RLMT: RLMT and the Hubble Tension: Primordial and Late-Time Ω-Layer Tests against Planck, ACT DR6, SPT-3G D1, DESI DR2, Pantheon+ and DES Supernovae, with Control Models and Methodological Corrections

This version corrects an omission in the distribution package accompanying the paper. Two files required to reproduce the numerical results were missing from the public release:

• rlmt_primordial.py

• rlmt_layer_de_tanh_r_background_theory.py

Both files are now included in the updated distribution archive. No scientific results, numerical values, or conclusions have changed. This update ensures full reproducibility of all MCMC runs and late‑time 22‑layer dark‑energy models presented in the paper.

RLMT:RLMT and the Hubble Tension: Primordial and Late-Time Omega-Layer Tests against Planck, DESI DR2 and Pantheon+, with Methodological Corrections — Version 2 (English and Japanese).

This paper tests whether R-layer Mode Theory (RLMT), in any form implemented so far, can move the cosmologically inferred Hubble constant toward the locally measured values (SH0ES, CCHP, SN H0pe). LambdaCDM is used as the comparison baseline, not as a model to be refuted. The primordial-spectrum model (RLMT-lite) fits Planck and Pantheon+SH0ES as well as LambdaCDM but cannot shift H0, because it never touches the background expansion. Phenomenological late-time Omega-layer dark-energy models (TH-021 provisional eigenvalues, CAMB PPF fluid) fit Planck 2018 + DESI DR2 BAO + Pantheon+ as well as w0waCDM, but do not move H0 toward the local values when SH0ES is not in the fit. The answer to the paper's question is negative for every RLMT implementation tested so far.

Version 2 changes (2026-10-09):
- Best fits from a minimizer (two-pass BOBYQA) replace the lowest sampled chi2 of version 1; the Planck calibration prior A_planck is kept in the likelihood.
- DESI DR1 full-shape (ShapeFit) growth data added (df, dm per tracer; DR1 BAO parameters marginalised), with and without SH0ES. The full-shape data do not distinguish the dark-energy models (ShapeFit chi2 differences < 0.3).
- With SH0ES and full shape, the Omega-layer models improve chi2 by 12.7–13.7 relative to LambdaCDM, as w0waCDM does (13.2).
- The exponential-relaxation variant is identified as a purely phantom history (w relaxing from -5/3 to -1 by z ~ 0.5). It accommodates SH0ES at a smaller cost (Delta Q_DMAP ~ -7 to -8, versus ~ -3 for w0waCDM and the CPL-type variant), robust to the treatment of the optical depth and to its prior range, but still leaves H0 <= 69.5.
- Robustness test: replacing Planck low-l EE by a Gaussian on tau changes every model comparison by less than 1.
- New methodological records: ERR-004 (the Gaussian tau prior duplicated the low-l EE information in the sampled posteriors of version 1) and ERR-005 (the calibration prior was dropped in a first round of minimizations; caught before publication, no published number affected).
- The conclusion of version 1 is unchanged.

Files: main_v2_20261009.pdf (English), main_v2_ja_20261009.pdf (Japanese), RLMT_Hubble_followup_CODE_v1.2.zip (theory classes, DESI DR1 ShapeFit wrapper, all MCMC and minimizer yaml files, best-fit files, and analysis scripts; see README.md and YAML_LIST.md inside).

Version 3 corrects the information criteria for RLMT-lite (error record ERR-006): the mode weights depend on (β_eff, c_1, c_2, c_3) only through β_eff(c_2 − c_1) and β_eff(c_3 − c_1), so two of the seven extra parameters are not identifiable and Δk = 5. Planck only: ΔAIC +14.92 → +10.92, ΔBIC +46.6 → +33.5; Planck+Pantheon+SH0ES: ΔAIC +15.31 → +11.31, ΔBIC +55.7 → +40.2. The χ² values, posteriors and conclusions are unchanged. Nothing else changes.

