Energy and a clock
still need a support scale.
A conditional theorem identifies the remaining normalization ambiguity and the readout certificate required for a native-action measurement.
Measured total energy contains the support factor.
On the declared active sector, a=‖x‖² and E=(A0/τ0)a. The signed full-state period is Tstate=πτ0. Thus E Tstate/π identifies A0a.
Rescaling x by c, its apparatus map M by 1/c, and A0 by 1/c² preserves the physical trajectory, energy and clock. Without an independent absolute support scale, the model class leaves A0 undetermined. A mathematical unit-ray convention does not supply that physical calibration.
The readout can repeat twice per state cycle.
The half-cycle state is −x. A nonconstant quadratic readout has period Tstate/2; a nonzero signed linear orbit projection has period Tstate. Conserved energy is constant and supplies no recurrence clock.
Certify ρ=Tread/Tstate from the actual readout and its phase/sign reference. Then A0=E Tread/(πρa). No physical spinor or quantum phase identity is inferred.
A finite interval needs a positive support floor.
Given jointly certified positive intervals, A0 lies between E−T−/(πρ+a+) and E+T+/(πρ−a−). Correlations are permitted; coverage needs measurement evidence.
An independently calibrated absolute reconstruction with ‖x−x̂‖≤δ bounds a between max(0,‖x̂‖−δ)² and (‖x̂‖+δ)². Unobserved modes and map errors belong in δ. Without a positive lower support bound, this argument provides no finite upper action bound.
The next return must establish the map.
Require the carrier and active-state domain; calibrated subsystem energy and baseline; intertwiner/time map; readout period ratio; absolute support evidence; joint uncertainty bounds; and a frozen comparison rule with non-constructor review. The contract is proposed, not preregistered. No experiment or dataset is admitted.
Exact rational checks reconstruct the 15-dimensional tetrahedral boundary and rank-14 active sector, verify five synthetic map rescalings, and test clock and interval controls. QPB1 remains quarantined.
Exact verification receipt →Read the preceding anchor-rank theorem →