The no-phantom bound w ≥ −1

The no-phantom bound is the program’s requirement that the dark-energy equation of state w never fall below −1, the value of a cosmological constant. It holds because the program’s dark-energy field τ is an ordinary (canonical) scalar field, whose energy density can fall or hold steady as space expands but never rise, while w < −1, the phantom regime, would make it rise. The bound makes the dark-energy sector a sharp test: a robust measurement of an intrinsic w < −1 would kill the program’s framework outright. The second data release of the Dark Energy Spectroscopic Instrument (DESI) favours a phase with w < −1 at redshifts above about 0.4, so the bound stands against the direct reading of current data and is under real pressure; it survives only because fits without a phantom phase remain compatible and because w is inferred rather than observed.

Status: Alive, under real pressure — [D] in #1 §7 item 1, and part of checkpoint #11 §10’s [D + T] paragraph — not machine-checked; its test is row 4, and DESI’s Data Release 2 (DR2) leans against it (DESI DR2’s phantom crossing; root CLAUDE.md: “under real pressure ahead of DR3”).

The statement#

#11 §10, in the paragraph tagged Dark energy [D + T]: τ is hilltop quintessence on the Maslov circle, with “strictly w ≥ −1”. The lab notebook states and tags the bound (#1 §7 item 1, transcribed from its LaTeX):

w ≥ −1 always. τ is a canonical scalar; the null energy condition holds; phantom crossing is impossible. [D] Any robust measurement of w < −1 (as an intrinsic equation of state, not an artifact of an assumed ΛCDM background) falsifies this sector outright.

The handoff carries it among the live results as “thawing, w ≥ −1 always” (HANDOFF.md §2). The field is τ: hilltop quintessence on the Maslov circle.

The derivation#

The notebook’s argument has three steps, each one sentence.

1. τ is a canonical scalar. Its action is S_τ = ∫√−g[½(∂τ)² − V(τ)] (#1 §3, transcribed from its LaTeX): a kinetic term of the standard sign plus a potential, with no coupling to curvature and no second field.
2. The null energy condition holds. For such a field, uniform in space, the energy density is ρ = ½τ̇² + V and the pressure p = ½τ̇² − V, so ρ + p = τ̇², which cannot be negative. More generally, for any light-like vector k the stress-energy contracts to a square, T_μν k^μ k^ν = (k^μ ∂_μτ)² ≥ 0, which is the null energy condition [T] (the phantom divide and the null energy condition).
3. No phantom phase, no crossing. With ρ > 0, ρ + p ≥ 0 means w = p/ρ ≥ −1. Equivalently, from the continuity equation ρ̇ = −3H(ρ + p), τ’s energy density never grows while the universe expands. A w(z) that dips below −1 at any epoch is therefore impossible for τ, and so is a crossing of −1 in either direction.

Step 2 is textbook physics [T]; the program’s contribution is the premise, that dark energy is the canonical scalar τ, and #1 tags the result [D]. The derivation is short, and its key step does not depend on the shape of the potential: no choice of V can make τ violate the null energy condition, because V cancels from ρ + p. Turning that into w ≥ −1 needs ρ > 0, as step 3 states. It does rely on τ being the whole of dark energy. #11 supports that premise with the vacuum-energy identity: “§7.3 guarantees the wedge sector contributes nothing that competes with τ” (#11 §10).

What counts as a violation#

The bound concerns the equation of state of τ itself. A survey does not measure that directly; it measures distances and fits a model of w(z) to them (DESI baryon acoustic oscillations). A fitted w below −1 can arise without any intrinsically phantom component, for example when the assumed background model is wrong (the phantom divide). The documents draw the line accordingly.

• #1 §7 item 1 counts only a w < −1 “as an intrinsic equation of state, not an artifact of an assumed ΛCDM background”, ΛCDM being the standard model of a cosmological constant plus cold dark matter.
• #11’s kill condition is a “robust intrinsic w < −1” (§13 row 4).
• #11 §10 adds, of the current data, that “w(z) is inferred from distances rather than observed”.

The kill condition#

Row 4 of #11’s prediction table (row 4: thawing dark energy, never phantom):

DR3 is DESI’s Data Release 3. The no-phantom bound is the second kill condition; the first, a constant w = −1, tests the thawing drift (τ: hilltop quintessence on the Maslov circle).

How much a violation kills#

The documents differ on the scope of the second kill.

#11 ranks above the others. Its row 4 sets “kills outright” against “kills τ-sector” for a constant −1, which reads as more than the sector, and the handoff spells it out as the whole framework; #1 §7 item 1 and the withdrawn #14 confine the kill to the sector. No document states why the scope is wider than the sector. The register of conflicts between sources lists the disagreement as settled by precedence in #11’s favour.

