Tags
- I
- Import. A standard result or measured datum imported without reproof.
- R
- Realisation. The row states what a physical term denotes in the substrate: constitutive, not interpretive, defended by the totality of its registered consequences.
- 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 reproduced by the validation package.
- E
- Prediction. A falsifiable consequence of the rows above, stated with its falsifier and its present confrontation; graded on the rows it consumes.
- Ω
- $\Omega$-hard. Decided by the totality; not closeable by a bounded observer.
| # | Description | Status | Source |
|---|---|---|---|
| A. Inputs: imported, not derived here | |||
| A1p21001 | The finite substrate: the torsor Carrier, coordinatized by a frame as $\F_\Omega$, its cardinality the Carrier identity $\Omega=4S+1$ with the measured $S\sim10^{122}$ (Planck units) locating the coherence horizon, with the phase cycle $C_{\Omega-1}$, the quarter-turn core $Q_4$, the drive (time is scale-dilation, each Subject’s own), the coherence horizon $\sqrt\Omega$, and the resolution floor $1/\sqrt\Omega$. | I | |
| A2p21002 | The three frame freedoms: the frame space is a torsor under $\mathrm{SL}_2(\F_\Omega)$, the spatial chart three-dimensional per frame, its continuum shadow the $3$-sphere. | I | [Akhtman & Voether 2026, §9.6] |
| A3p21003 | The quadratic extension $K$, the Frobenius involution, the Lorentzian norm form, and the boost cycle (the Dirac layer). | I | |
| A4p21004 | Composite coherence: a bound cluster is one synchronised drive orbit (offset confinement, the $m$-body synchronisation lemma), coupling with coefficient $m$ against an incoherent $O(\sqrt m)$. | I | |
| A5p21005 | Lattice potential theory and the discrete harmonic Green’s function, with Coulomb asymptotics $1/4\pi r$. | I | [Duffin 1953; Lawler & Limic 2010] |
| A6p21006 | The Kuramoto relaxation equation for coupled phase oscillators (the mathematical form of the drift dynamics). | I | [Kuramoto 1975; Strogatz 2000; Pantaleone 2002] |
| A7p21007 | The continuum uniqueness of the Fierz–Pauli stiffness and Deser’s energy–momentum self-coupling bootstrap, used as imported templates. | I | [Fierz & Pauli 1939; Deser 1970] |
| A8p21008 | Measured quantities used for comparison only: the GRAVITY mass–distance and EHT shadow for Sgr A$^*$, the SPARC acceleration scale, the classical-test values, and $H_0$. | I | measured [GRAVITY Collaboration 2022; Event Horizon Telescope Collaboration 2022; McGaugh et al. 2016; Will 2014] |
| A9p21009 | The minimal-instance results consumed: the two-face count of the drift (angular $\kap/S$, temporal $\kap/(2S)$, ratio $2$ the double cover), the registration dictionary $p_{\mathrm{sl}}=3(S/\kap)\,r_g$ with the forcing $2\gamma-\beta=1$, and the mass–energy channel $C_{\p-1}\cap C_{2(\p+1)}=Q_4$. | I | |
| B. Realisations: mathematics as physics (the constitutive moves) | |||
| B1p21010 | Gravity is the relaxation of forced frame drift: a field has no proper ideals, so every embedded massive shell is non-ideal and its frame drifts; reducing the mutual drift (raising coherence) is identically motion toward smaller separation. | R | § 1, 3 |
| B2p21011 | Mass is winding rate: the per-cell rate identification, one $Q_4$ datum in the channel $C_{\p-1}\cap C_{2(\p+1)}=Q_4$, tested against its falsifier at the instance [Akhtman & Voether 2026]; $E=mc^2=hf$ an identity; the composite coefficient $m$ is derived (C1), not constitutive. | R | Def. 3.1, 3.2 |
| B3p21012 | Distance is decoherence, counted in the relational steps of the observer’s chart. | R | Def. 3.3 master: 00:D3 |
| B4p21013 | Channel unity: the clock tick (temporal correlation) and the decoherence count (spatial correlation) are two readings of one capacity. | R | Lem. 3.5 |
