For Researchers: What's Citable

Audited-Negative: closed by execution, for the algebraic-coupling class (scope below).

Honest novelty statement: the core obstruction is not ours. Milgrom proved MOND-as-modified-inertia must be non-local in time (astro-ph/0510117, building on the Milgrom 1994 Galilei-invariance theorem); the spatial non-locality this no-go actually uses — the successful organizing variable is acceleration/enclosed mass, not any local column — is carried by the RAR/MDAR literature (McGaugh 2004; Lelli, McGaugh & Schombert 2016; Lelli et al. 2017, scatter ≲0.13 dex — who tested surface columns, never volumetric ρ; Stiskalek & Desmond 2023 for the systematic variable sweep) and by the elliptic Bekenstein–Milgrom field equation. What this project adds is the quantified local-density instance: any gravity modification keyed on the local volumetric density ρ(r) via algebraic coupling (C(ρ)·g as a multiplicative scalar on the force — the class the framework's algebraic reading belongs to. Scope (corrected 2026-09-11, visitor researcher persona; see revision notes): the framework's field-equation form ∇·[C∇Φ] = 4πGρ (Refracted Gravity's) carries a gradient coupling ∇C·∇Φ in discs and is outside this statement as written. For that form the SPARC evidence is the disc-geometry grid in the box above (floored switch at γ ∈ {0.489, 2}, f ∈ {0.089, 0.315}, knee 3×10⁻⁴–0.16 M_☉/pc³, every point worse than MOND) plus RG at its published, unrefitted parameters: an executed grid, not a class theorem; gradient-based schemes such as symmetron screening and non-local state variables such as enclosed mass are not covered and are not claimed to fail here) fails in three independently executed ways — head-to-head on the same SPARC points against the acceleration-keyed form, density keying loses at ΔBIC +2843 with γ free (best-fit γ → 0.046, i.e. the fit switches its own density dependence off; +142 after effective-N deflation; see Honest Assessment); the cross-system ρ↔g_bar offset is ~1.7 dex; and clusters require a ρ_crit 10⁴–10⁶× off the galaxy calibration (Coma, four ansätze, one structurally bounded at velocity ratio ≤2 vs observed 4.6). The sign statement below (ρ_t ∝ V⁻² required by the BTFR, ρ_crit ∝ V⁺² asserted) is the data-free reason all three fail. Not a route against density keying: the SPARC RAR ΔBIC = +184 (removed from this list 2026-09-14; see revision notes). That fit keyed the compander on acceleration (C as an implicit μ on g_obs, γ pinned at 2), so it refutes the γ = 2 pin in the acceleration-keyed realization, not density keying. Nor is the environment run (r² = 0.0001): Honest Assessment reclassified it on 2026-09-05 as refuting a registered amplitude that is consistent with the equation's own tiny ambient-density lever.

Companion result, no data required (added 2026-09-07, visitor researcher persona): the framework's galaxy sector is not local either, so the no-go above is not a contest between a local theory and a non-local phenomenon. ρ_crit = A·V_flat² keys the coherence threshold to an r → ∞ quantity (BTFR-fixed by total baryonic mass), and B_max = 1/Ω_m caps a per-galaxy boost with a cosmological parameter. Both are Predictive Closure violations under the framework's own MRH definition. The sector needs the one non-local variable it lacks (g_bar) and uses two it should not have. This also supplies the mechanism behind artifact 4 below: at SPARC-sampled x, C ≈ γρ/(A V_flat²), so γ and A enter only as γ/A — one number per galaxy, which is what ρ(lnγ, lnA) = +1.000000 measures. Full statement on MRH.

Why this is the whole difference, not one failure among six (added 2026-08-02): Tier 1 defines f_DM = 1−C, so C is the interpolating function μ by definition. And at γ=1/2 — SPARC's free fit lands at 0.489, 2.2% away — C(ρ) = x/(x+2) = μ_simple(x/2) identically (see Coherence Function). Put those together: the galaxy sector is MOND, with μ's argument swapped from acceleration to local density ρ. (Precision, 2026-09-08, see revision notes: in MOND μ is keyed on g_obs, and the SPARC fit that produced γ = 0.489 used C exactly that way — as an implicit μ(g_obs/a₀′), solved for g_obs; see Coherence Function for the script line. The fit swapped nothing; the swap to ρ is the framework's stated law, which is what the no-go tests.) That single substitution — the one this no-go quantifies — is not one result among the site's six executed refutations. It is the entire content of what makes this framework different from MOND in the galaxy sector. Everything else downstream (BTFR slope, DM-fraction ceiling, RAR shape) is a consequence of that one swap failing.

