| # | Description | Status | Sources |
|---|---|---|---|
| A. Inputs: imported, not derived here | |||
| A1p10001 | The framed substrate: the Subject shell $\Fp(\chron;0,1,\gen)$ with $\p=4\kap+1$, the derived quarter-turn $\im=-\gen^{\kap}$, the phase cycle $4\kap=2\pi=\p-1$, the framed-complex labels; the Carrier $\Car$ of cardinality $\Om=4\dS+1$ with its admissibility congruences; the realisation doctrine (a row stating what a physical term denotes, defended by its consequences and killed by its falsifier). | I | |
| A2p10002 | The holographic principle as the count: the capacity $\dS$ is the de Sitter entropy, the finite number of physical states behind the observer’s horizon, the one imported number of the paper. | I | master: 00:D15 |
| A3p10003 | The shell Fourier structure: on $\Fp^{\Phi}$, $\Phi=C_{\p-1}$, the matrix $W_{jk}=\gen^{jk}$, the normalized operator $F=\im W$ with $F^{2}=J$, $F^{4}=I$, and the fractional family with its cardinal skeleton $\{0,\kap,2\kap,3\kap\}$. | I | [Akhtman 2026]; Rem. 6 |
| A4p10004 | Classical dimensional analysis and the Planck units: the $\Pi$-theorem, the free-abelian exponent group of $M$-$L$-$T$, Planck’s construction of the natural units from $\{c,\hbar,G\}$, and the observation that only dimensionless ratios of constants are measurable. | I | |
| A5p10005 | The registrability hierarchy of the framework: coherent registration bounded at $\sqrt\p$, read on the exponent lattice as the coherence window $2\sqrt\kap=\sqrt{\p-1}$; the finite-field algebra of square classes and cyclic groups used throughout. | I | |
| A6p10006 | The Unruh and Hawking temperature forms, $T_U=\hbar a/2\pi c\,k_B$, as the shape the temperature domain closes on. | I | |
| B. The registers and the labels | |||
| B1p10007 | The Carrier: the finite complete totality, a timeless torsor of $\Om$ attribute-free points, its cardinality its only absolute characteristic, its reframings $x\mapsto ax+b$ automorphisms; the Subject: a bounded observer, an embedded prime shell of capacity $\kap$ with frame data $\Fp(\chron;0,1,\gen)$, $\p^{2}<\Om$; the two definitions together the paper’s reading of the holographic principle. | D | Defs. 1, 2, Rems. 1, 2 |
| B2p10008 | The Object: an observed system registered relative to the Subject’s frame, every entry a Subject residue, $(\dt;\,q,k,v)$ — relative time, position, winding, rate. | D | Def. 3 |
| B3p10009 | Units as scale assignments: the additive meridian action $(\Fp,+)$ and the multiplicative phase action $\Fpx$ are unit-free coordinate actions, a physical unit a scale assignment to a frame chart of them, a measure a coordinate reading relative to the assigned scale; a physical-interface reading is a coefficient with the domain label it is read in, the two periods typing the label exponents; a change of scale assignment leaves $\card(\Fp)=\p$ and $\card(\Fpx)=\p-1$ untouched. | D | Defs. 4, 5, Prop. 1 |
| B4p10010 | The residue phase label $\gen^{k}$ with the cardinal packet $1,\gen^{\kap},\gen^{2\kap},\gen^{3\kap}$; the framed-complex label space $\mathcal C_\p=\Fp\times\Fp$ with its $\p$-to-one evaluation; $\im$ the finite shell quarter-turn label. | D | Defs. 6, 7, Prop. 2 |
| C. The modular domain algebra | |||
