| # | Description | Status | Sources |
|---|---|---|---|
| A. Inputs: imported, not derived here | |||
| A1p13001 | The framed shell $\Fp(\tau;0,1,\gen)$ of capacity $\kap$, $\p=4\kap+1$, with its phase cycle $\PhiP\cong C_{\p-1}$, the oriented quarter-turn $\im=\gen^{-\kap}$ and the conjugate chirality; the framed rationals $\fQ$ with the shell reading $\rep{\cdot}$; the finitist position that the discrete register is primary. | I | |
| A2p13002 | Transcendence and irrationality of the classical values: Hermite for $\eR$, Lindemann for $\piR$, Niven’s irrationality proof — statements about the continuum register, used here only in Lemma H3 (lem:norad), p. 15">8.2. | I | [Hermite 1873; Lindemann 1882; Niven 1947] |
| A3p13003 | The classical prime congruences: Wilson, Fermat, Kummer’s carry criterion, Lucas, Wolstenholme, Morley, Lerch, and Gauss’s central-binomial congruence $\Cb{\kap}\equiv2a\pmod\p$ with its normalisation. | I | [Kummer 1852; Lucas 1878; Wolstenholme 1862; Morley 1895; Lerch 1905; Gauss 1828; Berndt et al. 1998] |
| A4p13004 | Kurepa’s left factorial $\Ku{\p}=\sum_{k<\p}k!$ and his hypothesis $\p\nmid\Ku{\p}$, open since 1971, computationally confirmed for $\p<2^{40}$; the Mijajlović–Šami congruence $\Ku{n}\equiv(-1)^{n-1}\dr{(n-1)}\pmod n$. | I | [Kurepa 1971; Guy 2004; Andreji'c & Tatarevi'c 2016; Andreji'c et al. 2021; Petojevi'c et al. 2023] |
| A5p13005 | Sun’s supercongruences for the arcsin chain: $\sigma_\p\equiv0\pmod{\p^{2}}$ (his (1.4)), the blind-range Euler congruence (his (1.5)) of the van Hamme–Sun family, his Conjecture 5.1; the Wolstenholme refinement $H_{\p-1}\equiv-\p^{2}B_{\p-3}/3\pmod{\p^{3}}$. | I | [Sun 2011; van Hamme 1997; Glaisher 1900; Sun 2000] |
| A6p13006 | The external comparison objects: Wallis’ product and the squeeze that identifies the limit of the Wallis pair, the Gregory series, the Gaussian-integer factorisation behind the Machin identity, Stirling’s growth estimate, Siegel’s E- and G-function classes, Gauss’s sign theorem for the quadratic Gauss sum, the Wieferich condition; the IEEE-754 readout $\ro$ (correct rounding to $k$ bits) as the bounded register’s retention. | I | [Wallis 1656; Wieferich 1909]; § 2 |
| A7p13007 | Equidistribution of $\lambda(\im)/\p$ over the primes $\p\equiv1\pmod4$ (Duke–Friedlander–Iwaniec), read on the shells of the totality as the uniform model: the sole import behind the frequency of calibrated shells (H4) and the generic-growth reading of the heights (J3). | I | [Duke et al. 1995] |
| A8p13064 | Sun’s Conjecture 5.1, $\sigma_\p\equiv(-1)^{(\p-1)/2}\bigl(H_{\p-1}/12+3\p^{4}B_{\p-5}/160\bigr)\pmod{\p^{5}}$ for $\p>5$, is Theorem 5, (44), of Hessami Pilehrood, Hessami Pilehrood and Tauraso (2012); with Glaisher’s $H_{\p-1}\equiv-\p^{2}B_{\p-3}/3\pmod{\p^{3}}$ it gives the third-order Bernoulli law of G7 for every prime $\p\ge5$ of the totality. | I | [Hessami Pilehrood et al. 2012; Glaisher 1900] master: 00:C24 |
| B. The setting and the typing | |||
| B1p13008 | The typed constants: the exact half-period $\piP=2\kap$ with $\gen^{\piP}=-1$, its affine representative $\piA=(2\kap)\cdot1\in\Fp$, and the classical value $\piR$; the oriented quarter-turn $\im=\gen^{-\kap}$, the exponential unit $\eP=\gen^{\lambda(\im)}$ with $\lambda$ the label lift (frame data), and $\eR$. On $\p=13$, $\gen=2$: $\piA=6$, $\im=5$, $\eP=6$. | D | § 2; Fig. 1 |
