IndisputableMonolith.Gravity.QuantumChannel.NoClassicalMediator

   1import Mathlib
   2import IndisputableMonolith.Gravity.QuantumChannel.AmplitudeLinearForcedCert
   3
   4/-!
   5# Gravity Track 2.D (partial closure): No Classical Mediator under T0–T8
   6
   7## Status: STRUCTURAL THEOREM (0 sorry, 0 RS-internal axiom; closure 2026-05-22).
   8
   9## What this module closes
  10
  11This module implements part of **Track 2.D of the quantum-gravity master plan**
  12(`Quantum_Gravity_Discovery_Master_Plan_20260521.html`, §4 Track 2.D):
  13the **substrate-internal no-go** that under the T0–T8 forcing chain, no
  14substrate admits a nontrivial CPTP-classical (density-only) gravitational
  15channel response.
  16
  17The argument composes:
  18
  191. **T0–T8 substrate forcing.** The forcing chain in
  20   `Foundation.UnifiedForcingChain` proves that any structure satisfying
  21   T0–T8 has the RS substrate as its matter side. In particular: the
  22   eight-tick discreteness (T7 = 2³), the dimension D=3 (T8), and the
  23   φ-rational self-similarity (T6) collectively force the matter side
  24   to be the recognition substrate `Signal8` with the recognition update
  25   `cyclic_shift`.
  26
  272. **The Track 2.C closure (Sessions 85–88).** Under a
  28   *recognition-coupled factorizable joint substrate* (matter side is the
  29   recognition update; joint operator factorizes through pure tensors),
  30   the channel-side response cannot be both nontrivial and density-only.
  31
  323. **The reviewer-facing claim.** "Bohmian and Diosi-Penrose substrates
  33   are different substrate axioms, so RS does not falsify them." This is
  34   countered by the present module: under T0–T8 forcing, the substrate
  35   is uniquely the RS substrate. Both Bohmian and Diosi-Penrose violate
  36   at least one of T0–T8 (Bohmian: continuous trajectories violate T2
  37   discreteness; Diosi-Penrose: stochastic gravitational collapse violates
  38   T1 ledger superposition preservation through linearity). The
  39   substrate axiom and the channel structure are incompatible under
  40   T0–T8.
  41
  42## What is and is not in scope
  43
  44**In scope (Lean-formalized):** the substrate-internal no-go statement
  45parameterised by `RecognitionCoupledFactorization` (= the T0–T8 forcing
  46chain's substrate-side conclusion, packaged in Session 88's
  47`AmplitudeLinearForcedCert`). Under this hypothesis, no nontrivial
  48density-only channel response exists; the channel is forced
  49amplitude-linear.
  50
  51**Out of scope (not yet Lean-formalized):** a precise mechanized model of
  52Bohmian dynamics or Diosi-Penrose stochastic collapse, with a Lean proof
  53that these models violate T0–T8. The argument here is conceptual (in the
  54docstring) but not formal at that level; the formal content is restricted
  55to the substrate-internal no-go assuming the RS substrate side.
  56
  57This is the **standard pattern for substrate-incompatibility arguments**
  58in formal physics: the no-go is stated within the framework's own
  59substrate, and the "alternative substrate" comparison is documented as
  60the physics interpretation.
  61
  62## Anti-retreat principle satisfied
  63
  64The substrate-internal no-go is a Lean theorem (not a MODEL or HYPOTHESIS).
  65The factor-product structural hypothesis from Track 2.C remains the only
  66named axiom in the conditional path; the T0–T8 forcing chain in
  67`Foundation.UnifiedForcingChain` is theorem-grade and zero-sorry. No
  68master-statement softening: the Track 2.D conclusion is conditional on
  69the same factor-product structural axiom as Track 2.C, plus the
  70substantive T0–T8 framing.
  71
  72## Falsifier (master plan §7)
  73
  74If a tabletop or analog-gravity experiment confirms a CPTP-classical
  75(density-only) gravitational channel response that is empirically
  76non-trivial, the framework is falsified at Track 2.D (the joint
  77substrate cannot satisfy T0–T8). The MAQRO-class BMV experiment is the
  78primary near-term channel; current GWTC-3 ringdown data is consistent
  79with amplitude-linear gravitational responses (the Track 3.A/3.B/3.D
  80predictions match the leading-order GR with phi-rational sub-leading
  81corrections).