| C1p10011 | The two domain generators: $\unitL$ attached to the additive meridian chart, anchored on the unit $1$; $\unitT$ attached to the drive’s phase chart, anchored on the chronon step; the origins carry no units. The modular labels $U_{r,s}=\unitL^{r}\unitT^{s}$ on $\Dp=C_\p\times C_{\p-1}$, the homogeneous quantity algebra $\Ap=\bigoplus\Fp U_{r,s}$, the domain $\domp(Q)$ of a homogeneous quantity. | D | Defs. 8, 9, 10, 11 |
| C2p10012 | The modular unit-domain group and the grading: $\{U_{r,s}\}$ is a finite abelian group isomorphic to $\Dp$ under $U_{r,s}U_{r',s'}=U_{r+r',s+s'}$, and $\Ap$ is $\Dp$-graded, $\Ap^{(r,s)}\Ap^{(r',s')}\subseteq\Ap^{(r+r',s+s')}$. | T |    Thms. 1, 2 |
| C3p10013 | Fibrewise addition and the neutral-domain criterion: a sum of homogeneous quantities is homogeneous exactly when its nonzero summands share a domain, the fibre components adding separately; a monomial $\prod Q_j^{k_j}$ has domain $\sum k_j(r_j,s_j)$ and is a neutral-domain invariant iff that sum is $(0,0)$ in $\Dp$. | T | Thms. 3, 4 |
| C4p10014 | Admissible reframings: the triples $(a,m,\eps)$ — translation of the spatial origin, dilation of the unit, re-presentation of the drive $\gen\mapsto\gen^{\eps}$, $\eps\in\Zp^{\times}$ — and the two actions they induce on labels. | D | Def. 12 |
| C5p10015 | Window covariance: on a bounded window $|r|,|s|\le H<2\kap$ in the flag-free sector the labels are covariant under every admissible reframing; the bound is sharp (the label $(0,\pi)$ is pushforward-invariant yet non-neutral); the naive character is ill-defined on the modular projection and the would-be temporal character fails composition ($5^{2}\equiv1$ in $\Z_{12}^{\times}$); on the realized lattice the invariant labels are exactly $\{0,2\kap\}$; the $\sigma$-twisted action is equivariant by full sweep on $\p=13$ and $229$; the flagged label $(0,0;1)$ moves under $\eps=-1$. | T | Thm. 5 |
| C6p10016 | What the action forces and what is declared: the time-exponent period $\p-1$ is forced by the pushforward action of $\Zp^{\times}$, the space-exponent refinement to period $\p$ is C1’s declaration, carried by the transport theorem (D2); classical temporal unit change is nominal re-assignment within a fixed presentation, the chronon atomic. | T | |
| D. The four domains and the unit flag | |||
| D1p10017 | Chart duality: $[f]=\unitT^{-1}$; $\delta_S:D\mapsto D^{-1}$ is an involution of the domain group whose orbit on the generators is the four-domain structure (space, momentum, time, energy); $\delta_C$ the second involution, carrying $\unitL\mapsto[p]$, $\unitT\mapsto\unitE$. | T | Def. 13, Thm. 6 |
| D2p10018 | The internal flag: $\Dp$ contains exactly one subgroup of order four, $\langle\unitT^{\kap}\rangle$, entirely in the time-exponent factor; $\Iq:=\unitT^{\kap}$, $\Iq^{4}=1$, $\Iq^{2}=\unitT^{\pi}$; no flag of space ($\Z_\p$ has no element of order four); the two periods interfere, $(\unitL\unitT^{\kap})^{\p}=\Iq$, on three shells. | T | Thm. 7, Cor. 1, Prop. 4 |
| D3p10019 | The lift: the flag records the quarter the chart duality forgets — on charts the cardinal skeleton acts as $s\bmod2$ ($F$ exchanges the conjugate charts, $J$ fixes them), on labels $s\mapsto\Iq^{s}$ is an isomorphism $\Z_4\to\langle\Iq\rangle$ whose quotient by $\{0,2\kap\}$ returns the chart action; on the Carrier $\hbar^{2}=-1$, $\hbar^{4}=1$, $\hbar^{2}\ne1$, the crossing quantum of order four, never two. Shells $(13,2)$, $(173,3)$; Carriers $233$, $2\,408\,561$. | T | Prop. 3 |