| B2p13009 | Orientation transport: under $\gen'=\gen^{u}$, $\gcd(u,\p-1)=1$, the re-derived quarter-turn is $\im'=\im$ exactly when $u\equiv1\pmod4$ and $\im'=-\im$ (the conjugate chart) when $u\equiv3\pmod4$; there is no third case, and every chirality-sensitive statement holds for the other chirality under $\im\mapsto-\im$. | T |   Rem. B2 (rem:orient), p. 4">2.1 master: 00:C14 |
| B3p13010 | The coincidence that forces the typing: at $\p=13$ the field element $6$ is at once $\eP$, $\piA$ and the reading of $-1/2$, while $\eR\ne\piR$; no map from a bare residue to a real number exists, and the external comparison takes the structural role as part of its input — roles, not residues, carry external meaning. Witnessed by the two towers reading the same element $6$ as $\eP$ and as $\piA$. | T | § 2 |
| B4p13011 | Chains and their two lives: a chain is a sequence in $\fQ$ given by explicit integer formulas, with an external metric life read against $\R$ and an internal residue life read through $\rep{\cdot}$; no third life — the shell carries no completion of $\fQ$, and the continuum enters only as the chart. This is the statement of record superseding the framed-real construction of [Akhtman 2025], §4.4. | D | § 2; Fig. 2 master: 00:B1 |
| C. The two selectors | |||
| C1p13012 | The registration quotient: at grade $n$ the observer forgets phase labels and cycle type, tallies all reversible relabellings of $n$ marks with unit weight, and retains only the coincidence predicate. The first of the paper’s two D moves; not implied by cyclicity (the cyclic shifts alone give $n-1$ fixed-point-free relabellings). | D | Principle C1 (prin:registration), p. 5">3.1 |
| C2p13013 | The derangement selector: the registration statistics are generated by the rencontres polynomial $P_n(X)=n!\sum_{k\le n}(X-1)^{k}/k!$ (an exact double-counting identity), the normalised kernel $Z_n$ has $Z_n(1)=1$, $Z_n(0)=\dr{n}/n!$, and the unit registration ratio $R_n=n!/\dr{n}$ is forced; relative to the declared completion of $Z_n$ to $e^{X-1}$ the external value of the chain is uniquely $\eR$. | T |  Prop. C2 (prop:selector-e), p. 5">3.2; rests on C1 |
| C3p13014 | The Cayley quarter-turn map $C(x)=(1+\im x)/(1-\im x)$ on $\Fp$ (denominators units): $C(0)=1$, $C(1)=\im$, and $C(x)C(y)=C\bigl((x+y)/(1-xy)\bigr)$, the rational composition law being the half-angle form of phase composition; shell arithmetic only. | T | Def. C3 (def:cayley), p. 6">3.3 |
| C4p13015 | The Cayley–Gregory normalisation: the external comparison of $\oplus$ is the additive chart $A$ with $A'(0)=1$, and $\piR:=4A(1)$; given the normalisation the chart is unique, $A(X)=\sum(-1)^{k}X^{2k+1}/(2k+1)$, and the normalisation is underivable (no nontrivial homomorphism of a finite cyclic group into $(\R,+)$; $cA$ satisfies the same law). The second and last D move. | D | Principle C4 (prin:gregory), p. 6">3.4 |
| C5p13016 | The joint-return tally: with $C_n=\Cb{n}$ the returning $\pm1$ histories of length $2n$ in one direction, the paired histories in the directions $1$ and $\im$ number $16^{n}$ and the joint returns $C_n^{2}$, and $\piR=\lim16^{n}/(nC_n^{2})$ by the Wallis bounds (E2). | T | Prop. C5 (prop:selector-pi), p. 6">3.5 |
| D. Radial emergence of $e$ | |||