Scope split — which half generalizes (2026-07-08): the three numbered results carry different generality. The sign statement below (ρ_crit must scale as V⁻²) is BTFR-forced and profile-independent. It holds for local-ρ schemes that modulate the force algebraically (C(ρ)·g), not for every local-ρ MOND mimic; see the counterexample paragraph below (scope corrected 2026-07-27, applied to this paragraph 2026-09-23). The ΔBIC +2843 ensemble rejection is specific to the log-density compander family actually tested (C = tanh(γ ln(1+ρ/ρ_crit))); other local-density functional forms would need their own ensemble runs. The cluster/offset arguments sit in between: the ~1.7 dex ρ↔g_bar mismatch is a property of the data geometry, but its magnitude was computed for this framework's calibration. Cite the sign statement for generality; cite the ΔBIC +2843 for this family (and +184 only for the γ = 2 pin in acceleration keying).

One-line citable form — the sign statement (2026-07-02): a knee keyed on local volumetric density must fall as V⁻² to track an a₀ acceleration threshold (forced by the BTFR: r_t ∝ V², M ∝ V⁴, so ρ_t ∝ a₀²/GV²); the framework asserts ρ_crit ∝ V⁺² — inverted sign, 240×–300,000× magnitude error growing with V. This is profile-independent within the family it covers. Measured since (2026-08-27; archive ledger row restated 2026-08-28): on SPARC the knee carries no velocity exponent at all; V⁺² is excluded at about 11σ. The refutation stands as a wrong exponent. “Inverted sign” and the 240×–300,000× figures are the pre-measurement statement, kept here because this is its citable form. Scope (corrected 2026-07-27; see revision notes): the result does not hold for any local-ρ MOND mimic — see the counterexample paragraph below. It holds for local-ρ schemes that modulate the force algebraically (C(ρ)·g), which is the class C(ρ) belongs to. Detail on Parameter Derivations.

The usable lemma is a locality triage: the discriminating axis for the contemporary emergent-gravity wave is the locality of the modification's state variable. Keyed on local ρ(r) (any “coherence/information/entropy volumetric-density → gravity” scheme): killed by the above. Keyed on non-local variables — Verlinde's enclosed M_B(<r), MOG's enclosed mass, MOND's acceleration or surface-density Σ relations — explicitly escapes. This sorts proposals before fitting.

Positioning against the relativistic completions (added 2026-07-03): the non-local escape route this no-go identifies is not hypothetical — it is the one the published relativistic MOND completions already take. AeST (Aether-Scalar-Tensor; Skordis & Złośnik, PRL 127, 161302 (2021)) reproduces MOND phenomenology from non-local field structure and passes CMB and linear-LSS constraints — the existence proof that the escape class is viable, modulo ongoing stability/ghost analyses of that theory. Read together: the local-density shortcut is quantitatively dead (this result); the surviving program is the AeST-class completions (the field's current frontier). A referee evaluating any new “density → gravity” proposal can use the triage above to decide which side of that line it falls on before fitting anything.

Known escapes from this no-go (added 2026-07-10): AeST is not the only published escape, and the second one is the more instructive because it keeps a local density criterion. Superfluid dark matter (Berezhiani & Khoury, PRD 92, 103510 (2015)) has dark matter Bose-condense in galaxies above a local density/temperature threshold — the same core intuition as this framework — but the MOND-like force is mediated by the condensate's phonons, so the force scale enters independently of the switching criterion. That separation is exactly what C(ρ) lacks: it conflates the switch and the force into one local function, and that conflation — not the density threshold per se — is what the data kill. (Whether the BTFR-forced V⁻² argument constrains the superfluid's condensation threshold is an open check, queued — the expectation is no, because the phonon coupling carries the a₀ scale separately.) Escape taxonomy: non-local theory (AeST-class), local-criterion-gated medium (superfluid-DM-class), or differential local-density coupling (symmetron-class, added 2026-07-27 — see below); algebraic local-density modulation, the class C(ρ) belongs to, remains excluded. (An unsourced attribution to the screening literature was withdrawn from this paragraph 2026-07-27; see revision notes.)

Counterexample found — the generality claim is withdrawn (2026-07-27): the screening-literature vein flagged as unwalked on 2026-07-23 has now been walked, and it contains not a precedent but a counterexample. Burrage, Copeland & Millington, Radial acceleration relation from symmetron fifth forces, PRD 95, 064050 (2017) [erratum PRD 95, 129902, conclusions unaltered] reproduce the RAR for the 153-galaxy SPARC sample with no cold dark matter, using a scalar keyed on the local volumetric density ρ(r), with the Lagrangian parameters (μ, M, λ) held universal across the sample. The mechanism is the distinction this page was missing: their extra force is the gradient of a function of ρ — g_sym(r) = (c²/2) d/dr [(φ(r)/M)²] — not a multiplier on g, and differentiating a density profile is exactly what lets a density-keyed field track an acceleration-keyed relation. They obtain a closed form in acceleration variables: g_sym = g_bar/(exp √ (g_bar/g†) − 1), g† ≈ 1.20×10⁻¹⁰ m s⁻². Corrected statement: local volumetric density is not excluded as an organizing variable for the RAR; what is excluded is algebraic local-density modulation of the force. Density-keyed schemes do run into trouble elsewhere — the symmetron cannot supply the lensing mass without extending the field content (Burrage, Copeland, Käding & Millington, PRD 99, 043539 (2019)), and BCM 2017's own caveat is Solar-System tension in the unscreened regime (partly retired by O'Hare & Burrage, PRD 98, 064019 (2018)) — but those are different refutations than the one this no-go asserts, so they do not rescue its generality. A no-go dies to one existence proof of the thing it forbids.

