| # | Description | Status | Sources |
|---|---|---|---|
| A. Inputs: imported, not derived here | |||
| A1p14001 | The de Sitter entropy $\dS$, the single quantitative import, fixing the cardinality exactly, $\Om=4\dS+1$. | I |   master: 00:A7 |
| A2p14002 | The Carrier congruences, relations certified by the realisations (the $c$-square $\dS$ even, the triality centre $\dS\equiv1\pmod3$, completeness $4\dS+1$ prime); instantiated on the laboratory Carrier as host-decided values. | I |  master: 00:A7 |
| A3p14003 | The containment argument: the Carrier is the totality, so there is no ensemble for $\dS$ to range over (the reductio, the audit of potential infinity, and finite incompleteness). | I | |
| A4p14004 | Unit reciprocity: a unit is a reciprocal cross-frame relation, so registered durations are two-way and one-way durations are gauge. | I | |
| A5p14005 | The equatorial-horizon identification in the de Sitter chart [approx]: the static-patch horizon of a pole observer at colatitude $\pi/2$. | I | |
| A6p14006 | Existence of the acceleration floor, the sub-floor amplitude regime, and the value form $a_0=cH/2\pi$: the synchronisation-threshold derivation lives upstream, imported here in value form, with this paper adding only the register typing. Upstream: the $2\pi$ is this paper’s dictionary (B2) applied to the registered drive period; the threshold is the realisation tagged in the dark-sector paper (row B3) and the gravity paper (Prop. afloor), with its discrete falsifier; the value is quoted at the channel-1 rate $H_0=67.4$, consistent with the octant to $0.1\%$ (C8). | I | master: 00:L1 |
| A7p14007 | Laboratory constants $\lP$, $\tP$, $c$ as unit conversions [I], a channel of their own. | I | — |
| A8p14008 | Observational data and the rival chart’s identities [ΛCDM], confrontation only: the Planck fit, the distance ladder, the RAR knee and its intermediate-redshift evolution, stellar ages (superseding release), the entropy budget. | I | [Planck Collaboration et al. 2020; Riess & others 2022; Freedman 2021; McGaugh et al. 2016; Ciocan & others 2026; Valcin et al. 2026; VandenBerg et al. 2013; Egan & Lineweaver 2010] |
| B. Realisations and conventions: mathematics as metrology | |||
| B1p14009 | The calibration faces: the observed $\Lambda\sim1/\Om$ and $H\sim\Om^{-1/2}$ [approx] are the two chart faces of the import, carrying no evidential weight of their own. | R | § 3 |
| B2p14010 | The angle–count dictionary: $2\pi$ the full cycle, $h=2\pi\hbar$; every angular numeral is the chart face of a count ($\pi/4\leftrightarrow\dS/2$ chronons). | D | |
| B3p14011 | Age as a totality datum: the measured age is the frame-invariant horizon depth, registered as a count in the Subject register; the datum is the supremum of closable depth — the saturation clause: saturated by the totality’s causal record, approached by material inventories. The registered depth is what the stellar-age ladder measures (channel 4); its $0.2\%$ degeneracy with the rival chart’s $t_0$ at the present chronon is a coincidence of charts (P2), not part of the identification. | R | § 3.1 |
| B4p14012 | The registered-rate role: the $H$ in the floor value is the observer’s registered rate, the Subject-only-frame doctrine, giving $a_0(z)\propto H(z)$ — falsifiable physical content, graded a realisation. | R | § 3.5 master: 00:P1 |
| B5p14013 | The circularity criterion: an estimate counts as a measurement only if its confrontation channel is disjoint from the channel that fitted the import; channel conflation and register conflation are one error class. Scoped to the metrology register; the explanation register is judged by zero-freedom derivation (§ 3.3). | D | § 3.3 master: 00:L2 |
