The Interior Observer Cosmological Framework: Paper 34 - The Hubble Tension Resolved: Six Methods, Three Rungs, Zero Fitted Parameters

The Hubble tension — the 5σ discrepancy between the Planck CMB measurement (67. 4 ± 0. 5 km/s/Mpc) and the SH0ES Cepheid distance ladder (73. 0 ± 1. 0 km/s/Mpc) — has resisted a decade of proposed resolutions. This paper resolves it with zero fitted parameters and no new physics beyond the Interior Observer (IO) framework. The resolution follows from a single recognition: different H₀ measurement methods belong to different observable classes on the IO α-ladder, and each class sees a different effective expansion rate dressed by the boundary transit factor f_Γ = 1/ (1+γ²). The inverse-projector law Hₑff (α) = H₀ × f_Γ^1−α is derived at theorem grade for the late-time homogeneous estimator family. Two new bridge extensions — a stellar-photometric bridge for standard-candle calibrator chains and a centered lens-normalization for time-delay cosmography — complete the classification for all major methods. Six independent H₀ measurements span three rungs of the α-ladder. Every method falls within 0. 57σ of its predicted value: Planck at +0. 35σ, gravitational wave sirens at −0. 57σ, TRGB direct at −0. 07σ, TDCOSMO at −0. 06σ, SH0ES at +0. 04σ, and the TRGB+SN ladder at −0. 15σ. The 5σ crisis was three rungs of a ladder and a photometric calibrator chain. The complete IO paper series is open-access at https: //zenodo. org/communities/interior-observer/ v2. 0 (May 2026): Hygiene republication. Non-canonical theorem labels migrated to the canonical taxonomy. First-use expansions added for cosmic microwave background (CMB), Supernovae H₀ for the Equation of State (SH0ES), Tip of the Red Giant Branch (TRGB), Time-Delay COSMOgraphy (TDCOSMO), Friedmann-Robertson-Walker (FRW), Kubo-Martin-Schwinger (KMS), and other abbreviations. Master-identity terminology harmonized. §6 audit reference clarified to point directly to the field-redefinition audit 9. Contributors paragraph harmonized with the IO series standard. §8 scope and closure ledger upgraded: §8. 6 added with Theorems 34. PC0/PC1/PC2 closing the Planck CMB row as a compound CMB inference; §8. 1–§8. 4 promoted from admissions to ten scoped extension theorems (S8. 1–S8. 2 baseline and inverse-projector placement, A1–A4 stellar-photometric extension, B1–B4 lens-normalization extension), with full proofs relocated to Appendix A. §8. 1 active baseline citation updated to the current Paper 29–Paper 34 active projection registry; the Schur 68. 91 km/s/Mpc package is documented as a retired diagnostic branch after the Paper 29 hidden-parameter audit. No numerical values change; the v1. 0 scorecard and the v1. 2 R4 audit outcome are preserved. See https: //github. com/dfife/io-framework-public/tree/main for claim-naming convention. v1. 2 (May 2026): R4 / FIRAS readout-normalization correction sweep inherited from Paper 17 v1. 5 — Paper 34 audit-only outcome. A field-redefinition audit was performed on the Paper 34 Hₑxt (α, n) formula against Paper 17 v1. 5's FIRAS-fixed optical readout normalization R4FIRAS = 1. 0031 ± 0. 0096. The audit confirmed no R4 damage in Paper 34 calculations: Hₑxt (α, n) = H₀ × f_Γ^ (1−α) × x^ ( (n/2) Kgauge) does not use the optical thermal readout law Tₒbs (R4) = TIO × x^ (R4 Kgauge). The (n/2) photometric half-leg payload is a discrete count of uncancelled stellar-photometric calibrator legs in the H₀ observable-class ladder, structurally distinct from the optical thermal R4 normalization slot. Inserting R4 into the photometric exponent would be a new error, not a correction (rejected counterfactual Hwrong (α, n) = H₀ × f_Γ^ (1−α) × x^ ( (n/2) R4FIRAS Kgauge) gives a maximum H₀ shift of 0. 0052 km/s/Mpc, included only as an audit diagnostic). The six-method H₀ scorecard is unchanged: max |residual| = 0. 57σ; SH0ES at +0. 04σ. R4FIRAS is recorded as a framework constant in the v1. 2 bundle for cross-paper provenance, with explicit non-dependency boundary. Status of Paper 34 results is unchanged: Hₑxt (α, n) is DERIVED/SCOPED within the stellar-photometric and centered lens-normalization extensions; classification of the Planck row is DERIVED/SCOPED H₀ method-class baseline, not a CMB-temperature prediction. Reproducibility bundle: paper34-v1. 2 (release tarball SHA256 bfbb1f74a9f98d684e0e5f0d72a926dcdbb956e4fccaabc1dd124aab435d2597; validator 16/16 PASS). See https: //github. com/dfife/io-framework-public/tree/main for claim naming convention. v1. 1 (May 2026): A formal field-redefinition audit of the Hₑxt (α, n) formula was performed (the standard methodology for detecting hidden parameters in scoped theories: replace each numerical or structural choice by a free variable, test whether existing theorems force the original value, classify accordingly). The audit found no hidden continuous fitted parameter and confirmed that no method’s (α, n) assignment is classified as fitted; the strongest anti-fit signal is that GW sirens are kept at (α, n) = (1, 0) even though (3/2, 1) would fit the measurement more closely — structural derivation overrides numerical fit. The audit recommended explicit conditional-visibility hygiene per IO Framework Conventions v2. 0 §2. 4: a new “Scope and Open Premises” section is added, listing the active Paper 10 legacy projected baseline H₀ = 67. 58 km/s/Mpc, the inverse-projector placement admission J_α = f_Γ^ (α-1), the stellar-photometric extension premises A1-A4, the centered lens-normalization extension premises B1-B4, and the explicit statement that Hₑxt is a result derived inside the two adopted extensions, not a universal all-method H₀ theorem. A new “Code and Data Availability” section is added, citing the public reproducibility bundle archived as the immutable GitHub Release paper34-v1. 1 with tarball SHA256 4f921fd3b5df7729fb3f887ba12b3e1621039d38dd0fbd0779d50a366b54ac62. The bundle contains the Hₑxt grid evaluator, the published-measurement comparison scorecard with σ-pulls, the anti-fit backstop verifier, frozen JSON outputs, the κ-audit report, and a frozen-output validator for reviewers without recomputing. Bibliography expanded with three new entries: the reproducibility bundle, the κ-audit report, and IO Framework Conventions v2. 0. Title refreshed to lead with the quantitative claim “Six Methods, Three Rungs, Zero Fitted Parameters, ” and Zenodo keyword list refreshed to add the named-method vocabulary that discovery scripts in the Hubble-tension community search on (SH0ES, Cepheid, TRGB, TDCOSMO, Pantheon+, GW standard sirens, Type Ia supernovae). No numerical values change; the v1. 0 scorecard is preserved. v1. 0 (April 2026): Publication release. Observable-class resolution of the Hubble tension. Two new bridge extensions: stellar-photometric bridge (SH0ES, TRGB) and centered lens-normalization (TDCOSMO). Full formula: Hₑxt (α, n) = H₀ × f_Γ^ (1−α) × x^ (n/2) Kgauge. Six methods classified. Maximum deviation 0. 57σ. SH0ES at +0. 04σ from α = 2, n = 2 prediction. All nine Cosmo upgrade requirements satisfied. DERIVED/SCOPED within extensions.