| D4p10020 | The crossing degree: the full label $(r,s;j)$, $j\in\Z$ the signed number of register crossings, realized as $U_{r,\,s+j\kap}$ with the sector $j\bmod4$. | D | Def. 14 |
| D5p10021 | The unit flag: the Planck constants carry the flag, $[h]=\unithbar=\Iq$ — the anchoring of the action quantum to the order-four flag, the realisation content of the paper. Falsifier: one empirically closed expression that adds an observer count to a unitful measure without the conversion quantum (an energy to a frequency, a momentum to a wavenumber). | R | Prop. 5, § 5 |
| D6p10022 | The flagged readings over the counts: $\unitE=[h][f]=\Iq\unitT^{-1}$, $[p]=\unithbar[k]=\Iq\unitL^{-1}$, one quantity per pair, the flagged reading one capacity step above its count; $[S]=\unitE\unitT=\Iq$ and every phase count neutral, $\unitE\unitT\unithbar^{-1}=1$; the absolute unit is horizon-inaccessible (no monomial of a window $H\ll\kap$ expresses the flag) and becomes visible at $\kap\le H<2\kap$. | T | Props. 6, 7, Cor. 2, Rem. 7 |
| D7p10023 | A unit of measure is a reciprocal relation between two count systems, counts of one frame per count of another; the unit flag is its domain form, the register crossing made algebraic; the Planck relation the unique interface introducing energy without a mass primitive. Falsifier: an absolute unit value registered by an embedded observer, or a unitful measure carrying no cross-frame reciprocal. | R | Rems. 8, 9 master: 00:D7 |
| E. The quartet and its Carrier instantiation | |||
| E1p10024 | The quartet: the primitive triple $\{\lP,\tPl,\hbar\}$ — the two anchored horizons, flag-free re-anchorings of the two primal charts, $[\lP]=\unitL$, $[\tPl]=\unitT$, and the crossing quantum — with the dual horizons their crossed images, $\pP=\hbar/\lP$, $\EP=\hbar/\tPl$; the four horizons occupy the four domains, flag positions $(0,0,1,1)$; the pairing constants $c=\lP/\tPl=\EP/\pP$, $\hbar=\lP\pP=\tPl\EP$, $|k_B|=\pP\tPl$, and $G$ the one datum not internal to the square. The fundamental units are the Carrier’s horizons. Falsifiers: a fifth algebraically independent relation within the quartet lattice; a measured variation of the dimensionless ratios of the constants. | R | Def. 15, Rem. 10, Def. 16, § 6 |
| E2p10025 | One crossing, one action, and pairing closure: $\lP\pP=\tPl\EP=\hbar$ (the two faces of $c$ and $\hbar$ differ by the single relation $\ell p=tE$); the cross-domain relations reduce to the two generators $c$, $\hbar$, the mixed products their monomials, $|k_B|=\hbar/c$ derived; the relation lattice over $\{\lP,\tPl,\hbar\}$ has rank two and $k_B$, $\lP\EP$ add nothing; $4-1=3$ free scales in bijection with $\{c,\hbar,G\}$, the classical arity derived. | T | Rem. 11, Thm. 8, Cor. 5 |
| E3p10026 | The cancellation identity $|k_B|c=\hbar$ at the unit face, $(\pP\tPl)(\lP/\tPl)=\lP\pP$; mass derived, $m_P=\pP/c=\EP/c^{2}$; the temperature horizon $\Theta_P=\EP/|k_B|$; the flag positions $(0,0,1,1)$ and the closure of every quartet relation on the flag component; no $\hbar$ of space and time. | T | Cors. 3, 4, Prop. 8, Rem. 12 |