| B6p14014 | The closure budget: the horizon depth of Lemma 1(i) is the total reciprocal-registration budget — out and back draw on one quarter-sector count and the exchange must close within it. An FRC closure realisation, not an import of chart geometry; its continuum shadow [approx] is the causal diamond of the static patch. Member of $B_{\mathrm{oct}}$; presently single-use (see the lemma remark). | R | § 3.4 |
| B7p14015 | The depth register rule: an instrument reads either a rate/curvature (a ratio of co-chained counts, the one-way gauge cancelling) or a depth (a registration whose count is the datum); the A4 halving binds depth registrations only, and no spatial instrument registers totality depth. The conversion of a spatial arc into temporal depth by $c$ is part of this rule — the reciprocal unit relation, declared, not chart-derived (proper distance over $c$ is not automatically radar time in curved geometry). Classifies every table row; the classification could fail empirically, so the rule is graded a realisation. | R | § 2 |
| B8p14016 | The effective-fluid correspondence behind P4: the rival chart’s continuity law $\dot\rho_\Lambda+3H(\rho_\Lambda+p_\Lambda)=0$ [ΛCDM] with $\dot\rho_\Lambda=0$ gives $w=-1$; and the single-component clause — the framework’s dark sector is derived with no second dark-energy component available, so an observed $w_{\mathrm{eff}}\neq-1$ cannot be reassigned to an extra evolving fluid. The clause makes P4’s falsifier total rather than escapable. | R | § 3 |
| B9p14017 | The area law: the projected horizon area in Planck cells is the full phase cycle, $A=4\dS$ [approx] at the horizon ($4\pi n_A=4\dS$, $n_A=(r_H/\lP)^{2}$ [approx]); the Bekenstein–Hawking quarter $\dS=A/4$ is the quarter-turn $Q_4$, the generative four of $\Om=4\dS+1$ and $\p=4\kap+1$; area is a Subject-projection property, the Carrier carrying none. | D | master: 00:L5 |
| B10p14044 | The capacity axis counts hydrogen units (one gram at $N_A$ to $0.003$ dex [approx]): a definition, not a registered shell. The least register window resolving the total winding of $N$ units is $3N$, a theorem about the observing Subject (B11); a phase-locked composite of like units is an aggregate, never one shell (B12; a shell of capacity $3N$ exists only when $12N+1$ is prime, B13); the largest shell any Subject registers whole is the nesting bound $\mathsf{q}<\Om^{1/4}$ (00:B8), $N<(\Om^{1/4}-1)/12$ units, $8\times10^{2}$ kg at $\dS=3.3\times10^{122}$ [approx] ($\Om^{1/4}$ units overcounts by $12$); everything heavier an aggregate of locks; the proton’s $1$ against $3$ compares a tower label with a chart count. | D | |
| B11p14046 | The additive window: for $N$ modes at resolvable heights $|\hat h_i|\le\kap_i$ (38:A8) the total winding satisfies $|\sum_i\hat h_i|\le\sum_i\kap_i$, attained, every total in the range occurring; the least register window resolving the total exactly is the capacity sum $\sum_i\kap_i$, $3N$ for $N$ hydrogen units — a capacity of the observing Subject, not a shell of the composite (B10). | T | |
| B12p14047 | The lock of like units: $N$ like units never fuse (00:F6, $\gcd(12,12)=12$): their joint registration (00:F2) recurs on $C_{12}$ with period $12$ for every $N$, one superselected $C_{12}$ offset per further unit — an aggregate, never one shell. Instances: the diagonal drive on $C_{12}^{2}$ has $12$ orbits of length $12$, on $C_{12}^{3}$ $144$. | T | |