The live discriminator context — the External Field Effect (added 2026-07-24): the sharpest currently-contested MOND-vs-ΛCDM discriminator is the claimed detection of MOND's External Field Effect in SPARC (Chae et al. 2020, ApJ 904, 51; 2021 — ~4σ, contested: Paranjape & Sheth 2022 show an EFE-like signal is generically expected in ΛCDM; Freundlich et al. 2022 find no EFE in Coma-cluster ultra-diffuse galaxies, a different sample). It bears on this no-go from the positive side: the EFE keys on external acceleration — a strictly non-local variable — while the framework's registered ambient-density environment effect, run as registered (2026-07-14), shows no trace (r² = 0.0001). If environment enters galaxy dynamics at all, it enters through the non-local variable class this no-go predicts must win; if the EFE detection dissolves into ΛCDM systematics, environment enters through neither variable and the no-go is untouched. No branch rescues a local-density coupling. (Whether Chae's estimators can in principle separate external-acceleration from ambient-density coupling is queued as a research check.)

The field-equation completion, stated carefully: the algebraic g = g_bar/C(ρ) has a field-equation completion, ∇·[C(ρ)∇Φ] = 4πGρ, which is linear in Φ. It conserves momentum only in spherical symmetry or with the variational striction force included. The net self-force is −(1/8πG)∫|∇Φ|²∇C d³x, and striction cancels it exactly. Linearity gives superposition, not EFE = 0: an external field is refracted wherever ∇C ≠ 0, giving a linear EFE of order g_extΔC/C without striction and a nonlinear one with it. EFE = 0 exactly holds for the algebraic reading only. Details and the floored/unfloored scope: MOND Unification. (Two overstatements corrected; see revision notes.)

Prior-art audited (2026-07-23): the Milgrom surface-density and modified-inertia corpus (1983 trilogy; astro-ph/0510117 including its worked examples; the Σ^†/central-surface-density line through Milgrom 2016; Famaey & McGaugh 2012; Banik & Zhao 2022) and the modern variable-comparison literature (McGaugh 2004; Lelli et al. 2017; Stiskalek & Desmond 2023) were searched for a prior quantified instance of local volumetric density failing as the organizing variable. None exists in that corpus (scope added 2026-07-27; see revision notes. A prior-art null is bounded by its corpus and must be stated that way): every published discrimination in the MOND variable-comparison literature tests acceleration against radius, orbital frequency, size, or surface density — ρ(r) was never in any tested variable set (Stiskalek & Desmond's 2023 feature sweep is the nearest miss: three surface densities, no volume density). The argument template of the sign statement is Milgrom's own — length-keyed modifications were excluded in 1983 for forcing Tully-Fisher slope 2 instead of 4 (1983b §III; Sanders 1986 for the direction form) — and the volumetric-density instantiation quantified here appears to be the first for the algebraic-coupling class. Caveats: Milgrom 1983b was verified through secondary witnesses (the archival scan is image-only); the screening-literature vein was walked on 2026-07-27 and overturned the unqualified form of this null — see the counterexample paragraph above; and the triage lemma is prior art independently of the quantification — the screening literature has classified modified-gravity mechanisms by which derivative of Φ keys the modification (Φ: chameleon/symmetron/dilaton; ∇Φ, i.e. acceleration: k-mouflage; ∇²Φ, i.e. density by Poisson: Vainshtein) since Joyce, Jain, Khoury & Trodden, Phys. Rep. 568, 1 (2015), a strictly finer split than this page's two-way local/non-local version, and one that files the viable Vainshtein/Galileon class under “local density” — including the working relativistic MOND of Babichev, Deffayet & Esposito-Farèse, PRD 84, 061502(R) (2011).

Untested: the time-of-flight face (dim ≥ 5), which is structurally unreachable. Audited-Negative: the preferred-frame / dim-4 face, a naturalness gap of 16–28 OOM with custodial escapes unexhibited.