| B5p21014 | The single realisation: the objects we weigh are coherent composites. | R | § 3 |
| B6p21015 | A cell-local change of frame data is the gauge symmetry, its compensating connection the field; the source’s conservation is that gauge invariance. | R | Prop. 5.9 master: 00:D5 |
| C. Derived: theorems and consequences within this paper | |||
| C1p21016 | The linearised dynamics is the gradient flow of a coherence free energy — its current derived from the pair tally conditional on channel unity (Lemma 3.9) — and a locked cluster of cardinality $m$ couples with coefficient exactly $m$. | T | 1/1 |
| C2p21017 | Newton’s law $F=-Gm_1m_2/r^2$ with $G=1/4\pi\varkappa$ [chart] (register value $G=2S$): the $r^{-2}$ from harmonicity in the three frame freedoms, the product $m_1m_2$ from coherent additivity. | T | 1/1 master: 00:E3 |
| C3p21018 | Universal attraction from the single arrow of the drive (a repulsive charge would need a time-reversed, Frobenius-branch clock). | T | Prop. 4.7 |
| C4p21019 | The equivalence principle as an identity; the bias field is clock rate, giving the redshift $\Delta f/f=Gm/rc^2$; the turnaround radius against the Hubble drive. | T | 1/1 |
| C5p21020 | The magnitude $G=\hbar c/m_P^2$, frame-covariant (a relativity principle for scale); gravity’s weakness is cardinality dilution. | T | § 4.4; grav.orderone 1/1 master: 00:B7 |
| C6p21021 | The conserved dust-tensor source as the Frobenius-symmetric square on the mass shell (cardinality conservation and torsor equivariance). | T | Prop. 5.1 |
| C7p21022 | Channel unity forces the spatial bias to equal the temporal one, $\gamma=1$, restoring the full light deflection $4Gm/c^2b$. | T | 2/2 |
| C8p21023 | The exponential isotropic metric by multiplicative composition, with $\beta=\gamma=1$: deflection, Shapiro delay, and perihelion at their observed values. | T | 2/2 |
| C9p21024 | The discrete Fierz–Pauli functional, exactly gauge-invariant and the unique adjacency-local stiffness on the shell (the spin-one case is unique-Maxwell). | T | 2/2 master: 00:E1 |
| C10p21025 | No preferred-frame leakage: the Lense–Thirring precession at its observed value, with $\alpha_1,\alpha_2=O(1/\sqrt\Omega)$. | T | Prop. 5.11; grav.rotating 1/1 |
| C11p21026 | The nonlinear completion forced: the exact cut-flux law (transfer antisymmetry) plus the winding grading forbid independent static self-sourcing, the clock-comparison cocycle makes the exponential reading unique (exactly the character $\gen^{-\Delta n}$, $e^{-u}$ its observer lift), and Deser’s bootstrap does not apply because what gravitates is winding rate. | T | Thm. 6.5; grav.branch 1/1 master: 00:E4 |
| C12p21027 | The exact saturating static profile and its slip core at $r_*=\sqrt{Gm}$: horizonless, operationally black at $r_f=r_s/\ln\Omega$ — the exact statement the cut form, $4\pi r^{2}$ its [chart] reading; the $r_f$-anchored claims conditional on the fold-transition programme. | T | 2/2 master: 00:E5 |
| C13p21028 | Area-law entropy $S=c_S'A_f/\ell_P^2$: the $\tfrac14$ the $Q_4$ gauge quotient counted on the shell ($S/A=\kap/\p $ exactly), the Carrier identity $S=(\Omega-1)/4$ at the coherence-horizon shell the same quarter (a consistency, not a closure); the count identified with the coordinate area at the operational cut (fold-conditional, the identification a realisation with the capacity prediction as its falsifier); an exact merger area law $\Delta A=2M_1M_2$ on the count face of the mass; a phase-slip (Hawking-scaling) emission suppressed below one quantum per Hubble time. | T R | 2/2 |