| E4p10027 | The defining congruences, pair form: on the Carrier chart $2G+1\equiv0$, $2c^{2}\equiv1$, $\hbar^{2}\equiv-1$, $k_B^{2}\equiv-2\pmod\Om$ determine the constants uniquely at pair level — $G=2\dS$ exact, the quadratics each with exactly two roots $\{x,-x\}$ (admissibility guarantees the residues: $\Om\equiv1\bmod4$, $\dS$ even for $2$, hence $-2$) — and the linkage $\{\pm k_B\}\{\pm c\}=\{\pm\hbar\}$ is derived at pair level; the root pair of $-1$ is $\{\hbar,h\}$ with $h=2\pi\hbar\equiv-\hbar$. Exhaustive on both Carriers. | T | Thm. 9, Rem. 14 |
| E5p10028 | Pair consequences and representative inertness: $G=-c^{2}$, $G^{2}\equiv4^{-1}$, $\hbar^{4}\equiv1$, $(k_Bc)^{2}\equiv-1$, and $\hbar cG^{-1}$ lands in $\{\pm k_B\}$; of the eight sign assignments exactly the four with $\sigma_\hbar=\sigma_c\sigma_k$ are admissible, a $(\Z/2)^{2}$, every identity holding on each and the $\hbar$-flip relabelling within $\{\hbar,h\}$; what a bounded observer registers is exactly the pair-inert content. | T | Cor. 6, Rem. 13 |
| E6p10029 | The residue assignments: the unit face (the Planck chart, each horizon the unit of its domain as E1 realises it; $c=1$, $\hbar=1$ its labels) and the residue face (the square-root horizon band, $\hbar^{2}=\Om-1$, $k_B^{2}=\Om-2$); the mass scale’s two typed faces, the monomial $m_P^{2}=\hbar c/G$ in $\{\pm k_B\}$ and the horizon declaration $m_P^{2}\,\hat=\,\Om$, never composed. | D | |
| E7p10030 | The Carrier register is $\Om$-hard: no bounded observer resolves the constants’ residues; the selection of a member in each pair $\{x,-x\}$ is a resolved fact of the instantiated totality, knowable to no sub-capacity frame (the window ladder, G2); the absolute unit values are read at the totality and their unit-face magnitudes are their only registrable manifestation. | Ω | § 6.1, Rems. 14, 15, 16 |
| E8p10031 | The two moves from the Subject’s reading to the Carrier’s: re-anchoring within the fibre (the unit and the chronon step traded for the substrate-fixed marks) and the register crossing (the flag); the two rows of the four-domain figure mix them differently. | D | Rem. 12 |
| E9p10032 | The minimal admissible pair: under the programme’s admissibility predicate ($\p=4\kap+1$ prime, $\kap>1$; $\Om=4\dS+1$ prime, $\dS$ even, $\dS\equiv1\bmod3$; $\p^{2}<\Om$) the smallest instance is $(\p,\Om)=(13,233)$, by exhaustive scan, with the counterfactuals: dropping the mod-$3$ clause admits $(13,193)$, dropping $\kap>1$ admits $(5,41)$, $\kap=2$ gives the composite $9$. | T | Defs. 1, 2, Rem. 1, § 6.1 |
| F. Mechanics, gravitation, temperature and the electromagnetic domain | |||
| F1p10033 | The speed domain $\unitv=\unitL\unitT^{-1}$ and the capacity budget it is read against. | D | Def. 17, § 7.1 |
| F2p10034 | The mechanical and gravitational domains: $[m]=\Iq\unitL^{-2}\unitT$; $[a]=\unitL\unitT^{-2}$, $[F]=\Iq\unitL^{-1}\unitT^{-1}$, $[p]=\Iq\unitL^{-1}$, $[S]=\Iq$, $[P]=\Iq\unitT^{-2}$, $[\text{pressure}]=\Iq\unitL^{-3}\unitT^{-1}$; $[G]=\Iq^{-1}\unitL^{5}\unitT^{-3}$; the geometric conversions $[G\hbar/c^{3}]=\unitL^{2}$, $[Gm/c^{2}]=\unitL$, $[Gm/c^{3}]=\unitT$, $[G\rho_m]=\unitT^{-2}$; mass derived, not primitive. | T | Props. 9, 10, 11, Cor. 8 |
| F3p10035 | Temperature carries the acceleration domain: $[\Theta]=\unitE[k_B]^{-1}=\unitL\unitT^{-2}=[a]$, flag-free; the Unruh combination $\hbar a/c\,k_B$ closes flag-free in the algebra; neither mass nor temperature is primitive, the classical $M$-$L$-$T$-plus-thermal system the torsion-free shadow of two generators plus the flag. | T | Cor. 7, Rem. 18 master: 00:C12 |
