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Paper 4 · Source & PDF papers/v600-cosmic-tensions Programme overview All five V600 papers Dependency · Paper 3 τσ involution Dependency · Paper 1 V600 + Dic5 + V24
What this paper does. Defines two K-class projector corrections to the identity on the 12-dim rational Tτ-cycle space C of V600: Ĥ = IC + P72 and Ĉ = IC − P0. Proves a K-saturated admissibility theorem: among rank-one K-class projector corrections, only ±P72 and ±P0 are admissible. Reports the exact trace ratios 13/12 and 11/12, then compares against published cosmology under an explicit Layer-3 sign-assignment hypothesis.
Epistemic status. Three layers, kept separate. The trace identity (Layer 1) is unconditional finite-group mathematics. The numerical match against published data (Layer 2) is a directional observation with full uncertainty bands stated. The coupling rule "H0 ↔ +P72, S8 ↔ −P0" (Layer 3) is an explicit hypothesis, falsifiable by future measurements, and is not asserted as a derivation. We do not claim to resolve the H0 or S8 tensions.

The Setup

Inherits V600, Dic5, the bulk subgroup V24, and the τσ involution from Papers 1 and 3. The cycle space C ≅ ℚ12 is the 12-dimensional rational Tτ-cycle space; on C, K-class projectors PK are diagonal rank-mK projectors indexed by the K-multiset of V600:

K-multiset = {72:1, 0:1, 52:5, 20:5}
Total dimension 1+1+5+5 = 12. The classes K=72 and K=0 are rank-one (a single cycle each); K=52 and K=20 are rank-five each.

The Two Operators

On C, with the ΛCDM-baseline taken as I12, the paper defines two rank-one K-class projector corrections:

Ĥ = IC + P72    Ĉ = IC − P0
P72: rank-one diagonal projector onto the K=72 max-K cycle (a "bulk anchor"). P0: rank-one diagonal projector onto the K=0 trivial-K cycle (a "bulk zero-mode").

The K-Saturated Admissibility Theorem

Layer 1 · Theorem

Within the diagonal, K-saturated operator algebra AK on C, the only rank-one K-class projector corrections to IC are ±P72 and ±P0. The hypotheses — diagonality in the cycle basis and K-class saturation — are stated explicitly. The paper does not claim that arbitrary rank-one operators or arbitrary τσ-invariant rank-one operators are excluded.

A phase-independence lemma covers all 32 = Z25 canonical lifts of τσ from Paper 3: the trace ratios and the admissibility classification are independent of the chosen lift.

The Trace Ratios

Once the operators Ĥ and Ĉ are fixed, the per-dimension trace ratios are exact rationals:

tr(Ĥ)/12 = 13/12 ≈ 1.0833
From IC with rank-1 P72 added: trace 12 + 1 = 13.
tr(Ĉ)/12 = 11/12 ≈ 0.9167
From IC with rank-1 P0 subtracted: trace 12 − 1 = 11.

Both are unconditional Layer-1 mathematics. They follow from the K-multiset structure of V600, with no fitting and no continuum approximation. The 1/12 modifications are a direct combinatorial consequence of rank-1 corrections on a 12-dim space.

Layer 3 — The Sign-Assignment Hypothesis

Stated as hypothesis H1, not as derivation. The paper proposes:
  • H0 ↔ +P72 (interpreted as a max-K, scale-setting coupling)
  • S8 ↔ −P0 (interpreted as a trivial-K, clustering-exclusion coupling)
Trace data alone does not distinguish P72 from P0 within the rank-1 admissible pair — that distinction is the entire content of H1. H1 is falsifiable by future H0 / S8 measurements; it is not derived in this paper.

Layer 2 — Empirical Comparison

Under H1, the 13/12 and 11/12 ratios act on the corresponding Planck 2018 baseline values and are then compared against late-time measurements:

Quantity Planck 2018 baseline VFD prediction Late-time observation Residual
H0 (km/s/Mpc) 67.36 13/12 × 67.36 = 72.97 SH0ES 2022: 73.04 ± 1.04 ≈0.06σ
S8 0.832 11/12 × 0.832 = 0.763 KiDS-1000 multi-probe: 0.766 (+0.020/−0.014) ≈0.24σ (directional)

S8 ≡ σ8√(Ωm/0.3) is reported because published weak-lensing intervals are reported as S8. The S8 residual is computed using the directional KiDS lower error because the predicted value sits below the central value. Both values lie inside the cited late-time uncertainty bands.