| B13p14048 | Shells of capacity $3N$ and fusion: the shell of capacity $3N$ is $\F_{12N+1}$ and exists iff $12N+1$ is prime ($N=1,3,5,6,8,9$ below $10$; none at $N=2$, $25=5^{2}$), never the lock’s register (B12); fusion of coprime cycles is multiplicative, $\kap=n_{1}n_{2}/4$: hydrogen $(4,3)\to3$, one orbit of length $12$ (00:F6). | T | |
| C. Derived: theorems and consequences within this paper | |||
| C1p14018 | Two faces: a frame supplies one spatial and one temporal datum, so every instrument reading the size of the totality is one of exactly two faces, converted by $c$. | T | § 3.2 |
| C2p14019 | The non-channels: the constants correspondence carries no dimensional readout, the horizon-quarter closure is an internal consistency loop, the registered entropy budget is a floor eighteen orders below $\dS$ [approx]. | T | § 3.1 |
| C3p14020 | The same-channel demotion: $t_0H_\Lambda=\tfrac23\artanh\sqrt{\Omega_\Lambda}$ [ΛCDM] depends on the one fitted parameter, so the $0.2\%$ landing on $\pi/4$ restates the fit (by B5). | T | § 3.3 |
| C4p14021 | The concordance: five evidence readings return $\sqrt{\dS}=(1.3\text{--}1.8)\times10^{61}$ [approx], $\pm17\%$ half-range on the count face, factor $1.9$ on $\dS$; the gauge reading 4a is displayed and excluded from every statistic. | T | § 3.2 master: 00:L2 |
| C5p14022 | The octant lemma: the measured age is $\dS/2$ chronons, $(\pi/4)\,r_H/c$ in the chart [approx]; the halving and the sector are exact through the enumerated realisation set $B_{\mathrm{oct}}=\{$A4, A5, B2, B3, B6, B7$\}$, the same class as the constants correspondence. | T | Lem. 1 master: 00:L3 |
| C6p14023 | The octant sector: $C_{8}\subset C_{4\dS}$ exists exactly when $\dS$ is even, its character $\zeta_8$ the residue the Tsirelson identity carries; the octant count $\dS/2$ is an integer on every admissible Carrier. | T | § 2, [Akhtman & Voether 2026] |
| C7p14024 | The dark-energy output: $\Omega_\Lambda=\tanh^{2}(3\pi/8)=0.6837$ [ΛCDM] against the fitted $0.685\pm0.007$, a $0.19\sigma$ landing — a zero-freedom derivation, not an independent measurement (two-register separation, § 3.3). | T | master: 00:L3 |
| C8p14025 | The entailed-rate consistency: with the channel-1 $\Lambda$ the age–rate locus returns the channel-1 $H_0$ (to $0.1\%$), same-channel by the definition $\Lambda=3\Omega_\Lambda H_0^{2}/c^{2}$, a corollary of C7 and not a prediction — the floor value itself consumed as import (A6) under the role realisation (B4), not derived here; the two-cluster ratio is the chart identity $(H_{\mathrm{rate}}/H_0)^2/\Omega_\Lambda$, displayed, not claimed. | T | § 3.5; rests on A8 |
| C9p14026 | The registrable triangle [approx]: width $\log_{10}(r_H/\lP)=61.0$ dex — the count face $\sqrt{\dS}$ in its chart dress $\sqrt{\dS/\pi}$, the solid angle the declared transport; the thirteen-object diagonal, slope $3.03$ (intercept $1.00\times10^{3}$ at the metre pivot); the constrained $k=3$ cubic coefficient $\tilde c=m/R^{3}=0.97\times10^{3}\,\mathrm{kg\,m^{-3}}$ (not a density: a sphere’s is $0.62$ dex lower); the Compton entry returns its input, the Schwarzschild exit is the chart’s one derived quantity, the over-closure bound $1.8\times10^{8}M_\odot$ within the $(1.4\text{--}2.8)\times10^{8}$ sensitivity band — a bound, no observed scale predicted (X5); classified a consistency chart, not a channel. | T | § 4 master: 00:L4 |
| C10p14027 | The capacity axis [approx]: holographic ring capacity $\log_{10}\kap_{q}$, zero at hydrogen ($\kap_H=3$; $\kap_{q}=2$ has no shell), one dex per chart dex, terminating at the count face $\sqrt{\dS}$ — the triangle’s own sixty dex; the one-gram horizontal lands on $N_A$ to $0.0035$ dex (the $1.008$ u offset), asserted before drawing. | T | § 4 |