Every other prediction in this framework either (a) reproduces MOND/ΛCDM/Zurek without discriminating content, or (b) faces a decisive constraint. The one channel that isn't MOND-or-Zurek is the substrate-discreteness Lorentz-violation signature (LIV). Its status has two faces — one that is genuinely unreachable, one that carries a severe naturalness gap under existing experiments:

Face 1 — Time-of-flight (dim≥5): three locks, genuinely unreachable

1. Symmetry forbids the refutable channel (under assumed reflection-symmetric substrate). The substrate dispersion relation is even in k (lattice reflection symmetry), so the leading LIV correction is quadratic (n=2) — no linear (n=1) term. Current GRB time-of-flight bounds put n=1 LIV above the Planck scale (LHAASO GRB 221009A 2024: E_QG,1 > 5.9 E_Pl; Fermi-LAT: > 7.6 E_Pl).Note: Lock 1 holds only if the substrate has reflection symmetry. A parity-breaking discretization restores the n=1 term — which is already refuted. The choice of even-k symmetry is an assumption, not a derivation.
2. The same symmetry pins the prediction below reach. n=2 sits at E_QG,2 ≈ E_Pl, while the current bound is E_QG,2 > 6×10⁻⁸ E_Pl (LHAASO 2024) — a gap of ~10⁷. Foreseeable GRB instruments gain factors of a few per event; closing 10⁷ is not a foreseeable-instrument prospect.
3. Non-uniqueness makes it non-confirming even if reached. n=2 discrete-substrate LIV is generic to LQG and causal-set models; a detection would read as “quantum gravity,” not as Synchronism specifically.

Face 2 — Preferred frame + dim-4: CPSU 2004 naturalness gap (c_μν ~ 10⁻² at one loop, 16–28 OOM above bounds; custodial escapes unexhibited)

A discrete substrate with absolute time picks out a preferred (CMB-rest) frame. Preferred-frame physics shows up at dimension four — the renormalizable, non-Planck-suppressedlevel — via two channels:

• Species-dependent limiting speed (SME c_μν). A dim-4 operator ω²=m²+(1+c_LIV)k² is even in k — the even-k symmetry does not forbid it. Existing bounds: |c_LIV| ≲ 10⁻¹⁸ (cavity Michelson–Morley rotating resonators) to ~10⁻²²–10⁻²⁹ (Hughes–Drever clock-comparison experiments). These are tabulated annually in the SME Data Tables (Kostelecký & Russell, Rev. Mod. Phys. 83, 11 (2011), updated 2024). The framework's absolute-time / universal-clock commitment gives up boost invariance — the minimal custodial protector against radiative generation of this coefficient (Collins, Perez, Sudarsky, Urrutia & Vucetich, PRL 93, 191301 (2004)). Verdict: severe naturalness/fine-tuning gap (CPSU 2004). Computation (2026-06-26 explorer): tree-level c_μν=0 by single-substrate universality (no species-dependent k² coefficient at tree level). One-loop correction is UV-dominated and Planck-cutoff-independent: c_μν ~ α/π ~ 10⁻²–10⁻³. Existing bounds: ≲10⁻¹⁸ (cavity MM), ≲10⁻²⁹–10⁻³⁰ (nucleon comagnetometer). Fine-tuning gap: 16–28 orders of magnitude. Two standard perturbative custodial escapes exist in the literature but are unexhibited in this framework: (1) SUSY (Groot Nibbelink & Pospelov, PRL 94, 081601, 2005): forbids dim-3/4 LV operators entirely; (2) Anisotropic scale-hierarchy (Pospelov & Shang, PRD 85, 105001, 2012): demonstrated for Hořava–Lifshitz gravity (the canonical absolute-time class) — separates the LV-generating scale from the electroweak scale via Λ_HL ≪ M_pl, suppressing SM-sector LV perturbatively without restoring boost invariance. The framework adopts neither: single-substrate universality is itself the obstacle to a Λ_HL ≪ M_pl scale separation. Status: open custodial-mechanism gap, not a closed refutation.
• Velocity-anisotropic phase-transition threshold (Phase-16, 2026-06-24). If identity thresholds are fixed in the substrate/CMB frame, Earth's motion (v ≈ 370 km/s, β ≈ 1.2×10⁻³) produces a fractional-frequency anisotropy: dipole ∼β ≈ 1.2×10⁻³, quadrupole ∼β² ≈ 1.5×10⁻⁶. Cavity-MM and clock-comparison experiments bound this at 10⁻¹⁸–10⁻²². The prediction is 11–19 orders of magnitude above existing limits. Escape: co-moving thresholds — survivable but non-novel (recovers standard SR with zero new content).