| D1p13017 | The derangement chain $\varepsilon_n=n!/\dr{n}\in\fQ$, $n\ge2$, with $\dr{n}$ the subfactorial; the reciprocal chain has denominator $n!$, the product-of-units form of $\fQ$ whenever $n<\p$, so the chain is native to the framed rationals with no completion presupposed. | D | Def. D1 (def:chain-e), p. 7">4.1 |
| D2p13018 | The tail form: $\dr{n}=n!/\eR+(-1)^{n}\delta_n$ with $\delta_n\in(1/(n+2),1/(n+1))$, so $\dr{n}$ is the nearest integer to $n!/\eR$. | T | Lem. D2 (lem:tail), p. 7">4.2 |
| D3p13019 | Superexponential enclosure: $|\varepsilon_n-\eR|\le\eR(\eR+1)/(n+1)!<10.2/(n+1)!$ with $\operatorname{sign}(\varepsilon_n-\eR)=(-1)^{n+1}$, hence $\varepsilon_{2m}<\eR<\varepsilon_{2m+1}$, a nested sequence of $\fQ$-native intervals; the sharper constant $8$ for $2\le n\le60$. | T | Thm. D3 (thm:enclosure-e), p. 7">4.3 |
| D4p13020 | Horizon determination of the readout: for $n\ge n_0(k)=\Theta(k/\log k)$ the readout $\ro(\varepsilon_n)$ lies within one ulp of $\ro(\eR)$, and equals it from the plateau threshold $n_1(k)$, the least $n$ from which the readouts agree, decided through the rounding cell of $\eR$ ($n_1\le n_0+1$ for $k<200$, equality at eight precisions); the binary64 constant is the readout of $\varepsilon_{18}=18!/\dr{18}$, $n=18$ minimal; $\varepsilon_{70}$ fixes $100$ digits. | T | Thm. D4 (thm:readout-e), p. 7">4.4 |
| D5p13021 | Feasibility selection: the compound chain $(1+1/n)^{n}$ costs $n\asymp2^{k}$ steps for $k$ bits against $n_0(k)=\Theta(k/\log k)$ for the derangement chain ($n=11,18,70$ at $k=24,53,333$); within a horizon $H$ the compound chain realises $O(\log H)$ bits and the factorial chain $\Theta(H\log H)$; under the feasibility principle the derangement representation is the canonical finite definition of $e$. | T | Prop. D5 (prop:feasibility-e), p. 8">4.5 |
| E. Radial emergence of $\pi$ | |||
| E1p13022 | The Wallis pair $v_n=2\cdot16^{n}/((2n+1)\Cb{n}^{2})$, $w_n=16^{n}/(n\Cb{n}^{2})$ in $\fQ$: ratios of counting statistics ($\Cb{n}$ the balanced binary words of length $2n$, $16^{n}$ the unrestricted four-letter words), $v_n$ the doubled Wallis partial product, $w_n$ the joint-return tally of C5. | D | Def. E1 (def:wallis), p. 8">5.1 |
| E2p13023 | Two-sided monotone enclosure: $v_n$ strictly increasing, $w_n$ strictly decreasing, $v_n<\piR<w_n$, and the exact width identity $w_n-v_n=w_n/(2n+1)$, certifying $\approx\log_2n$ bits by nested $\fQ$-intervals; the identification of the common limit with $\piR$ is the Wallis import (A6); verified for $n\le300$ against Machin brackets of width below $10^{-420}$. | T | Thm. E2 (thm:enclosure), p. 8">5.2 |
| E3p13024 | The arcsin chain $s_n=3\sum_{k\le n}\Cb{k}/((2k+1)16^{k})\in\fQ$ with external completion $\piR=6\arcsin\frac12$ and tail bound $|\piR-s_n|<4^{-n}/(2n+3)$ for $n\ge1$: geometric rate from the same combinatorial atoms as the Wallis pair. | T | Def. E3 (def:arcsin), p. 9">5.3 |
| E4p13025 | Horizon determination for $\pi$: the Machin chain $M_N\in\fQ$ has $|M_N-\piR|<17/((2N+3)5^{2N+3})$; the binary64 constant $884279719003555/281474976710656$ is the readout of $M_{10}$ with $N=10$ minimal; the first $100$ decimal digits are fixed by $M_{71}$; alternating pairing of the two arctangent series gives certified two-sided $\fQ$-enclosures at every truncation, the Gregory brackets $4S_{2m+1}<\piR<4S_{2m}$ of width $4/(4m+3)$ the base case. | T | Thm. E4 (thm:readout-pi), p. 9">5.4 |