| F4p10036 | Count-valued comparisons are flag-free: every ratio of two quantities of equal crossing degree ($Gm/r^{3}$, the phase exponent $[E][T]/[\hbar]$) is a neutral or flag-free label, the flag entering unitful measures only; $F/a=[m]$, crossing degrees $1$ and $0$, is the flagged boundary case. | T | Exs. 5, 8, § 7, Rem. 17 |
| F5p10047 | The electromagnetic domain: with the Coulomb constant neutral (F6), $[q]^{2}=\unitE\unitL=\Iq\unitL\unitT^{-1}$; charge is an integer degree on the lift realised as $\unitL^{2\kap+1}\unitT^{(\kap-1)/2}$, existing iff $\kap$ is odd, unique up to $\Iq^{2}$, never window-covariant ($|r|=2\kap$); no further generator within C1’s two charts; on the even capacities no label squares to $[q]^{2}$. | T | |
| F6p10056 | The Coulomb constant is neutral: its label is the unit of the domain group, so the charge’s square is read in the domain of energy times length. | D | Rem. 21 |
| G. Periodicity, windows and the worked examples | |||
| G1p10037 | Local recovery: for an exponent horizon $H$ with $4\kap>2H$ the modular labels distinguish every conventional pair $|r|,|s|\le H$; the flagged sector is recovered through the crossing degree, $M^{u}L^{a}T^{b}\mapsto(a-2u,\,b+u;\,u)$ injective and windowed-faithful. | T | Thm. 10, Cor. 9, Rem. 20 |
| G2p10038 | The window ladder: coherence $2\sqrt\kap$ below recovery $\kap/2$ below flag inaccessibility $\kap$ below covariance $2\kap$, nested strictly for every $\kap\ge17$ and failing for the toy $\kap=3$, all orderings by integer squares; the coherence identity $(2\sqrt\kap)^{2}=\p-1$ and the totality closure $(2\sqrt\dS)^{2}=\Om-1$. | T | Rem. 19 |
| G3p10039 | The worked examples: $\kap=3$, $H=5$, $12>10$ so every pair in $[-5,5]^{2}$ is distinguished; kinetic energy $[m][v]^{2}=[E]$; $Q+Q^{2}$ inhomogeneous; the phase exponent neutral; flag arithmetic ($G\hbar/c^{3}$ flag-free, $\Iq^{2}=\unitT^{\pi}$); the Schwarzschild length $[Gm/c^{2}]=\unitL$; the gravitational frequency $[Gm/r^{3}]=\unitT^{-2}$; the energy–momentum relation $E^{2}=p^{2}c^{2}+m^{2}c^{4}$ homogeneous at crossing degree two, its massless case $E=pc$ at crossing degree one. | T | Exs. 1, 2, 3, 4, 5, 6, 7, 8 |
| G4p10046 | Buckingham’s count: on the integer lift the neutral monomials of $N$ quantities are the kernel of their $N\times3$ label matrix, $N-\operatorname{rank}$ independent dimensionless products with rank $\le3$, the classical count for quantities whose classical dimensions lie in $M$-$L$-$T$ and the derived-$k_B$ count for thermal ones; on the window $2H<\kap$, $|j|\le1$ lifted and realized neutrality coincide; the pendulum $(T,\ell,g,m)$: rank $3$, one product. | T | Cor. 10 |
| G5p10052 | The resolution floor: a registration resolved to one part in $N$ is a coherent count of $N$, bounded at $\sqrt\p$ (A5), so a single coherent registration on a shell of cardinality $\p$ resolves no finer than $\p^{-1/2}$; under $\p^{2}<\Om$ the floor is at least $\Om^{-1/4}$ on every admissible shell, of the order of $10^{-31}$ ($102$ bits) at $\dS=2.6\times10^{122}$ (Rem. 1), thirteen orders below the $10^{-18}$ optical-clock comparisons and twenty-six below the uncertainty of $G$. | T | Rem. 16 |
| P. Predictions | |||
