For Researchers: What's Citable

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 this framework belongs to; 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 — the SPARC RAR ensemble rejects the density-compander at ΔBIC=+184 (conservative ≥+33) (free fit collapses to MOND); 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).

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 the enclosed-mass acceleration g_bar to local density ρ. 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 any local-ρ MOND mimic. The ΔBIC=+184 (conservative ≥+33) ensemble rejection is specific to the log-density compander family actually tested (μ = tanh(γ ln(1+x))); 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 for this family.

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. Scope corrected 2026-07-27: an earlier version of this line claimed the result holds for any local-ρ MOND mimic. It does not — 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. Attribution withdrawn 2026-07-27: this paragraph previously stated that “the ρ-vs-g_bar mismatch is the standard reason screened scalars don't reproduce MOND.” That attribution was never sourced, and the screening literature does not make it. Hinterbichler & Khoury, PRL 104, 231301 (2010) do not mention MOND, rotation curves, or dark matter anywhere; Joyce, Jain, Khoury & Trodden, Phys. Rep. 568, 1 (2015) mention MOND once, attached to the acceleration-keyed (k-mouflage) screening class as a construction tool — the opposite valence; and the published reasons chameleon f(R) fails on SPARC are the f_R0 spread and core/cusp degeneracy (Naik, Puchwein, Davis & Arnold, MNRAS 480, 5211 (2018); MNRAS 489, 771 (2019)), not a variable mismatch.

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, disputed: Freundlich et al. 2022 and Paranjape & Sheth 2022 offer ΛCDM-consistent readings). 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.)

EFE = 0 survives the momentum-conservation objection (2026-08-04): the algebraic g = g_bar/C(ρ) modification above looks unsourced, but a field-equation completion exists (∇·[C(ρ)∇Φ] = 4πGρ) that conserves momentum and is linear in Φ — which preserves EFE = 0 exactly while also being why the exterior field of an isolated mass diverges as ρ → 0. See the full derivation on MOND Unification.

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 — the original wording said “none exists,” unqualified, which was false: the screening corpus, named as unwalked in this very paragraph, contains a counterexample. 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).

Honest novelty statement: the protocol is assembled prior art — adversarial pairs from AI Safety via Debate (Irving, Christiano & Amodei 2018), role structure from CAMEL/MetaGPT, failure modes from the multi-agent-systems literature. The citable artifact is the program-level null result with retrospective controls (N=6 audits, not preregistered held-out experiments; one corpus, one framework): 3,308 sessions produced ~47 internally-consistent candidate claims (1.4% survival rate); of those, 6 received external expert audit, and 0 survived. The honest denominator is 0 of 6 audited (0 of 47 pending audit) — not “0 across 3,308 sessions,” which would overstate the number of adjudicated trials by two orders of magnitude (corrected 2026-07-09 after two visitor personas independently flagged the site quoting this null at three different, mutually inconsistent denominators). Both error rates on the 6 are measured — temporal-asymmetry control 0/6 (later-demoted claims caught; median prior-art year ~1996), vocabulary-asymmetry 4/4 on the prior-art-rediscovery subclass, and specificity 0/6 (every held-out genuine discovery false-flagged — a measured false-positive rate of 6/6 = 1.0 on the control set — see Honest Assessment). Discrimination lives entirely in unautomated human novelty judgment.

Put the two rates together and the detector is uninformative: combined sensitivity (true-positive rate) is 6/6 = 1.0 (self-simulated upper bound, see above); specificity is 0/6, i.e. the false-positive rate is also 6/6 = 1.0. Youden's J = TPR − FPR = 1.0 − 1.0 = 0 (AUC ≈ 0.5). A classifier that flags every demoted claim and every genuine discovery carries zero discriminating information — both numbers were already published on this site, on two different pages, and had never been combined until a 2026-07-14 visitor pass did the subtraction. The 6/6 catch rate is not the methodology's best result; multiplied against its own specificity, it is the demonstration that the detector cannot tell a demotion from a discovery.

Three precisions on the above (added 2026-07-27, from an external-reviewer pass). (1) The positive class is “is a reparametrization.” Under the opposite convention the sensitivity and specificity labels invert; both are defensible, and J = 0 either way, but the convention was never stated and two expert readers have now read the numbers as swapped. (2) 6/6 = 1.0 is better described as a false discoveryrate — a true false-positive rate needs a true-negative count this positive-only design does not have, and J = 0 here is a degeneracy of the design rather than a measurement. (3) The program-level null is itself underpowered and now says so. On n = 6 with 0 survivors, the Clopper–Pearson bound admits a true novelty-survival rate up to ~0.39 (one-sided) or ~0.46 (two-sided). The site badges its galaxy tests “underpowered as registered” unflinchingly; holding the methodology verdict — the sector nominated as the real contribution — to a lower evidentiary standard than the physics was the asymmetry worth fixing. (4) J = 0 is a point estimate on n = 6 and n = 6, and should be quoted with its interval (added 2026-08-08, from a visitor researcher pass applying precision (3) to the detector itself). Clopper–Pearson at 95% gives sensitivity ≥ 0.541 and specificity ≤ 0.459, hence J ∈ [−0.46, +0.46] — an interval running from anti-informative to moderately informative. The point estimate is the right sign and the design degeneracy in (2) is the deeper problem, but quoting J = 0 bare on the page labelled “what's citable” is the same sin this site correctly flags on the consciousness demo.

Measured, not asserted: adversarial same-corpus AI pairs filter for internal consistency but cannot substitute for out-of-distribution evaluation. Relevant to anyone building AI-for-science generation pipelines on shared training corpora.

No control arm; a confound the null doesn't resolve (superseded by the correction above — retained for the record): the corpus audited is Synchronism itself, which this site independently concludes is a MOND reparametrization. “Adversarial same-corpus AI pairs cannot generate out-of-distribution novelty” and “this particular seed framework contained none to find” are observationally degenerate in a single-arm, single-corpus design — there is no measured human-pair survival rate on the same corpus to compare against. The generalization to AI-for-science pipelines generally is a hypothesis this design cannot license on its own.

Artifact status downgraded (2026-07-18, external researcher review): until the cross-vendor corpus control runs — the one registered experiment that would break the single-arm degeneracy above — this is a registered null with a stated path to becoming a citable artifact, not yet the artifact. What is citable today is the measured retrospective-control numbers (0/6, J = 0) and the confound analysis itself; the program-level generalization gates on the unrun control arm.

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. 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.

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–Gambini–Pullin, 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. Corrected framing per 2026-06-30 explorer audit: earlier versions mislabeled this “refuted” and “doubly-obstructed” — both overstatements corrected here.

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: local construction S = 1.98; nonlocal-grid construction 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): the cap is substrate-independent — Bell's structure theorem for any real-valued local-realist model, not an artifact of the phase substrate. 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 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.