This is a companion paper to "RLMT: Lindblad Cosmogenesis, SU(2) Geometry, and a Numerical Confrontation with Planck 2018 Data" (Tohi 2026). Its aim is narrow: to test whether R-layer Mode Theory (RLMT), in any form implemented so far, can move the cosmologically inferred Hubble constant H0 toward the locally measured values (SH0ES, CCHP, SN H0pe). ΛCDM is used throughout as the comparison baseline, not as a model to be refuted. RLMT's primordial-spectrum parametrization ("RLMT-lite") cannot do this: it fits Planck and Pantheon+SH0ES as well as ΛCDM (Δχ² = +0.9, +1.3; ΔAIC = +10.9, +11.3 from its five identifiable extra parameters alone) and never touches the background expansion. We then test the late-time route: a family of phenomenological Ω-layer dark-energy models in which the TH-021 provisional layer eigenvalues populate an effective equation of state w_eff(a), implemented as a CAMB PPF fluid and fitted to Planck 2018 + DESI DR2 BAO + Pantheon+, with and without the SH0ES calibration, and also with the DESI DR1 full-shape (ShapeFit) growth information. Best fits are obtained with a minimizer. With SH0ES and the full-shape data, the Ω-layer models improve χ² by 12.7–13.7 relative to ΛCDM, exactly as w0wa does (13.2); the improvement is that of evolving dark energy in general, not of RLMT specifically. The full-shape data do not distinguish the dark-energy models (their ShapeFit χ² differ by < 0.3). Without SH0ES, no model moves the best-fit H0 toward the local values by more than 0.6; the CPL-type variant behaves like w0wa and lowers it. One variant, exponential relaxation, is a purely phantom history (w relaxing from −5/3 to −1) that accommodates SH0ES at a smaller cost than the others: its tension statistic ΔQ_DMAP ≃ −7 to −8, versus ≃ −3 for w0wa and the CPL-type variant, robust to the treatment of the optical depth and to its prior range. It still leaves H0 ≤ 69.5 and a tension equivalent to 8.5σ in this statistic. A test of the TH-021 eigenvalue position (r = 0.2) is uninformative. The answer to the paper's question is therefore negative for every RLMT implementation tested so far, and we state why. Five methodological errors found along the way (three in the first version, one affecting its sampled posteriors, one caught before publication) are documented, and a sixth (ERR-006) is corrected in this version. Reproduction code is provided.

Version 4 answers the question left open in version 3: whether the smaller cost at which the exponential-relaxation variant ("exprelax") accommodates SH0ES belongs to the RLMT construction or only to the shape of its history. A non-RLMT control model with the same time axis and the same range of w — a phantom component decaying exponentially in time to a cosmological constant, w(a) = −1 + w_early exp(−t/τ) — was fitted with the same likelihoods and the same minimizer (calibration prior kept, ERR-005). It reproduces most of the effect: ΔQ_DMAP = −6.1 against −7.8 for exprelax and ≃ −2.4 for w0waCDM and the CPL-type variant, and like exprelax it does not lower H0 without SH0ES. The remainder (about 1.6 ± 1 in ΔQ, and a χ² advantage of 2.65 with SH0ES and 1.0 without) comes from the plateau produced by the saturating softmax map, which a single exponential cannot reproduce; since any saturating amplitude would produce such a plateau, it is not counted as evidence for RLMT. Version 4 also repeats the two runs of version 1 that reached only practical convergence, sampling ln(10^10 A_s); they do not reach the multivariate criterion either, and their results are unchanged. Nothing else changes.

