diff --git a/formal/KUOS/Architecture/QiWuxingGK5BPNorthRawStirlingV4_35.lean b/formal/KUOS/Architecture/QiWuxingGK5BPNorthRawStirlingV4_35.lean new file mode 100644 index 000000000..90336972b --- /dev/null +++ b/formal/KUOS/Architecture/QiWuxingGK5BPNorthRawStirlingV4_35.lean @@ -0,0 +1,227 @@ +import KUOS.Architecture.QiWuxingGK5BONorthFactorialDifferenceV4_34 +import Kuu.F112.PolygonNativeC60AQFullPentagonPhysicalNativeGK5BPRawNewtonStirling +import Mathlib.Tactic + +/-! +Qi v4.35 — genuine NORTH C60 RAW observations / Newton +signed first-kind Stirling certificates. + +For the first time this sequence reads ONLY integer-indexed +observations of real original typed F55/P92B histories, rather +than assuming original monomial coefficients are supplied as +reconstruction inputs. Independent GK5BP proves an exact +rational Newton interpolation and a finite signed-Stirling +coefficient-lift criterion. A true original F55/P77 quotient +class is used only to supply its genuinely DESCENDED Int[X] +polynomial value; equality of such values never identifies +source Type-valued P77 classes. + +The north primitive (P=1), genuine double inverse plus original +F55 right unit (P=X²), and chosen P92B loops A=X³+2 and +B=2X²+X are checked on the canonical nonnegative integer +grids {0,1,2} and {0,1,2,3}; the latter is a NEW grid, not the +old {-1,0,1,2} four-point Lagrange set. + +Physical fifth-around C60 gauge and conditional P89 BadPair +reflection remain independent, as do the original-phi local +seam existence/obstruction results. No new Type-valued F55 +source, P94A four-cell, universal GK5AY seam, GK5BA kit, +unconditional P89 reflection or Gray restoration. + +Strict 空 authority; independent Kuu→Qi; original F54 +two Nat/two categories/Blocks and F55 Type history retained. +Append-only root107, #1558 Draft/open/unmerged, pinned +Lean v4.30.0-rc2 / mathlib +5450b53e5ddc75d46418fabb605edbf36bd0beb6. +No new axioms, sorry/admit/native_decide. +Copyright (c) 2026 Hidetoshi Itakura / 板倉英俊. Apache-2.0. +-/ + +namespace KUOS.Architecture.WuxingPentagonUnified +open KUOS.Architecture.WuxingCyclicPaths +open Kuu.F112.Physical +open CategoryTheory +set_option autoImplicit false +set_option warningAsError true +set_option maxRecDepth 8192 +set_option maxHeartbeats 4000000 +noncomputable section + +private def fivefoldGK5BPNorthFirst := + KuuNativeF45Tree.pentagon0 northNativeF45FirstStage + northNativeF45SecondStage northNativeF45ThirdStage + northNativeF45FourthStage + +private def fivefoldGK5BPNorthLast := + KuuNativeF45Tree.pentagon3 northNativeF45FirstStage + northNativeF45SecondStage northNativeF45ThirdStage + northNativeF45FourthStage + +/-- The finite raw signed-Stirling integrality certificate. +Only finite observations values(0),...,values(d), their genuine +forward differences, and signed falling-factorial coefficients +appear, not coefficients of an unknown original source. -/ +def fivefoldGK5BPRawStirlingCertificate + (d : Nat) (values : Nat → Rat) : Prop := + ∀ j ∈ Finset.range (d + 1), ∃ z : Int, + (z : Rat) = + ∑ k ∈ Finset.range (d + 1), + (nativeGK5BPRawForward values k / (k.factorial : Rat)) * + (nativeGK5BPSignedFirst k j : Rat) + +/-- Raw north integer-indexed observation data have a genuine +Int[X] Newton polynomial source IFF the FINITE signed-Stirling +certificate holds. This does not realize source Type witnesses. -/ +theorem fivefoldGK5BPNorthRaw_iff_integerPolynomial + (d : Nat) (values : Nat → Rat) : + (∃ P : Polynomial Int, + P.map (Int.castRingHom Rat) = + nativeGK5BPNewtonFromRaw d values) ↔ + fivefoldGK5BPRawStirlingCertificate