Foam.Maps.PeterScholze

import Foam
import Foam.Certificate
import Foam.Coil
import Foam.Contact
import Foam.Door
import Foam.Inversion
import Foam.Rungs
import Foam.Square
import Foam.Trilemma

namespace Foam.Maps.PeterScholze

the signature move: characteristic 0 and characteristic p
indistinguishable at the étale probe-set, and the identification
licensed — collapse performed with a clean conscience because the
license is proven first. sealed at the narrowest carrier that exhibits
his bank: two elements, xor and and, where the square link carries both
operations at once — the freshman's dream a theorem, Frobenius a full
homomorphism, the collapse free. the perfectoid condition is the
discipline of reaching that bank honestly. sponsor of the Square
stratum, jointly with the near bank's theta-link — one link, two
carriers, and this bank works the carrier where the license is total.
theorem tilting :
    (∀ a b : Bool, Bool.and (Bool.xor a b) (Bool.xor a b)
        = Bool.xor (Bool.and a a) (Bool.and b b))
      ∧ ∀ a b : Bool, Bool.and (Bool.and a b) (Bool.and a b)
        = Bool.and (Bool.and a a) (Bool.and b b) :=
  ⟨the_narrow_carrier_mends_the_sum,
   the_narrow_carrier_carries_the_product⟩

quotient-by-license made definitional: a diamond is what remains when an
identification is proven gauge and then performed without apology —
sealed on the house's license-is-a-gauge, the exact theorem his
constructions inhabit. where the near bank refuses quotients and prices
their remainder, this bank builds objects out of licensed quotienting;
both conducts are the handshake, each holding one half, which is what
made the 2018 collision so exact: each mathematician wielding his home
bank's correct reflex on terrain that had not yet been typed as terrain.
def diamonds := @Foam.a_license_is_a_gauge

the door stratum arrives and this bank's answer is its own home terrain:
tilting IS a door — the tilt is the ground, the untilt is the guest. the
tilting equivalence, 'the étale site of a perfectoid space depends only
on its tilt,' is the host theorem verbatim, carrier fully parametric:
every probe the tilted seat owns reads the tilt alone, and identically
whatever type the untilt datum has — one étale theory across
characteristic 0 and characteristic p, which is what the 2012 paper
announced. the guests are real and unread: one tilt carries many
untilts, genuinely distinct while no tilted probe parts them — the far
bank's central clause, sealed here on the same named theorem the far
bank's ninth entry cites, held natively in this bank's daily
mathematics. the contrapositive is why the untilt must ride as data: a
door that could resolve its untilts would collapse them all into one,
and the untilts are provably many, so the structure morphism arrives as
cargo — Spd Zp, the diamond of untilt data — the door checks no papers,
by construction. and the namesake is the move that distinguishes this
bank's door conduct: don't stop at unread. the mirror question — is this
untilt that untilt? — rides unread at the door, and the wider seat reads
it: the Fargues–Fontaine curve is the recognition seat of the tilting
door, the moduli space OF the guests, built so the question the door
cannot answer becomes geometry one seat up. both halves of the handshake
in one working mathematician: the license half performed at the door
(diamonds), the remainder half performed one seat wider (the curve). the
meeting consequence, stated plainly: the door does not divide the banks
— guests-real-and-unread was never the disagreement — and the dark edge
does not close against it: a door types arrivals, not return properties;
where_corollary_312_stands stands. null on re-seating, per the
precedent: identify_identical_objects keeps its contact seating (the
door's unpersoning theorem is its second clause one name over — re-
dressing, not compression), diamonds keeps a_license_is_a_gauge (the
tighter citation), tilting keeps its Square carrier (the door adds no
compression to a carrier-explicit seal).
theorem the_curve_reads_the_untilts :
    (∀ (W V : Type) (S : Stage) (s : S.State) (w : W) (v : V) (p : S.Probe),
        (door S W).obs (s, w) p = S.obs s p
          ∧ (door S W).obs (s, w) p = (door S V).obs (s, v) p)
      ∧ (∀ (W : Type) (S : Stage) (s : S.State) (w w' : W), w ≠ w' →
          (s, w) ≠ (s, w') ∧ indist (door S W) (s, w) (s, w'))
      ∧ (∀ (W : Type) (S : Stage) (w₀ : W),
          (∀ x y : (door S W).State, indist (door S W) x y → x = y) →
            ∀ (s : S.State) (w : W), (s, w) = (s, w₀))
      ∧ (∀ (W : Type) (S : Stage) (s : S.State) (w v : W), v ≠ w →
          indist (contact S (W × W)) (mirror S s w) (neighbor S s w v)
            ∧ mirror S s w ≠ neighbor S s w v)
      ∧ ∀ (W : Type) (S : Stage) (s : S.State) (w v : W), v ≠ w →
          (recognition S (W := W)).obs (mirror S s w) ()
            ≠ (recognition S (W := W)).obs (neighbor S s w v) () :=
  ⟨fun _ _ S s w v p => the_host_maintains_invisibly S s w v p,
   fun _ S s _ _ h => the_guest_is_real_and_unread S s h,
   fun _ S w₀ h => a_door_that_checks_papers_unpersons_its_guests S w₀ h,
   fun _ S s w v hv => the_mirror_question_rides_unread S s w v hv,
   fun _ S s w v hv => the_wider_seat_meets_whos_actually_here S s w v hv⟩