This is a companion paper to "RLMT: Lindblad Cosmogenesis, SU(2) Geometry, and a Numerical Confrontation with Planck 2018 Data" (Tohi 2026). Its aim is narrow: to test whether R-layer Mode Theory (RLMT), in any form implemented so far, can move the cosmologically inferred Hubble constant H0 toward the locally measured values (SH0ES, CCHP, SN H0pe). ΛCDM is used throughout as the comparison baseline, not as a model to be refuted. RLMT's primordial-spectrum parametrization ("RLMT-lite") cannot do this: it fits Planck and Pantheon+SH0ES as well as ΛCDM (Δχ² = +0.9, +1.3; ΔAIC = +10.9, +11.3 from its five identifiable extra parameters alone) and never touches the background expansion. We then test the late-time route: a family of phenomenological Ω-layer dark-energy models in which the TH-021 provisional layer eigenvalues populate an effective equation of state w_eff(a), implemented as a CAMB PPF fluid and fitted to Planck 2018 + DESI DR2 BAO + Pantheon+, with and without the SH0ES calibration, and also with the DESI DR1 full-shape (ShapeFit) growth information. Best fits are obtained with a minimizer. With SH0ES and the full-shape data, the Ω-layer models improve χ² by 12.7–13.7 relative to ΛCDM, exactly as w0wa does (13.2); the improvement is that of evolving dark energy in general, not of RLMT specifically. Without SH0ES, no model moves the best-fit H0 toward the local values by more than 0.6; the CPL-type variant behaves like w0wa and lowers it. The exponential-relaxation variant is a purely phantom history that accommodates SH0ES at a smaller cost than the others (ΔQ_DMAP ≃ −7 to −8 versus ≃ −3), and a non-RLMT control model of the same shape reproduces most of this. A test of the TH-021 eigenvalue position (r = 0.2) is uninformative. The answer to the paper's question is therefore negative for every RLMT implementation tested so far, and we state why. Six methodological errors found along the way are documented. Reproduction code is provided.

1. New section "Version 4: a non-RLMT control model of the exprelax shape" (Table tab:control, Figure fig:control): ΔQ = −6.1 (control) vs −7.8 (exprelax); the SH0ES property belongs to the shape; the remainder comes from the saturation of the map.
2. The two runs of version 1 judged by the practical convergence criterion (tanh_r on dataset C, exprelax on dataset D) were repeated with ln(10^10 A_s); results unchanged (Sec. "Practical convergence for two runs").
3. ERR-005 (calibration prior dropped in a first round of minimizations) recurred in a first minimization of the control model and was again caught before publication; recorded in Sec. ERR-005.
4. Abstract, development history, discussion ("The exprelax thread"), limitations, conclusion and the appendix "changes from version 1" updated; the abstract now counts six documented errors (ERR-001 to ERR-006). Title: a space added after "RLMT:".
5. Code package v1.3 (v1.2 + the files of version 4).

Version 5 adds three tests to the comparison of version 4.
(i) A second non-RLMT control model of the shape of the exponential-relaxation variant ("exprelax"), the CPL form with w0 fixed to −1 (one extra parameter). It gives the same tension statistic as the control of version 4, ΔQ_DMAP = −6.2 against −6.1, versus −7.8 for exprelax and −2.4 for w0waCDM, and with the DESI full-shape data it reproduces 95% of the exprelax value. The SH0ES property of exprelax therefore belongs to its shape — phantom in the past, w = −1 today — independently of the form of the control.
(ii) The whole late-time comparison repeated with ACT DR6 in place of the Planck high-ℓ data (P-ACT-lite + DESI DR2 + Pantheon+, with and without SH0ES), with one theory code for all models, validated against the published Planck best fits to 0.001 in χ². The ΔQ of the late-time models change by at most 1.3 and no conclusion changes; without SH0ES no late-time model raises the best-fit H0 by more than 0.5.
(iii) Standard axion-like early dark energy (EDE), fitted to the same data with the same minimizer as a reference for the early-universe route. ΔQ = −22.1 (Planck) and −24.4 (ACT DR6), and the best-fit H0 ≃ 71 even without SH0ES (f_EDE ≃ 0.09–0.11), in agreement with published ACT DR6 + DESI DR2 analyses. A first search had found a local minimum without SH0ES (H0 ≃ 69); the extra searches are documented. On the same data and with the same statistic, the late-time route — RLMT or not — does three to four times less than the standard early-universe model. EDE is a reference, not an RLMT model. No earlier result changes. The statement "a tension equivalent to 8.5σ" of the earlier abstracts is removed, since Q itself has no absolute meaning in this paper.

