The Euclid case for $a_0(z)$ — the decisive test of OBT’s one distinctive prediction

Reviewer/webmaster mode, cool head. This is the argumentaire for the single near-term measurement that decides OBT’s distinctive lever. It is built on the audited honest scope (steps 1→3→2 + anchor dig): the EVOLUTION is robust and Euclid-decisive; the RATE is systematics-limited and needs the cross-lever; OBT claims the FORM $a_0\propto H(z)$ and the evolution, not the $a_0$ coefficient.

1. The claim, in one line

OBT derives the MOND acceleration scale from the cosmic horizon, so it must evolve: \(a_0(z) = \frac{cH(z)}{2\pi} = a_0(0)\,E(z),\qquad E(z)=\sqrt{\Omega_m(1+z)^3+\Omega_\Lambda}\ \ (E(1)=1.76,\ E(2)=2.97).\) Milgrom’s standard MOND takes $a_0$ to be a universal constant. A measured evolution of $a_0$ with redshift falsifies constant-$a_0$ MOND and is the unique, currently-near-testable signature separating OBT from it.

2. Why this is THE test (and not another consistency check)

Of OBT’s signatures, $a_0(z)$ is the only one that is simultaneously distinctive (not shared with $\Lambda$CDM or constant-MOND), already hinted (three datasets: $a_0$ roughly doubles by $z\sim1$ — MUSE-DARK III RAR, Übler+2017 BTFR zero-point, KROSS), and near-term decisive (Euclid). Audited honest status:

• the evolution ($a_0\neq$ const) is robust;
• the rate ($cH(z)/2\pi$ vs the observed $\sim1.5\times$-faster) is undetermined — the three current hints are all kinematic (rotation/dispersion), so they share the $V_c^4$ “4× lever” ($a_0=V_c^4/GM_{\rm bar}$) and a shared high-$z$ velocity/baryonic systematic inflates them identically. A coherent $\sim10\%$ $V_c$ bias alone fakes the $1.5\times$. So the rate needs an orthogonal-lever measurement.

3. What Euclid uniquely provides — the lensing leg (the orthogonal lever)

• Euclid Wide Survey: $\sim14{,}000$ deg², shapes of $\sim1.5\times10^9$ galaxies, weak lensing to $z\sim2$; DR1-Foundation Nov 2026, weak-lensing products mid-2027 (ESA Euclid DR1 timeline).
• The weak-lensing RAR measures $g_{\rm obs}$ from the excess surface density $\Delta\Sigma$ around lens galaxies — no $V_c$. Its systematic lever is $g_{\rm obs}$ (2×; 2-halo / intrinsic-alignment / photo-z), orthogonal to the kinematic $V_c$ (4×; beam-smearing / asymmetric-drift / inclination). Proof of concept: Brouwer et al. 2021 measured the lensing RAR with KiDS-1000 two decades below $a_0$ (lens $\langle z\rangle\sim0.2$).
• Euclid extends this to redshift bins to $z\sim1.5$ with $\gg$KiDS statistics → the first $a_0(z)$ from lensing.
• The joint kinematic + weak-lensing RAR method already exists (e.g. arXiv:2310.15248) — the cross-lever is methodologically ready; Euclid supplies the high-$z$ lensing leg it currently lacks.

4. The measurement design (the cross-lever)

1. Euclid lensing-$a_0(z)$ — isolated-lens RAR in $z$-bins over $z\sim0.3$–$1.5$: $\Delta\Sigma\to g_{\rm obs}$; baryonic $g_{\rm bar}$ from Euclid+ground photometry (stars) + cold gas (the dominant $g_{\rm bar}$ systematic — pair with HI/CO subsamples or marginalize a calibrated gas prior).
2. Matched kinematic-$a_0(z)$ — MUSE-DARK / KROSS / JWST / ELT H$\alpha$ rotation at the same $z$ (the $V_c$ leg).
3. The comparison — lensing-$a_0(z)$ vs kinematic-$a_0(z)$ at matched $z$ (and matched host size/mass).
4. Orthogonal bonus legs — $\Sigma_\dagger(z)=a_0/G$ (critical surface density, no $V_c$) and $r_t(z)$ from the same lensing RAR: an over-determined set ($a_0$, BTFR, $\Sigma_\dagger$, $r_t$ have powers $+1,-1,+1,-\tfrac12$ of $E(z)$ — a single $E(z)$ must fit all).
5. Clean anchor — at $z\sim0.2$ the lensing RAR (Brouwer) already agrees with the kinematic $a_0\sim1.2$ → the lensing-vs-kinematic method offset is $\sim0$ at $z=0$, so a high-$z$ split is a clean systematic signal, not a zero-point artifact.

