IndisputableMonolith.Gravity.QuantumChannel.AmplitudeLinearForcedCert

   1import Mathlib
   2import IndisputableMonolith.Gravity.QuantumChannel.AmplitudeLinearForced
   3import IndisputableMonolith.Gravity.QuantumChannel.AmplitudeLinearForcedJoint
   4import IndisputableMonolith.Gravity.QuantumChannel.AmplitudeLinearForcedSubstrate
   5
   6/-!
   7# Gravity Track 2.C: Master Certificate (binary-tensor model)
   8
   9This module aggregates the Track 2.C closure from Sessions 85–87:
  10
  11* Session 85 (`AmplitudeLinearForced`) — single-factor substrate dichotomy:
  12  on `Signal8`, no nontrivial channel response is simultaneously
  13  amplitude-linear and density-only.
  14
  15* Session 86 (`AmplitudeLinearForcedJoint`) — joint-substrate lift on
  16  `JointSubstrate := Signal8 ⊗[ℂ] Signal8`: a `ℂ`-linear joint operator
  17  that factorizes on pure tensors through factor-wise responses must have
  18  each factor amplitude-linear under nontriviality of the other.
  19
  20* Session 87 (`AmplitudeLinearForcedSubstrate`) — substrate-side closure:
  21  under the substrate recognition update `cyclic_shift` on the matter side
  22  and pure-tensor factorization, the channel side is forced amplitude-linear.
  23
  24The aggregated **Track 2.C master theorem (binary-tensor model)**: under
  25a *recognition-coupled factorizable joint substrate* — the binary tensor
  26product `Signal8 ⊗[ℂ] Signal8` with a `ℂ`-linear joint operator that
  27factorizes on pure tensors and has the substrate recognition update on the
  28matter side — the gravitational-channel response must be amplitude-linear.
  29A density-only (CPTP-classical density-matrix) channel response collapses
  30to the trivial zero response.
  31
  32This **upgrades paper IV's T2 from `MODEL` to `STRUCTURAL THEOREM`**:
  33the physical identification of the channel response with the amplitude-linear
  34extension is no longer a modeling choice but a substrate consequence,
  35conditional on the named `FactorizableJointSubstrate` structural axiom
  36(matter and channel sectors evolve under independent dynamics that do not
  37mix at the operator level). The lift to general joint operators on
  38`Signal8 ⊗[ℂ] Signal8` without the factor-product assumption — the truly
  39unconditional Track 2.C closure — remains future work and would either
  40rederive factorization from a stricter substrate axiom or eliminate it
  41entirely from the joint-operator side.
  42
  43The master-statement requirement that no MODEL-tag step survive (anti-retreat
  44principle §2.5) is not yet met: the binary-tensor STRUCTURAL THEOREM is
  45strictly stronger than MODEL but strictly weaker than unconditional THEOREM.
  46The integrated discovery claim still requires the unconditional lift.
  47
  48Zero `sorry`. Zero new RS-specific axioms. The factor-product structure is
  49an explicit structural hypothesis, named and visible at the type level of
  50every conclusion.
  51-/
  52
  53namespace IndisputableMonolith
  54namespace Gravity
  55namespace QuantumChannel
  56namespace AmplitudeLinearForced
  57
  58open scoped TensorProduct
  59
  60/-- **Named structural hypothesis: factorizable joint substrate.** The joint
  61operator on `JointSubstrate` factorizes on pure tensors through factor-wise
  62responses on matter and channel. This captures the master-plan §4 Track 2.C
  63step 1 setup `JointSubstrate := MatterLedger ⊗ ChannelLedger` with
  64`R_joint = R_matter ⊗ R_channel`: matter and channel sectors evolve under
  65independent dynamics that do not mix at the operator level.
  66
  67Equivalently, the joint operator carries no cross-sector coherence: its
  68action on a pure tensor never produces entanglement between the matter and
  69channel factors. (A general `ℂ`-linear endomorphism of
  70`Signal8 ⊗[ℂ] Signal8` does not have this property; the swap operator is
  71the standard counterexample.) -/
  72structure FactorizableJointSubstrate where
  73  R_J : JointSubstrate →ₗ[ℂ] JointSubstrate
  74  R_M : Signal8 → Signal8
  75  R_C : Signal8 → Signal8
  76  factor : PureTensorFactorization R_J R_M R_C
  77
  78/-- The **canonical recognition-coupled factorization**: matter and channel
  79sides are both the substrate recognition update `cyclic_shift`, joint
  80operator is `TensorProduct.map cyclicShiftLinear cyclicShiftLinear`. -/
  81noncomputable def canonicalRecognitionFactorization :
  82    FactorizableJointSubstrate where
  83  R_J := canonicalCyclicJointOperator
  84  R_M := recognitionUpdate
  85  R_C := recognitionUpdate
  86  factor := canonicalCyclicJointOperator_pureTensorFactorization
  87
  88/-- A factorizable joint substrate is **recognition-coupled** when its
  89matter side is the substrate recognition update. This adds the master-plan
  90constraint that matter dynamics is fixed by the T0–T8 forcing chain:
  91single-site Schrodinger linearity forces the matter factor to be
  92`cyclic_shift`. -/
  93structure RecognitionCoupledFactorization extends FactorizableJointSubstrate where
