| # | Description | Status | Sources |
|---|---|---|---|
| A. Inputs: imported, not derived here | |||
| A1p32001 | The finite substrate $\F_\Om$, cardinality $\Om\sim10^{122}$ (Planck units), fixed by the de Sitter entropy. | I | |
| A2p32002 | The Hubble datum $\Hzero$, the one measured number entering $\az$. | I | measured |
| A3p32003 | The synchronisation premise (nearest-neighbour phase coupling) and unit channel capacity $G=\hbar c/m_P^2$. | I | |
| A4p32004 | The amplitude/Born rule: amplitude is the inter-subsystem projection, the squared amplitude a coincidence count. | I | |
| A5p32005 | The exact finite Fourier (FrFT) rotation between the coordinate and conjugate spectral charts. | I | |
| A6p32006 | The comprehension horizon: a bounded observer registers a signal only within its horizon in a chart. | I | [Akhtman & Voether 2026] |
| A7p32007 | The measured relations used for comparison only: the RAR/McGaugh function, $\az\!\approx\!1.2\times10^{-10}$, the $0.11$ dex scatter, the BTFR, the external-field downturn, the merging-cluster offset, the cluster-core residual; the SPARC tables (the package’s data/) as the confrontation data. | I | [McGaugh et al. 2016; Lelli et al. 2016; Lelli & others 2017; Chae & others 2020; Clowe & others 2006] |
| B. Realisations and conventions: mathematics $\to$ physics (the forced identifications, each with its falsifier, and one definition) | |||
| B1p32008 | Gravitation is the synchronisation of elementary clocks; distance is decoherence. Falsifier: a gravitating species supplying the galactic and cosmological abundance (§ 8), or entanglement induced off the Newtonian rate. | R | § 2.1 master: 00:D3 |
| B2p32009 | Mass is winding rate, $E=hf$; an acceleration advances phase at the rate $g/c$. Falsifier: a knee that does not track $H(z)$ (P1), or a rest mass with no winding. | R | § 2.1, 2.3 |
| B3p32010 | The acceleration floor is the phase-coherence threshold $g/c=\Hzero/2\pi$: the winding frequency reaches the drive’s cyclic frequency (one radian of phase per Hubble time). Falsifier: a measured floor off $c\Hzero/2\pi$ beyond the $H_0$ uncertainty, or a floor that is directional or density-dependent (P6). | R | § 2.3 |
| B4p32011 | Below the floor the registered force is read as an amplitude in the conjugate chart, the conserved flux as a count. Falsifier: a sub-floor law other than the geometric mean $\sqrt{\gb\az}$ at $x\ll1$, e.g. the wide-binary knee failing P3. | R | § 3.1, 3.2 |
| B5p32012 | The intrinsic scatter is the spread $\delta\alpha$ of the source’s collective phase, read through the chart angle; its identification with the disk’s dynamical temperature is the conjecture Y2. Falsifier: an intrinsic scatter that does not follow $\tan\alpha(x)$, i.e. no rise into the deep regime at fixed $\delta\alpha$ (P2). | R | § 5.1 |
| B6p32013 | The lensing enhancement is the coherence-weighted amplitude, following the bulk-coherent components. Falsifier: a merger lensing peak tracking the shocked gas; the cluster-core residual off the amplitude-sum form over the core’s composites is the exposure of the conjecture Y1. | R | § 6.1 |
| B7p32014 | The cosmological constant is the curvature of the wrapped finite chart. Falsifier: $w\neq-1$, or a negative or scale-varying $\Lambda$ (C11). | R | § 7.1 |
| B8p32015 | The masking barrier is the coincidence amplitude: registration fails when the killed noise walk first traverses the signal’s amplitude $\sqrt x$ in the metric $\delta_A$ the floor rate induces, coefficient one. Falsifier: an approach to Newton departing from $1-e^{-\sqrt x}$ beyond the binned-RAR uncertainty (the simple rational form differs by up to $0.05$ in $\gobs/\gb$ at $\gb\approx5\az$), or a knee displaced from $c\Hzero/2\pi$ ($\kappa^{2}\neq1$). Present confrontation: the binned SPARC relation departs from $1-e^{-\sqrt x}$ by less than the bin scatter in every bin of $2<x<10$, the rational form disfavoured in $\chi^2$ (dark.rar_shape). | R | |