This is a companion paper to "RLMT: Lindblad Cosmogenesis, SU(2) Geometry, and a Numerical Confrontation with Planck 2018 Data" (Tohi 2026). Its aim is narrow: to test whether R-layer Mode Theory (RLMT), in any form implemented so far, can move the cosmologically inferred Hubble constant H0 toward the locally measured values (SH0ES, CCHP, SN H0pe). ΛCDM is used throughout as the comparison baseline, not as a model to be refuted. RLMT's primordial-spectrum parametrization ("RLMT-lite") cannot do this: it fits Planck and Pantheon+SH0ES as well as ΛCDM (Δχ² = +0.9, +1.3; ΔAIC = +10.9, +11.3 from its five identifiable extra parameters alone) and never touches the background expansion. We then test the late-time route: a family of phenomenological Ω-layer dark-energy models in which the TH-021 provisional layer eigenvalues populate an effective equation of state w_eff(a), implemented as a CAMB PPF fluid and fitted to Planck 2018 + DESI DR2 BAO + Pantheon+, with and without the SH0ES calibration, and also with the DESI DR1 full-shape (ShapeFit) growth information. Best fits are obtained with a minimizer. With SH0ES and the full-shape data, the Ω-layer models improve χ² by 12.7–13.7 relative to ΛCDM, exactly as w0waCDM does (13.2); the improvement is that of evolving dark energy in general, not of RLMT specifically. Without SH0ES, no model moves the best-fit H0 toward the local values by more than 0.6; the CPL-type variant behaves like w0waCDM and lowers it. The exponential-relaxation variant is a purely phantom history that accommodates SH0ES at a smaller cost than the others (ΔQ_DMAP ≃ −7 to −8 versus ≃ −3); two non-RLMT control models of the same shape reproduce most of this (ΔQ = −6.1, −6.2), with Planck and with ACT DR6. Standard early dark energy, fitted to the same data, gives ΔQ = −22 to −24 and H0 ≃ 71 without SH0ES. A test of the TH-021 eigenvalue position (r = 0.2) is uninformative. The answer to the paper's question is therefore negative for every RLMT implementation tested so far, and we state why. Six methodological errors found along the way are documented. Reproduction code is provided.

Changes in version 5:
1. New section "Version 5: a second control, early dark energy and ACT DR6" (Tables tab:fixw0, tab:v5):
   - CPL control with w0 = −1 (ΔQ = −6.18; −7.61 with the full-shape data);
   - axion-like EDE on Planck and ACT DR6 (ΔQ = −22.1 and −24.4; best-fit H0 ≃ 71 without SH0ES; minimizer uncertainty about ±0.5);
   - the late-time comparison on P-ACT-lite + DESI DR2 + Pantheon+ (no conclusion changes).
2. Title: "ACT DR6" added. Abstract, development history, discussion, limitations, conclusion and code list updated. The "8.5σ" statement of the abstract removed (Q itself has no absolute meaning; only ΔQ is used).
3. References: ACT DR6 (Louis et al.; Calabrese et al.), EDE (Karwal & Kamionkowski; Poulin et al. 2019), Planck SRoll2 (Pagano et al.), Poulin et al. 2025 (ACT DR6 + DESI DR2 and EDE). The reference to Appendix X-II now carries its DOI (10.5281/zenodo.23130084).
4. Code package v1.4 (v1.3 + the files of version 5: hact_20261020/, ede_20261020/, fixw0/, chains/min*/ best fits).

Version 6 closes the items left open in version 5 and adds two CMB data sets.
(i) A non-RLMT control model with a saturating amplitude, w = −1 + (2/3) tanh[s_early exp(−t/τ)], on the same time axis and with the same range of w as the exponential-relaxation variant ("exprelax"). It fits as well as exprelax on all four data combinations, and the two tension statistics ΔQ_DMAP agree within 0.1 (Planck) to 1.3 (ACT DR6).
(ii) Two more minimizer passes for exprelax without SH0ES lower its minima by 1.2 (Planck) and 0.6 (ACT DR6); its ΔQ becomes −6.55 and −6.54 instead of −7.76 and −7.12. Together with (i), nothing in the SH0ES property of exprelax remains specific to RLMT; the remainder that version 4 attributed to the saturating map is withdrawn.
(iii) ACT DR6 + Planck PR4 CMB lensing, and (iv) the SPT-3G D1 temperature and polarization band powers (SPT-lite, through candl), added to P-ACT-lite + DESI DR2 + Pantheon+, with and without SH0ES. No conclusion changes: without SH0ES no late-time model raises the best-fit H0 by more than 0.5 over ΛCDM; with Planck, ACT and SPT combined the late-time models give ΔQ = −3 to −8 and early dark energy −25 (H0 = 71.1 without SH0ES); the lensing and SPT data alone mildly prefer the late-time models (Δχ² −0.2 to −3) and mildly disfavour early dark energy (+0.2 to +2.1).
(v) A review of early-universe routes: two known classes fix the onset of an early-dark-energy-like injection without a free time parameter, but no RLMT ingredient fixes its amplitude without a new scale, so no RLMT early-universe model is attempted.
The answer of the paper is unchanged and now holds against all three CMB experiments: RLMT, in the two routes implemented, does not resolve the Hubble tension.