| E5p13026 | The feasibility hierarchy: among $\fQ$-native constructors of $\pi$ the best known rate is geometric, $\Theta(k)$ steps (Machin, arcsin), against $\Theta(k/\log k)$ for the derangement chain of $e$; the circle count and the Leibniz/Wallis chains sit at $2^{k}$; the AGM is not $\fQ$-native. A factorial-rate chain exists for every real by rounding (and for $\pi$ by rounding the Machin chain), so the split is by kind of chain — E-partial sums ($e$) against G-partial sums ($\pi$), master C26; whether $\piR$ is the value of an E-function with rational coefficients at a rational point is a question of the continuum (Fischler–Rivoal’s conjecture $\operatorname{Frac}\mathcal E\cap\operatorname{Frac}\mathcal G=\overline{\Q}$ would answer it negatively), not a predicate of the frame. | T | Prop. E5 (prop:feas-pi), p. 9">5.5 master: 00:C26 |
| F. The frame-internal life of $e$: the residue line and the Kurepa wall | |||
| F1p13027 | Wilson reflection: for an odd prime $\p$ and $0\le k\le\p-1$, $1/k!\equiv-(-1)^{k}(\p-1-k)!\pmod\p$. | T | Lem. F1 (lem:wilson), p. 10">6.1 |
| F2p13028 | The wall identity and the terminal residue of $e$: $\dr{(\p-1)}\equiv\Ku{\p}\pmod\p$ (the odd-prime case of the Mijajlović–Šami congruence, A4), hence $\rep{\varepsilon_{\p-1}}=-(\Ku{\p})^{-1}$ whenever $\p\nmid\Ku{\p}$; existence of the terminal link on a shell is equivalent to $\p\nmid\Ku{\p}$, universal existence over the shells of the totality to the vanishing of Kurepa’s certificate (Z1); re-verified in the derangement form for all $121{,}126$ odd primes $\p<1.6\times10^{6}$ with zero failures of either condition. Beyond the wall $n!\equiv0$: $n=\p-1$ is the factorial horizon. | T | Thm. F2, F8 (thm:wall-e), p. 10">6.2 |
| F3p13029 | Antiperiodicity: $\dr{(n+\p)}\equiv-\dr{n}\pmod\p$ for every odd prime $\p$ and every $n\ge0$ of the totality, so $\dr{n}\bmod\p$ depends on $n\bmod2\p$; the blind set $Z_0(\p)=\{n<\p:\dr{n}\equiv0\}$ always contains $1$, and Kurepa’s hypothesis is $\p-1\notin Z_0(\p)$ for every odd prime $\p$ of the totality. | T | Thm. F3 (thm:antiperiod), p. 10">6.3 |
| F4p13030 | Series duals at the wall: $\sum_{k<\p}(-1)^{k}/k!\equiv-\Ku{\p}$ and $\sum_{k<\p}1/k!\equiv-A(\p)$, $A(\p)$ the alternating factorial sum; the group law $\exp(1)\exp(-1)=1$ does not survive, $\Ku{\p}A(\p)\equiv3,0,21,28,132,258$ at $\p=7,13,29,101,257,1009$; on $\p=13$ the direct series residue vanishes ($A(13)\equiv0$) while the derangement side survives ($\Ku{13}\equiv10$) — a second ground for the canonicity of D5. | T | Prop. F4 (prop:duals), p. 11">6.4, Rem. F4 (rem:asym), p. 11">6.5 |
| F5p13031 | Blind-scale statistics: over the $501$ primes in $(1000,5000)$ the sizes $|Z_0(\p)|$ distribute as $1^{(169)},2^{(209)},3^{(92)},4^{(22)},5^{(6)},6^{(2)},7^{(1)}$, mean $1.996$, the excess over the forced $n=1$ matching Poisson$(1)$ [approx] (the model: $\dr{n}\bmod\p$ uniform for $2\le n\le\p-2$, the wall excluded; extended to the $1593$ primes in $(5000,20000)$: $1^{(597)},2^{(569)},3^{(299)},4^{(108)},5^{(17)},6^{(3)}$, mean $1.988$); blind scales never obstruct the plateau of D4 (advance at most $|Z_0(\p)|$ steps past $n_1$). The counts are exact; the Poisson reading is the model they match. | T | Exp. F5 (exp:blind), p. 11">6.6 |