  94  matter_eq_recognitionUpdate : R_M = recognitionUpdate
  95
  96/-- Canonical witness: cyclic-shift on both factors is recognition-coupled. -/
  97noncomputable def canonicalRecognitionCoupled : RecognitionCoupledFactorization where
  98  toFactorizableJointSubstrate := canonicalRecognitionFactorization
  99  matter_eq_recognitionUpdate := rfl
 100
 101/-- **Track 2.C master theorem (forcing direction).** Under a
 102recognition-coupled factorizable joint substrate, the channel-side response
 103is amplitude-linear. The substrate dynamics on the matter side, combined
 104with the joint linearity of `R_J` and the factor-product structure,
 105propagates amplitude-linearity to the channel factor. -/
 106theorem track2C_channel_isAmplitudeLinear
 107    (F : RecognitionCoupledFactorization) :
 108    IsAmplitudeLinear F.R_C := by
 109  have hFact : PureTensorFactorization F.R_J recognitionUpdate F.R_C := by
 110    have hM := F.matter_eq_recognitionUpdate
 111    have := F.factor
 112    rw [hM] at this
 113    exact this
 114  exact isAmplitudeLinear_channel_of_recognitionUpdate hFact
 115
 116/-- **Track 2.C closure (density-only impossibility).** Under a
 117recognition-coupled factorizable joint substrate, no nontrivial density-only
 118channel response is admissible: a CPTP-classical density-matrix readout is
 119forced to the trivial zero response. -/
 120theorem track2C_channel_eq_zero_of_density_only
 121    (F : RecognitionCoupledFactorization)
 122    (hDen : IsDensityOnly F.R_C) (φ : Signal8) :
 123    F.R_C φ = 0 := by
 124  have hFact : PureTensorFactorization F.R_J recognitionUpdate F.R_C := by
 125    have hM := F.matter_eq_recognitionUpdate
 126    have := F.factor
 127    rw [hM] at this
 128    exact this
 129  exact channel_eq_zero_of_density_only_of_recognitionUpdate hFact hDen φ
 130
 131/-- **Track 2.C closure (existence-form no-go).** There is no
 132recognition-coupled factorizable joint substrate whose channel response is
 133both density-only and nontrivial. -/
 134theorem track2C_not_exists_nontrivial_density_only_channel :
 135    ¬ ∃ (F : RecognitionCoupledFactorization),
 136      IsDensityOnly F.R_C ∧ (∃ φ : Signal8, F.R_C φ ≠ 0) := by
 137  rintro ⟨F, hDen, φ, hCφ⟩
 138  exact hCφ (track2C_channel_eq_zero_of_density_only F hDen φ)
 139
 140/-- **Headline Track 2.C theorem (binary-tensor model).** Under a
 141recognition-coupled factorizable joint substrate, the channel-side response
 142must be amplitude-linear, and any density-only (CPTP-classical) candidate
 143collapses to the trivial zero response.
 144
 145This is paper IV's T2 *forced from substrate*, under the named binary-tensor
 146factor-product joint-substrate axiom: STRUCTURAL THEOREM. The unconditional
 147lift to arbitrary joint operators (without factorization) remains future
 148work. -/
 149theorem track2C_headline (F : RecognitionCoupledFactorization) :
 150    IsAmplitudeLinear F.R_C ∧
 151      (IsDensityOnly F.R_C → ∀ φ : Signal8, F.R_C φ = 0) :=
 152  ⟨track2C_channel_isAmplitudeLinear F,
 153   fun hDen φ => track2C_channel_eq_zero_of_density_only F hDen φ⟩
 154
 155/-- **Master cert structure.** Aggregates the Sessions 85–87 closures
 156into a single inhabited Prop bundle: every clause is theorem-grade. -/
 157structure Track2CCert where
 158  /-- Session 85: single-factor substrate dichotomy. -/
 159  single_factor_dichotomy :
 160    ∀ (R : Signal8 → Signal8),
 161      IsAmplitudeLinear R → IsDensityOnly R → ∀ ψ : Signal8, R ψ = 0
 162  /-- Channel amplitude-linearity under recognition coupling (Sessions 86–87). -/
 163  channel_amplitude_linear :
 164    ∀ (F : RecognitionCoupledFactorization), IsAmplitudeLinear F.R_C
 165  /-- Density-only impossibility under recognition coupling (Sessions 85–87). -/
 166  density_only_impossible :
 167    ∀ (F : RecognitionCoupledFactorization),
 168      IsDensityOnly F.R_C → ∀ φ : Signal8, F.R_C φ = 0
 169  /-- Existence-form no-go (Sessions 85–87). -/
 170  no_nontrivial_density_only :
 171    ¬ ∃ (F : RecognitionCoupledFactorization),
 172      IsDensityOnly F.R_C ∧ (∃ φ : Signal8, F.R_C φ ≠ 0)
 173  /-- Hypothesis space is nonempty: canonical recognition coupling exists. -/
 174  canonical_witness_exists : Nonempty RecognitionCoupledFactorization
 175
 176/-- **Master cert inhabitant.** Wraps the Sessions 85–87 anchors. -/
 177noncomputable def track2CCert : Track2CCert where
 178  single_factor_dichotomy _ hLin hDen ψ :=
 179    eq_zero_of_isAmplitudeLinear_isDensityOnly hLin hDen ψ
 180  channel_amplitude_linear := track2C_channel_isAmplitudeLinear
 181  density_only_impossible := track2C_channel_eq_zero_of_density_only
 182  no_nontrivial_density_only :=
 183    track2C_not_exists_nontrivial_density_only_channel
 184  canonical_witness_exists := ⟨canonicalRecognitionCoupled⟩
 185
 186theorem track2CCert_inhabited : Nonempty Track2CCert := ⟨track2CCert⟩
 187
 188end AmplitudeLinearForced
 189end QuantumChannel
 190end Gravity
 191end IndisputableMonolith
 192