frc/ledgers/p38_s13.py that decides the row, opened at its marker; the tokens lean and leanML name the row’s declaration in the Lean core (no axioms) and on Mathlib. The ledger is the paper’s checklist: § 2.1 carries block A, § 6 block B, § 7 block C, § 8 block D; block X collects what the construction explains (§ 9.2), block P its falsifiable content (§ 10); § 10.2 describes the ledger file and the two libraries. Rows are cited across the corpus as 38:XN.
| # | Description | Status | Sources |
|---|---|---|---|
| A. The pair: the instance arithmetic | |||
| A1p38001 | The pair: $\p=13$, $\kap=3$, $\Om=233$, $\dS=58$; $\gen=2$, $\im=5=-\gen^{\kap}$. | T |   § 2.1 |
| A2p38002 | The tower identity: $C_{12}=C_{4}\times C_{3}$, $\gen=3\cdot5=2$; unit exponents exactly at $\kap=3$; general form $4a-\kap b\equiv1\pmod{4\kap}$, solvable for every odd $\kap$ (Theorem 8 inside the Subject). | T | § 2.1 |
| A3p38003 | The channel at the instance: $\gcd(12,14)=2$, $\gcd(12,28)=4$; $N(\im)=-1$ (Theorem 8). | T | Theorem 8 master: 00:C20 |
| A4p38004 | The Carrier quarter roots, arithmetic: $78^{58}=89=h$, $h^{2}=-1$, $\hbar=144$, $\hbar h=1$, $\hbar+h=\Om$; octant $\zeta_{8}=97$, $\zeta_{8}^{2}=h$; $233=13^{2}+8^{2}$ with $h=13/8$, $\hbar=8/13$ on $\F_{233}$. | T | Theorem 3 master: 00:B3 |
| A5p38005 | The naming $\{\hbar,h\}$ and the constant loci of Fig. 4: the registration reading of [Akhtman 2026; Akhtman 2026]; consumed once, by the registration orientation $\iota:\im\mapsto\hbar$ (Proposition 9); proofs consume A4 alone. | R | Fig. 4 |
| A6p38006 | The FRC algebra half [Akhtman 2025]: $\tfrac12=7$, $-\tfrac12=6$; the seam between rows $6$ and $7$. | T | § 2.1 |
| A7p38007 | The frame counts: $2184=156\times14=14\times13\times12$; the norm-one sphere is $\SL_{2}$ (its quaternion determinant), the spin cover of $\mathrm{PSL}_{2}$ ($1092$); $\SL_{2}$ obstructs at $-1$ (Theorem 6). The elements counted are the frame moves of the Subject’s chart, not registrations of the Carrier (00:C22). | T | Theorem 6 |
| A8p38008 | The resolvable spectrum (Proposition 15): symmetric heights, window $|\hat k|\le\kap\Leftrightarrow4|\hat k|<\p$, energy the magnitude with the sign as orientation, pairwise-exact conservation below capacity; instance window $\{0,\pm1,\pm2,\pm3\}$. | T | Proposition 15 |
| A9p38050 | The two fibrations and the frame triple (Theorem 12): $G\cong\Prj^{1}\times\F_\p\times C_{\p-1}$ for every odd $\p$ of the totality, $14\times13\times12$; non-split fibres $C_{\p+1}$ over $\p(\p-1)=156$ with the cone-chart section, split fibres $C_{\p-1}$ over $\p(\p+1)=182$ without; the unit $\im$ is split at every $\p=4\kap+1$. | T | Theorem 12 master: 00:C22 |
| A10p38039 | The drawing chart: the radial ladder $R\sin(2a\pi/13)$, the chart angle unit $\pi/232$ on $C_{464}$, the Hopf leaves; incidence, ordering, winding, anchoring and flow exact, the connecting curves the declared chart, each constant [approx]. | D | § 3 |
| B. The sky: the drawn observable | |||
| B1p38009 | The node field: $72=3\times2\times12$; retarded labels, one chronon of lookback per row. | T | § 6 |
| B2p38010 | The radial ladder: shells at $R\sin(2a\pi/13)$ [chart]; capacity $4a<\p\Leftrightarrow a\le\kap$. | T |  § 6; rests on A10 |
| B3p38011 | The mounting: $\times3=\gen^{4}$, $120^{\circ}=8$ steps, quarter-blind. | T | § 6 |
| B4p38012 | The sky chronon: $\chron\bmod24$, the fibre product of $C_{12}$ and $C_{8}$ over $C_{4}$; antipodal at $+12$, closed at $+24$. | T | § 6 master: 00:C16 |
| B5p38013 | The recurrence data: $696=\dS\cdot12$, spinor $1392$ [proxy-chart]; the pair registers only the monotone dephasing. | T | Theorem 4 |
| B6p38014 | The Hopf representation: node-on-fiber exact; foliation depth $=$ rung ladder; the drawn leaves are the continuum foliation by the unit $\im$, the chart face of the count of A9, distinguishing neither fibration [chart]. | T | § 3; rests on A10 |
| B7p38015 | The frame leak: halved tick $\pi/232$; $\pi/2$ at the quarter station; continuous wrap on $C_{464}$ (Theorems 3, 4). | T | § 6 |