| X. Explicability dividends: the explanation register | |||
| X1p14028 | The cosmological-constant problem dissolves: $\Lambda$ is the curvature face of the one cardinality, not a vacuum energy, so there is no mode sum to cancel and the $10^{120}$-order discrepancy does not arise. | T | § 3, [Weinberg 1989] |
| X2p14029 | Milgrom’s coincidence $a_0\approx cH_0/2\pi$, on record since 1983, becomes a value with zero freedom: the $2\pi$ is the fixed cycle convention, the reverse rebooking of $h=2\pi\hbar$, and the $H$ is the observer’s registered rate. | T | master: 00:L1 |
| X3p14030 | The age coincidence $t_0H_\Lambda\approx\pi/4$ becomes the octant: $\Omega_\Lambda=\tanh^{2}(3\pi/8)=0.6837$ [ΛCDM] is an output of the realisation set, landing at $0.19\sigma$ on the fitted $0.685\pm0.007$ — never an independent measurement of $\dS$ (same-channel, X6), a zero-freedom derivation in full standing; the $(\Lambda,H_0)$ entailment at the channel-1 $\Lambda$ is this same landing read on $H_0$, not a further prediction. | T | |
| X4p14031 | The congruence $\dS$ even acquires a second physical face: it is the existence of the octant sector, hence the well-definedness of the measured age — a structure of the Carrier’s cycle lattice, certified by realisation and read as an observational condition, never an arithmetic property of the numeral. | T | Lem. 1(iv) master: 00:C16 |
| X5p14032 | The termination of the single-object sequence is a consistency of the wall structure, the exit of C9 read as a bound: the typical-density diagonal over-closes at $1.8\times10^{8}M_\odot$ [approx] (sensitivity band $(1.4\text{--}2.8)\times10^{8}$ across fit variants), so every heavier single object continuing the condensed-density sequence is necessarily wall-bound; the supermassive holes are that continuation, not an anomaly beyond it. Classified with the triangle: consistency, not derivation of an observed scale. | T | § 4 |
| X6p14033 | The celebrated sub-percent age agreement is judged: same-channel by the circularity criterion, a restatement of the fitted dark-energy fraction — consistency, not evidence — and the criterion locates exactly which confrontations do carry weight. | T | § 3.3 |
| X7p14034 | The universe-as-black-hole puzzle dissolves: the Hubble radius sits on the black-hole line identically ($\rho_{\mathrm{BH}}(r_H)=\rho_c$ [ΛCDM]), but the Schwarzschild reading consumes an exterior, and the containment argument (A3) makes an exterior incoherent — the totality on its own horizon is the chart’s closing point, not a hole in a background. | T | § 4, [Lineweaver & Patel 2023] |
| X8p14035 | The interior completions weighed: black-hole cosmology keeps the same saturation identity by consuming a parent exterior and a singularity-replacing mechanism (torsion; exclusion pressure); the finite reading consumes neither: the containment argument (A3) closes the chart at $\Om$, and no singularity is available over a finite totality. The registered record-depth bound separates the completions empirically (P2). | T | |
| P. Falsifiable predictions: the exposures | |||