Under pressure#

DESI’s DR2 reports that in every dataset combination “the favored w(z) shows a phase of w > −1 at low redshifts and a phantom crossing to w < −1 above redshifts z ≃ 0.4” [21S] (#11 §10). The preferred w(z) thus contains the phase the bound excludes. #11 §10, “Under pressure (v2.1)”:

The framework’s strict w ≥ −1 therefore stands against the direct reading of the data. It survives on two points: non-phantom quintessence fits remain compatible [19S], and w(z) is inferred from distances rather than observed, as [21S] also notes. DR3 is the decisive test (§13, row 4).

The fits without a phantom phase are “non-phantom compatible at ~2σ” (#11 §10). The root CLAUDE.md states the position: “DESI DR2 favors a phantom crossing above z ≈ 0.4, against the framework’s strict w ≥ −1 (§10). The dark-energy sector is under real pressure ahead of DR3.” #11 §15 lists “DESI DR2’s phantom crossing against §10’s strict w ≥ −1” among the program’s degeneration risks.

The bound is alive because the data do not yet meet the kill condition: the crossing is a preference of fitted models, not a robust measurement of an intrinsic w < −1. If DR3 confirms an intrinsic phantom phase, the bound fails and, by #11’s row 4, the kill is outright. How the finding entered the program, and the italic passage of #11 §10 that still reads DESI’s hints as “exactly the required drift”, are on DESI DR2’s phantom crossing.

The popular gloss#

Both popular articles explain the bound through the sign of kinetic energy (the popular articles). #12: phantom behaviour “would require the orientation’s kinetic energy to be negative, and kinetic energy is the square of a rate of change; a square of a real quantity cannot be negative” (‘Dark energy as the orientation of the split’). #10: dark energy “can never strengthen past the constant line, because exceeding it would require negative kinetic energy, a square of a real number coming out below zero” (‘The referee is already on the field’). These are explanations of step 2 above, not sources for the bound’s status.

History#

• v0.1 (#1 §7 item 1; §9 row 5). The bound is derived [D] and made a kill condition, with the intrinsic-versus-artifact qualification.
• v0.2. After Debt 3, the notebook lists “row 5 (thawing, no-phantom, DESI DR3)” among the program’s distinctive, still-live content.
• v1 (#9 §8; §9 row 4). “Strictly w ≥ −1”, and the requirement “never phantom”, with the kill “robust intrinsic w < −1 (kills outright)”.
• #11 v2.0. §10 “strictly w ≥ −1”; row 4 as in v1.
• Round 3. The soliton manuscript #14 v3, since withdrawn, described the parent framework as predicting “thawing, never-phantom dark energy, currently consistent with hints of weakening dark energy in recent baryon-acoustic-oscillation data” (§4). The round-3 referee found the DESI-preferred solution phantom-crossing, and ruling 11 accepted it: “The parent framework’s ’never phantom’ needs the checkpoint’s own nuance: non-phantom fits remain compatible at ~2σ.” (round-3 rulings).
• #11 v2.1 (2026-10-19). §10 gains the “Under pressure (v2.1)” paragraph, row 4 the clause “DR2 already favors a crossing above z ≈ 0.4 (§10)”, and §15 the degeneration risk. The bound itself is unchanged.

Sources#

• papers/broken-born-checkpoint-v2.md §10 — “strictly w ≥ −1” and the “Under pressure (v2.1)” paragraph
• CLAUDE.md ‘Since the handoff’ — “under real pressure ahead of DR3”
• papers/broken-born-checkpoint-v2.md §13 row 4 — the requirement and the kill conditions
• papers/broken-born-checkpoint-v2.md §15 — DR2’s crossing among the degeneration risks
• review/round3-adjudication.md ‘Rulings’ 11 — the finding accepted
• HANDOFF.md §2 — “w ≥ −1 always”; “kills the framework outright”
• notes/broken-born-theory.md §3 — the canonical action; §7 item 1 — the derivation [D] and the intrinsic-versus-artifact qualification; §9 row 5 — the v0.1 kill condition; ‘Addendum (v0.2)’ — row 5 among the live content
• notes/broken-born-program-v1.md §8, §9 row 4 — v1’s “Strictly w ≥ −1” and kill condition
• notes/knot-dark-matter-standalone.md §5 — exclusion condition 6 (withdrawn manuscript)
• popsci/universes-other-half.md ‘Dark energy as the orientation of the split’, popsci/angle-of-the-mirror.md ‘The referee is already on the field’ — the plain-language gloss (popular articles)