w Evolving: Λ Topological Resolution

The Dark Energy Spectroscopic Instrument (DESI) reports mounting evidence (2.8-4.2σ) that the dark energy equation of state w evolves with redshift. Mode Identity Theory (MIT), which derives cosmological observables from topological boundary conditions, appears vulnerable; however, the framework holds Λ fixed by construction. 

The cosmic standing wave Ψ(t) = cos(t/2) imprints phase-dependent structure on observable geometry. When observers fit distance data using fluid-based models, they recover apparent w(z) evolution, not because dark energy changes, but because the fitting framework cannot capture phase position. The "Phantom Crossing" emerges naturally as an inference artifact, not exotic physics. 

MIT distinguishes constant Λ (the surface mode) from varying w_eff (the observer's inference signature). The phase offset δ = −1.06 rad is locked by pre-registration, placing observers approximately 2.8 Gyr before the geometric turnaround.

time-stamped paper can be found here

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I. CONTEXT

THE OBSERVATIONAL LANDSCAPE

The DESI DR2 results present the most precise baryon acoustic oscillation measurements to date, drawing from over 14 million galaxies and quasars across 11 billion years of cosmic history. Building on initial Y1 indications, the expanded dataset more than doubles the statistical power. When combined with cosmic microwave background data and supernova compilations, the results indicate that dark energy's equation of state w (the ratio of pressure to energy density) may not equal −1 as assumed in ΛCDM.

The emerging picture suggests dynamic evolution: present value w₀ ≈ −0.85 to −0.70 (less negative than Λ); past behavior w < −1 at higher redshifts (phantom-like); and a "Phantom Crossing" through w = −1 near z ≈ 0.5.

This presents sharp tension for any framework treating the cosmological constant as fundamental.

MIT'S POSITION

Mode Identity Theory derives Λ from surface-mode projection on bounded topology. The cosmological constant emerges not as a free parameter but as a geometric consequence of boundary conditions at dimensional index n = 2:

    Λ_top · ℓ_P² = Ω⁻¹ · C(60/120)

where Ω ≈ 10¹²² is the cosmic-to-Planck hierarchy (the number of Planck areas on the cosmic horizon, an empirical input), R_Λ = √(3/Λ) is the de Sitter radius, and C(60/120) = 2 is the phase coefficient at the antinode. The relation Ω ≈ (R_Λ/ℓ_P)² is a consistency condition, not the source of Ω; MIT derives Λ from the topology and horizon scale, then verifies the de Sitter relation holds. If Λ itself evolved, MIT would face direct structural contradiction.

However, DESI constrains the inferred equation of state w(z), not vacuum energy Λ directly. While related in standard cosmology, these quantities separate in a topological framework. MIT's cosmic wave structure naturally produces apparent w(z) evolution while Λ remains constant, resolving the tension without adding new fields or exotic physics.

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II. DERIVATION

THE COSMIC STANDING WAVE

MIT's anti-periodic boundary conditions admit a standing wave solution on the bounded domain. The anti-periodic boundary condition requires two full traversals of the domain to return to the original field configuration (an anti-periodic holonomy with a π phase flip under one traversal). The resulting period is 4π:

    Ψ(t) = cos(t/2)

where t is the cosmic phase. The full cycle spans t ∈ [0, 4π] (approximately 33 Gyr). The geometric turnaround occurs at t = 2π (~16.6 Gyr), where the wave reaches its minimum value (Ψ = −1).

PHASE-REDSHIFT MAPPING

To connect the cosmic wave to observational coordinates, we relate the topological phase t to the cosmological scale factor a:

    t(a) = (2π + δ) · a = (2π + δ)/(1 + z)

The offset δ calibrates the present phase relative to the geometric turnaround (t = 2π). For δ = 0, observers are exactly at turnaround.

MIT locks δ = −1.06 rad.

The boundary conditions are: z → ∞ (early universe) ⟹ t → 0; and z = 0 (present) ⟹ t = 2π + δ ≈ 5.22 rad.

Justification for linear mapping: The choice t ∝ a is the simplest monotonic function satisfying both boundary conditions. More complex mappings would introduce additional free parameters without physical motivation. We treat this as the minimal ansatz; Euclid/Rubin reconstructions can test it by checking whether a single δ explains the full w_eff(z) shape across the observable window.

FROM TOPOLOGY TO INFERENCE

MIT holds Λ fixed by boundary conditions, a surface mode invariant under phase evolution. However, observational cosmology does not measure Λ directly. Distance indicators (BAO standard rulers, supernovae) are interpreted through parametric models, typically assuming an equation of state w = P/ρ for a dark energy fluid.

