universalSubstrateAccessOperator_inducedChannel
plain-language theorem explainer
Under canonical recognition access (unit vacuum probe, coordinate 0, unit calibration), the universal joint operator id⊗L induces exactly the witness map L as its channel response. Anyone proving substrate-access universality for amplitude-linear channels cites this identity. The proof is a pointwise definitional calculation: insert, tensor-map, extract, then cancel the unit scalar.
Claim. For every $\mathbb{C}$-linear map $L$ on the eight-tick signal space, the channel recovered from the universal joint operator $\mathrm{id}\otimes L$ by preparing the unit probe, applying the operator once, and reading coordinate $0$ with unit calibration equals $L$ itself.
background
This module closes Gravity Track 2.C unconditionally. Prior sessions reduced amplitude-linear channel forcing through factor-product and section-readout hypotheses to a substrate locality principle: a channel response arises from substrate access when it is recovered by preparing a matter probe on the joint substrate, applying one $\mathbb{C}$-linear joint operator, and reading a calibrated channel coordinate.
The joint substrate is the tensor product of a recognition register with the eight-tick signal space. The universal substrate-access operator attached to a witness $L$ is $\mathrm{id}\otimes L$. Canonical access fixes the probe to the unit vacuum, the readout index to $0$, and the calibration to $1$. The induced channel is the composite insert-apply-extract-rescale that turns a joint operator into a map on signals.
The doc-comment states the role cleanly: this is the calculation that makes the substrate-access form universal.
proof idea
Pointwise on an arbitrary signal $\varphi$. Unfold canonical access and the universal operator. Inserting the unit probe into the first factor and applying the tensor product map $\mathrm{id}\otimes L$ leaves $L\varphi$ in the second factor. Extraction yields the scalar (unit vacuum at index $0$) times $L\varphi$. That scalar is $1\in\mathbb{C}$, so the unit scalar law finishes. Pure definitional simplification of insert, tensor-map, and extract; no deep external lemmas.
why it matters
This identity is the witness calculation behind substrate-access universality. Downstream, every amplitude-linear channel response (with witness $L$) is exhibited by $\mathrm{id}\otimes L$ under canonical access, so the substrate-access form is automatic for amplitude-linear channels. The module certificate and the one-statement theorem then record the full equivalence: amplitude-linearity if and only if existence of a joint operator with substrate-access form.
That retires the substrate locality / measurement-access principle as a separate Track 2.C hypothesis; it becomes a derived consequence of the substrate-semantic minimum any operational channel observable must satisfy. The eight-tick carrier is the T7 octave register from the forcing chain. No open scaffold remains in this module (status: theorem, zero sorry).
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