IndisputableMonolith.Gravity.QuantumChannel.AmplitudeLinearForcedSectionReadout
Defines nonzero matter-section readout: a channel response recovered by injecting a fixed matter reference into a joint linear operator, projecting one channel coordinate, and rescaling. Proves such readouts inherit amplitude-linearity and force density-only channels to vanish, strictly weaker than global pure-tensor factorization. Gravity Track 2.C substrate work cites it; arguments reduce the section to the single-factor dichotomy.
claimA channel response $R_C$ is a nonzero matter-section readout of a joint linear operator $R_J$ when $R_C(\varphi)=\chi^{-1}\pi_{i_0}(R_J(\psi_0\otimes\varphi))$ for a fixed matter reference $\psi_0$, channel coordinate $i_0$, and scalar $\chi\neq 0$. Under this section hypothesis, amplitude-linearity of $R_J$ implies amplitude-linearity of $R_C$, and any density-only $R_C$ is identically zero. The same conclusions hold for the recognition-specialized section variant. Pure-tensor factorization implies section readout.
background
Gravity Track 2.C closes the substrate side of the quantum-channel analysis of recognition responses. Upstream work (Sessions 85–86 and the substrate module) established the single-factor dichotomy: a response that is both amplitude-linear and density-only must vanish, and the joint-operator lift of that dichotomy.
A joint operator $R_J$ acts on a matter–channel tensor product. Global pure-tensor factorization would constrain $R_J$ on every pure tensor. Matter-section readout is weaker: only the operational slice $\psi_0\otimes\varphi$ is constrained. One injects a fixed matter reference $\psi_0$, applies $R_J$, extracts the channel factor at a fixed coordinate $i_0$, and divides by a nonzero scalar $\chi$. The resulting $R_C$ is the channel response actually measured.
The module also records a recognition-specialized section readout and a certificate packaging the forcing conclusions for downstream locality arguments.
proof idea
The module is theorem-bearing, not a pure definition dump. Joint section readout is introduced as the operational recovery map above. Amplitude-linearity of the channel response is obtained by transporting the joint operator's linearity through injection, projection, and nonzero rescaling on the fixed section.
Density-only vanishing follows by reducing the section readout to the single-factor substrate dichotomy: if the recovered channel were density-only and nonzero, the joint side would violate the already-proved amplitude-linear/density-only implies zero theorem. Nonexistence of a nontrivial density-only channel admitting a section readout is the packaged form of that reduction.
Pure-tensor factorization is shown to imply section readout (specialize the global factorization to $\psi_0$), so the earlier factorization corollaries factor through this weaker hypothesis. Recognition section readout repeats the same pattern under the recognition-specialized interface, and the forcing certificate bundles the conclusions.
why it matters in Recognition Science
This module is Session 111's section-readout layer in Gravity Track 2.C. It retires the need for full pure-tensor factorization when forcing amplitude-linearity of operational channel responses: only a nonzero matter-section readout is required.
Downstream, SubstrateLocalAccess imports it as the step beyond that retirement. That parent module is marked a structural theorem (zero sorry, zero RS-internal axiom) and develops the substrate locality / measurement-access principle. Without section-readout forcing, locality would still depend on a global factorization hypothesis stronger than what measurement actually supplies.
In the broader Recognition stack, the result keeps the gravity-side quantum channel aligned with the amplitude-linear substrate forced by the recognition composition law and the forcing chain, while matching the operational content of a local readout.
scope and limits
- Does not constrain the joint operator on pure tensors outside the fixed section $\psi_0\otimes\varphi$.
- Does not by itself prove substrate locality; that is discharged downstream in SubstrateLocalAccess.
- Does not claim section readout without a nonzero scalar $\chi$ and a well-defined channel coordinate projection.
- Does not address continuum or field-valued channels beyond the formal linear-operator setting used here.
- Does not re-prove the single-factor amplitude-linear/density-only dichotomy; it applies that upstream result.
used by (1)
depends on (1)
declarations in this module (14)
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structure
JointSectionReadout -
theorem
isAmplitudeLinear_channel_of_sectionReadout -
theorem
channel_eq_zero_of_density_only_of_sectionReadout -
theorem
not_exists_nontrivial_density_only_channel_with_sectionReadout -
def
sectionReadout_of_pureTensorFactorization -
theorem
isAmplitudeLinear_channel_of_pureTensorFactorization_via_sectionReadout -
structure
RecognitionSectionReadout -
theorem
isAmplitudeLinear_channel_of_recognitionSectionReadout -
theorem
channel_eq_zero_of_density_only_of_recognitionSectionReadout -
theorem
not_exists_nontrivial_density_only_channel_with_recognitionSectionReadout -
structure
SectionReadoutForcingCert -
def
sectionReadoutForcingCert -
theorem
sectionReadoutForcingCert_inhabited -
theorem
factor_product_retirement_one_statement