| B9p32051 | The bulk-coherence weight of a component $c$, $w_c=\bigl|\langle e^{i\theta}\rangle_c\bigr|^{2}/\langle|e^{i\theta}|^{2}\rangle_c\in[0,1]$: unity for a virialised clump whose constituents share a bulk phase, near zero for gas a merger shock has randomised. | D | Eq. (11) |
| C. Derived: theorems and consequences within this paper | |||
| C1p32016 | Newton’s inverse-square law from the discrete Gauss law of the synchronisation flux. | T | Prop. 1; dark.flux_exact master: 00:E3 |
| C2p32017 | The acceleration floor $\az=c\Hzero/2\pi$, parameter-free, $13\%$ below the fitted value ($0.7\sigma$ of its systematic band). | T | |
| C3p32018 | Amplitude is the square root of the count (exact in $\Z[i]$, by cross-term cancellation). | T | Prop. 2; dark.born_exact |
| C4p32019 | The deep-regime geometric mean $\geff=\sqrt{\gb\az}$, given the amplitude reading B4; the mean force of the link is Newtonian at every noise level (dark.deep_regime; exactly, dark.deep_regime_fp), so the law is registration, not force modification; its instantiation by a transport run generating the registration rate is the sector’s open gate. | T | master: 00:L1 |
| C5p32020 | Flat rotation curves and the baryonic Tully–Fisher relation $v^4=GM\az$. | T | Eq. (4); dark.deep_mond master: 00:L1 |
| C6p32021 | The registration theorem: the finite coincidence count, its first-passage law, and flux conservation give the radial acceleration relation given B8; the fitting function of [McGaugh et al. 2016] is this form with $g_\dagger$ for $\az$. | T | master: 00:L1 |
| C7p32022 | The finite-cycle first-passage law: $e^{-\sqrt x}$ is carrier-exact (amplitude identity) and the resolvable reading of the exact transform, to $O(\Om^{-1/2})$. | T | |
| C8p32023 | The intrinsic-scatter relation, smallest at high acceleration and rising as $x^{-1/4}$. | T | Eq. (9); dark.rar_scatter |
| C9p32024 | The external-field effect from the total-field dependence of the registration. | T | Eq. (10) |
| C10p32050 | The merging-cluster lensing offset onto the coherent component, the enhancement sourced by $\sum_c w_c\Sigma_c$ (B6, B9). | T | |
| C11p32026 | The cosmological constant $\Lambda\sim1/\Om$, sign positive, $w=-1$ de Sitter section; no $120$-order cancellation. | T | |
| X. What the construction explains: the surplus over the galactic and cosmological phenomenology | |||
| X1p32027 | Newton’s inverse-square law and the equivalence principle follow from the discrete Gauss law of the synchronisation flux, the acceleration of a test cluster independent of its mass and composition. | T | Prop. 1; dark.flux_exact master: 00:E3 |
| X2p32028 | The acceleration scale $\az=c\Hzero/2\pi$ is the phase-coherence threshold, the acceleration whose winding frequency reaches the drive’s cyclic rate, the $2\pi$ the radian-to-cycle conversion of the angular Hubble rate, parameter-free, $13\%$ below the fitted value at the entailed $\Hzero$. | T | Eq. (2); dark.deep_mond |
| X3p32029 | Flat rotation curves and the baryonic Tully–Fisher relation $v^{4}=GM\az$ are the deep-regime geometric mean $\geff=\sqrt{\gb\az}$, the amplitude reading of the field below the floor. | T | Eq. (4); dark.deep_mond |
| X4p32030 | The radial acceleration relation carries no fitted function: its interpolation is the finite first-passage registration given one declared identification, the amplitude barrier (B8), and it is the curve the data selected. | T | |
| X5p32031 | The relation is tight, its intrinsic scatter ($0.04$ dex at the knee, SPARC at fixed $\az$) smallest at high acceleration and rising as $x^{-1/4}$ into the deep regime, the spread of the collective phase. | T | Eq. (9); dark.rar_scatter |
| X6p32032 | The external-field effect breaks the strong equivalence principle, the internal dynamics depending on the total field through the registration. | T | Eq. (10) |
| X7p32033 | The merging-cluster lensing offset is a selection: the coherence-weighted enhancement follows the collisionless galaxies and drops the shock-decohered gas ($C_{ij}\to0$), not a coincidence. | T | |
