Boundary Time Readout, Condensation Progress, and the Hubble Tension in AQM: A Conditional Closure from the Center-Complement Wilson Line to T = φ

This paper establishes the boundary time-readout theorem in the Algebraic Quantum Morphogenesis (AQM)framework and gives a conditionally closed explanation of the Hubble tension. Previous papers have separately established the finite-dimensional spectral neck algebra,measurement as center splitting,the K-flow three-step condensation path,the Standard-Model algebra,boundary-center response,the dark-energy suppression chain,the boundary spectral-closure functor,and the emergence of a three-dimensional spatial spectral dimension.The present paper addresses the next layer:after the threedimensional low-energy spatial spectral shell has been closed,how does the time field arise from boundary readout,and why does the present local Hubble reading appear rescaled relative to the CMB-inferred value?

The paper first defines the center projection of the third-step completed condensation sector,

P₃ ∈ Z (Aeff),

and defines the third-step condensation progress by

ϕ = ρ(P3).

The bulk Wilson line W is defined as the readout of the two-boundary connecting channel that has not yet been absorbed by the third-step condensation center. Under the minimal center-readout normalization,

⟨W ⟩ = ρ(1 −P₃ ) = 1 −ϕ

For multi-branch nonuniform weights,this formula naturally generalizes to

⟨W ⟩ = 1 −ϕ_eff.

The boundary entanglement entropy is then defined as the information cost of the boundary transmission amplitude,

S_EE := −S_Pl ln⟨W ⟩,

and the AQM time field is defined by

T := 1 −e ^{−SEE /SPl} .

The main theorem is therefore

T = ϕ.

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The revised version adds an explicit dynamical closure for the present condensation progress.Define the third-step effective condensation integral

_{I3 =} ∫_ttst0a_{rt Γ}e3^ff _{(t)dt, ϕ0 = 1 −e} −I3 

In boundary-center record readout,the full stable record length after last scattering is

^Θ_full := ln(1 + zdec).

The low-redshift Hubble readout couples only to one manifest center component among the three Z₃ centers,hence

IH3 = Θ_H = 13Θ_full , TH0 = ϕ_H = 1 −e ^−ΘH = 1 − (1 + zdec)^−1/3 . (0.10)

If the local boundary clock τ and the bulk/CMB modular time t satisfy

dτ = T dt

then

H_local = ^HCT^MB .

With z_dec = 1089 .92 and

H₀ ,_CMB = 67 .4 

one obtains

TH0 ≃ 0.90286,     HA0,QMloca_l ≃ 74.65 

This paper does not claim to derive a complete Lorentzian spacetime,the Einstein equation,or a full reconstruction of all cosmological parameters from finite algebra alone.Its scope is the conditional closure of the AQM boundary time-readout mechanism and its conditional explanation of the Hubble tension.