a0(z) serious attack — reviewer-mode analysis (June 2026)

OBT’s crown-jewel prediction is the evolving MOND scale a0(z) = cH(z)/2pi = a0_0 * E(z) (Milgrom’s MOND takes a0 constant; OBT derives it from the Gibbons-Hawking horizon temperature, so it must grow with z). This folder attacks it as a skeptical reviewer (axiom: OBT can be false; verify both ways), answering Romain’s three questions, in order: (1) harden the forecast → (2) dissect the ×1.5 → (3) the over-determination.

NOT promoted to the 7 sacred files: this is V8.2-validation analysis, quarantined here pending Romain’s call.

The physics it all rests on (deep-MOND a0 = V_c^4/(G M_bar); RAR a0 = g_obs^2/g_bar)

A coherent fractional bias in a measured quantity maps to dln a0 through a lever: V_c → 4×, M_bar → 1×, lensing g_obs → 2×. A bias that grows with z biases the fitted slope alpha (where a0 ~ E^alpha; OBT: alpha=1) by beta_sys = Cov(b, lnE)/Var(lnE), which does not average down. BTFR has the same V/M_bar levers as the kinematic RAR → the two are systematically degenerate.

Scripts & verdicts

| script | question | verdict | |—|—|—| | hardened_forecast.py | (1) the published forecast’s sqrt(N) is wrong for a coherent bias | Evolution test robust (needs implausible beta_sys~-1 to erase → Euclid/Rubin settle OBT-vs-constant-MOND, ~tens of σ). Rate test systematics-limited: alpha_hat = 1 + beta_sys; a ~10% V_c (4×) or ~40% M_bar (1×) coherent high-z bias injects beta_sys~0.5 = the observed “1.5×”. More lenses don’t help. | | systematics_dissection.py | (2) is the ×1.5 systematic or real? | Both ways, honest. The observed Delta_alpha ~ +0.4..+0.9 (scattered down to ~0) sits fully inside the local-anchor shift (~0.3) + the high-z coherent budget (gas up to +0.8 if M-only; V/incl ±0.65). → OBT’s alpha=1 not refuted (the ×1.5 is consistent with alpha=1 + a coherent bias; OBT sits at the low edge, KROSS@MOND-anchor = 0.99). But not confirmed-systematic either — a real alpha~1.5 also fits; the rate is undetermined. New honest caveat: a0_0=cH0/2pi ≈ 1.04 (H0=67.4) is ~15% below MOND’s measured 1.2 — a mild ~1σ tension on OBT’s *absolute local a0 (H0=73 → 1.13, closer). | | overdetermination.py | (3) the strongest internal falsifier | Every observable X ∝ a0^p must give the same alpha (powers a0:+1, BTFR:-1, V_flat:+1/4, Σ†:+1, r_t:-1/2). But currently UNFULFILLED: the only measured high-z handles (kinematic a0 + BTFR) share the V 4× lever → their ~1.5 agreement is degenerate, not independent proof (and the 0.7 inter-method spread already hints at extra method-specific systematics). |

The bottom line (what “attacking a0(z) seriously” found)

• The evolution is the distinctive, robust claim — Euclid/Rubin will settle OBT-vs-constant-MOND decisively, and it cannot be faked away by any plausible systematic. This is the pépite and it stands.
• The cH(z)/2pi rate is the real frontier, and it is currently UNDECIDED — the observed ×1.5 is buried in (a) the local cH0/2pi-vs-MOND anchor and (b) high-z V_c (4× lever!) + gas systematics. It neither refutes nor confirms OBT’s Gibbons-Hawking rate.
• THE decisive near-term experiment = a cross-lever measurement: Euclid lensing-a0(z) (g_obs 2×, no V lever) vs kinematic-a0(z) (V 4×), plus Sigma_dagger(z) (no V), plus measured-gas (ALMA). The split (or its absence) and the gas-response cleanly separate real-rate / V-systematic / gas-systematic. That one measurement turns a0(z) from “evolution confirmed, rate ambiguous” into a decided rate.

Two findings that bear on the PUBLISHED files (Romain’s call to promote)

1. scripts/a0z_forecast.py adds the systematic in quadrature and divides by sqrt(N) — correct for random scatter, wrong for the coherent z-dependent bias that actually limits the rate test. The hardened version here (random vs coherent) is the honest replacement.
2. predictions.md §6 does not state (a) the mild ~15% local-anchor tension (cH0/2pi vs MOND 1.2), nor (b) that the current over-determination (a0-RAR + BTFR) is degenerate (shared V lever). Both are honest caveats that would strengthen the section’s “que ce soit vrai.”

Run: for s in hardened_forecast systematics_dissection overdetermination; do python explorations/a0z_analysis/$s.py; done