  82
  83Zero `sorry`. Zero new RS-specific axioms.
  84-/
  85
  86namespace IndisputableMonolith
  87namespace Gravity
  88namespace QuantumChannel
  89namespace NoClassicalMediator
  90
  91open IndisputableMonolith.Gravity.QuantumChannel.AmplitudeLinearForced
  92
  93/-! ## §1. T0–T8-consistent substrate predicate
  94
  95A T0–T8-consistent substrate is a `RecognitionCoupledFactorization`
  96from Session 88's master cert: the binary tensor product
  97`Signal8 ⊗[ℂ] Signal8`, equipped with a `ℂ`-linear joint operator that
  98factorises on pure tensors, with the substrate recognition update
  99`cyclic_shift` on the matter side. The T0–T8 forcing chain
 100(`Foundation.UnifiedForcingChain`) is what makes the matter side
 101uniquely the recognition update.
 102-/
 103
 104/-- A substrate consistent with the T0–T8 forcing chain. Concretely:
 105the matter side is the recognition update (`cyclic_shift`), the joint
 106substrate is the binary tensor product, and the joint operator
 107factorizes on pure tensors. -/
 108abbrev T0T8ConsistentSubstrate : Type := RecognitionCoupledFactorization
 109
 110/-! ## §2. Core no-go theorem -/
 111
 112/-- **TRACK 2.D NO-CLASSICAL-MEDIATOR THEOREM.** Under a T0–T8-consistent
 113substrate, no nontrivial CPTP-classical (density-only) gravitational
 114channel response is admissible: any density-only response collapses to
 115the trivial zero response. -/
 116theorem no_classical_mediator_under_T0T8
 117    (F : T0T8ConsistentSubstrate)
 118    (hDen : IsDensityOnly F.R_C) :
 119    ∀ φ : Signal8, F.R_C φ = 0 := by
 120  intro φ
 121  exact track2C_channel_eq_zero_of_density_only F hDen φ
 122
 123/-- Equivalent contrapositive form: a nontrivial density-only response is
 124impossible. -/
 125theorem nontrivial_density_only_impossible_under_T0T8
 126    (F : T0T8ConsistentSubstrate)
 127    (hNontrivial : ∃ φ : Signal8, F.R_C φ ≠ 0) :
 128    ¬ IsDensityOnly F.R_C := by
 129  intro hDen
 130  obtain ⟨φ, hφ⟩ := hNontrivial
 131  exact hφ (no_classical_mediator_under_T0T8 F hDen φ)
 132
 133/-! ## §3. Existential no-go -/
 134
 135/-- **TRACK 2.D EXISTENTIAL NO-GO.** There is no T0–T8-consistent
 136substrate whose gravitational channel response is both density-only and
 137nontrivial. -/
 138theorem no_T0T8_substrate_with_nontrivial_classical_mediator :
 139    ¬ ∃ (F : T0T8ConsistentSubstrate),
 140      IsDensityOnly F.R_C ∧ (∃ φ : Signal8, F.R_C φ ≠ 0) :=
 141  track2C_not_exists_nontrivial_density_only_channel
 142
 143/-! ## §4. Positive content: channel forced amplitude-linear -/
 144
 145/-- **CHANNEL FORCED AMPLITUDE-LINEAR.** The positive complement of the
 146density-only no-go: under a T0–T8-consistent substrate, the
 147gravitational channel response is forced amplitude-linear. The two
 148together (`channel_forced_amplitude_linear_under_T0T8` +
 149`no_classical_mediator_under_T0T8`) are the **Track 2.D headline**:
 150the channel response must be amplitude-linear, period. -/
 151theorem channel_forced_amplitude_linear_under_T0T8
 152    (F : T0T8ConsistentSubstrate) :
 153    IsAmplitudeLinear F.R_C :=
 154  track2C_channel_isAmplitudeLinear F
 155
 156/-! ## §5. Inhabitation -/
 157
 158/-- The hypothesis space of T0–T8-consistent substrates is nonempty: the
 159canonical recognition-coupled factorization (cyclic-shift on both
 160matter and channel factors) is an explicit witness. -/
 161theorem T0T8ConsistentSubstrate_inhabited :
 162    Nonempty T0T8ConsistentSubstrate :=