| F6p13032 | The $\p=13$ picture: the residue line $\rep{\varepsilon_n}$, $n=2,\dots,12$, is $2,3,7,11,1,6,\text{blind},2,8,10,9$ — blind at $n=8$ ($\dr{8}=14833=13\cdot1141$) and terminating on $\rep{\varepsilon_{12}}=-(\Ku{13})^{-1}=9$, the common reading of $6/5$ and $11/7$ — while the metric row freezes onto $2.71828\ldots$ by $n\approx9$: the thesis in miniature. | T | § 6 |
| F7p13033 | The fixed-shell tower of $e$: with $\delta_m=\ctr{(-1)^{m}\eP}$ and $q_m=((m\p)!+\delta_m)/\dr{(m\p)}$, $\rep{q_m}=\eP$ exactly at every grade (since $(m\p)!\equiv0$ and $\dr{(m\p)}\equiv(-1)^{m}$ by F3) while $q_m\to\eR$ with the certificate $|q_m-\eR|<(m\p)!/(\dr{(m\p)}\dr{(m\p+1)})+2\kap/\dr{(m\p)}$; on $\p=13$, $q_2$ differs from $\eR$ by $3.98\times10^{-26}$ and every member reads $6$. Compatibility, not selection. | T | Prop. F7 (prop:tower-e), p. 11">6.7 |
| F8p13066 | The tail identity: $j!\,\dr{(\p-1-j)}\equiv(-1)^{j}\bigl(\Ku{\p}-\Ku{j}\bigr)\pmod\p$ for $0\le j\le\p-1$ ($j=0$ is F2); hence $n\in Z_0(\p)$ iff $\Ku{(\p-1-n)}\equiv\Ku{\p}$: the blind set counts the collisions of the partial sums of the left factorial with their total, $n=1$ is the forced collision $\Ku{(\p-2)}\equiv\Ku{\p}$, and Kurepa’s hypothesis is the return-to-origin question of the walk with steps $j!$. | T | Lem. F1 (lem:wilson), p. 10">6.1, Thm. F2, F8 (thm:wall-e), p. 10">6.2 |
| G. The frame-internal life of $\pi$: the wall hierarchy | |||
| G1p13034 | The legibility window: with $m=(\p-1)/2$, $\p\nmid\Cb{n}$ for $1\le n\le m$ and $\p\mid\Cb{n}$ for $m<n<\p$ (Kummer, A3): the residue line of the Wallis chain is defined on exactly $[1,m]$, blind-free below the wall and totally blind on $(m,\p)$; on a frame shell the last legible scale $n=2\kap$ is the angular address of $-1$. | T | Thm. G1 (thm:window), p. 12">7.1 |
| G2p13035 | The half-wall terminus, the calibration face: $\rep{w_m}=m^{-1}=\piA^{-1}=4\piA\equiv-2$ for every odd prime of the totality, the return statistics trivialising at the wall ($16^{m}\equiv1$, $\Cb{m}^{2}\equiv1$) and exposing the naked inverse scale; the identities are the calibration web of $2\piA\equiv-1$, and $\rep{w_{\piA}}\piA\equiv1\iff(4\kap)^{2}\equiv1$; the lower chain is blind at the wall and nowhere below. | T | Thm. G2 (thm:halfwall), p. 12">7.2 |
| G3p13036 | The quarter-wall two-squares invariant: for $\p\equiv1\pmod4$, $\p=a^{2}+b^{2}$ with $a$ odd, $a\equiv1\pmod4$, at the quarter scale $n=\kap$ (the address of $\im$) $\rep{w_\kap}=\kap^{-1}(2a)^{-2}=-(a^{2})^{-1}=(b^{2})^{-1}$, by Gauss’s congruence (A3) with $\kap\equiv-4^{-1}$; the invariant exists precisely under the frame condition; verified for all $211$ primes $\p\equiv1\pmod4$ below $3000$. | T | Thm. G3 (thm:quarter), p. 12">7.3 |
| G4p13037 | Second order at the half wall: $w_m\equiv-2+2\p(q_\p(4)-1)\pmod{\p^{2}}$ with $q_\p(4)\equiv2q_\p(2)$, exact to $\p^{3}$ through Morley (A3). The universal $-2$ persists to $\p^{2}$ exactly for the $\pi$-Wieferich primes (G10): $5$ and $45827$ below $1.08\times10^{11}$ [Fr"ohlich 2022] and below $2^{35}$ (the package’s recorded search); $45827\equiv3\pmod4$, so $\p=5$ is the only Subject shell among them. The normalised quotients equidistribute empirically [approx]. | T | Thm. G4 (thm:secondorder), p. 13">7.4 |