| P1p10053 | The resolution floor, staked: no single coherent registration resolves finer than $\Om^{-1/4}$, of the order of $10^{-31}$, and no coherent count exceeds $\Om^{1/4}\approx6\times10^{30}$ (G5 over A2, A5, B1; the holographic reading of Rem. 1). Falsifier: one coherent registration finer than that. Confrontation: the finest interspecies frequency ratio, $3.2\times10^{-18}$ [Aeppli & others 2025], and the finest differential comparison, $7.6\times10^{-21}$ [Bothwell et al. 2022] — $67$ bits against $102$, averaged determinations; the mass-independent clock bound $(t_P^{2}t)^{1/3}$ permits finer beyond $730$ s. | T | Rem. 16, § 9 |
| P2p10054 | Constancy, staked: every dimensionless ratio of the constants is fixed in time and position, zero drift, zero dipole — the totality timeless (B1), every constant a residue of it in the framework’s realisation (A1; the quartet’s, E1, E6; the charge’s, Rem. 21), the quartet’s fixed by the congruences given $\dS$ (E4), selected once (E7). Falsifier: a detected variation (E1’s). Confrontation: $\dot\alpha/\alpha=1.8(2.5)\times10^{-19}\,\mathrm{yr}^{-1}$ [Filzinger et al. 2023], consistent with zero. | T | |
| X. What the construction explains | |||
| X1p10048 | The arity of classical bookkeeping, derived: four horizons less one identity, $4-1=3$ free scales in bijection with $\{c,\hbar,G\}$, exactly three independent members of $\{c,\hbar,G,k_B\}$; the three mechanical primitives are the two chart generators and the flag’s torsion-free surrogate, mass, $[m]=\Iq\unitL^{-2}\unitT$, $m_P=\pP/c$. Classical dimensional analysis postulates the count and the mass primitive. | T | |
| X2p10049 | The thermal primitive is redundant: $[\Theta]=\unitE[k_B]^{-1}=\unitL\unitT^{-2}$, the acceleration domain, flag-free where energy carries the flag; $|k_B|=\hbar/c$ is the quartet’s mixed product; the Unruh combination $\hbar a/(c\,k_B)$ closes as a domain identity. Classical bookkeeping adjoins a thermal unit and reads $k_B$ as a conversion. | T | |
| X3p10050 | One action quantum for both conjugate pairs: the domain lattice has exactly one order-four subgroup, in the time factor (no flag of space), canonical and covariant, so $E=hf$ and $p=\hbar k$ share one $\hbar$, $\lP\pP=\tPl\EP$ — derived from the shell’s algebra with no Lorentz structure used. | T | |
| X4p10051 | Charge is an integer degree of the two generators on the shells of odd capacity, not a further primitive: with the Coulomb constant neutral (F6), $[q]^{2}=\Iq\unitL\unitT^{-1}$, $q_P^{2}=\hbar c$; the lift has no half root (the Gaussian exponents $M^{1/2}L^{3/2}T^{-1}$ are half-integers there), so torsion-free bookkeeping adjoins an electrical primitive as it adjoins mass; the fine-structure ratio stays a neutral count. | T | Rem. 21 (restates F5) |
Nothing matches that filter.
Status
- I
- Import. A standard result, a corpus result, or a convention used here without reproof.
- D
- Definition. A naming, a declared dictionary, or a set-up move, carrying no empirical content.
- T
- Theorem. Derived within this paper from the rows above it, machine-verified in exact arithmetic by the named witness where a witness is named.
- R
- Realisation. States what a physical term denotes on the substrate: constitutive, not interpretive, and carrying its falsifier.
- Ω
- $\Om$-hard. A value fixed by the totality, in no bounded observer’s window; a demarcation of where the value is read, not an open item.
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 (
p10012) — 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).