If the underlying reality includes phase-dependent topological structure, observers fitting data with fluid-based models will infer apparent w evolution, not because dark energy changes, but because the fitting framework cannot capture phase position.

THE PARAMETERS

Coupling strength ε: The amplitude of the inference deviation. Determined by how strongly the phase structure imprints on distance observables.

Phase offset δ: The observer's position within the cosmic cycle, measured from the geometric turnaround (t = 2π). For δ = 0 (present at turnaround), the crossing condition gives z_cross = 1. For δ < 0: approaching turnaround, so observers infer w₀ > −1 (for ε > 0) with the crossing at lower redshift.

Topology fixes δ; current data calibrate ε via w₀.

These parameters are not independent. Given δ, the coupling ε is fixed by the observed w₀. At z = 0, using cos(π + x) = −cos(x):

    ε = (w₀ + 1) / cos(δ/2)

With δ = −1.06 rad and w₀ ∈ [−0.85, −0.70] (DESI), ε ∈ [0.17, 0.35] (note cos(δ/2) > 0 for |δ| < π). The representative value ε ≈ 0.25 corresponds to mid-band w₀ ≈ −0.78. At ε ~ 0.3, O(ε²) corrections are ~10%, comparable to current observational uncertainties; the linear approximation suffices for present data but higher-order terms may become relevant as precision improves. The topology sets δ; observation reveals ε through the w₀ it returns.

THE MECHANISM

The cosmic standing wave Ψ(t) modulates the effective expansion history:

    H_eff(z) = H_Λ(z) [1 + (ε/2)Ψ(t(z))]

where H_Λ(z) is the standard ΛCDM Hubble rate. This modulation propagates to the luminosity distance:

    d_L(z) = (1 + z) ∫₀^z c dz' / H_eff(z')

When observers fit this d_L(z) using standard w₀w_aCDM templates, the recovered equation of state absorbs the phase structure. The inference template to leading order in ε:

    w_eff(z) = −1 − ε cos[(2π + δ) / (2(1 + z))]

This functional form is an ansatz for how phase modulation projects onto w₀w_a reconstructions. The exact coefficients depend on the fitting procedure; we adopt this template as the minimal projection consistent with the H_eff modulation structure. It is not a physical property of dark energy; it is the inference signature of observers at phase position δ fitting constant-Λ reality with evolving-w tools.

THE PHANTOM CROSSING

Standard cosmology treats "phantom" behavior (w < −1) as exotic, requiring negative kinetic energy or ghost fields. In MIT, no such machinery is needed.

The inferred w passes through −1 because the cosine function does. At some redshift z_cross, the phase argument produces cos = 0, yielding w_eff = −1. The crossing condition cos[·] = 0 requires the argument to equal π/2:

    (2π + δ) / (2(1 + z_cross)) = π/2   ⟹   z_cross = 1 + δ/π

With δ = −1.06 rad: z_cross = 1 + (−1.06)/π ≈ 0.66. This prediction follows from δ given the linear phase-redshift mapping.

For z > z_cross, the cosine term changes sign, producing w_eff < −1. This is not phantom dark energy. It is what the fit returns when constant-Λ reality is projected through phase-dependent topology.

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III. OBSERVATIONAL CONSTRAINTS

LOCKED PARAMETERS

The phase offset δ = −1.06 rad is pre-registered. Combined with DESI observations, this yields:

δ ............... −1.06 rad ........ Status: Locked

z_cross ...... 0.66 ............... Status: Deterministic from δ

Phase t... ≈ 5.22 rad .......... Status: Derived

w₀ ........... −0.85 to −0.70 .. Status: DESI input

ε ............... 0.17-0.35 ......... Status: Derived

Note: 1 radian ≈ 2.6 Gyr on the ~33 Gyr cycle. The z_cross value 0.66 follows deterministically from δ = −1.06 via z_cross = 1 + δ/π. The falsification bounds (< 0.4 or > 0.9) represent a conservative 2σ observational tolerance, not uncertainty in the prediction itself.

QUALITATIVE ALIGNMENT

MIT naturally produces all qualitative features reported by DESI: present-day w₀ > −1 (phase approaching turnaround); past w < −1 (cosine sign change at z > z_cross); and phantom crossing near z ≈ 0.5 (z_cross = 0.66 in MIT).

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IV. DISCUSSION

Λ VERSUS w_eff

The distinction is essential. The cosmological constant Λ is fixed by boundary conditions in MIT. The inferred equation of state w_eff varies as a phase signature.

MIT holds Λ constant. What varies is what observers infer.

The apparent tension between MIT's gentler w_a and DESI's steeper slope is expected. CPL fits a linear template to curved data; the effective slope steepens to compensate for the missing curvature. This mismatch is diagnostic of the cosine form, not a failure of the prediction.