| C. The flow | |||
| C1p38016 | One retarded flow law: $\varphi(k)=-(b_{C}-k)\pi/116$; the Subject draws the gauge-free curl (Theorem 4). | T | § 7 |
| C2p38017 | One retarded flow across the three representations (its Dirac reading with D10, D11); the toggle switches fibers only. | T | § 7 ; rests on C1 |
| C3p38018 | Ownership: the origin never dressed; rays anchor at the Observer at every clock state. | T | § 7 |
| C4p38019 | The half-turn exact: odd chart, point reflection; the rim kiss (Theorem 8). | T | § 7 |
| C5p38020 | The $C_{4}$ stations: parity image at $b_{C}=116+k$, home at $b_{C}=k$, gauge-free. | T | § 7 |
| C6p38021 | The dilation instance $3/58$; the $(53,13)$ kill test (Theorems 4, 8). | T | Theorem 4 |
| C7p38022 | The two faces of the leak: angular $\kap/\dS$ (the curl), temporal $\kap/(2\dS)$ (the ring, on $C_{464}$); ratio exactly $2$, the double cover. | T | Statement 14 |
| C8p38023 | The face-ratio identity, orbit-free (both sides’ scales cancel): the pair’s two faces (C7) have ratio exactly $2$; read in the comparison chart of the laboratory dictionary (C12), the parametrised isotropic metric [approx: continuum chart], the identity forces $2\gamma-\beta=1$, a combination measured to $10^{-4}$ (Cassini, lunar laser ranging) and so discriminating no metric the data have not already selected; Brans–Dicke’s $2\gamma-\beta-1=-2/(2+\omega)$ is the same fact read in one family. | T | |
| C10p38044 | The per-chronon leak rate: $(\kap/\dS)/(\p-1)=1/(4\dS)=1/(\Om-1)$ exactly, $\kap$-free — one chart tick per chronon (the temporal face on the cover $1/(2(\Om-1))$); at the instance $(3/58)/12=1/232$, at $(5,233)$ $(1/58)/4=1/232$. The per-revolution fraction $\kap/\dS$ (C7) is this rate times the shell period $\p-1$. | T | § 9.2 |
| C11p38045 | The floor reading: the leak completes one cycle per drive period $\Om-1$ chronons $=2\pi/H_{0}$ (the T16 dictionary of 21-grav), so its cyclic rate is $H_{0}/2\pi$ and its acceleration face $cH_{0}/2\pi=a_{0}$, the synchronisation floor (32-dark B3, master L1; at the instance $a_{0}=c/(\Om-1)$, 37-sim). The pair’s gravitational face is the registration floor, $\kap$-free, hosted at every scale; the apsidal reading (C7, Reading 5) is the same leak on one shell’s cycle. Falsifier: L1’s, a measured floor off $cH_{0}/2\pi$. | R | § 9.2 master: 00:L1 |
| C12p38052 | The laboratory dictionary: on a hosted pair $p_{\mathrm{sl}}=3(\dS/\kap)\,r_{g}$ (Statement 14) is one relation, homogeneous of degree $0$, between two chart lengths, fixing their ratio $3\dS/\kap$ ($58$ at the instance, $174$ at $(5,233)$) and consuming no chart datum: a definition of the chart’s $r_{g}$ per $(\kap,\dS)$, host-decided, under which $3\,r_{g}/p_{\mathrm{sl}}=\kap/\dS$ is C7’s identity; no hosted pair confirms or refutes it. | D | Statement 14 |
| C13p38053 | The apsidal bound: a nested Subject registers $\kap/\dS=(\p-1)/(\Om-1)<1/(\sqrt\Om+1)$, the nesting condition $\p^{2}<\Om$ itself; on the physical Carrier no Subject registers an observed precession (Mercury’s $8.0\times10^{-8}$ per revolution sits $54$ orders above the bound $10^{-61}$ [approx]), and the pair reads per orbit of period $T$ the floor $TH_{0}/2\pi$ (C10, C11; § 9.2). | T | |
| C14p38054 | The joint cycle (Theorem 4): for every $\kap$-odd pair the chart count advances by $(\p-1)/(\Om-1)=\kap/\dS$ of its cycle per shell revolution; the joint cycle closes at $\mathrm{lcm}(\p-1,\Om-1)$ at (home, home), and the exact midpoint is (home, $-1$). | T | Theorem 4; rests on A1 |
| D. The light cone: the meridians and the covering | |||
| D1p38025 | The capacity stations: $4d<13$ selects $\{1,2,3,10,11,12\}$; the flow law threads every station at all $232$ clock states. | T | § 8 |
| D2p38026 | The step law $\tfrac{11\pi}{12}+\tfrac{\pi}{13}+\tfrac{\pi}{232}$ and the leak $\pi/156-\pi/232$ on the space trace: exact rationals of the chart angle, the angle itself the declared chart unit (A10). | T | |