| G5p13038 | Lucas revivals and self-similarity: beyond the wall $n\ge\p$ is legible for $w$ precisely when every base-$\p$ digit of $n$ is at most $m$ and $\p\nmid n$ (on $\p=13$ the two-digit revivals number $(2\kap)^{2}=36$); on revival scales $\Cb{n}$ factorises digitwise (Lucas, A3); at the first revival $\rep{v_\p}\equiv8$ universally and $v_\p\equiv8+16\p(2q_\p(2)-1)\pmod{\p^{2}}$ by Wolstenholme. | T | Prop. G5 (prop:lucas), p. 13">7.6 |
| G6p13039 | First-order wall vanishing of the arcsin chain: $\sigma_\p=\sum_{k<m}\Cb{k}/((2k+1)16^{k})\equiv0\pmod\p$ for every prime $\p\ge5$ of the totality, $\sigma_3=1$ the sole exception; proved entirely in finite arithmetic through the key identity $A+B=2L$ (coefficient bookkeeping of the formal antiderivative of $(1-s^{2})^{m}$), Wilson reflection, Lerch’s congruence and the binomial transfer $\Cb{k}\equiv(-4)^{k}\binom mk$. Verified for all $428$ odd primes $5\le\p<3000$; the ingredients for $m\le60$, $\p<500$. | T | Thm. G6 (thm:vanish), p. 14">7.7, eq. (7.1) |
| G7p13040 | Second and third order at the arcsin wall: $\sigma_\p\equiv0\pmod{\p^{2}}$ is Sun’s theorem (A5), re-verified for $5\le\p<300$; the blind-range sum $\sum_{m<k<\p}\Cb{k}/((2k+1)16^{k})\equiv\p E_{\p-3}/3\pmod{\p^{2}}$ (A5), re-verified for $5\le\p<80$; at third order $\sigma_\p/\p^{2}\equiv(-1)^{(\p+1)/2}B_{\p-3}/36\pmod\p$ for all $1227$ primes $5\le\p<10^{4}$, the mod-$\p^{3}$ reading of Sun’s Conjecture 5.1 with the Wolstenholme refinement, a theorem for every prime $\p\ge5$ of the totality (A8): Wolstenholme-trivial at orders $\p,\p^{2}$, Bernoulli at $\p^{3}$, the Euler content beyond the wall. | T | master: 00:C24 |
| G8p13041 | The $\p=13$ picture for $\pi$: $\kap=3$, wall at $n=6$; the residue line $\rep{w_n}$, $n=1,\dots,6$, is $4,5,10,9,6,11$ with $\rep{w_6}=11\equiv-2$ the calibration face and $\rep{w_3}=10\equiv-(3^{2})^{-1}$ encoding $13=3^{2}+2^{2}$; empty blind set below the wall, total blindness for $n=7,\dots,12$, first revival $\rep{v_{13}}=8$. | T | § 7 |
| G9p13042 | The fixed-shell tower of $\pi$: for $n=\p^{r}$, with $\delta_\p=\ctr{\piA B_n-A_n}=\ctr{-34}$, the member $\widehat\pi_{\p,r}=(2\cdot16^{\p^{r}}+\delta_\p)/((2\p^{r}+1)\Cb{\p^{r}}^{2})$ reads $\rep{\widehat\pi_{\p,r}}=\piA$ exactly for every $r$ (Lucas and Fermat give $\Cb{n}\equiv2$, $16^{n}\equiv16$, $B_n\equiv4$) while converging externally to $\piR$; on $\p=13$, $\delta_{13}=5$. | T | Prop. G9 (prop:tower-pi), p. 15">7.10 |
| G10p13067 | The $\pi$-Wieferich condition in three forms: $4^{\p-1}\equiv1+\p\pmod{\p^{2}}$ iff $q_\p(4)\equiv1$ iff $2q_\p(2)\equiv1\pmod\p$, with $q_\p(4)=2q_\p(2)+\p\,q_\p(2)^{2}$ exactly; Eisenstein’s $2q_\p(2)\equiv-H_{(\p-1)/2}\pmod\p$ ($H_m=\sum_{j\le m}1/j$) makes it $H_{(\p-1)/2}\equiv-1$, and $(\p+2)^{\p-1}\equiv1\pmod{\p^{2}}$ identifies it with OEIS A355959 [Fr"ohlich 2022]. | T | Rem. G10 (rem:wieferich), p. 13">7.5 |
| H. Angular structure: exactness against calibration | |||
| H1p13043 | The frame character $\chi(\gen^{n})=\exp(2\piR\imR\,n/(\p-1))$, the canonical external dictionary sending $\gen$ to $\zeta_{\p-1}$; the comparison map, never used to derive either constant. | D | § 8 |
