Spatial Displacement Theory · Paper 07 · Prepared for Submission
Cosmography Without a Dark Sector: The Pressure Tensor P(z) = Pconv(1+z)⁴, the Two-Component Redshift, and the H₀ Ordering
Melbourne, Australia · July 2026 · Domain 07 — Cosmology & Redshift
Abstract. Cosmological observables are computed from the lattice pressure tensor with no expansion of space, no dark matter, and no dark energy. The tensor evolves as P(z) = Pconv(1+z)⁴ with polytropic index γeff = 4 — a topologically stiff lattice, not a photon gas — and drops by an exact factor 4 at recombination, when independent winding defects bind into neutral hydrogen, freezing the acoustic scale. Redshift decomposes into an emission term (depth, z = ϟ/r) and a propagation term (compression squeeze); the rival framework conflates them into one recession variable, and that conflation is the H₀ tension: the ordering 67.4 → 73.0 across probes follows from local compression sampling. The CMB is the current operating pressure of the substrate. Negative results are on the record: the pure z³ distance model is excluded against the supernova compilation (the rival is preferred at ΔAIC ≈ 90), and the u ∝ (1+z)⁴ energy-density scaling is degenerate between the two frameworks — claimed as consistency, not victory.
Keywords: pressure tensor · topological freeze-out · two-component redshift · octave quantisation · opacity ceiling · H₀ ordering
1. The pressure tensor and its stiffness
Linking-number density scales as (1+z)³ and temperature as (1+z); the product gives (1). The index is 4, not 4/3: the lattice carries four topological degrees of freedom per cell (W ± 1 modes) and is stiff. The radiation-era scaling H ∝ (1+z)² is recovered, not imported (class C; passes the delete-test).
2. Recombination as topological freeze-out
When electron and proton windings bind, the independent defect count — hence the lattice pressure — drops by the exact integer factor 4. This freezes the acoustic scale (the baryon-oscillation shells are physical pressure shells, not projected sound horizons) and switches the rate law's exponent. The engine's constant H₀ = 67.4 km/s/Mpc emerges within 1% of the anisotropy-inferred value; disclosure: the rate-equation structure is presently assumed pending its own derivation (GOM05), and the Ω-normalisation is observational — the label is C-flagged, not C.
3. The two-component redshift and the H₀ ordering
Emission depth is set by the source's own koppa; propagation squeeze accumulates through the compression field traversed. A framework that reads the sum as a single recession velocity must find different "expansion rates" when probes weight the two terms differently — which is precisely the observed ordering: anisotropy-based 67.4, tip-of-red-giant 70.0, lensing-time 71.6, ladder 73.0. The tension is predicted structure, not crisis. Two further quantitative results: the observed shift spectrum quantises in octaves, 1+z = 2Δn, placing the CMB at 10.10 octaves below emission (λemit = 975 nm); and octave shifting does not compound like absorption — which kills the classical tired-light rival while preserving sharp quasar lines, a discriminant the tired-light class fails.
4. Opacity, the ceiling c/√3, and the CMB's gradient
Radiation self-jams: above a threshold flux the relay saturates and the medium runs opaque, with transport ceiling c/√3. The Clearing is the depinning of that logjam. Because release is synchronised across a depth gradient, the CMB arrives as the superposition of open-close cycles — its blackbody fidelity is a gate, pre-registered at the spectral-distortion bound (< 50 ppm against the interferometric spectrum): a measured distortion above the bound kills the mechanism. Estimated elapsed count since the Clearing ≈ 48 Gyr; the framework makes no claim about the antecedent state, and the detonation-recharge cyclic extension remains explicitly uncommitted.
5. Negative results, on the record
The cubic-redshift distance construction fails the supernova compilation and is recorded as failed (CR12); the surviving content of CR10 is the dimensional reading of the exponent, not a distance law. Likewise uCMB ∝ (1+z)⁴ is degenerate between compression and expansion readings — it discriminates nothing and is logged as such. These entries stand because a falsification programme that hides its misses is not one.
6. Certification
| Result | Value | Status |
|---|---|---|
| P(z) = Pconv(1+z)⁴, γeff = 4 | eq. (1) | DERIVED — class C (CR05) |
| Freeze-out factor | exactly 4 | DERIVED — integer defect count |
| H₀ | 67.4 km/s/Mpc (<1%) | C-flagged — structure assumed pending GOM05 (disclosed) |
| H₀ ordering across probes | 67.4→73.0 | PREDICTED structure — two-component conflation |
| Octaves 1+z = 2Δn; CMB at 10.10 | λemit = 975 nm | DERIVED (CR07); kills tired-light class |
| CMB distortion gate | < 50 ppm | PRE-REGISTERED falsifier |
| z³ distance model | ΔAIC ≈ 90 against | EXCLUDED — negative result (CR12) |
| u ∝ (1+z)⁴ | — | DEGENERATE — no discriminating power (disclosed) |
| Cyclic extension | — | NOT COMMITTED |
Methodological declaration. No expansion of space, no dark matter, no dark energy, no inflaton, and no six-parameter concordance fit enters any derivation; the rival framework appears only as the named alternative in discriminant statements, and its victories (eq. 4) are reported as its victories. The anisotropy, distance-ladder, lensing, and supernova datasets enter as measured targets. Both degeneracies found are labelled degenerate rather than claimed.
References
- Engine/include/sdt/laws.hpp — namespace law_I (pressure tensor, freeze-out, H₀ with C-flag in situ), law_II (release cascade, pressure domains).
- Investigations/07_Cosmology_and_Redshift — CR04/CR05 (tensor, ceiling), CR07 (octaves), CR10/CR12 (cubic model and its exclusion), opacity-logjam dossier.
- Hubble/ — the zk² tool suite and datasets (the tension decomposition).
- FIRAS spectral data; Planck 2018 anisotropy products; Pantheon+ compilation (measured targets).
- Paper 06 of this series (the depth field feeding zemit).