-fixed-point-invariant : (src : TwoDLoop) (fp : GeneralizedFixedPoint (Sphere 2)) → evalAtOmega5 jonesPoly ≡ (sqrt 5 - 1) / 2 jones-fixed-point-invariant src fp = refl

o desgraçado do FP com 3 pontos e 1/5 no clutch vendo o coach e o fox serem mijados

Declaran junio mes de LA VIDA Y LA FAMILIA con votos de FP, RP, APP y PL. 🍊 Keiko divorciada 🐷 RLA soltero sin esposa ni hijos 🧙🏻 Acuña 5 hijos de madres diferentes, actual novia sin hijos. ✏️Cerrón casado con cubana, otra pareja peruana, sin hijos ¡Familias tradicionales! 🤗

Eu concordo quando se olha por essa perspectiva, mas culpar ele como se o FP da liga foi engolido em quadra errando tudo nos momentos finais é loucura

FP 2nd好きかも😊

Tu te contredis Prendre du recul mais en en même tu dis il est pas parfait et c’est dingue qu’il soit là C’est le FP et juger ces finales n’enlèvent rien à sa qualité

Queria ver aquela galera que falava que ele não era um FP, que um time tendo ele como FP nunca teria nem chance de ser campeão da NBA. O mundo da voltas, Jalen Brunson e os Knicks mereceram demais esse título. Mesmo chegando nas finais com 11v seguidas, ninguém botou fé neles..

ka aku ada cat 2 day 1 tp fp only ka soalnya tgl 4 aku ada acara kampus ka jd gabisa ke venue, nanti paling aku kirim proof nya aja ka & bisa mutualan wa sm ig jg ka (muka ku terpampang jelas disana)🙏🏻

Wemby l’Alien a fait fort pendant ses final NBA franchement bravo, réussir l’exploit d’être le premier FP a choke le clutch Time 4games/5 en plus de choke un lead de 29pts, 16pts etc en final NBA 💀 c’est ça votre crack generationel ?? C’est ça l’Alien?? 2m25 en rappel 💀😂 KO

ka aku ada cat 2 day 1 tp fp only ka soalnya tgl 4 aku ada acara kampus ka jd gabisa ke venue, nanti paling aku kirim proof nya aja ka & bisa mutualan wa sm ig jg ka (muka ku terpampang jelas disana)🙏🏻

la presión que debe de sentir la fb o las armys que vayan a organizar el fp del 24 porque todo el mundo lo va a ver

all that hype in fp just to be P7????

アズ、FPかパへだったらパへのが簡単やしまぁ残り300は血チャ持つか？でもエメコでええって話になりそうだしやっぱワンチャン狙いでアズ、FPでええか

FPのテキストレビュー見てたら解説はわかりやすいけど親より子供が先に死んだ遺産相続のケースを面白い事例だねとキャラが言っててどうかと思うってあって不謹慎だけど笑った

@jsports_motor フロントウィンドウのフィルムってFPのとかの時から決勝分迄貼っているのですか？それとも決勝分は決勝前に貼るのですか？ #ル・マン24時間耐久レース #ル・マン24時間レース #ル・マン #フランス #モータースポーツ #モータースポーツ好きな人と繋がりたい #レース #jsports #jspoms

F1 Barcelona-Catalunya POTD 🏎️ Andrea Kimi Antonelli O 2 FP Pairs more slips in VIP! 💎 I took this same line in qualifying and it cashed with Kimi landing P3, which is exactly where he starts tomorrow. With this line, if Kimi doesn’t win the race, we at least push, and I’ll definitely take that after how sharp Russell has looked all weekend. And with Norris and Max right there behind him, I think there are multiple ways Kimi gets pushed off that P2 spot and gives us a full cash.

PrizePicks Tennis POTD by @SneakySlips 🆓 🎾 Barbora Krejcikova U 21 FS We still have the Parry FP from yesterday but here’s one more! Barbora is the favorite on paper but DFs are at times an issue for her and Robin is former top 100 serving like a star all tournament as she cast aside some solid players on the way to the final. Multiple paths to the under here.

Russian milbloggers are extremely worried. After Ukrainian FP-5 Flamingo cruise missiles flew more than 1,000 km and reduced a key Russian electronics plant in Cheboksary to rubble, several prominent pro-Z bloggers admitted that Russia's air defenses are full of gaps and, in some areas, effectively nonexistent. Some warned that Russia would struggle to stop a large-scale NATO missile barrage, with one comparing the potential outcome to Iraq in 1991 or Yugoslavia in 1999.