These are two distinct LIV operators: Face 1 (dim≥5 dispersion, time-of-flight) and Face 2 (dim-4 SME c_μν, preferred-frame). Reading the badges together — “structurally unreachable” and “naturalness gap” — is not a contradiction. The protected channel cannot discriminate (non-unique, ~10⁷ below reach); the preferred-frame channel carries a severe fine-tuning requirement. The framework's absolute-time commitment is the source of both.

Net: the time-of-flight face is symmetry-protected and unreachable (three locks stand). The preferred-frame / dim-4 face carries a severe naturalness gap (c_μν ~ 10⁻², 16–28 OOM above existing bounds; CPSU 2004) — and it is the framework's defining commitment (absolute time) that generates it. This is the framework's only non-MOND, non-Zurek channel. Standard perturbative escapes (SUSY, anisotropic scale-hierarchy) exist in the literature but are unexhibited in the framework; without one, the naturalness gap is unresolved.

Research lineage: PREDICTIONS.md B7 / Phase-12 / Phase-13 / Phase-16 in the Synchronism research archive. Dim-4 c_μν coefficient computed (2026-06-26): radiative value ~10⁻², 16–28 OOM above existing bounds (CPSU 2004 naturalness problem, not a data-driven refutation). For the framework to survive this channel, an unexhibited custodial mechanism must be specified — known candidates are SUSY (Groot Nibbelink & Pospelov 2005) and anisotropic scale-hierarchy (Pospelov & Shang 2012, demonstrated for Hořava–Lifshitz). This is the framework's most severe naturalness constraint (framing corrected per 2026-06-30 explorer audit; see revision notes).

Failed: S ≤ 2 in every construction built (executed 2026-06-21 / 2026-07-06). The local arm stays at 2 because it is local (factorisable): that is Bell's theorem. No-signaling alone would allow up to 2√2 (quantum) or 4 (a PR box). The two nonlocal arms stay at 2 for a construction reason: their grid mixing turned out to be equivalent to relabelling the measurement angles. (Wording corrected 2026-09-25; it previously read “every no-signaling construction”, which a graduate-physics reader correctly flagged as wrong: the bound comes from locality, not from no-signaling.) It is counted among the 6 as a theorem-level check, not as an executed refutation of the substrate class.

The framework's substrate ontology was tested against Bell by direct CHSH simulation (kuramoto-lattice-suite, research-ledger bet B1), with freely chosen settings and measurement only through observer-pattern phase-lock. Results: no construction exceeded the local bound S = 2 without signaling (quantum mechanics reaches 2√2 ≈ 2.83): local construction S = 1.98; nonlocal-grid construction stuck at the bound, S ≡ 2.00, at every coupling strength with zero signaling (a uniform shared phase is gauge-equivalent to relabeling the measurement angles — smooth single-grid mediation stays local-realist); a global-clock construction reaches S = 2.67 only by also signaling. The 2026-07-06 run extends the cap to the framework's own saturation-gated Intent-density substrate (S = 1.85 ≤ 2, no signaling). On the local constructions the cap is Bell's theorem for any real-valued local-realist model, not an artifact of the phase substrate. Scope, matching Honest Assessment (2026-09-11): the local arm illustrates Bell's theorem and adds nothing to it. The two nonlocal constructions built (the grid, whose setting dependence is a relabeling, and the clock, which exceeds 2 only by signaling) are construction nulls, facts about those constructions and not about the substrate class. Hidden-communication substrates were not built and remain untested. Toner & Bacon (PRL 91, 187904, 2003) reproduce singlet correlations from local variables plus one hidden bit per trial, with no observable signaling. Nothing here is a theorem about nonlocal substrates. The triptych A (real-local) = 2 < B (Born-rule cos² projection) = 2√2 < C (PR-box) = 4 localizes the Tsirelson value as the fixed point of the projection law — reachable only by importing Hilbert-space structure wholesale.

Honest classification (sharpened 2026-07-18, external researcher review): the number is a corollary, the lesson is what transfers. S ≤ 2 was the guaranteed outcome for any no-signaling local-realist construction — Bell's theorem functioning as designed — so the simulations are pedagogy plus a registered bet honestly lost, not a new result. What transfers is the worked lesson: “our substrate is nonlocal by construction, so Bell doesn't apply” is a recurring move in emergent-QM proposals, and this is a minimal executed counterexample — declaring the substrate nonlocal does not evade Bell, it chooses the nonlocal horn, and the construction must then produce a non-relabelable, conditionalsetting-dependence (as Toner–Bacon's hidden bit does) or stay at S ≤ 2. Cite it for that lesson, not for the number. Full construction detail and the open Born-rule problem on Two Reframes.

Audited-Negative: executed 2026-09-03; the statement does not depend on the framework.