d values := + nativeGK5BPNewton_integral_iff_rawStirling d values + +/-- The same necessary-and-sufficient certificate applies to +the ACTUAL GK5BL rational Lagrange interpolant of raw north +observations on the canonical integer grid. -/ +theorem fivefoldGK5BPNorthLagrange_iff_rawStirling + (d : Nat) (values : Nat → Rat) : + (∃ P : Polynomial Int, P.map (Int.castRingHom Rat) = + nativeGK5BLRationalInterpolant (nativeGK5BKIntegerGrid d) + (fun z : Int => values z.toNat)) ↔ + fivefoldGK5BPRawStirlingCertificate d values := + nativeGK5BPLagrange_integral_iff_rawStirling d values + +/-- For an actual Int[X] polynomial of degree ≤d, its +observed integer-indexed values supply the raw signed-Stirling +finite certificate, by actual Lagrange recovery. This theorem +does NOT assume that arbitrary raw values admit Int[X] lifts. -/ +theorem fivefoldGK5BPIntegerSource_rawStirling + (P : Polynomial Int) (d : Nat) (hdegree : P.natDegree ≤ d) : + fivefoldGK5BPRawStirlingCertificate d + (fun n : Nat => ((P.eval (n : Int) : Int) : Rat)) := by + apply (fivefoldGK5BPNorthLagrange_iff_rawStirling d + (fun n : Nat => ((P.eval (n : Int) : Int) : Rat))).1 + refine ⟨P, ?_⟩ + have hdata : + nativeGK5BLRationalInterpolant (nativeGK5BKIntegerGrid d) + (fun z : Int => ((P.eval (z.toNat : Int) : Int) : Rat)) = + nativeGK5BLRationalInterpolant (nativeGK5BKIntegerGrid d) + (fun z : Int => ((P.eval z : Int) : Rat)) := by + apply Lagrange.interpolate_eq_of_values_eq_on + intro z hz + change z ∈ (Finset.range (d + 1)).image + (fun k : Nat => (k : Int)) at hz + rcases Finset.mem_image.mp hz with ⟨k, hk, hzk⟩ + subst z + simp + rw [hdata] + exact (nativeGK5BLRationalInterpolant_recovers_int P d + (nativeGK5BKIntegerGrid d) (nativeGK5BKIntegerGrid_card d) hdegree).symm + +/-- Genuine ORIGINAL north F55 source f: its true integer +P92B depth polynomial evaluations at 0,...,d pass the +complete finite signed-Stirling integrality certificate. -/ +theorem fivefoldGK5BPNorthOriginalF55_rawStirling + (f : NativeGK5AINorthPentagonCells) (d : Nat) + (hdegree : + (nativeGK5AIOriginalToP92Structural f).gk5bgDepthPolynomial.natDegree ≤ d) : + fivefoldGK5BPRawStirlingCertificate d + (fun n : Nat => + (((nativeGK5AIOriginalToP92Structural f).gk5bgDepthPolynomial.eval + (n : Int) : Int) : Rat)) := + fivefoldGK5BPIntegerSource_rawStirling + (nativeGK5AIOriginalToP92Structural f).gk5bgDepthPolynomial d hdegree + +/-- A genuine ORIGINAL F55/P77 four-cell quotient q has the +same finite raw observation certificate for its DESCENDED +polynomial value, not a newly reconstructed Type proof/class. -/ +theorem fivefoldGK5BPNorthP77_rawStirling + (q : NativeF55RawVerticalFourClasses + fivefoldGK5BPNorthFirst fivefoldGK5BPNorthLast + northNativeF45PentagonLong northNativeF45PentagonShort) + (d : Nat) + (hdegree : (nativeGK5BGPolyOnP77Class + fivefoldGK5BPNorthFirst fivefoldGK5BPNorthLast + northNativeF45PentagonLong northNativeF45PentagonShort q).natDegree ≤ d) : + fivefoldGK5BPRawStirlingCertificate d + (fun n : Nat => (((nativeGK5BGPolyOnP77Class + fivefoldGK5BPNorthFirst fivefoldGK5BPNorthLast + northNativeF45PentagonLong northNativeF45PentagonShort q).eval + (n : Int) : Int) : Rat)) := + fivefoldGK5BPIntegerSource_rawStirling + (nativeGK5BGPolyOnP77Class + fivefoldGK5BPNorthFirst fivefoldGK5BPNorthLast + northNativeF45PentagonLong northNativeF45PentagonShort q) + d hdegree + +/-- True original north primitive P=1 has the raw +three-integer-observation signed-Stirling certificate. -/ +theorem fivefoldGK5BPNorthPrimitive_rawStirling : + fivefoldGK5BPRawStirlingCertificate 