| C14p21029 | Dark energy as the curvature of the finite chart, $\Lambda\sim1/\Omega$ (the fine-tuning problem unasked), and the CMB large-angle anomaly as the wrapped-chart signature. | T R | § 8.1 |
| C15p21030 | The radiative sector: the quarter-turn conjugate momentum makes the dynamics a wave equation: two helicity-$\pm2$ gravitons at speed $c$, quadrupole radiation; gauge leaves two polarisations. | T R | 1/1 |
| C16p21031 | The order-one constants are determined: $c_S'=\tfrac14$ (the $Q_4$ gauge quotient, counted on an instantiated shell and carried by the Carrier identity), the floor’s $2\pi$ the full cycle of the angle–count dictionary [D], its threshold the realisation [R], the recurring constant the solid angle $4\pi$, reducing in the Carrier register to the calibration congruence $(4S)^{2}\equiv1$ (one residue relation). | T R | 1/1 |
| C17p21032 | The rotating solution as the gravitomagnetic (quarter-turn) dual: frame-dragging $=$ Lense–Thirring, horizonless, shadow split at $O(a)$; the ergoregion is an $O(a^2)$ feature of the rapid-rotation residue. | T R | Prop. 7.3; grav.rotating 1/1 master: 00:T5 |
| C18p21033 | The primordial spectrum is structural: $n_s=1$ as the scale-dilation fixed point, the red tilt an asymptotic conjecture with committed protocol ($-\pi^{2}/\ln\Omega$, the angle structural, the linearity condition named), the wrapped chart fixing the low large-angle power. | T R | Prop. 7.8; grav.primordial 1/1 master: 00:T2 |
| C19p21034 | The registration crossover made exact: the killed walk is the noise sector, first passage the masking event, the window attribution the framed-rational tally ratio $g_N/f$, and the orbital response now derived (Theorem 4.6 with the registered-inertia identity); the remaining declared content is the masking-barrier identification (barrier $=$ coincidence amplitude $\sqrt x$) and the corpus’s unified sampling clause, shared with [Akhtman & Voether 2026]. | T R | 3/3 master: 00:L1 |
| C20p21035 | The two-shift update law: kick (finite Fourier shift by the source character) composed with metaplectic transport gives $\Delta^{2}_{\tau}q=-\nabla u$ with $m$ cancelling — registered inertia and the equivalence principle inside one derived law. | T | 2/2 |
| C21p21036 | The defect chain: relative-cycle defect with exact recurrence; the compression identity (relaxation $=$ unresolved leakage, recurrence retained); the pair-tally current (sine derived, conditional on channel unity); binding bookkeeping decided — substrate winding sources the field, the defect registered-side, $\eta_{\mathrm{Nordtvedt}}=0$. | T | 3/3 |
| C22p21037 | Induced relaxation, chart-relative form: given the registered adjacency chart with unit capacity, the weak-defect short-memory reading of $K_{\mathrm S}=P_{\mathrm S}U_{\mathrm C}P_{\mathrm S}$ is the gradient flow of the coherence free energy with the pair-tally current — proof by composition of the defect pillars, the forced leading contraction, and the return chains. | T | Thm. 3.10; grav.defect master: 00:T1 |
| C23p21038 | The 1PN equivalence lemma: through first post-Newtonian order the two-body metric coefficients coincide with general relativity’s isotropic form on the superposed potential, cross term included, so the Einstein–Infeld–Hoffmann sector is shared exactly. | T | 2/2 master: 00:T4 |
| C24p21039 | The fold-return floor identity at dev scale: the period-averaged exterior return through the fold, $\overline{P_{\mathrm{ext}}F\,U_{\mathrm C}^{\,n}F^{-1}P_{\mathrm{ext}}}$, equals the uniform floor operator exactly — no stationary exterior structure above $1/N_I$; the no-prompt-echo criterion verified at toy scale. | T | Rem. 6.7; grav.fold_echo 1/1 master: 00:T3 |