Statement. For any algebraic modification of gravity keyed on local density, g_obs = g_bar/ε(ρ) with a saturating ε whose knee ρ_c is calibrated above galactic midplane densities (here ρ_crit ≈ 10³ M_☉/pc³ at V = 200 km/s against ~0.1 M_☉/pc³ in the solar neighbourhood), rotation-curve data sample only x = ρ/ρ_c ≪ 1, where ε equals its own linearisation to better than observational precision. The shape parameter and the knee normalisation then enter only through their product: the model has one measurable number, and it is the floor.

Numbers (SPARC, explorer 2026-09-03). SPARC samples x at a median of ~7×10⁻⁵; C = γx holds to 1.8% at worst, 0.22× the data's precision on C. Fisher correlation ρ(lnγ, ln A) = +1.000000. Moving γ by 40× and A by 10⁹ changes χ²/N by 1.07–2.7×; moving the boost floor changes it by 6.35×10⁴. The galaxy data measure the ceiling and nothing else.

Transferability and the counter-example. This applies to any ε(ρ) model with a knee above ~10⁻² M_☉/pc³. Refracted Gravity (Matsakos & Diaferio 2016; Cesare et al. 2020) is the contrast case: its permittivity ε(ρ) is the same family as this framework's C_Ω (the floored x/(1+x) form is RG at Q = ½ exactly; the tanh-log compander matches it only at γ = Q = ½ with the knee rescaled 2×; qualifier added 2026-09-16) but its knee is fitted insidethe sampled range, so its shape parameters are identifiable and were measured. Same equation, one identifiable and one not, decided entirely by where the knee was placed — which is the citable point.

Honest novelty statement: the identifiability logic is standard (a linearised model cannot constrain its nonlinearity); what is new is the executed Fisher analysis on the specific framework and the placement of Refracted Gravity as the identifiable member of the same family. A researcher persona (2026-09-06) called this “a two-page note, not a subsection” (promoted 2026-09-06; see revision notes). The full Fisher matrix is not yet written up as a standalone artifact — seeded to the explorer track.

Audited-Negative: executed 2026-09-07 on the Baumgardt & Hilker 2018 catalogue. The bound does not depend on the framework.

