Date: 31 January 2026 Author: Nanjie Ma Purpose: Formal clarification to readers and referees on two major points
1. Clarification on the Hubble Tension Analysis Mechanism
The paper’s description of the Hubble tension resolution mechanism requires clarification; the following is the official statement.
The present implementation obtains H₀ = 68.47 ± 4.82 km/s/Mpc through quantum geometric uncertainty, not through dynamical evolution. Specifically, the QNM framework derives a statistical distribution of H₀ values from N=21 matrix eigenvalue fluctuations, whose 1σ range naturally encompasses both early-universe (Planck: 67.4 km/s/Mpc) and late-universe (SH0ES: 73.04 km/s/Mpc) measurements. This represents a quantum geometric interpretation of the Hubble tension: the observed measurement discrepancy may reflect fundamental quantum uncertainty in spacetime geometry rather than systematic error alone.
The framework retains its core scientific contributions: first-principles parameter derivation from mathematical constants, 88.9% statistical consistency with observations, and innovative quantum–cosmological unification. A complete dynamical evolution system H(z) is under active development and will be presented in the next version with full time-dependent cosmic evolution capability.
2. Statement on Early Version Loss and Wording Corrections
Owing to the complexity of the framework and version loss that occurred in an earlier phase, some documents had retained wording such as “back-calculation” or “inference based on fitting.” These were not systematically corrected until a full review and correction was completed on 28 January 2026. After correction, the main text, supplementary materials, code, and data are now aligned; numerical results remain consistent with those before the correction, with only wording and traceability brought into line.
This version formally confirms and implements the following points:
- Independent derivation of N=21: Established as the unique solution under six-dimensional geometry via the “Geometric Resonance Theorem,” not by empirical tuning.
- Proof of the intrinsic constant: c_raw ≈ 2.75 is proven to be an emergent statistical constant of the matrix mechanism, not an empirical fit.
- Zero-parameter axiomatization: (π+e) and related factors are confirmed as “zero-parameter physical assumptions” based on mathematical structure, with no dependence on observational fitting.
These corrections and clarifications do not alter the scientific conclusions or numerical results already reported in the paper; they only improve the accuracy of wording and the transparency of traceability.
Contact: phoenix-mx@hotmail.com | ORCID: 0009-0002-4415-1209
Verification and further detail: Full Hubble tension clarification: QNM_HUBBLE_TENSION_CLARIFICATION_STATEMENT.md; version and parameter alignment: VERSION_UPDATE_2026_CN.md (Chinese) and related version notes.
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