Tags
- 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.
- O
- Open. Genuinely unresolved.
| # | Description | Status | Source |
|---|---|---|---|
| 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 $\It=-\gen^{\kap}$, 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 | |
| A3p10003 | The shell Fourier structure: on $\Fp^{\Phi}$, $\Phi=C_{\p-1}$, the matrix $W_{jk}=\gen^{jk}$, the normalized operator $F=\It W$ 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. 4 |
| 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, Rem. 1 |
| 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; $\It$ 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 | 1/1 |
| 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 | 1/1 |
| 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 | 4/4 |
| 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 declared and carried by the transport theorem (D2); classical temporal unit change is nominal re-assignment within a fixed presentation, the chronon atomic. | T D | Rems. 2, 3, Prop. 4 |
| 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 | 1/1 |
| 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 | 2/2 |
| 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 | 5/5 · lift.L5 |
| 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 | 1/1 |
| 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. 6, 7 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. 8, 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 | 1/1 |
| 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 | 1/1 |
| 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 | 3/3 master: 00:C8 |
| 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 | 2/2 master: 00:B10 |
| E6p10029 | The residue assignments: the unit face (the Planck chart, each horizon the unit of its domain; $c=1$, $\hbar=1$ 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 R | § 6.1 (the residue assignments) |
| 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, Rem. 12 |
| 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. 10 |
| 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 | 1/1 master: 00:C13 |
| F. Mechanics, gravitation and temperature | |||
| 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 | 1/1 |
| 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 | 1/1 master: 00:C12 |
| F4p10036 | Count-valued comparisons are flag-free: every ratio of two quantities of equal crossing degree ($F/a$, $Gm/r^{3}$, the phase exponent $[E][T]/[\hbar]$) is a neutral or flag-free label, the flag entering unitful measures only. | T | 1/1 |
| 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 | 2/2 |
| 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. 14; dom.G 1/1 master: 00:C13 |
| 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 $[Gm/r^{3}]=\unitT^{-2}$. | T | 1/1 |
V. Machine verification (the validation package finite-ring-space/src/10-dimensions) | |||
| V1p10040 | The exact suite: eight layers, $200$ micro-checks — A the shell datum and domain algebra on $\F_{13}$ ($32$), B the quartet at the unit face ($30$), C the defining congruences on the laboratory Carrier ($13$), D both Carriers, faces, minimality, temperature ($50$), E covariance ($18$), F window bound, twisted action, dualities ($15$), G realized action, ladder, transport, pair canonicity ($15$), H the pair layer ($27$). | T | |
| V2p10041 | The lift: L1–L5 on the shells $(13,2)$, $(173,3)$ and the Carriers $233$, $2\,408\,561$, $10$ micro-checks. | T | |
| V3p10042 | The source gates: $105$ required tokens and verbatim bans on sections/*.tex and mdpi.tex — markup residue, the $\kap/\chron/\dt$ roles, the corpus blacklist, the constructive register, the required forms, the doctrine gates of rounds 02–06 — run in the corpus tree, where the manuscript lives. | T | |
| V4p10043 | The driver: two suites, one notebook run in the browser, $13$ family checks over $210$ exact micro-checks, a record per check written to results.json keyed to these rows. | T | run_all; the notebook |
| O. The open front | |||
| O1p10044 | The electromagnetic domain: charge and the electromagnetic units are not constructed in the two-generator-plus-flag system; whether they enter as a further flag, a further generator, or a monomial of the existing ones is the paper’s stated deferral. | O | § 9 |
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Ledger history
Ledger history. 2026-09-13: ledger added (the paper’s Table 2 tagged sixteen headline claims; the ledger states them one predicate per row, the machine-verified rows citing the package’s check identifiers). One open row, O1, the electromagnetic domain; when it closes its content is re-homed in block F and the row leaves. No row has been retired.