| C25p21040 | The PPN triangle closed by two routes: channel unity gives $\gamma=1$ (C7), the instance cover forces $2\gamma-\beta=1$ (A9), jointly $\beta=1$ — Proposition 5.6 reached independently, the deviation $\tfrac23(2\gamma-\beta-1)$ its falsifier. | T | Rem. 5.7; [Akhtman & Voether 2026] |
| X. What the construction explains: the explanation register | |||
| X1p21041 | The cosmological-constant magnitude. $\Lambda\sim1/\Omega$ in Planck units is the curvature of the closed chart, the totality datum, not a vacuum energy requiring cancellation across $122$ orders. The fine-tuning problem is not solved but unasked. | T R | § 1 |
| X2p21042 | No dark-matter particle. The radial acceleration relation is a registration crossover of the resolution floor, not evidence for undetected mass: galactic dynamics require no dark sector beyond the floor already fixed by $\Omega$. | T R | |
| X3p21043 | No horizons, no singularities, no information loss. Collapse ends in a horizonless body, operationally black at $r_f$ (conditional on the fold-transition programme, C12), with an exact saturating interior profile — and information conservation is a theorem citation, not an absence claim: interior data live in offset sectors the drive confines exactly [Akhtman & Voether 2026]: the theory never forfeits predictivity. | T R | Prop. 6.4, 6.2 |
| X4p21044 | Misner’s standing objection dissolves. The exponential metric is derived, not posited as a self-coupled field equation, so the field-equation pathology Misner identified in exponential-metric gravity does not arise here. | T | Rem. 6.7 |
| X5p21045 | No graviton quantisation problem. The radiative field is collective phase synchronisation, not a propagating fundamental quantum: there is no separate field left over to quantise. | T R | § 5.5 |
| X6p21046 | The large-angle microwave anomaly. The anomalously low large-angle correlation ($p\sim1\%$ [Schwarz et al. 2016]) is the signature of the wrapped chart: super-horizon modes are identified, not populated, truncating correlations beyond the horizon scale. This consonance is a number, the primordial spectrum being structural, not transported. | T R | Prop. 7.8; grav.primordial 1/1 |
| X7p21047 | The equivalence principle is an identity, not a postulate. | T | Cor. 4.8 |
| X8p21048 | The weakness of gravity is cardinality dilution, not a hierarchy to be tuned. | T | § 4.4 |
| P. Falsifiable predictions: the exposures | |||
| P1p21049 | Strong-field signatures at or outside the photon sphere: shadow $+4.6\%$, ringdown $-4.4\%$, ISCO frequency $-6.9\%$ with accretion efficiency $5.48\%$ (against $5.72\%$): one origin, jointly falsifiable, conditional on Theorem 6.5's inputs alone. | E | 1/1 |
| P7p21050 | Floor-surface signatures: no prompt echoes (the round-trip delay to $r_f$ carrying $e^{2u(r_f)}=\Omega$) and no evaporation bursts; conditional on the fold-transition programme (C12, C24; the no-prompt-echo criterion a theorem at dev scale); a reported echo detection at the stated precision falsifies the construction. | E | 2/2 |
| P2p21051 | Preferred-frame effects at $O(10^{-61})$: the drive’s in-principle signature, far below current $10^{-5}$ bounds. | E | Prop. 5.11 |
| P3p21052 | The weak-acceleration floor $a_0=cH_0/2\pi=1.04\times10^{-10}\,\mathrm{m\,s^{-2}}$ at the entailed $H_0=67.4$ [Akhtman & Voether 2026], against the fitted $1.20\pm0.24\times10^{-10}$: $13\%$ low, $0.7\sigma$ of the systematic-dominated error, no free parameter; the $2\pi$ the dictionary’s full cycle [D], the one-cycle-per-period threshold the realisation [R] with its discrete falsifier. | E | Prop. 7.4; grav.rar 1/1 master: 00:L1 |