| X8p32034 | The cluster-core residual is referred to the coherent amplitude addition $\sqrt{N_{\mathrm{eff}}}$ over the core’s separately virialised composites, the one law that also gives the merger selection (C10); the quantitative closure, with $N_{\mathrm{eff}}$ uncomputed, is the conjecture Y1. | T | |
| X9p32035 | The cosmological constant is small, $\Lambda\sim1/\Om$, the curvature of the finite chart and a reading of the one cardinality rather than a mode-sum vacuum energy, so no $120$-order cancellation arises. | T | § 7.1 |
| X10p32036 | The dark-matter and dark-energy scales coincide because both are readings of the one cardinality $\Om$, the floor $\az=c\Hzero/2\pi$ and the curvature $\Lambda\sim1/\Om$ fixed together. | T | § 7.1 |
| X11p32037 | The dark sector adds no particle: the galactic discrepancy is the amplitude reading of the baryonic field, so the direct-detection and axion nulls are consistent and no dark subhaloes exist. | T | § 8 |
| P. Falsifiable predictions | |||
| P1p32038 | The redshift law $\az(z)=cH(z)/2\pi$: the knee shifts and the Tully–Fisher zero-point evolves, $v_{\mathrm{flat}}\propto E(z)^{1/4}$ ($+7\%$, $+15\%$, $+31\%$ at $z=0.5,1,2$), parameter-free; shape resolvable, value in Z1. Falsifier: a high-$z$ knee away from $cH(z)/2\pi$, or a non-evolving Tully–Fisher zero-point. | T | master: 00:P1 |
| P2p32039 | The scatter is the two-variable form $\sigma=(2\tan\alpha/\ln10)\,\delta\alpha$, vanishing for cold high-$x$ systems and rising as $x^{-1/4}$; SPARC at fixed $\az$: intrinsic $0.038$ dex overall, $0.04$ at the knee, $0.13$ at $x\simeq0.02$, bounding $\delta\alpha\lesssim3.5^{\circ}$. Falsifier: a deep-regime intrinsic scatter that does not rise, or a measured collective spread the scatter does not track. | T | |
| P3p32040 | Wide binaries ($\gtrsim10$ kAU) follow $\gb/(1-e^{-\sqrt{x}})$ with the global $\az$ and the vector Galactic field $\gext\simeq1.8\az$: a $10$–$30\%$ velocity enhancement ($12$–$16\%$ at $10$–$40$ kAU on the collinear estimate), the exponential knee distinct from Newtonian and rational forms. Falsifier: Newtonian wide-binary motion, or a knee of the wrong shape. | T | |
| P4p32041 | Pressure-supported systems (ellipticals, dSphs, UDGs) lie below the cold-disk relation by the coherence factor $\sqrt{w}$, $w=1/(1+(\sigma_v/v)^{2})$, by $0.05$–$0.15$ dex for $\sigma_v/v=0.5$–$1$, correlating with $\sigma_v/v$. Falsifier: pressure-supported systems on the cold-disk relation. | T | |
| P5p32042 | Coherence-state cluster gas: the X-ray gas gravitates only when bulk-coherent, contributing in a relaxed cluster ($C_{ij}\to1$) and dropping out of a merger ($C_{ij}\to0$), the lensing displacement growing with the shock Mach number; the quantitative core residual of the same law is the conjecture Y1. Falsifier: a merger lensing peak tracking the shocked gas, or no relaxed-versus-merging difference. | T | |
| P6p32043 | A global, isotropic scale and no dark substructure: the floor is the global drive rate, identical in every direction and environment once the external-field correction is made; no new dark-matter state and no dark subhaloes, so stream perturbations and the satellite abundance trace baryons alone. Falsifier: a directional or density dependence of $\az$ beyond the external-field effect, or stream gaps and substructure lensing with no baryonic counterpart. | T | master: 00:P4 |
| V. Machine verification | |||
| V1p32044 | The validation package finite-ring-space/src/32-dark: the twelve scripts and the figure script as written — dark.flux_exact, dark.born_exact, dark.firstpassage_finite, dark.meridian_walk, dark.deep_regime, dark.deep_regime_fp, dark.interpolation, dark.rar_shape, dark.deep_mond, dark.rar_scatter, dark.cluster_coherent, dark.predictions, dark.make_figures — run through one registry, one family per script, its micro-checks the script’s own verdict lines together with the registry’s predicates on the script’s namespace and output (nine scripts print without asserting; every headline numeral is pinned by a labelled check in darkcommon.py), each family naming the rows it witnesses; the notebook 32-dark-main.ipynb executes them in the browser and the driver run_all.py writes results.json. | T | App. A; run_all |
