The Compressed Microwave Background: The Shift Parameter Forgives, the Acoustic Scale Does Not

Two numbers compress nearly everything the microwave background says about the expansion history: the shift parameter R = 1.7502 ± 0.0046 and the acoustic scale l_A = 301.63 ± 0.15. This paper computes both for four histories in the Hubble-tension dispute, evaluating the sound horizon at recombination by direct integration and the distance to last scattering with radiation included. The Planck history reproduces both (pulls of −0.14 and +1.87 sigma, the latter within the tolerance of the approximations used here). The shift parameter turns out to forgive almost everything: the SH0ES rate on the Planck shape matches R at +0.13 sigma, because R is constructed to be nearly degenerate along the tension's direction. The acoustic scale forgives nothing. That same history misses l_A by 75 sigma, because its self-consistent sound horizon shrinks by only 4.1 percent while its distance to last scattering shrinks by the full 7.7 percent. The small-ruler combination, which shrinks the ruler by fiat, misses l_A by 137 sigma in the opposite direction: its distance to last scattering falls only 1.6 percent, not the required 7.8. Restoring the acoustic scale at the ladder's expansion rate with the shrunk ruler forces the matter fraction to 0.3165 — a physical matter density of 0.1688, eighteen percent above the Planck value the construction set out to preserve — and at that forced point the shift parameter is recovered automatically (+0.27 sigma). The evolving-dark-energy variant, finally, misses l_A by 5.5 sigma at fixed Planck normalisation. The compressed microwave background thereby identifies the precise joint move any resolution must make: ruler, distance, and matter density cannot be chosen separately. This study is computed from published values; it is not a measurement.