2 + (fun n : Nat => + (((nativeGK5AIOriginalToP92Structural + nativeGK5AGNorthOriginalCell).gk5bgDepthPolynomial.eval + (n : Int) : Int) : Rat)) := + fivefoldGK5BPNorthOriginalF55_rawStirling + nativeGK5AGNorthOriginalCell 2 + fivefoldGK5BKNorthPrimitiveDouble_degree_le_two.1 + +/-- True original north double inverse WITH real right F55 +identity has an independent raw 3-site certificate. -/ +theorem fivefoldGK5BPNorthDoubleInverse_rawStirling : + fivefoldGK5BPRawStirlingCertificate 2 + (fun n : Nat => + (((nativeGK5AIOriginalToP92Structural + fivefoldGK5BFNorthDoubleInverseRightUnit).gk5bgDepthPolynomial.eval + (n : Int) : Int) : Rat)) := + fivefoldGK5BPNorthOriginalF55_rawStirling + fivefoldGK5BFNorthDoubleInverseRightUnit 2 + fivefoldGK5BKNorthPrimitiveDouble_degree_le_two.2 + +/-- The two genuinely chosen north P92B endomorphism loops +each have a separately typed raw four-observation certificate +on the NEW grid {0,1,2,3}, not an original-to-phi seam claim. -/ +theorem fivefoldGK5BPNorthTripleLoops_rawStirling : + fivefoldGK5BPRawStirlingCertificate 3 + (fun n : Nat => + ((fivefoldGK5BGNorthTripleLoopA.gk5bgDepthPolynomial.eval + (n : Int) : Int) : Rat)) ∧ + fivefoldGK5BPRawStirlingCertificate 3 + (fun n : Nat => + ((fivefoldGK5BGNorthTripleLoopB.gk5bgDepthPolynomial.eval + (n : Int) : Int) : Rat)) := by + obtain ⟨ha, hb⟩ := fivefoldGK5BKNorthTripleLoops_degree_le_three + exact ⟨fivefoldGK5BPIntegerSource_rawStirling + fivefoldGK5BGNorthTripleLoopA.gk5bgDepthPolynomial 3 ha, + fivefoldGK5BPIntegerSource_rawStirling + fivefoldGK5BGNorthTripleLoopB.gk5bgDepthPolynomial 3 hb⟩ + +/-- The coefficient/arithmetic observation layer remains +separate from the TRUE physical north C60 fifth-gauge closure, +existing original positive/negative seams and conditional +original P89 BadPair source-reflection obstruction. -/ +theorem fivefoldGK5BPNorthPhysicalAndP89Boundary + (frame : NativeC60FivePhaseGauge) (i : FivefoldIndex) : + Nonempty (nativeGK5AZSeamAt nativeGK5AGNorthOriginalCell) ∧ + (¬ Nonempty (nativeGK5AZSeamAt + fivefoldGK5BFNorthDoubleInverseRightUnit)) ∧ + (fivefoldGK5AGNorthPhysicalFifthDarts frame i = + (nativeC60GaugeDart frame .north (i + 4), + nativeC60GaugeDart frame .north i)) ∧ + ((¬ Function.Injective + (fivefoldGK5ANP94NorthPhysicalFourClassMap frame i)) ↔ + fivefoldGK5AQNorthReflectionObstruction) := + fivefoldGK5BONorthPhysicalAndP89Boundary frame i + +#print axioms fivefoldGK5BPNorthRaw_iff_integerPolynomial +#print axioms fivefoldGK5BPNorthLagrange_iff_rawStirling +#print axioms fivefoldGK5BPIntegerSource_rawStirling +#print axioms fivefoldGK5BPNorthOriginalF55_rawStirling +#print axioms fivefoldGK5BPNorthP77_rawStirling +#print axioms fivefoldGK5BPNorthPrimitive_rawStirling +#print axioms fivefoldGK5BPNorthDoubleInverse_rawStirling +#print axioms fivefoldGK5BPNorthTripleLoops_rawStirling +#print axioms fivefoldGK5BPNorthPhysicalAndP89Boundary + +end +end KUOS.Architecture.WuxingPentagonUnified diff --git a/formal/KuuOSFormal.lean b/formal/KuuOSFormal.lean index 07e1bb75a..f398d0ce1 100644 --- a/formal/KuuOSFormal.lean +++ b/formal/KuuOSFormal.lean @@ -530,6 +530,7 @@ import KUOS.Architecture.QiWuxingGK5BNNorthFiniteDivisibilityV4_33 import Kuu.F112.PolygonNativeC60AQFullPentagonPhysicalNativeGK5BOForwardDifferenceFactorial import KUOS.Architecture.QiWuxingGK5BONorthFactorialDifferenceV4_34 import Kuu.F112.PolygonNativeC60AQFullPentagonPhysicalNativeGK5BPRawNewtonStirling +import KUOS.Architecture.QiWuxingGK5BPNorthRawStirlingV4_35 /-! Public aggregate: retain every previously published proof and its original