This is a companion paper to "RLMT: Lindblad Cosmogenesis, SU(2) Geometry, and a Numerical Confrontation with Planck 2018 Data" (Tohi 2026). Its aim is narrow: to test whether R-layer Mode Theory (RLMT), in any form implemented so far, can move the cosmologically inferred Hubble constant H0 toward the locally measured values (SH0ES, CCHP, SN H0pe). ΛCDM is used throughout as the comparison baseline, not as a model to be refuted. RLMT's primordial-spectrum parametrization ("RLMT-lite") cannot do this: it fits Planck and Pantheon+SH0ES as well as ΛCDM and never touches the background expansion. The late-time route — a family of phenomenological Ω-layer dark-energy models in which the TH-021 provisional layer eigenvalues populate an effective equation of state, implemented as a CAMB PPF fluid — fits Planck 2018 + DESI DR2 BAO + Pantheon+ as well as w0waCDM, but without SH0ES moves the best-fit H0 by at most 0.6. Its one distinctive variant, exprelax, owes its smaller tension statistic to its shape and saturation, which three non-RLMT control models reproduce. Standard early dark energy, fitted to the same data as a reference, does three to four times more. A test of the TH-021 eigenvalue position is uninformative. Six methodological errors found along the way are documented. Reproduction code is provided.
Changes in version 6:
1. New section "Version 6: a saturating control, deeper minima, CMB lensing and SPT" (Tables tab:v6, tab:lens, tab:spt; verdict of version 6).
2. The exprelax entries without SH0ES in Tables tab:cd-min, tab:control and tab:v5 are superseded (notes added); part (b) of the verdict of Sec. "Version 4" is withdrawn.
3. Discussion: the early-universe route reviewed (onset fixed by matter domination or by a curvature trigger; amplitude needs a new scale in RLMT). Limitations: saturating-control item closed, flat-valley uncertainty (±0.5) added, CMB item updated (SPT lensing not included), ghost-free coupled-quintessence route for a phantom w noted.
4. Abstract, development history, conclusion and title updated ("SPT-3G D1" and "Control Models" added).
5. References: ACT DR6 lensing (Madhavacheril et al.; Qu et al.), Planck PR4 lensing (Carron et al.), SPT-3G D1 (Camphuis et al.; Khalife et al.), candl (Balkenhol et al.), dark-matter-triggered EDE (Lin et al.), curvature-triggered EDE (Jing et al.).
6. Code package v1.5 (v1.4 + the files of version 6: tanhexp, exprelax refinement, ACT lensing, SPT; theory class with tanhexp, raw lensing spectrum and copied spectra).
7. Technical note: before version 6 our theory class returned an incorrect lensing-potential spectrum for requests without the ℓ factor, and returned stored spectra by reference; no likelihood used in versions 1–5 was affected.