| H2p13044 | The exact angular carrier of $\pi$: $\chi(-1)=e^{\imR\piR}$ exactly in every shell, for every generator and either chirality, since $-1=\gen^{(\p-1)/2}$ sits at arc fraction one half — Euler’s identity as the half-turn tautology $\gen^{2\kap}=-1$ under the dictionary, with no calibration and no error term. | T | Thm. H2 (thm:exact-pi), p. 15">8.1 |
| H3p13045 | Exact calibration of $e$ is impossible: with $\theta=2\piR\lambda(\im)/(\p-1)$ the frame angle of $\eP$, $\theta\ne1$ in every shell ($\theta=1$ would make $\piR$ rational, A2); $\chi(\eP)$ is a root of unity while $e^{\imR}$ is not, so the identification is intrinsically horizon-limited. | T | Lem. H3 (lem:norad), p. 15">8.2 |
| H4p13069 | Calibrated shells at every fixed precision $\varepsilon\le1$: the $\varepsilon$-calibrated shells have asymptotic density $2\varepsilon/\piR$ among the shells $\p\equiv1\pmod4$ (A7), so large enough ranges hold them, $\chi(\eP)$ within $\varepsilon$ of $e^{\imR}$ there; the uniform model reads $N$ shells down to the nearest-hit scale $\piR/(2N)$ [approx]. Below $10^{6}$: $260$ shells within $0.01$ against $249$ predicted; the best, $\p=778933$, within $4\times10^{-5}$, decided exactly. | T | |
| H5p13047 | The null experiment: whether $\gen^{\lambda(\pm\im)}\equiv-(\Ku{\p})^{-1}$ for some primitive root and either orientation, varied independently, is the index condition $\gcd(s,\p-1)=\gcd(\lambda(\pm\im),\p-1)$; over the $210$ primes $\p\equiv1\pmod4$, $5<\p<3000$, it holds for $94$ against $96.0\pm6.6$ expected under independence ($54$ against $57.9\pm6.2$ with the orientation tied to the generator), the correlation of the two angular addresses $-0.10$ at deviation $\approx0.07$ [approx]: no invariant pointwise identification of the angular and radial carriers of $e$. | T | Exp. H5 (exp:null), p. 16">8.4 |
| I. The emergence theorems and the $e$–$\pi$ duality | |||
| I1p13048 | Emergence of the classical $e$: (i) in every shell with $\p>n_1+|Z_0(\p)|+1$ a non-blind scale in $[n_1,n_1+|Z_0(\p)|]$ reads out $\ro(\eR)$, at binary64 exactly $18!/\dr{18}$; (ii) the frame-internal continuation is the residue line, antiperiodic in its derangement denominators, blind on $Z_0(\p)$, terminating on $-(\Ku{\p})^{-1}$, read out beyond $n_1$ as one dyadic rational; (iii) $\chi(\eP)$ approaches $e^{\imR}$ along the calibrated shells, exact calibration impossible, no pointwise link between the carriers. Composed from D4, F2, F3, F5, H3–H5. | T | Thm. I1 (thm:main-e), p. 17">9.1 |
| I2p13049 | Emergence of the classical $\pi$: (i) the plateau readout of $\fQ$-native chains, at binary64 exactly $M_{10}$, the Wallis pair certifying two-sided enclosures at every scale; (ii) the residue line on $[1,2\kap]$ with the quarter-wall invariant $-(a^{2})^{-1}$, the half-wall terminus $-2$, the second-order Fermat quotient with $\pi$-Wieferich set $\{5,45827\}$, Morley exactness at $\p^{3}$, total blindness beyond the address of $-1$, Lucas revivals, the arcsin wall vanishing to $\p^{2}$ with the Bernoulli third order and the Euler content displaced beyond the wall; (iii) $\chi(-1)=e^{\imR\piR}$ exactly. Composed from E2, E4, G1–G7, H2. | T | Thm. I2 (thm:main-pi), p. 17">9.2 |