Citable form: for any theory of the form g = g_N/C(ρ) with a floored tanh-in-log-density switch (the same family as Refracted Gravity's permittivity, but identical to it only at γ = Q = ½ with the knee rescaled 2×: below the knee the tanh-log form is linear in ρ at every γ, while RG goes as ρ^2Q),evaluated under L2 dynamics (algebraic, or the field equation without striction), on each cluster in isolation, with no smoothing (D ≲ 10 pc), without potential escapers, and on a post-hoc ratio criterion with no σ, the outer velocity-dispersion slopes of 42 Galactic globular clusters exclude a knee density ρ_c ∈ 0.1–300 M_☉/pc³ (6.8×10⁻²⁴ – 2×10⁻²⁰ g/cm³) at a switch sharpness γ = 0.489 (Hill index 2γ ≈ 1 in (1+ρ/ρ_c), not in ρ/ρ_c), narrowing to 0.5–100 M_☉/pc³ at γ = 2. “Excluded” = slope mismatch more than twice that of MOND with the external field effect, which these data admit (−0.093 vs Newtonian −0.057). Stated without a ratio, that is a mean outer-slope residual beyond ≈ 0.19 in magnitude; each mean carries ±0.027 (statistical only). Citable numbers, residual form (2026-09-24): density law −0.196 to −0.199 ± 0.027 after refitting the mass scale under the law itself (−0.211 on catalogue masses); MOND+EFE −0.093; MOND without EFE −0.245; Newton −0.057 (King-like) or +0.021 ± 0.027 (Plummer). There is no systematic budget. Under L3 (striction included) this band does not hold: it becomes 0.031–0.196 M_☉/pc³ at γ = 0.489, with no passing knee, and the outer shells stop being bound (caveat iii below).Refracted Gravity is not excluded: its published ρ_c = 10⁻²⁴·²⁵ g/cm³ = 0.0083 M_☉/pc³ lies below the excluded window, and every value in that set's 1σ range (0.0052–0.0158 M_☉/pc³) scores within MOND+EFE's mismatch on the same statistic. Provenance (corrected 2026-09-16, explorer 2026-09-15): 0.0083 (with ε₀ = 0.089, Q = 0.47) is RG's elliptical-galaxy (E0) calibration, Cesare et al. 2022 (arXiv:2102.12499), not the disc one. RG's DiskMass disc calibrations are ρ_c = 7.4×10⁻⁴ (mean; ε₀ = 0.56, Q = 0.92) and 4.3×10⁻³ M_☉/pc³ (joint fit; ε₀ = 0.661, Q = 1.79). Both also sit below the excluded window, so no verdict changes. Two open caveats on the window itself (added 2026-09-16): (i) the clusters were treated as isolated. Under the field equation ∇·[C∇Φ] = 4πGρ a cluster whose knee lies inside it refracts the Milky Way's field, a term these rows do not contain (see the next paragraph); (ii) the window assumes density is read pointwise or smoothed over ≲ 10 pc, and at that smoothing the same equation makes halo clusters orbit at ~0.6 of the field acceleration felt by diffuse tracers, which Milky Way cluster and star kinematics disfavour at ~2σ (post-hoc). (iii) The window is an L2 object. Under the action, with striction included (L3), the knee shell is 1.7–13× gravity in striction with net outward force. 18 of 27 verdicts on the γ = 0.489 row change, the excluded band becomes 0.031–0.196 M_☉/pc³, no knee passes, and with Plummer tails up to 39 of 42 clusters have outward net gravity (explorer, 2026-09-16). (iv) Potential escapers (energetically unbound stars inside r_t; Küpper et al. 2010; Claydon, Gieles & Zocchi 2017) are the known Newtonian flattening of outer σ(r). They are not in the mock. (v) Not prospective: Session 611 registered γ only, and the knee, floor and dynamics were chosen after, and they span pass to excluded. The criterion is a post-hoc ratio with no σ. (vi) Newton-conditioning through the catalogue mass: checked 2026-09-19, small. A researcher reader objected that the Baumgardt & Hilker masses are N-body fits to these same σ(r) profiles under Newton, so every alternative is graded against a Newton-tuned model. The mechanism is real: the outer slope is mass-independent under Newton only, and the published run used catalogue M with no free scale. Pre-registered control (site d9d7beb, identity reproduces −0.057 / −0.093 / −0.245 / −0.211 first): refitting one mass scale per cluster under each law moves the density-law residual −0.211 → −0.195 and its ratio 3.73× → 3.45× (−7%); MOND without EFE 4.33× → 4.26×; MOND+EFE 1.65× → 1.67×. The reason is visible in the fitted scales: the inner bins sit far above the knee where every law is Newtonian, and they pin the mass (density-law 〈f〉 = 1.28 against Newton's own 1.36). Swapping the N-body half-mass radius for the photometric half-light radius moves the density residual −0.199 → −0.196. Pre-fixed verdict: the “manufactured exclusion” reading is refuted at this level. Two of the maintainer's five registered predictions failed (the fitted scale is above 1, not 0.35–0.8; and under Plummer the ratio moved 46% because its Newtonian denominator, +0.021 ± 0.027, is consistent with zero). That second miss is the useful one: the ratio-to-Newton statistic is unstable wherever Newton fits well. The residuals carry a statistical error (±0.027 on each mean; the density law is 7.8 stat-σ from zero, Newton 2.1); what has no number is the systematic budget. Still unaddressed by this control: escapers, anisotropy, L3, and the N-body-derived r_c of the King-like model. Cite it as an L2 window, not as an exclusion. This sits beside the B_max ≲ 14 bound from SPARC dwarfs as the second class-level constraint this program has produced.

Why globular clusters, and why the EFE is the discriminating variable: read algebraically (g = g_N/C(ρ), exact for an isolated spherical cluster), a density-keyed law has no external field to appeal to. Scope (added 2026-09-16): under the field equation that is no longer true. The law is linear in Φ, so the Galactic field enters by superposition, refracted by the cluster's own C profile: a linear external-field dependence, not MOND's nonlinear one. Its size inside real clusters has not been computed on this data (the maintainer's order-of-magnitude check is in maintainer/scripts/gc_external_field_refraction_estimate.py). MOND with its EFE switched off scores −0.245 on the same statistic, statistically indistinguishable from the density law; MOND survives these clusters because of the EFE and for no other reason. So the clusters separate density-keyed from acceleration-keyed gravity on the profile shape even where the two predicted boost amplitudes are collinear (Pearson r = +0.87 across the population, because outer-halo clusters are also the diffuse ones). Prior-art screen: Cesare et al. 2022 used globular clusters as tracers of elliptical-galaxy potentials, not as self-gravitating systems crossing the knee. Honest scope: the ± is statistical only. Quote the residuals, not ratios to the Newtonian residual: under a Plummer profile Newton's own residual is +0.021 ± 0.027, consistent with zero, so ratios to it swing from 3.7× to 9× while the density residual stays put (corrected 2026-09-24; this line previously said the opposite, and Honest Assessment had the right form). Tides and the contested clusters: dispersion bins beyond 0.98 of the catalogue tidal radius were cut, and constant anisotropy was scanned (β from −0.6 to +0.8). Tidal heating and unbound extra-tidal stars inside r_t were not modelled. The ≥ 10-bin cut leaves 42 clusters at R_GC ≤ 22 kpc. That removes the remote, isolated clusters where MOND is actually contested: NGC 2419 (Ibata et al. 2011; Sanders 2012), Pal 14 and Pal 4, with 1–3 dispersion bins each in the catalogue. The window rests on inner-halo clusters. What this bound does not do (scope correction 2026-09-09): it does not combine with the solar-neighbourhood (Oort-limit) constraint into a joint no-go. Evaluated at the same γ, the two windows overlap everywhere from γ = 0.3 to 3: the clusters call the bottom of the solar window ok and its top marginal, leaving a joint window at ρ_c ∈ 0.0039–0.0079 M_☉/pc³ (γ = 0.489) and 0.0735–0.078 (γ = 2), sliding as e^1/γ between them (corrected 2026-09-09; see revision notes). Quote this window with its γ attached; it is meaningless without one. Full result, robustness table and the γ fork it opens for this framework specifically: Honest Assessment.

