The axion, originally proposed to resolve the strong CP problem, has emerged asone of the most compelling dark matter candidates. For over four decades, haloscopeexperiments—including ADMX, CAPP, HAYSTAC, and ORGAN—have pursued a detection strategy predicated on scanning microwave cavities over broad frequency ranges, searching for a single, unknown axion signal peak. This approach treats the axion massas a free parameter and relies on empirical discovery rather than deductive prediction. The present work demonstrates that this paradigm can be replaced by a fixed-frequencystaring protocol applicable to any axion haloscope. The transition from blind search totargeted verification is enabled by the Dicke Radiometer Equation, which provides thequantitative bridge between theoretical power levels and experimental detectability The deductive chain originates in the topological invariant ∆ = 4 ln 99, which emergesfrom the ergodic dissipation of the Collatz parity-block map and its topological completionvia the Dalvi Dictact. From this invariant, the Primordial Action Product Law (PAPL) Γ (∆) = M2Ple−5∆ follows as the necessary dimensional completion, seeding all dimensionalconstants. Within the G¨odel–Kripke–Mahapatra (GKM) Mathesis, the hyperbolic chargekqcd = 1 − e−1 ≈ 0. 632 acts as a topological regulator, locking the strong-interactionsector of the vacuum and fixing the axion mass to ma = 6. 32 µeV. This corresponds toa resonant frequency fa = ma/h = 1. 528 GHz, invariant across the three cosmologicalphases Λ ∈ 9, 6, 3. The KUTE Axion Product Law (KAPL), ma × fa =√Λ × Γ (∆), deduces the axiondecay constant as a phase-dependent quantity: fa = 9. 54 × 1012 GeV for Λ = 9, 7. 79 ×1012 GeV for Λ = 6, and 5. 51 × 1012 GeV for the current vacuum phase Λ = 3. Becausethe axion-photon coupling scales as gaγγ ∝ 1/fa ∝ 1/√Λ, the haloscope signal powerscales as P ∝ 1/Λ. Consequently, a haloscope tuned to the fixed frequency 1. 528 GHzwill observe not a single power level but a discrete set of amplitudes corresponding to thethree phases, with power ratiosP9: P6: P3 =19: 16: 13= 1. 00: 1. 49: 2. 95. This multi-amplitude signature serves as a smoking-gun discriminator. Unlike thermalnoise or electronic interference, which produce random fluctuations, the predicted signalmanifests as a step-like power hierarchy at the fixed frequency. The Dicke RadiometerEquation, Pmin = kBTsysr∆ft, provides the quantitative framework for resolving this structure. The bandwidth ∆fis not a search window but a high-resolution spectral bin centered at 1. 528 GHz; theintegration time t must be sufficient to distinguish the Λ = 3 (strongest) from theΛ = 9 (weakest) phases. Numerical simulations incorporating MedRes (200Hz) andHiRes (20mHz) channels, Earth’s Doppler modulation, and the quantum limit of SQUIDamplifiers demonstrate that all three phases lie above the noise floor for existing haloscopeplatforms. The Savitzky-Golay filter is identified as the computational embodiment of the DalviDictact, performing topological completion on experimental data. The predicted signal-to-noiseratios exceed current sensitivity limits, ensuring that the Λ = 9 state remains visiblewithin the quantum noise floor. This framework establishes a binary test for any axionhaloscope experiment: 1. If the experiment scans 1. 528 GHz and observes a single Lorentzian peak, thestandard axion model holds. 2. If the experiment applies the Dicke equation to resolve the sub-structure and observes three distinct power plateaus in the ratio 1/9: 1/6: 1/3, the KUTE AxionProduct Law (KAPL) is verified, and the GKM logical chain is validated. By adopting this fixed-frequency staring protocol, haloscope experiments transition fromsearching for an unknown particle to verifying the topological mandate of the vacuum. Discovery at 1. 528 GHz would not merely confirm dark matter—it would validate theDalvi-Ramanujan deductive architecture of the universe, closing the singularity of empirical uncertainty.
Dillip Kumar Mahapatra (Mon,) studied this question.