IndisputableMonolith.Gravity.QuantumChannel.AmplitudeLinearForcedCert
Certificate module packaging Track 2.C of the quantum-gravity plan: the gravitational channel is forced to be amplitude-linear, and no nontrivial density-only channel exists on a factorizable joint matter-channel substrate. Gravity and QG auditors cite it when discharging the amplitude-linear forcing hypothesis into the master theorem. The argument assembles single-factor, joint, and substrate dichotomies into an inhabited cert record.
claimOn a factorizable joint substrate $J = M \otimes C$ (matter and channel ledgers with independent responses $R_J = R_M \otimes R_C$), any channel that is both amplitude-linear and density-only vanishes. Equivalently, there is no nontrivial density-only gravitational channel; the forced channel is amplitude-linear. The module exports an inhabited certificate recording this dichotomy and the canonical recognition factorizations.
background
Track 2.C upgrades paper IV's T2 from modeling assumption to theorem: the amplitude-linear gravitational channel must be forced from substrate linearity. Session work on a single factor (Signal8) already shows that amplitude-linearity and density-only response cannot hold together nontrivially. The joint lift models matter plus channel as a binary tensor product of ledgers.
A factorizable joint substrate means the joint operator acts on pure tensors by factor-wise responses: no cross-sector coherence, so pure tensors never become entangled between matter and channel. A general $\mathbb{C}$-linear endomorphism of $\mathrm{Signal8} \otimes \mathrm{Signal8}$ need not factorize (the swap is the standard counterexample). Recognition-coupled factorizations are the canonical RS instances of that structural hypothesis.
Upstream modules close the single-factor dichotomy, its joint-substrate lift, and substrate-side closure; this cert module only packages those results.
proof idea
Not a free-standing proof development. It imports the single-factor dichotomy, the joint-substrate lift, and substrate-side closure, then defines structural hypotheses (factorizable joint substrate, recognition-coupled factorization) and the canonical RS instances.
Lemmas restate that the Track 2.C channel is amplitude-linear, vanishes if density-only, and that no nontrivial density-only channel exists. A headline theorem and an inhabited Track2CCert record bundle those facts for downstream discharge. Argument shape: assemble prior dichotomies into a cert, not re-prove the forcing from scratch.
why it matters in Recognition Science
Feeds three parents: the Gravity Master Theorem (Track 7.A conditional master statement), the Amplitude-Linear Forcing Structural Witness (Track 2.C/2.D structural input AmplitudeLinearForcedUnconditional), and No Classical Mediator (Track 2.D partial closure under T0–T8).
In the master plan this is Track 2.C step packaging: joint substrate as matter ledger tensor channel ledger with factorized response, so gravity cannot hide in a classical density-only mediator once substrate linearity is granted. Ties the forcing chain's discrete ledger structure (eight-tick octave, D = 3) to the claim that the gravitational channel is amplitude-linear by necessity. Closes the cert interface those parents import rather than leaving bare lemmas.
scope and limits
- Does not re-prove the single-factor or joint dichotomies; only certifies them.
- Does not claim every endomorphism of the joint space factorizes; only the named structural class.
- Does not derive Einstein equations or a full quantum gravity dynamics.
- Does not remove conditionality from the master theorem beyond the Track 2.C cert interface.
- Does not address non-factorizable or cross-sector-coherent joint operators.
used by (3)
depends on (3)
declarations in this module (11)
-
structure
FactorizableJointSubstrate -
def
canonicalRecognitionFactorization -
structure
RecognitionCoupledFactorization -
def
canonicalRecognitionCoupled -
theorem
track2C_channel_isAmplitudeLinear -
theorem
track2C_channel_eq_zero_of_density_only -
theorem
track2C_not_exists_nontrivial_density_only_channel -
theorem
track2C_headline -
structure
Track2CCert -
def
track2CCert -
theorem
track2CCert_inhabited