THE ROLE OF δ

The phase offset δ differs fundamentally from typical cosmological parameters: it is bounded (|δ| ≪ 2π), has meaningful sign (negative = pre-turnaround), and encodes a structural claim (observers are near turnaround, not at an arbitrary phase).

MIT locks δ = −1.06 rad prior to Euclid DR1. This is not a fit; it is a pre-registered prediction.

STRUCTURAL VERSUS TEMPORAL MIDPOINT

MIT identifies two distinct midpoints: the structural midpoint (√Ω_H ≈ 10⁶¹, where Ω_H = (R_H/ℓ_P)² is the Hubble-scale hierarchy), where observers are equidistant from Planck and Hubble scales; and the temporal midpoint (t = 2π), the geometric turnaround.

With δ = −1.06 rad, observers are at t ≈ 5.22 rad, approaching but not yet at the turnaround. At 1 rad ≈ 2.6 Gyr, this places us ~2.8 Gyr before turnaround.

Cortês & Liddle note that the phantom crossing occurring at the center of the observable window is "a substantial and unsettling coincidence." MIT offers a resolution: the structural midpoint (√Ω_H) and the temporal midpoint (t ≈ 2π) nearly coincide at the present epoch, suggesting that observers at this structural position would naturally find the crossing centered in their window.

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V. FUTURE OBSERVATIONS

As DESI accumulates data and complementary surveys (Euclid, Rubin, Roman) come online, constraints on w(z) will sharpen. The theory identifies three observational targets: (1) precision on z_cross; (2) w(z) curvature; and (3) high-z behavior at z ≈ 1-2.

FALSIFICATION CRITERIA

All predictions are pre-registered prior to Euclid DR1 (October 2026):

z_cross — MIT Value: 0.66 — Falsified if observed < 0.4 or > 0.9 (2σ tolerance)

w(z) form — MIT Value: Cosine — Falsified if linear preferred at ΔAIC > 4

Λ — MIT Value: Constant — Falsified if Λ(z)/Λ(0) ≠ 1 at 2σ

The observed w₀ is not an independent MIT prediction; it is the DESI observable from which ε is derived. The hard geometric prediction is z_cross, which follows deterministically from δ given the linear phase-redshift mapping. For the w(z) functional form comparison: CPL freely fits two parameters (w₀, w_a), while MIT locks δ by pre-registration and determines ε from the observed w₀, leaving zero free parameters for the shape prediction beyond the anchor point. ΔAIC penalizes CPL for this extra freedom; if CPL is still preferred after the complexity penalty (ΔAIC > 4), the cosine inference signature is falsified.

Critical test: MIT predicts w(z) has curvature. With sufficient precision:

    d²w/dz² ≠ 0

A purely linear w(z) at high precision would falsify the cosine inference signature.

Inverse prediction: The epoch-dependent a₀ supplement predicts a₀(z) ∝ H(z) while Λ remains constant. Joint confirmation would strongly support MIT's dimensional hierarchy.

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VI. CONCLUSION

Mode Identity Theory welcomes DESI's evidence for dark energy evolution while maintaining Λ constant. The resolution lies in distinguishing the physical constant from the inferred parameter.

The cosmic standing wave Ψ(t) = cos(t/2) imprints phase-dependent structure on observable geometry. When observers fit distance data using fluid-based w(z) models, they recover apparent evolution, not because Λ changes, but because the fitting framework cannot capture phase position.

The framework yields: (1) Phantom Crossing at z = 0.66; (2) Derived Coupling: given DESI's w₀, MIT implies ε ∈ [0.17, 0.35]; (3) No new free parameters in the Euclid test (δ is pre-registered, ε is calibrated from current w₀ and then held fixed); (4) Testable Curvature: cosine form distinguishable from linear CPL.

These features emerge from topology, not tuning. The cosmological constant remains what MIT always claimed: a surface mode fixed by boundary conditions. What "evolves" is not Λ, but perspective.

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REFERENCES

1. DESI Collaboration (2025). arXiv:2503.14738

2. Shatto, B. (2025). Mode Identity Theory: Modal Realization on Nested Topology. DOI: 10.5281/zenodo.18064856

3. Chevallier, M. & Polarski, D. (2001). Int. J. Mod. Phys. D 10, 213.

4. Linder, E. V. (2003). Phys. Rev. Lett. 90, 091301.

5. Shatto, B. (2025). High-Redshift Galaxy Mass Anomalies. DOI: 10.5281/zenodo.18072995

6. Cortês, M. & Liddle, A. R. (2025). arXiv:2504.15336

7. Team Cosine (2026). Pre-registration. DOI: 10.5281/zenodo.18189079