An Intrinsic Big Bang Nucleosynthesis Fine-Tuning in Conserved-Charge Dark Matter from Relativistic MOND Clocks

A class of relativistic completions of Milgrom's dynamics keeps the modified-inertia sector inside a clock — a scalar φ entering only through Q = nμ∂μφ, with n the unit gradient of a primordial clock field — and supplies cosmological dark matter as the conserved Noether charge of the clock-scalar's shift symmetry, φ → φ + const. This architecture is attractive: the dark matter is then pressureless (cs2=0) and clusters like cold dark matter, it does not decay, and the same clock-internal structure lets the theory evade a preferred-frame post-Newtonian kill and a deep-MOND gradient instability. We prove this architecture carries an intrinsic Big Bang Nucleosynthesis (BBN) fine-tuning. On a Friedmann background the eliminated energy density is quadratic in the conserved charge C: ρ = B + 3M2H2 − (M2/3f2)(A + C/a3)2, so the same charge that sources the a−3 dust (cross term ∝ AC) necessarily sources a stiff a−6 component (square term ∝ C2). The stiff term grows as a−2 relative to radiation toward early times; BBN bounds it to Ωstiff,0 ≲ 4×10−25. Demanding the dust be the observed dark matter while the stiff term respects BBN forces |C|/|A| ≲ 3×10−24 — about twenty-four orders of magnitude, along a direction no symmetry protects. The fine-tuning is proved not removable within the healthy class: the ghost-free degeneracy that keeps the clock's propagating sector free of a Boulware–Deser mode forces the constitutive potential K(Q) to be exactly quadratic, which forces the density to be quadratic in C; a linear-in-charge (stiff-free) dust does not exist in the family; the tuning-free corner (A=0) has no dark matter; and enlarging A merely relocates an equal ~23-order tuning into the cosmological-constant sector (δCC = 2(ΩΛ/Ωc)(|C|/|A|)). Every algebraic step is reproduced in exact symbolic arithmetic by committed scripts, and the two load-bearing identities — the charge-quadratic density decomposition and the nonvanishing stiff coefficient −M2/3f2 — are additionally machine-checked in the Lean 4 proof assistant against its standard foundational axioms {propext, Classical.choice, Quot.sound}. What is not claimed: this is not a falsification — the theory is viable at the tuned point — and it is not a statement about whether nature is MONDian. It is a naturalness cost, one hierarchy worse than ΛCDM's, that any healthy conserved-charge MOND clock must pay unless a protecting mechanism absent from the present class is supplied. The dust's benefits are stated alongside the cost: it clusters like cold dark matter at linear sub-horizon order (σ8=0.811, no structure deficit), it cannot decay or annihilate, and the clock-internal structure that generates it also passes the preferred-frame post-Newtonian gate that excludes AeST. The construction whose fine-tuning is analysed here was developed collaboratively in the associated public repository; this paper isolates and proves the cost. Every dimensional number is carried on both a0 footings; the fine-tuning is a0-independent. AI-assisted research draft; not peer reviewed.