5. The forecast (this work, a0z_forecast.py, verified vs MUSE-DARK 16$\sigma$)

• Euclid lensing RAR, $z\sim0.3$–$1.5$, $\sim1$–$2\%$ per-$z$-bin on $a_0$ → $\sigma_\alpha\approx0.02$.
• Evolution ($\alpha\neq0$): decisive — $>5\sigma$ on $\alpha\neq0$ even with $20\%$ per-bin systematics; constant-$a_0$ MOND settled.
• Rate ($\alpha=1$ vs $1.5$): only the cross-lever decides it — statistically the gap is $\sim24\sigma$ if there were no coherent bias, but a coherent $\beta_{\rm sys}\sim0.5$ (the $\sim10\%$ $V_c$ / $\sim40\%$ gas budget) sits on the kinematic leg and does not average down. The lensing leg does not carry the $V_c$ lever, so the kinematic-vs-lensing difference isolates it (worked example: a true $\alpha=1$ with a $10\%$ $V_c$ + $10\%$ gas bias reads $\hat\alpha_{\rm kin}=1.57$ but $\hat\alpha_{\rm lens}=1.29$ — a $0.28$ split).

6. The decision tree — what Euclid will actually tell us

| Euclid lensing-$a_0(z)$ vs kinematic-$a_0(z)$ | meaning | |—|—| | both $\approx0$ (no evolution) | OBT refuted, constant-$a_0$ MOND vindicated | | both $\approx1.5$, robust to measured gas | the rate is real → OBT’s $cH(z)/2\pi$ coefficient/rate refuted (but $a_0$ still evolves — MOND-constant still dead; a faster rate would need a new horizon-coefficient origin) | | kinematic $\approx1.5$ but lensing $\approx1.0$ | the $V_c$ systematic inflated the kinematic hints → OBT’s $cH(z)/2\pi$ safe | | both $\approx1.5$ but shrink under measured gas | gas-census systematic → $cH(z)/2\pi$ safe | | both $\approx1.0$ | OBT’s $cH(z)/2\pi$ confirmed |

Every branch is informative — there is no null outcome. This is what makes it a real test.

7. Honest caveats (the cool head)

• The lensing RAR has its own systematics (2-halo, intrinsic alignments, photo-$z$, the baryonic/gas $g_{\rm bar}$). The cross-lever’s power is precisely that these are different from the kinematic $V_c$ systematics: a shared real $a_0$-evolution shifts both legs together, while orthogonal systematics shift them apart. The comparison is robust even though each leg individually is systematics-limited.
• OBT predicts the FORM, not the coefficient. The thermodynamic content is $a_0\propto H(z)$ (a horizon-set scale) + its evolution; the $1/2\pi$ coefficient is an O(1) prior-art coincidence (Milgrom/Verlinde/McCulloch), consistent with the measured $a_0=1.20\pm0.24_{\rm syst}$ at $\le0.7\sigma$. So the sharp Euclid tests are (i) is $a_0$ evolving? and (ii) is the rate consistent with $E(z)$ ($\alpha\simeq1$)? — not the absolute value.
• $E(z)$ vs $(1+z)$ are nearly degenerate below $z\sim1.5$; pushing the kinematic leg to $z\sim2$–3 (JWST/ELT) breaks it. Use MOND-regime ($g_{\rm bar}\sim a_0$) tracers only — never the compact, $a_0$-blind disks.

8. The ask

A $z$-binned Euclid weak-lensing RAR (extend Brouwer-KiDS to $z\sim1.5$) + matched kinematic-$a_0(z)$ (MUSE/JWST/ELT) + measured cold gas (HI/CO subsamples). This single, feasible-now programme converts $a_0(z)$ from “evolution hinted, rate ambiguous” into a decided rate, and over-determines the horizon origin of $a_0$ via $\Sigma_\dagger(z)$ and $r_t(z)$. Euclid DR1 (2026/2027) + the existing joint-RAR pipeline make it the sharpest near-term play on physics through the horizon — and OBT has staked, in advance and in the open, exactly what each outcome means.

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Sources: Euclid DR1 timeline & Wide Survey (ESA/cosmos.esa.int; Euclid preparation forecasts arXiv:1910.09273, 2512.09748); Brouwer et al. 2021 A&A 650 A113 (KiDS-1000 lensing RAR); joint kinematic+lensing RAR arXiv:2310.15248; MUSE-DARK III arXiv:2604.22613; Übler et al. 2017 ApJ 842 121; Harrison et al. 2017 MNRAS 467 1965 (KROSS); McGaugh, Lelli & Schombert 2016 PRL 117 201101 ($a_0=1.20\pm0.02\pm0.24$). Forecast & cross-lever: this folder (a0z_forecast.py [= scripts/], overdetermination.py, systematics_dissection.py, step2_crosslever_feasibility.md, step3_real_papers.md, anchor_tension.py).