| D3p38027 | The scale-tower circuit: the period at the FRC algebra half; the radius $R\lvert\sin(2a\pi/13)\rvert$ the declared drawing chart (A10, [approx]). | T | |
| D4p38028 | The photon travels scales: absorption is the completed decoherence cascade; nothing crosses the Observer. | R | § 8 |
| D5p38029 | Four rays realise the four quarter classes (Theorem 8), each a fiber geodesic to the quarter fold $a=13/4$. | T | Theorem 8 |
| D6p38030 | The momentum dual $M_{3}$ at winding $5$: stations, tent shells, $C_{26}$ fold; framed-rational passages $13/5$, $52/5$. | T | § 8 |
| D7p38031 | The covering: $144$ slots cover $72/72$; multiplicity $1$ odd $/$ $3$ even; $144=36+108$. | T | Theorem 16 |
| D8p38032 | The half-cell ramification: $t_{1/2}=13/(2m)$; $M_{0}$'s at the origin. | T | § 8 |
| D9p38033 | The horizon fold fusions: $5\cdot39\equiv-13\ (104)$; eight ends, four points; $E\leftrightarrow B$. | T | § 8 |
| D10p38034 | The quadrature reading: $M_{0}/M_{3}$ the two quadratures of $F=E+iB$ [Akhtman 2025]. | R | § 8 |
| D11p38035 | The headline realisation: $i\hbar\,\partial_t\psi=\hat H\psi\ \rightleftharpoons\ \psi_{\chron+1}=\gen\psi_{\chron}=i(3\psi_{\chron})$. | R | § 5 master: 00:D1 |
| D12p38036 | The covering parity theorem, closed at every shell (Theorem 16, Lemma 1): $\mu\in\{1,3\}$ by the odd four-sum; decider the row parity and position by the eight-case table; completeness a corollary; the carrier the sheet parity: even rows $\gen$-divisible (thrice), odd rows the half-cell obstruction of D8 (once). | T | Theorem 16 |
| D13p38037 | The family law: the one meridian law generalized (baseMW; $M_{0}$, $M_{3}$ its windings $1$, $5$), the covering arithmetic fully certified; the drawn pair is the family’s representative, a drawn ten-sibling layer retracted as washing the sky. | T | § 3 |
| X. What the construction explains | |||
| X1p38046 | The weakness of gravity is the $1/\dS$ suppression of the frame leak: one chart tick per chronon, $1/(4\dS)$ of the cycle — $1/232$ at the instance, of order $10^{-123}$ at physical capacity; the denominator that makes the world look continuous makes gravity look weak (C10, § 9.2). | T | § 9.2; rests on C10 |
| X2p38047 | The continuum is the large-capacity look: what changes with capacity is resolution, not structure; the instance stands to the physical pair as exact instantiation, every structural theorem holding verbatim at $\dS\approx10^{122}$ by the $\Om$-blindness license (§ 9.1): the theorems of § 4 consume no numeral of $\dS$ or $\kap$. | T | § 4, § 9.1 |
| X3p38048 | The acceleration floor $a_{0}=cH_{0}/2\pi$ is the drive seen in the chart: the leak’s cyclic rate is the drive’s own, one cycle per drive period $\Om-1$ chronons $=2\pi/H_{0}$, and its acceleration face is the floor (C10; C11, the realisation); at the instance $a_{0}=c/(\Om-1)$. | T | master: 00:L1 |
| X4p38051 | The shell is holographic: a Subject’s frame datum is one projective direction, one shell cell, one drive phase, and the bulk frame is that boundary datum with nothing left over (Theorem 12; the realisation reading and its falsifier, Reading 13). | T | Theorem 12; rests on A9 master: 00:C22 |
| P. Predictions and falsifiers | |||
| P1p38049 | The registration floor $a_{0}=cH_{0}/2\pi$, read from the pair’s $\kap$-free per-chronon leak rate (C11): parameter-free, tracking $H(z)$. Falsifier: a measured floor off $cH_{0}/2\pi$ beyond the $H_{0}$ uncertainty, or one not tracking $H(z)$ (§ 10, item 5; the dark sector’s L1). | T | master: 00:L1 |
Nothing matches that filter.
Status
- T
- Theorem. Derived and machine-verified in exact arithmetic by the named witness.
- D
- Definition. A naming or declared chart convention, carrying no empirical content.
- R
- Realisation. The row states what a physical term denotes in the substrate: constitutive, not interpretive, defended by the totality of its registered consequences.
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 (
p38001) — 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).