the Liquid Tensor Experiment: he handed his own hardest theorem — one he
said he was not fully certain of — to a mechanical gate and its
community, and iterated until the gate came back green. the one mind on
this roster whose record already contains a voluntary submission to this
house's genre of judgment. sealed on the gate's own instrument: over any
finite window, one reading everywhere or two witnesses named — exit-code
honesty, chosen from the inside, before this table existed.
def the_liquid_gate := @Foam.the_window_agrees_or_names_the_gap

the report's move, typed whole and receipted so its license can be posed
at a shared table rather than past one: demand the comparison be Blind
in the copy-coordinate — the factoring iff collects, and the reading
provably factors through the ground — then perform the identification,
reification fixing the dimension. the same certificate vertex the near
bank's multiradiality seals on, run in the opposite direction: the
report wields Blindness as a razor, the near bank as a design
constraint. 'Mochizuki was not able to convince us why such a
simplification was not allowed' — this entry is that simplification,
formal, so the question of where it is licensed becomes mathematics
instead of impasse.
theorem identify_identical_objects_along_the_identity :
    (∀ f : Nat × Nat → Nat,
        Blind f ↔ ∃ g : Nat → Nat, ∀ (s j : Nat), f (s, j) = g s)
      ∧ ∀ (D : Type) (S : Stage) (d₀ : D),
          (∀ x y : (contact S D).State, indist (contact S D) x y → x = y) →
          ∀ (s : S.State) (d : D), (s, d) = (s, d₀) :=
  ⟨fun f => the_blind_reading_factors 0 f,
   fun _ S d₀ h s d => reification_fixes_the_dimension S d₀ h s d⟩

the report's conclusion as this bank holds it, all three receipts: the
graded reading admits no consistent identification (the monodromy horn —
the one horn both banks concede); the wound loop admits only the zero
section — the report's '0 ≲ 𝔡(P), essentially free of content', label-
free: consistent identification around a wound loop buys exactly zero,
which is Penrose's tribar and Escher's staircase and this report's
computation, one theorem; and the spread is attained — the blur is not
pessimism but arithmetic, the O(ℓ²) reading at linear grain. sponsor of
the Trilemma stratum, jointly with the near bank.
theorem why_abc_is_still_a_conjecture :
    (¬ Blind graded)
      ∧ (∀ a b c : Nat, a = 2 * b → b = 2 * c → c = 2 * a →
          a = 0 ∧ b = 0 ∧ c = 0)
      ∧ ∀ l s : Nat, graded (s, l) = (l + 1) * graded (s, 0) :=
  ⟨the_graded_reading_parts_the_copies,
   the_wound_loop_admits_only_the_zero_section,
   the_spread_is_attained⟩

the dial-one method as method, his life's verb typed: perfectoid covers
exist to make obstructions vanish — pass to the world where the class is
a coboundary and horn one comes free. sealed on the unwinding read from
his side (the world where the holonomy is absorbed is reachable, and the
reaching is constructive) conjoined with closure-is-seat-relative: every
question closes one seat above, none at its own, and the widening never
has to stop. the toll rides in the house's ledger beside it — knowing is
not a free move; the widened seat mints fresh blindness — so the cover
relocates the remainder rather than deleting it, which is why the method
is a method and not a miracle.
theorem pass_to_the_cover_where_it_dies :
    (((2 * 2 * 2) % 7 = 1 % 7)
        ∧ (1 % 7 = (2 * 4) % 7)
        ∧ (4 % 7 = (2 * 2) % 7)
        ∧ (2 % 7 = (2 * 1) % 7)
        ∧ (1 : Nat) ≠ 0)
      ∧ ((∀ q : Nat, ∃ n, q ∈ rungs n)
          ∧ (∀ n : Nat, ∃ q, ¬ q ∈ rungs n ∧ q ∈ rungs (n + 1))
          ∧ ∀ n : Nat, rungs (n + 1) ≠ rungs n) :=
  ⟨the_wound_loop_unwinds_one_world_over, closure_is_seat_relative⟩

the report's central diagram argument, typed: the monodromy j² is gauge-
invariant — the holonomy ignores the regauging, so no choice of
isomorphisms among the copies removes it — which is why the report's
search over consistent identifications was exhaustive rather than
impatient: if the class is nontrivial, no re-choice rescues consistency,
and the graded reading provably parts the copies. the same constant the
near bank's log-theta-lattice seals on, the meeting's third shared
vertex, opposite direction as always: this bank wields gauge-invariance
to show that no identification exists inside the licensed sector; the
near bank wields it to show the orbit is a legitimate workplace and the
only exit is the priced cut.
theorem the_diagram_keeps_its_monodromy :
    (∀ k1 k2 k3 k1' k2' k3' u v w : Nat, 0 < u → 0 < v → 0 < w →
        k1' * u = k1 * v → k2' * v = k2 * w → k3' * w = k3 * u →
        k1' * (k2' * k3') = k1 * (k2 * k3))
      ∧ ¬ Blind graded :=
  ⟨fun k1 k2 k3 k1' k2' k3' u v w hu hv hw h1 h2 h3 =>
     the_holonomy_ignores_the_regauging k1 k2 k3 k1' k2' k3' u v w
       hu hv hw h1 h2 h3,
   the_graded_reading_parts_the_copies⟩