| P1p14036 | (Lead.) The floor runs with the registered rate, $a_0(z)\propto H(z)$, the role statement of the floor closure, against both a constant fundamental $a_0$ and feedback-emergent scaling. Status: the constant-floor branch is excluded — four redshift bins rising from $1.99$ to $2.71\times10^{-10}\,\mathrm{m/s^2}$ over $z=0.50$–$1.28$ [Ciocan & others 2026], $\Delta\chi^2=23$ against $A\,H(z)/H_0$ [ΛCDM] — and the running form fits the bins at free amplitude ($A=1.39\pm0.03$, $\chi^2=5.4/3$); the binned rate sits on the $\propto H(z)$ chord while the authors’ global linear fit runs above it, so the shape waits on the likelihood refit of the resolved tracks, and the amplitude’s $0.12$ dex excess over $cH_0/2\pi$ lies inside the $0.2$ dex gas systematic. Falsifier: a likelihood refit or a wider-redshift sample excluding the $\propto H(z)$ form after the baryonic-mass and pressure-support systematics close. | T | master: 00:P1 |
| P2p14037 | The registered record depth is bounded by $(\pi/4)\,r_H/c=13.79$ Gyr [approx] (channel-1 anchor) at every observational frame chronon, against a $\Lambda$CDM $t_0$ that grows with chart time: degenerate to $0.2\%$ now, separating at ${\sim}7\%$ per Gyr; the octant carries zero parameter freedom at fixed $\Lambda$. Status [Valcin et al. 2026]: the oldest cluster population $0.5\sigma$ below the bound, the inferred age straddling it within $0.1\sigma$ (§ 3.5). Falsifier: one well-dated object older than the octant value, in any frame — fit-independent, fatal; the same bound excludes interior-bounce black-hole cosmology (X8). | T | master: 00:L3 |
| P3p14038 | The age–rate locus $t_{\mathrm{age}}H_0=(\pi/4)/\tanh(3\pi/8)=0.950$, exact for the realisation set. Read on the fit-independent stellar age ($13.61\pm0.34$ Gyr): $H_0=68.2\pm1.7$ km/s/Mpc [approx], $0.5\sigma$ from Planck, $0.65\sigma$ from TRGB, $2.4\sigma$ from the Cepheid ladder — the framework favours the CMB/TRGB value at that significance, and forbids the $\Omega_\Lambda\approx0.76$ by which the rival chart would accommodate the ladder value at the same age. Read on the channel-1 $\Lambda$ it returns the channel-1 $H_0$ (to $0.1\%$): a consistency, not a prediction (C8). Falsifier: a confirmed ladder $H_0$ with the stellar age standing ($t_\star H_0\ge1.0$ against $0.950$). | T | master: 00:L3 |
| P4p14039 | (Lead.) The dark-energy equation of state is rigid: $\Lambda$ is the curvature face of the fixed cardinality, constant, its effective-fluid reading $w=-1$ exactly with no $w_0w_a$ freedom; the fluid statement consumes the conservation-law correspondence and the single-component clause (B8). Status: DESI-era combined fits prefer evolving dark energy at $3$–$4.2\sigma$ [DESI Collaboration 2025], contested by reanalyses (§ 3). Falsifier: confirmed evolution of $w$, which kills $\Lambda\sim1/\Om$ outright and exposes P2 through $\Lambda$. | T | master: 00:L3 |
| Z. Horizon: the residue beyond the bounded observer | |||
| Z1p14043 | The numeral of $\dS$: the prototypical $\Om$-hard residue, measurable through its two faces, expressible from within by no observer; import its permanent status, this metrology the complete epistemic access to it. | Ω | § 3.5 master: 00:L2 |
Nothing matches that filter.
Status
- I
- Import. A standard result or measured datum imported without reproof.
- R
- Realisation. The row states what a physical term denotes in the structure: constitutive, not interpretive, defended by the totality of its registered consequences and confronted through its declared exposures.
- D
- Definition. A naming or set-up move, carrying no empirical content.
- T
- Theorem. Derived within this paper from the rows above it; where the source column links the validation package, its numerals are reproduced by the linked check.
- O
- Open. Genuinely unresolved; its cost stated, not consumed.
- Ω
- $\Om$-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 (
p14002) — 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).