Foundation Pair Kernel Gap2a Source Calorimeter Faithful Readout

/-- **The instrument.**  A faithful readout bound to a channel-block energy
attachment: per-block gains that reconstruct the at-cut (zero-block)
reading from the through-carrier (`n`-block) reading.  Fields are
target-blind production dynamics only: positivity and transfer inversion
against the attachment's own committed fields.  No `sourceScale`, `hbar`,
`nativeActionQuantum`, cotangent coordinate, or desired readout equality
occurs in any field; the faithfulness right-hand side is the attachment's
own empty-block value, not a literal unit. -/
structure FaithfulChannelReadout {N : ℕ} [NeZero N]
    (channelEnergy : PostingEventChannelEnergy3 N) where
  gain : RealizedPostingEvent3 N → ℕ → ℝ
  gain_pos : ∀ event n, 0 < gain event n
  gain_undoes_attenuation :
    ∀ event n,
      gain event n * channelEnergy.blockEnergy event n =
        channelEnergy.blockEnergy event 0

/-- **Mechanical candidate-A exclusion.**  A faithful readout of any
nonempty block must strictly amplify: the carrier attenuates, so unit gain
would misreport the posted content.  The proof never evaluates `φ` and
never cites `1 ≠ φ⁵`; it holds for every attenuation constant. -/
theorem gain_gt_one_of_attenuating
    {N : ℕ} [NeZero N]
    {channelEnergy : PostingEventChannelEnergy3 N}
    (readout : FaithfulChannelReadout channelEnergy)
    (event : RealizedPostingEvent3 N) (m : ℕ) :
    1 < readout.gain event (m + 1) := by
  have hlt := blockEnergy_succ_lt_one channelEnergy event m
  have hpos := channelEnergy.blockEnergy_pos event (m + 1)
  rw [gain_eq_inv_blockEnergy readout event (m + 1)]
  exact (one_lt_inv₀ hpos).mpr hlt

/-- **The instrument class is rigid**: two faithful readouts of the same
attachment agree everywhere.  Zero residual instrument freedom, in contrast
with the free-`q` heat-carrier wall. -/
theorem faithful_readout_unique
    {N : ℕ} [NeZero N]
    {channelEnergy : PostingEventChannelEnergy3 N}
    (readout₁ readout₂ : FaithfulChannelReadout channelEnergy)
    (event : RealizedPostingEvent3 N) (n : ℕ) :
    readout₁.gain event n = readout₂.gain event n := by
  rw [gain_eq_inv_blockEnergy readout₁ event n,
    gain_eq_inv_blockEnergy readout₂ event n]

/-- On a coherence-complete physical carrier, the faithful gain is exactly
the reciprocal native action quantum `φ⁵`.  The dimension `5` arrives from
the same completeness theorem arrow A used
(`completeCarrier_dimension_eq_configDim`), and the value from
`gain_forced`; neither is posited here. -/
theorem completeCarrier_gain_eq_nativeActionQuantumInv
    {N : ℕ} [NeZero N]
    {physical : PostingPhysicalChannelCarrier3 N}
    {channelEnergy : PostingEventChannelEnergy3 N}
    (readout : FaithfulChannelReadout channelEnergy)
    (hcomplete : PostingCarrierCoherenceComplete3 physical)
    (event : RealizedPostingEvent3 N) :
    readout.gain event
        (physicalPostingCarrierDimension3 physical event) =
      nativeActionQuantumInv := by
  rw [gain_forced readout event
      (physicalPostingCarrierDimension3 physical event),
    completeCarrier_dimension_eq_configDim hcomplete event,
    GapDerivation.configDim_at_D3,
    nativeActionQuantumInv_eq_constants_phi_pow_five,
    zpow_natCast]

/-- **The selective inhabitation the D2 revival demands:** on a
coherence-complete carrier the readout heat is a posted-record-flux carrier
at exactly `nativeActionQuantumInv`, with the quantum instrument-forced
(`gain_forced`) rather than free. -/
theorem readoutHeat_selects_nativeActionQuantumInv
    {N : ℕ} [NeZero N]
    {physical : PostingPhysicalChannelCarrier3 N}
    {channelEnergy : PostingEventChannelEnergy3 N}
    (readout : FaithfulChannelReadout channelEnergy)
    (hcomplete : PostingCarrierCoherenceComplete3 physical)
    (event : RealizedPostingEvent3 N) :
    HeatIsPostedRecordFlux nativeActionQuantumInv
      (readoutHeat readout physical event) := by
  intro c c'
  show readout.gain event
      (physicalPostingCarrierDimension3 physical event) *
      (stepHeatCell c c' : ℝ) =
    nativeActionQuantumInv * (stepHeatCell c c' : ℝ)
  rw [completeCarrier_gain_eq_nativeActionQuantumInv
    readout hcomplete event]

/-- Candidate-A rejection on a coherence-complete carrier: the dimension
theorem supplies only nonemptiness (`configDim = 5 > 0`). -/
theorem readoutHeat_rejects_unit_quantum
    {N : ℕ} [NeZero N]
    {physical : PostingPhysicalChannelCarrier3 N}
    {channelEnergy : PostingEventChannelEnergy3 N}
    (readout : FaithfulChannelReadout channelEnergy)
    (hcomplete : PostingCarrierCoherenceComplete3 physical)
    (event : RealizedPostingEvent3 N) :
    ¬ HeatIsPostedRecordFlux 1 (readoutHeat readout physical event) := by
  apply readoutHeat_rejects_unit_quantum_of_pos_dimension
  rw [completeCarrier_dimension_eq_configDim hcomplete event,
    GapDerivation.configDim_at_D3]
  norm_num

/-- **Conditional closure.**  Interface premise + faithful instrument +
coherence-complete carrier force the residual atom.  This is a theorem
about what the typed interface buys; it is not an unconditional derivation
of the adopted law. -/
theorem sourceCalorimeter_forces_remainingPhysicalEquality
    {N : ℕ} [NeZero N]
    {physical : PostingPhysicalChannelCarrier3 N}
    {channelEnergy : PostingEventChannelEnergy3 N}
    (readout : FaithfulChannelReadout channelEnergy)
    (hcomplete : PostingCarrierCoherenceComplete3 physical)
    {sourceScale : ℝ} {event : RealizedPostingEvent3 N}
    (hinterface :
      SourceScaleReadsCompleteCarrierReadout
        physical readout sourceScale event) :
    RemainingPhysicalEquality sourceScale := by
  change sourceScale = uniqueCotangentCoordinate
  exact hinterface.trans
    (completeCarrier_gain_eq_uniqueCotangentCoordinate
      readout hcomplete event)