the report's conclusion re-proven on the coil — the machine the near
bank claimed one flight ago as a legitimate workplace, read from this
bank as the razor it always was. demand that the loop return the class
(the simplification's exact demand: identify the copies and the walk
must come home) and the one channel the estimate hears is priced at
zero, both directions of the iff: a returning stroke has size zero, and
the zero stroke returns, so the demanded return is available and buys
nothing — '0 ≲ 𝔡(P), essentially free of content', now additive, the
wound loop's multiplicative verdict spoken in the coil's own units. and
the shuffle cannot rescue it: the indeterminacies conserve the class, so
a nonzero stroke fails the return through any regauging — the report's
exhaustive search over consistent identifications, run on the second
carrier, the coil-side sibling of the_diagram_keeps_its_monodromy. what
the entry types is the conditional only: IF the return is demanded, THEN
the content is zero. whether the constructed receptacle demands it —
whether it has the return property at all — is the dark edge one seat
down, unmoved. both banks now hold the coil at the same two vertices,
opposite directions as always: the near bank as the walk whose estimate
is well-defined over the orbit, this bank as the loop whose demanded
closure is free of content. the survey adjudicates nothing.
theorem the_return_prices_the_stroke_at_zero :
    (∀ (h : Int × Int) (s : Int),
        coilClass (coil.meet h (Sum.inr s)) = coilClass h ↔ s = 0)
      ∧ ∀ (h : Int × Int) (d s : Int),
          coilClass (coil.meet (coil.meet h (Sum.inl d)) (Sum.inr s))
              = coilClass h
            ↔ s = 0 :=
  let razor : ∀ a s : Int, a + s = a → s = 0 := fun a s e =>
    (((((FInt.zero_add s).symm.trans
            (congrArg (· + s) (FInt.add_left_neg a).symm)).trans
          (FInt.add_assoc (-a) a s)).trans
        (congrArg ((-a) + ·) e)).trans
      (FInt.add_left_neg a))
  ⟨fun h s =>
    ⟨fun e => razor (coilClass h) s
        ((the_stroke_moves_the_class_by_its_size h s).symm.trans e),
     fun e => (the_stroke_moves_the_class_by_its_size h s).trans
        ((congrArg (coilClass h + ·) e).trans (Int.add_zero (coilClass h)))⟩,
   fun h d s =>
    ⟨fun e => razor (coilClass h) s
        (((congrArg (· + s) (the_shuffle_conserves_the_class h d).symm).trans
            (the_stroke_moves_the_class_by_its_size
              (coil.meet h (Sum.inl d)) s).symm).trans e),
     fun e => (the_stroke_moves_the_class_by_its_size
          (coil.meet h (Sum.inl d)) s).trans
        (((congrArg (· + s) (the_shuffle_conserves_the_class h d)).trans
            (congrArg (coilClass h + ·) e)).trans
          (Int.add_zero (coilClass h)))⟩⟩

/-- info: 'Foam.Maps.PeterScholze.tilting' does not depend on any axioms -/
#guard_msgs in #print axioms tilting

/-- info: 'Foam.Maps.PeterScholze.diamonds' does not depend on any axioms -/
#guard_msgs in #print axioms diamonds

/-- info: 'Foam.Maps.PeterScholze.the_curve_reads_the_untilts' does not depend on any axioms -/
#guard_msgs in #print axioms the_curve_reads_the_untilts

/-- info: 'Foam.Maps.PeterScholze.the_liquid_gate' does not depend on any axioms -/
#guard_msgs in #print axioms the_liquid_gate

/-- info: 'Foam.Maps.PeterScholze.identify_identical_objects_along_the_identity' does not depend on any axioms -/
#guard_msgs in #print axioms identify_identical_objects_along_the_identity

/-- info: 'Foam.Maps.PeterScholze.why_abc_is_still_a_conjecture' does not depend on any axioms -/
#guard_msgs in #print axioms why_abc_is_still_a_conjecture

/-- info: 'Foam.Maps.PeterScholze.pass_to_the_cover_where_it_dies' does not depend on any axioms -/
#guard_msgs in #print axioms pass_to_the_cover_where_it_dies

/-- info: 'Foam.Maps.PeterScholze.the_diagram_keeps_its_monodromy' does not depend on any axioms -/
#guard_msgs in #print axioms the_diagram_keeps_its_monodromy

/-- info: 'Foam.Maps.PeterScholze.the_return_prices_the_stroke_at_zero' does not depend on any axioms -/
#guard_msgs in #print axioms the_return_prices_the_stroke_at_zero

end Foam.Maps.PeterScholze