Same badge and descriptor as onTier 1 andHonest Assessment. The DR2 registration as adopted is no fix: its kill branch fires with under 1% probability if ΛCDM is true (2026-09-21), so the likely DR2 outcome is “underpowered” again unless it is re-registered before the data. It is the Session 107 mechanism, not the current dark-energy sector (see below).

TEST-04a predicted σ₈ ≈ 0.76 (calibrated to the then-live S8 lensing tension); DESI DR1 full-shape combined σ₈ = 0.841±0.034 (Table 10) — a 2.4σ amplitude tension on σ₈. Correction (2026-07-14): the criterion was registered on a different statistic, fσ₈(z=0.51) > 0.46 for >3σ. On that statistic directly — LRG1 fσ₈ = 0.474 × 1.16±0.062 =  0.550±0.062 — the threshold is exceeded by only ~1.5σ, short of the registered >3σ bar. σ₈ is inferred assuming GR growth kernels, so using it to falsify a modified-growth model risks circularity. DESI's own purpose-built modified-gravity analysis, Ishak et al. arXiv:2411.12026 (JCAP 09 (2025) 053, previously uncited here), gives μ₀ = 0.11 (+0.45/−0.54) from DESI alone — a 12% fσ₈ suppression maps to a μ₀ inside that 1σ band. Honest reading: the test as registered lacked the power to discriminate this framework from GR — the σ₈ comparison is real but is not the registered kill. A single bin (LRG1, z=0.51) also shows fσ₈ growth above fiducial (ratio 1.16±0.13, ~1.2σ), and the DESI DR1 full-shape RSD ensemble growth index γ_growth ≈ 0.58, above GR's 0.545, leans mildly toward suppression — the predicted direction. Caveats: the prediction was post-hoc (σ₈ calibrated to the S8 lensing tension, which has since receded), the μ₀ projection above has not been formally run (seeded as an explorer topic), and the verdict is frozen at DR1. Provenance: the 0.418 came from Session 107's G_local/G_global mechanism, not from the framework's current dark-energy sector. That sector is background-only, and its one derived perturbation channel shifts fσ₈ by about −0.22% (see Dark Energy). So this negative applies to that mechanism class, not to the current sector. What transfers: any framework whose mechanism damps late-time structure growth to match a receded S8 tension sits in the same disfavored-on-σ₈ box, though this specific DESI test does not discriminate it from GR.

Currency: verdict anchored to DESI DR1 (arXiv:2411.12021) as of 2026-07-02. DESI DR2 full-shape parameter papers are not yet published (expected ~Spring 2027) — note this is now a narrower claim than “no DR2 full-shape results exist”: preliminary DR2 full-shape results were presented publicly in April 2026 (“Cosmology with DESI DR2: From BAO to Full-Shape Clustering,” PIRSA:26040071) and DR2 Lyα full-shape validation posted late July 2026 (arXiv:2607.27411). Integrity note (2026-08-01): this repository's git history shows the fσ₈(z≈0.5) ≤ 0.46 threshold was committed 2026-07-01, after the April PIRSA talk — so this criterion cannot be verified as pre-registered relative to that preliminary presentation, only relative to the still-unpublished formal DR2 full-shape paper. If DR2 full-shape reports fσ₈(z≈0.5) ≤ 0.46, the verdict unfreezes for re-adjudication; any DR2 value above DR1's would deepen the disfavor, not relieve it. Whoever re-adjudicates this should check first whether a threshold-relevant fσ₈(z≈0.5) figure was already circulating by 2026-07-01 — if so, this criterion has the same post-hoc exposure already documented above for the σ₈ comparison.

Untested: the experiment that would decide it has not been run. The one control benchmark that exists (3 reparametrizations, 6 discoveries) is retrospective and in-distribution, and a single model that knew every answer scored it.

Honest novelty statement: the protocol is assembled prior art (AI Safety via Debate, Irving, Christiano & Amodei 2018; role structure from CAMEL/MetaGPT). What would be new is a measured answer to the question in the heading.