Version 7 asks whether relaxation dynamics can fix the shape of the dark-energy equation of state w(a), which in all earlier Ω-layer variants had a free onset and amplitude and therefore did no more than w0waCDM.
(i) Theory. For a hypothetical ultralight three-level sector with the level pattern (1,1,2) of the Ω layer, coupled to the cosmological horizon at T = H/2π — an assumption not derived from RLMT, and not the electroweak-scale realization of the Ω layer, whose level splitting vanishes after inflation — Lindblad relaxation from the maximally mixed state needs no onset parameter, and its shape is fixed: the departure grows as 1 − exp(−Γt) in cosmic time, with d ln s / d ln a = 1/(H0 t0) ≈ 1.05 today. Only the amplitude is free, bounded by the Born–Markov condition. For every map from populations to w that we tested (equation-of-state and density maps), the CPL parameters lie on the line w_a ≃ −(1 + w0) (slope −0.8 to −1.2; −1.6 for maps starting at second order). The sign of 1 + w is not predicted. We also note that the softmax populations with the non-degenerate TH-021 eigenvalues used in the earlier variants are not a solution of the relaxation dynamics.
(ii) Data. Three one-parameter models (the TH-023 map with its coefficient, phantom; a generic equation-of-state map; a density map) were fitted with new MCMC runs and minimizers to Planck 2018 + DESI DR2 BAO with Pantheon+, with Pantheon+ and the DESI DR1 full shape, with the recalibrated DES-Dovekie supernovae, and, as a stress test chosen in advance, with the original DES-SN5YR calibration; no SH0ES. Each model reproduces the ΛCDM χ² exactly at zero amplitude.
(iii) Result. The line fits as well as or better than ΛCDM with every data set (Δχ² = −0.6 to −2.4 with Pantheon+, the full shape and DES-Dovekie; −5.3 to −6.4 with DES-SN5YR) and worse than free w0waCDM by 8 to 15. The more a sample prefers w0waCDM, the more the line gains and the wider the gap. With the phantom map the best fit is ΛCDM itself and the data bound the amplitude to half the theory range (|1 + w0| < 0.056). No model raises H0 (67.0–68.6).
The answer of the paper is unchanged: RLMT, in the routes implemented, does not resolve the Hubble tension; a history fixed by relaxation dynamics is consistent with the data but does not reach the w0waCDM preference.

This is a companion paper to "RLMT: Lindblad Cosmogenesis, SU(2) Geometry, and a Numerical Confrontation with Planck 2018 Data" (Tohi 2026). Its aim is narrow: to test whether R-layer Mode Theory (RLMT), in any form implemented so far, can move the cosmologically inferred Hubble constant H0 toward the locally measured values (SH0ES, CCHP, SN H0pe). ΛCDM is used throughout as the comparison baseline, not as a model to be refuted. RLMT's primordial-spectrum parametrization ("RLMT-lite") cannot do this: it fits Planck and Pantheon+SH0ES as well as ΛCDM and never touches the background expansion. The late-time route — a family of phenomenological Ω-layer dark-energy models implemented as a CAMB PPF fluid — fits Planck 2018 + DESI DR2 BAO + Pantheon+ as well as w0waCDM, but without SH0ES moves the best-fit H0 by at most 0.6. Its one distinctive variant, exprelax, owes its smaller tension statistic to its shape and saturation, which three non-RLMT control models reproduce. Standard early dark energy, fitted to the same data as a reference, does three to four times more. A test of the TH-021 eigenvalue position is uninformative. Six methodological errors found along the way are documented. Reproduction code is provided.

Changes in version 7:
1. New section "Version 7: a dark-energy history fixed by Lindblad relaxation of a hypothetical ultralight three-level sector" (Tables 14 and 15): ingredients, relaxation history, maps from populations to w, comparison with the data (Gaussian projection and full fits), assumptions.
2. New MCMC runs and minimizer best fits for three one-parameter relaxation models with Pantheon+, DESI DR1 full shape, DES-Dovekie and DES-SN5YR supernovae (no SH0ES). ΛCDM references recomputed with the same theory class agree with the earlier values to 0.05–0.12 in χ².
3. Abstract, development history, limitations (hypothetical sector) and conclusion updated; "DES Supernovae" added to the title.
4. References: Lindblad-map paper (Tohi, Zenodo 10.5281/zenodo.23363409), Ω-layer U(1)_T paper (Tohi, Zenodo 10.5281/zenodo.23415734), Gibbons & Hawking (1977), Birrell & Davies (1982), DES-Dovekie (Popovic et al., arXiv:2511.07517), DES-SN5YR (DES Collaboration, ApJL 973, L14, 2024).
5. Code package v1.6 (v1.5 + late_omega_relax_20261026: theory classes with and without P(k), yaml generators, one-point check, analysis and minimizer scripts, theory checks, best fits and convergence records).
No earlier result changes.