| # | Description | Status | Sources |
|---|---|---|---|
| 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] master: 00:E3 |
| 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 | master: 00:C20 |
| 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 |
| B7p21065 | The count of the registration set at the operational cut is the coordinate area in Planck cells, $|R_\partial|=4\pi r_f^{2}/\ell_P^{2}$ up to the solid-angle recalibration of the chart. Falsifier: a collapsed object whose externally resolvable capacity is measured at the Bekenstein–Hawking value. | R | Prop. 6.8 |
| B8p32014 | The cosmological constant is the curvature of the wrapped finite chart (32:B7). Falsifier: $w\neq-1$, or a negative or scale-varying $\Lambda$. | R | § 8.1 |
| B9p32015 | The masking barrier is the coincidence amplitude: registration fails when the killed noise walk first traverses the signal’s amplitude $\sqrt x$, coefficient one (32:B8). Falsifier: an approach to Newton departing from $1-e^{-\sqrt x}$ beyond the binned-RAR uncertainty, or a knee displaced from $cH_0/2\pi$. | R | Prop. 7.7 |
| 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 | Lem. 4.1, 4.2, 3.9 |
| 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 | Thm. 4.4, Lem. 4.3 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 master: 00:E3 |
| 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 | Cor. 4.8–4.10 |
| 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 |
| 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 | Lem. 5.3, Prop. 5.4 |
| C8p21023 | The exponential isotropic metric by multiplicative composition, with $\beta=\gamma=1$: deflection, Shapiro delay, and perihelion at their observed values. | T | Lem. 5.5, Prop. 5.6 |
| 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 | Prop. 5.10; rests on 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 |
| 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 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 stated in the substrate coordinate, $r_*<r_f<M$ for $M>(\ln\Omega/2)^{2}$ Planck masses (master E10). | T | Prop. 6.2, 6.4 |
| C13p21062 | 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 read as the coordinate area at the operational cut (B7); the identification of the entropy with the channel count (master D15); 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 | |
| C14p21063 | The magnitude of dark energy, $\Lambda\sim1/\Omega$ in Planck units, the curvature of the finite chart (B8), the fine-tuning problem unasked; the CMB large-angle anomaly the wrapped-chart signature (C18). | T | |
| 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 | |
| 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 (14:B2), its threshold the realisation 32:B3, the recurring constant the solid angle $4\pi$, reducing in the Carrier register to the calibration congruence $(4S)^{2}\equiv1$ (one residue relation). | T | |
| 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 | master: 00:E5 |
| C18p21033 | The primordial spectrum: $n_s=1$ the scale-dilation fixed point; the tilt’s committed form $-\pi^{2}/\ln\Omega$, $\ln\Omega$ the e-fold count of the frame’s scale range, the realisation of master L8, with the measured tilt its falsifier ($\tfrac12\ln\Omega$ excluded at $8.2\sigma$; the linearity condition undefined on a shell); the wrapped chart fixing the low large-angle power. | T | master: 00:L8 |
| C19p21064 | 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); it rests on the masking barrier (B9) and the corpus’s unified sampling clause (master D10), shared with [Akhtman & Voether 2026]. | T | 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 | Thm. 4.6 |
| 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 | Lem. 3.7, Prop. 3.8, Lem. 3.9 |
| 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; rests on 00:E11 master: 00:E11 |
| 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 | § 7.2 master: 00:E4 |
| 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 | |
| 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 | |
| X3p21043 | No horizons, no singularities, no information loss. Collapse ends in a horizonless body, operationally black at $r_f$ (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 | master: 00:D15 |
| 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 | |
| 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 | |
| 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 $+4.6\%$, eikonal ringdown $-4.4\%$ (the $(2,2,0)$ reading open, GW250114 the exposure; master E13, E14), ISCO frequency $-6.9\%$ with accretion efficiency $5.48\%$ (against $5.72\%$): one origin, jointly falsifiable, conditional on Theorem 6.5's inputs alone. | T | master: 00:E14 |
| P7p21050 | Floor-surface signatures: no prompt echoes (the exact profile horizonless with no reflecting surface, C12; the round-trip delay to $r_f$ carrying $e^{2u(r_f)}=\Omega$) and no evaporation bursts; a reported echo detection at the stated precision falsifies the construction. | T | |
| P8p21060 | The primordial tilt $n_s-1=-\pi^{2}/\ln\Omega=-0.0348$ [approx] at $\ln\Omega=283.5$ (the master’s scale import), against Planck 2018 $-0.0351\pm0.0042$ ($+0.07\sigma$); $0.24\%$ per factor two in $S$; the neighbouring count $\tfrac12\ln\Omega$ excluded at $8.2\sigma$. | T | master: 00:L8 |
| P2p21051 | Preferred-frame effects at $O(1/\sqrt\Omega)=O(10^{-62})$ ($3\times10^{-62}$ at $\Omega=1.3\times10^{123}$): the drive’s in-principle signature, far below current $10^{-5}$ bounds. | T | |
| 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 (14:B2), the one-cycle-per-period threshold the realisation 32:B3 with its discrete falsifier. | T | 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 barrier identification (B9), 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 $v^4=GMa_0$. | T | |
| 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. | T | master: 00:P1 |
| 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. | T | |
| 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 |
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Status
- 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 py token is named, its content is reproduced by the framework’s ledger file; where a lean token is named, proved in Lean.
- Ω
- $\Omega$-hard. Decided by the totality; not closeable by a bounded observer.
Sources
- Gold: the python check that decides the predicate, at its line in the script (the source, rendered on this site); on hover the deciding family and the families that corroborate it, and the verdict of the last run (red when it fails). The script page's bar runs the predicate's cell of the package's notebook on Google Colab (one cell per predicate; the whole notebook from the header badge).
- The predicate as one Lean declaration, named by its accession key (
p21001) — the conjunction of the theorems that prove it — at its line in the rendered module, one mark per library that proves it, the core's first. Green: the core (lean/FrcCore) — the statement on the shell ring, by computation and induction, no Mathlib, no axioms: the finitist witness. Violet: Mathlib (lean/FrcLedger) — the classical statement for every shell on Mathlib's hierarchy (its axioms,propext,Classical.choice,Quot.sound, on hover). Red: a proof the gate rejects (an unfinished proof,native_decide) — not a witness. A predicate proved in both libraries shows both marks; one mark, one proof. The module page's bar opens the module as one executable file in Lean Live (the core on plain Lean, no Mathlib; the file elaborates in the browser and prints every predicate's axioms at its end); the header's two Lean badges count the marks of each colour and open the two executables. - 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).