IndisputableMonolith.Gravity.QuantumChannel.AmplitudeLinearForced
On the eight-tick complex signal space, no nontrivial channel response is both amplitude-linear and density-only. The module defines those response classes and proves their only common element is the zero map. Gravity Track 2.C cites this single-factor substrate dichotomy as the Session 85 closure. The argument routes amplitude-linearity through phase equivariance, then forces vanishing under the density-only constraint.
claimOn the eight-tick carrier $S_8:=\{0,\ldots,7\}\to\mathbb{C}$, any response that is amplitude-linear (homogeneous under complex scaling of the signal) and density-only (depends only on pointwise $|\psi|^2$) must be identically zero. Equivalently, no nontrivial map on $S_8$ is simultaneously amplitude-linear and density-only.
background
Recognition Science forces an eight-tick octave (T7) and a cyclic shift on the ledger state. Upstream, Complex Structure Forcing records that this shift cannot be diagonalized over $\mathbb{R}$, so complexification is algebraically forced: the analytic carrier is $\mathrm{Fin},8\to\mathbb{C}$ (Signal8).
This module works on that carrier. Amplitude-linearity means the channel response scales linearly with complex amplitudes. Density-only means the response sees only local intensities $|\psi|^2$, not phase. Phase equivariance is the intermediate property implied by amplitude-linearity: a global phase rotation of the signal rotates the response accordingly.
The setting is Gravity Track 2.C (quantum-channel and mediator constraints): classical or density-only mediators are incompatible with amplitude-linear response on the forced complex substrate.
proof idea
The module introduces predicates for amplitude-linearity, phase equivariance, and density-only response on maps out of Signal8. A short lemma shows amplitude-linearity implies phase equivariance. Combining amplitude-linearity with density-only forces the response to vanish identically. Corollaries restate the dichotomy: a nonzero amplitude-linear map cannot be density-only, and no nontrivial map satisfies both properties at once.
why it matters in Recognition Science
Session 85 of Gravity Track 2.C closes here as the single-factor substrate dichotomy on Signal8. The master certificate module aggregates this closure for the binary-tensor model. The joint-substrate module lifts the same no-go to matter-plus-channel modelled as a binary tensor product. MediatorUniversalityBoundary cites the vector-level result without re-proving it: a vector response on Signal8 cannot be amplitude-linear, density-only, and nonzero. Downstream of T7 and Complex Structure Forcing, the dichotomy blocks classical density-only mediators on the forced complex carrier.
scope and limits
- Does not treat joint matter-plus-channel substrates (lifted elsewhere).
- Does not classify nonlinear or non-vector channel responses.
- Does not re-derive complexification; imports the forcing result.
- Does not assert a unique physical mediator, only a no-go for one pair of properties.
- Does not address carriers beyond the eight-tick Signal8.
used by (3)
depends on (1)
declarations in this module (8)
-
abbrev
Signal8 -
def
IsAmplitudeLinear -
def
IsPhaseEquivariant -
def
IsDensityOnly -
theorem
isPhaseEquivariant_of_isAmplitudeLinear -
theorem
eq_zero_of_isAmplitudeLinear_isDensityOnly -
theorem
not_isDensityOnly_of_isAmplitudeLinear_of_ne_zero -
theorem
not_exists_nontrivial_isAmplitudeLinear_and_isDensityOnly