| P4p21053 | The radial acceleration relation, $g_{\mathrm{obs}}=g_N/(1-e^{-\sqrt{g_N/a_0}})$, no fitted function beyond the declared barrier identification (C19), the form the data selected across the four observed acceleration decades, its approach to Newton the discriminant ($0.05$ at $g_N\approx5a_0$ against the simple rational interpolant); the deep limit gives the baryonic Tully–Fisher relation $v^4=GMa_0$. | E | Prop. 7.7; grav.deepregime 1/1 |
| P5p21054 | The floor tracks the expansion, $a_0(z)\propto H(z)$ (BTFR zero-point $\propto E(z)^{1/4}$); high-$z$ rotation curves decisive. | E | § 7.4, [Akhtman et al. 2026] master: 00:L7 |
| P6p21055 | The 2PN periastron excess $\delta\dot\omega/\dot\omega\approx1.5\times10^{-6}$ for PSR J0737$-$3039, today degenerate with the mass refit; decisive either way once an independent observable fixes $M_{\mathrm{tot}}$ at $10^{-6}$. Independent of the coherent-fraction falsifier of [Akhtman & Voether 2026]; the ledger carries both. | E | § 7.2; grav.2pn 1/1 |
| V. Machine verification | |||
| V1p21056 | The validation package finite-ring-space/src/21-gravity: the paper’s twenty-one scripts as written, run through one registry as twenty-one families — the twelve of the original suite (grav.newton, grav.ppn, grav.strongfield, grav.fp_gauge, grav.branch, grav.fluxnoise, grav.rar, grav.deepregime, grav.radiative, grav.orderone, grav.rotating, grav.primordial), the eight of the revision rounds (grav.2pn, grav.1pn_eih, grav.binding, grav.counting, grav.defect, grav.inertia, grav.deepregime_orbit, grav.fold_echo) and the uniqueness proof (grav.fierz_pauli) — each family the script’s verdict over its labelled checks and printed PASS/FAIL lines, each naming the rows it witnesses; the notebook 21-gravity-main.ipynb executes them in the browser and the driver run_all.py writes results.json. | T | Reproducibility; run_all |
| V2p21057 | The kinds, recorded per family: EXACT — decidable identities over finite fields, cyclotomic extensions, exact rationals, or symbolic identities with no tolerance in the verdict (fourteen families); CHART — a continuum reading (a limit, a series coefficient, a numerical extremum) or a comparison with measured data, tagged [approx]/[data] in the script, verdict by stated tolerance (grav.newton, grav.strongfield, grav.branch, grav.rar, grav.rotating); SIM — the seeded Kuramoto simulation of grav.fluxnoise, verdict by stated tolerance on time averages. The figures are regenerated by the package’s make_fig_*.py outside the run. | T | Reproducibility |
| V3p21058 | What the run does not carry, stated: the comparable-mass $2$PN $\nu$-sector reduction (the failed first route is not shipped, per the corpus discipline; the $1$PN sector closed by grav.1pn_eih, the test-particle sector by grav.2pn); the imports and realisations of blocks A and B, which the suite consumes and does not check; the $\Omega$-hard residue Z1, absent by construction. | T | Reproducibility; § 7.2 |
| Z. Horizon: the residue beyond the bounded observer | |||
| Z1p21059 | The cross-scale running of $a_0$, the ratio $(\Omega-1)/d$: reduces to the factorisation of $\Omega-1$, the rate face of the Riemann scale operator. | Ω | master: 00:Z7 |
Nothing matches that filter.
Ledger history
Ledger history. 2026-06-23: the status section with three tables (what the construction explains X1–X8, the predictions P1, P1’, P2–P6, the predicate ledger A–C, Z); rounds 01–02 (June–July 2026) added C19–C25 and the revision scripts. 2026-09-14 (T26): the three tables merged into one ledger in the corpus format — X and P are blocks of it, V added, every machine-verified row citing the family identifiers of the validation package (finite-ring-space/src/21-gravity), accession keys assigned; the row P1’ relabelled P7 (labels are a letter and a number). No row has been retired.