Mapping Numerics to Claims

Computed

Exact rationals

13/12 and 11/12. Trace identities are unconditional mathematics, independent of any cosmological interpretation. Verified by certificate verify.py.

Observed

Numerical comparison

72.97 vs 73.04 ± 1.04 km/s/Mpc; 0.763 vs 0.766 (+0.020/−0.014). Reported as directional σ-residuals with bibliography pinned in references.bib.

Hypothesis

Layer 3 — not derived

Sign assignment H0 ↔ +P72, S8 ↔ −P0. Stated as falsifiable hypothesis H1, never as a theorem.

Reviewer Risk Register

  • "The coupling rule is hand-picked to fit the data." Mitigation: §4 classifies rank-one corrections inside AK — the only rank-one elements are ±P72, ±P0. The sign assignment is stated as Layer-3 hypothesis H1, not as derivation.
  • "Why IC as baseline?" Mitigation: IC is the K-blind uniform cycle weighting — unit weight on every Tτ cycle. This is a baseline choice, not a τσ-uniqueness theorem.
  • "What's the dynamical equation?" Mitigation: this paper presents observable-level ratios, not a Friedmann analogue. Tier-2 work, explicitly out of scope.
  • "Two benchmark matches from one hypothesis is too good." Mitigation: the layered presentation is the answer. Layer 1 is theorem-grade; Layer 2 is observation; Layer 3 is hypothesis. Numerical match alone does not constitute derivation.
  • "What about higher-rank corrections?" Mitigation: an ablation in §4 shows rank-5 corrections (P52, P20) give trace ratios 17/12 and 7/12, far outside the cited H0/S8 bands.

What Is Explicitly Out of Scope

  • CMB acoustic peaks, recombination physics, full ΛCDM 6-parameter derivation (Paper 5 territory)
  • Black-hole thermodynamics and Hawking spectrum (Papers 1+2)
  • The τσ construction itself (Paper 3)
  • Cosmic dipole 1+k/2 ladder — reserved for future synthesis
  • Ontology / metaphysics — no claim about V600 being THE substrate of physics
  • Cascade-unit to physical-unit calibration — tier-2
  • Cosmological perturbation theory inside V600 — tier-2
  • Pre-registered DESI/Euclid/eROSITA predictions — would require H1 to be promoted to derivation

Verification

Reproduction

The certificate verify.py constructs the K-class projectors PK from the K-multiset enumeration on V600's Tτ-cycle space, builds the operators Ĥ = I + P72 and Ĉ = I − P0, and computes the trace ratios in pure Python rationals. The K-saturated admissibility theorem is checked by exhaustive enumeration over rank-1 candidates inside AK. The phase-independence lemma is checked over all 32 elements of Z25.

Empirical baselines are pinned in references.bib: Planck 2018 (1807.06209), SH0ES 2022 (Riess et al. 2112.04510), KiDS-1000 multi-probe (Heymans et al. 2007.15633). The numerical comparisons in the table use these published central values and uncertainties verbatim.

The undismissable spine. Two operators on a 12-dim cycle space; one rational identity each; one hypothesis labelled clearly. The reviewer is offered: an operator-trace identity, a finite-group forcing argument, an empirical match within published bands, and an honestly-marked interpretive layer. The paper makes no attempt to "solve" cosmological tensions — the entire point is what survives without that ambition.

Two ratios. Twelve dimensions. Three layers, kept separate. No tension is claimed to be resolved.

Related
Programme V600 Programme Overview All five papers in one place: incidence, spectrum, involution, trace ratios, synthesis. Paper 3 τσ Involution The phase-independence lemma over the 32 canonical lifts depends on this construction. Paper 5 Structural-Spine Synthesis Imports this paper by citation only and names the shared structural tuple ΣV600.
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