SigmaLayer/LiftedMeasurement: the concrete de-isolation model from the LF5 pointer machinery #
Category: 7-SigmaLayer (the projective-sector layer (Paper C)).
The concrete DeisolationModel for Tranche 2b, on the dilated projective sector CP^{M}
(M + 1 = N * N). The physical interaction is the LF5 von Neumann de-isolation flow
measurementFlow (a genuine measure-preserving unitary map), and the contextual readout is the LF5
per-microstate pointer outcome vnPointerOutcome. The outcome regions are the pointer fibres, so the
readout is a genuine function (hence the outcome is unique), the fibres are pairwise disjoint, and the
readout equals some i exactly on the i-th region. Almost-everywhere totality (the outcome is defined
off an FS-null set, transferred through the measure-preserving interaction) is
vnDeisolationModel_ae_total.
Crucially, the model reproduces the Born STATISTICS, not merely a defined outcome: for the dilated
system state ψ' and i.i.d. FS-typical trials, the frequency of trials whose de-isolation readout is
pointer i converges almost surely to ‖⟨eᵢ, ψ⟩‖² (vnDeisolationModel_born_frequency). This is the
LF5 outcome-frequency capstone measurement_flow_outcome_frequency transferred through the
measure-preserving interaction, so the frequency is about the model's OWN outcome (readout after the
interaction), not the raw microstate. lifted_projectiveSector_measurement_born_capstone bundles the full
measurement: measure preservation, unique outcome a.e., record establishment, and Born frequencies.
This is a theorem-backed construction, not an assumed instance: every field is discharged by an LF5
lemma. The isolated dynamics is trivial here (trivialDynamics); the physical content is the
de-isolation interaction.
The record signature of the von Neumann pointer measurement: a single fixed context, outcome type
Fin N (the pointer index).
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The record semantics: the event of pointer outcome i is the pointer fibre
vnPointerOutcome ⁻¹' {some i} on CP^{M}.
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The concrete de-isolation model. Interaction = the LF5 measurement flow (measure-preserving
unitary); readout = the LF5 pointer outcome; outcome regions = the pointer fibres. Every field is
discharged by an LF5 lemma. Preparation type is Unit (the reference state ψ' is fixed).
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The readout records the established outcome (B5 proved for the model). If the pointer readout
after the interaction is some i, the post-interaction state lies in the record event for i.
Almost-everywhere unique outcome (T6 for the model). For almost every initial ontic state
(Fubini-Study measure), the pointer readout after the de-isolation interaction is defined: the outcome
is established a.e. Uniqueness is automatic (the readout is a function; its fibres are pairwise
disjoint). Transfers bornOutcome_ae_isSome through the measure-preserving interaction.
The lifted projective sector measurement capstone. For the concrete de-isolation model on CP^{M}
(M + 1 = N * N), with the LF5 measurement flow as the physical interaction and the LF5 pointer outcome
as the contextual readout, the following hold with no open hypotheses beyond a unit reference state:
- the interaction is measure-preserving (
DeisolationModel.interaction_preserves); - the outcome regions (pointer fibres) are pairwise disjoint, so the recorded outcome is unique;
- the readout records the established outcome (bridge B5);
- the outcome is established for almost every initial ontic state (target T6).
This is the contextual pointer readout and almost-everywhere unique outcome that the product forward
capstone product_projectiveSector_forward_capstone explicitly did not claim: the measurement content, delivered
from a genuine de-isolation interaction rather than an assumed instance.
The model reproduces the Born statistics (the measurement content). For the dilated system state
ψ' (the von Neumann dilation of a unit system state ψ) and i.i.d. FS-typical trials, the frequency
of trials whose de-isolation readout is pointer i converges almost surely to the Born weight
‖⟨eᵢ, ψ⟩‖². The event (interaction ∘ trial)⁻¹' (readout⁻¹' {some i}) is exactly "the model reads
pointer i on this trial", the readout applied AFTER the model's interaction.
This is the LF5 outcome-frequency capstone measurement_flow_outcome_frequency transferred through the
measure-preserving interaction: the composed trial process measurementFlow ∘ fsTrial still samples the
Fubini-Study law (measure preservation), and its per-trial indicators are still independent (a fixed
deterministic map of independent trials), so the Born weights are unchanged. Hence the frequency is a
genuine statistic of the model's own outcome, not of the raw microstate.
The full lifted projective sector measurement capstone (with Born statistics). For the concrete
de-isolation model on the dilated sector, with the system state ψ' the von Neumann dilation of a unit
state ψ, the following hold with no open hypotheses beyond the dilation data:
- the interaction is measure-preserving;
- the outcome regions (pointer fibres) are pairwise disjoint, so the recorded outcome is unique;
- the readout records the established outcome (bridge B5);
- the outcome is established for almost every initial ontic state (target T6);
- the frequency of pointer-
ireadouts converges almost surely to the Born weight‖⟨eᵢ, ψ⟩‖².
This is the genuine measurement: a defined, unique outcome a.e. AND the Born statistics, delivered from a de-isolation interaction rather than an assumed instance.