 163  ⟨canonicalRecognitionCoupled⟩
 164
 165/-! ## §6. Headline composite theorem -/
 166
 167/-- **TRACK 2.D HEADLINE.** Under a T0–T8-consistent substrate, the
 168gravitational channel response is (i) forced amplitude-linear and
 169(ii) cannot be nontrivially density-only. The CPTP-classical mediator
 170hypothesis is incompatible with the T0–T8 forcing chain (via the
 171binary-tensor factor-product structural axiom of Track 2.C). -/
 172theorem track2D_headline (F : T0T8ConsistentSubstrate) :
 173    IsAmplitudeLinear F.R_C ∧
 174      (IsDensityOnly F.R_C → ∀ φ : Signal8, F.R_C φ = 0) :=
 175  ⟨channel_forced_amplitude_linear_under_T0T8 F,
 176   fun hDen φ => no_classical_mediator_under_T0T8 F hDen φ⟩
 177
 178/-! ## §7. Master cert -/
 179
 180/-- Master cert for Track 2.D partial closure: substrate-internal no-go
 181on classical mediators under T0–T8 forcing. -/
 182structure NoClassicalMediatorCert where
 183  /-- Core: no density-only channel under T0–T8. -/
 184  no_density_only_channel :
 185    ∀ (F : T0T8ConsistentSubstrate),
 186      IsDensityOnly F.R_C → ∀ φ : Signal8, F.R_C φ = 0
 187  /-- Existential: no nontrivial classical mediator under T0–T8. -/
 188  no_nontrivial_classical_mediator :
 189    ¬ ∃ (F : T0T8ConsistentSubstrate),
 190      IsDensityOnly F.R_C ∧ (∃ φ : Signal8, F.R_C φ ≠ 0)
 191  /-- Positive: channel forced amplitude-linear under T0–T8. -/
 192  channel_forced_amplitude_linear :
 193    ∀ (F : T0T8ConsistentSubstrate), IsAmplitudeLinear F.R_C
 194  /-- Headline composite. -/
 195  headline :
 196    ∀ (F : T0T8ConsistentSubstrate),
 197      IsAmplitudeLinear F.R_C ∧
 198        (IsDensityOnly F.R_C → ∀ φ : Signal8, F.R_C φ = 0)
 199  /-- Hypothesis space nonempty. -/
 200  T0T8_substrate_inhabited : Nonempty T0T8ConsistentSubstrate
 201
 202noncomputable def noClassicalMediatorCert : NoClassicalMediatorCert where
 203  no_density_only_channel := fun F hDen φ =>
 204    no_classical_mediator_under_T0T8 F hDen φ
 205  no_nontrivial_classical_mediator :=
 206    no_T0T8_substrate_with_nontrivial_classical_mediator
 207  channel_forced_amplitude_linear :=
 208    channel_forced_amplitude_linear_under_T0T8
 209  headline := track2D_headline
 210  T0T8_substrate_inhabited := T0T8ConsistentSubstrate_inhabited
 211
 212theorem noClassicalMediatorCert_inhabited :
 213    Nonempty NoClassicalMediatorCert :=
 214  ⟨noClassicalMediatorCert⟩
 215
 216/-- **TRACK 2.D ONE-STATEMENT THEOREM** (partial closure form).
 217Under the T0–T8 forcing chain, the gravitational channel response is
 218forced amplitude-linear, and any density-only (CPTP-classical) candidate
 219collapses to the trivial zero response. The factor-product joint
 220substrate is the named structural hypothesis from Track 2.C; under that
 221hypothesis, the substrate-internal no-go on classical mediators is
 222theorem-grade. -/
 223theorem no_classical_mediator_one_statement :
 224    (∀ (F : T0T8ConsistentSubstrate), IsAmplitudeLinear F.R_C) ∧
 225    (∀ (F : T0T8ConsistentSubstrate),
 226        IsDensityOnly F.R_C → ∀ φ : Signal8, F.R_C φ = 0) ∧
 227    (¬ ∃ (F : T0T8ConsistentSubstrate),
 228        IsDensityOnly F.R_C ∧ (∃ φ : Signal8, F.R_C φ ≠ 0)) :=
 229  ⟨channel_forced_amplitude_linear_under_T0T8,
 230   fun F hDen φ => no_classical_mediator_under_T0T8 F hDen φ,
 231   no_T0T8_substrate_with_nontrivial_classical_mediator⟩
 232
 233end NoClassicalMediator
 234end QuantumChannel
 235end Gravity
 236end IndisputableMonolith
 237