| V2p32045 | The discipline, recorded per family as its kind: EXACT — exact rationals, $\Z[i]$ or $60$-digit identities, no tolerance in the verdict (the Gauss law, the amplitude identity, the first-passage law); SIM — a seeded stochastic simulation, verdict by stated tolerance (the noisy link, the killed walk); CHART — a continuum reading or a comparison with data, tagged [approx] in the text (the interpolation, the RAR and BTFR, the SPARC scatter, the predictions, the cluster illustration, the figures). | T | App. A; dark.make_figures |
| V3p32046 | What the run does not decide: the $\Om$-hard running (Z1) enters no check; the cluster illustration of dark.cluster_coherent evaluates the law $N_{\mathrm{eff}}=(\sum\sqrt{g_i})^2/\sum g_i$ on stated cases, not $N_{\mathrm{eff}}$ from a core population (Y1); the identification $\delta\alpha=\arctan(\sigma_v/v)$ (Y2) is bounded, not tested, by the SPARC residuals; the named test of B8 (dark.rar_shape) prefers the exponential form by $\chi^2$ but the two forms differ by at most $0.02$ dex per bin against a bin scatter of $0.12$ dex, so the binned relation does not exclude the rational form; the transport run generating the registration rate (C4) is not in the suite. | T | App. A |
| Z. Residue: $\Om$-hard | |||
| Z1p32047 | The cross-scale running $\az(d)/\az(\text{carrier})=(\Om-1)/d$; its uniform certificate closes only at carrier scale (master ledger Z7). | Ω | § 7.2 master: 00:Z7 |
| Y. Hypotheses: the conjectures stated ahead of derivation | |||
| Y1p32048 | The cluster-core residual as the coherent amplitude addition (12) over the core’s separately virialised composites, $\sqrt{N_{\mathrm{eff}}}$ over the smooth boost, the law that also gives the Bullet selection. Exposure: the coherence criterion is not derived; the radial acceleration relation forces $N_{\mathrm{eff}}=1$ within a galaxy (dark.rar_scatter), while the relaxed-core residual requires $N_{\mathrm{eff}}\simeq4$, uncomputed (§ 6.2). Deciding computation: $N_{\mathrm{eff}}$ from the core galaxy populations of [Sanders 2003; Pointecouteau & Silk 2005]. | O | § 6.2; dark.cluster_coherent |
| Y2p32049 | The collective phase spread as the disk’s dynamical temperature, $\delta\alpha=\arctan(\sigma_v/v_{\mathrm{circ}})$. Exposure: at $\sigma_v/v\simeq0.1$–$0.2$ the law gives two to four times the SPARC intrinsic scatter at every $x$; the knee bounds $\delta\alpha\lesssim3.5^{\circ}$; the within-galaxy scatter of $116$ disks shows no correlation with $1/V_{\mathrm{flat}}$ ($\rho=-0.02$). Test: the scatter against measured dispersions. | O | § 5.1; dark.rar_scatter |
Nothing matches that filter.
Status
- I
- Import. A standard result or measured datum used here without reproof.
- R
- Realisation. A forced identification of a physical term with the substrate object it denotes: constitutive, not interpretive; defended by its registered consequences; killed by the falsifier stated in its row.
- D
- Definition. A naming or set-up move, carrying no empirical content.
- T
- Theorem. Derived within this paper from the rows above it; where a family identifier is named, its content is verified by the validation package.
- Ω
- $\Om$-hard. Decided by the totality; not closeable by a bounded observer.
- O
- Open (conjecture). Stated ahead of derivation, carrying its exposure; a derivation re-homes the row.
Sources
- section
- The statement in the paper's body that carries the predicate: a proposition, a remark or a section.
- rests on
- The predicates it rests on, by label: realisations, definitions and imports of this ledger, predicates of other ledgers (
32:B3) and of the master (00:D15).