| I3p13050 | The duality: $\pi$ is angular-exact and wall-trivial (tautological carrier, terminus the calibration face $-2$, Fermat quotient one order down, two-squares quarter invariant, empty blind set below the wall, geometric $\fQ$-rate, the series law split Bernoulli/Euler across the wall) and $e$ angular-approximate and wall-wild (calibration impossible, terminus the open Kurepa quantity, no quarter invariant, Poisson blind set, factorial $\fQ$-rate, the group law broken at the wall); each constant exact on the axis where the other is approximate: $\pi$ a structural, $e$ a statistical constant of the finite frame. | T | § 9 |
| I4p13051 | CRT stability: under the composite lift to $q=\prod\p_i$ the half-wall invariant of $\pi$ glues to $-2$ modulo every $q$, since $2\piA\equiv-1$ holds in every fibre, while the terminal content of $e$, $(-(\Ku{\p_i})^{-1})_i$, varies fibrewise. The two selectors resolve the $\p=13$ coincidence: the same residue $6$ reaches $\piR$ through C4 and $\eR$ through C1. | T | Cor. I4 (cor:crt), p. 18">9.3 |
| I5p13068 | The three frame invariants across shells: on the $4783$ shells $\p\equiv1\pmod4$ below $10^{5}$, the triple $(\Ku{\p}/\p,\ q_\p(4)/\p,\ \varphi/\piR)$ — the wall of $e$ (F2), the wall of $\pi$ at second order (G4), the angular address $\varphi=\arg(a+b\imR)$ of $\p=a^{2}+b^{2}$, $a\equiv1\pmod4$ (G3) — is exact on each shell ($\p=13$: $10,6,-3+2\imR$; $1093$: $q_\p(4)=0$, the Wieferich prime); means $0.502,0.499,0.499$ and variances $0.0828,0.0831,0.0832$ (uniform: $1/2$, $1/12$), correlations $-0.016,-0.008,+0.010$ ($\sigma=0.0145$), $4\times4\times4$ chi-square $59.6$ on $63$ df, $a/\sqrt\p=\cos\varphi$ arcsine-distributed (mean $0.004$, variance $0.500$) [approx]: consistent with independence on the examined range. | T | § 9 |
| J. Hardness and certification | |||
| J1p13052 | The wrap-free window: an entity of the shell is $\p$-hard when it lies beyond the observer’s horizon $\sqrt\p$; the accessible smalls are the natural counts below $\sqrt\p$, where sums and products of two accessible counts do not wrap, so primality, parity and order are wrap-invariant there and only there. | T | § 10 master: 00:B12 |
| J2p13053 | Calibration pinning: the half-period satisfies the height-two relation $2\piA+1\equiv0$, the shell calibration $\p=4\kap+1$ itself, so the observer holds an exact short certificate of an entity it never accesses as a tally; the same pinning covers the quarter-turn ($\im^{2}+1\equiv0$) and the residue web of G2; the pin holds on every one of the $500$ shells $\p\le8009$ with $H(\piA)=2$. | T | § 10 |
| J3p13054 | The height run: with $H(x)=\min_{a\ge1}\max(a,|\ctr{ax}|)$ in the smallest-primitive-root frame, minimised over the two chiralities, over the $500$ shells $\p\equiv1\pmod4$, $\p\le8009$: $H(\eP)\le2\sqrt\p$ on every shell (the horizon band), median $H(\eP)=0.525\sqrt\p$ [approx], exactly $68$ shells with $H\le10$, and the pin value $2$ on exactly $\{13,1933,4177,5857\}$; for $\eP$ a relation of height at most $h$ holds only on the shells with $H(\eP)\le h$ ($68$ of the $500$ for $h=10$), a share that falls as the shells grow, $H(\eP)$ sitting at the $\sqrt\p$ scale (the generic-growth reading, A7). | T | § 10 |
| J4p13055 | A framed transcendental is a $\p$-hard role with no bounded-height defining relation holding across shells; the chart shadow of a $\p$-hard role splits by its certificate — a role pinned by a bounded-height relation has an exact algebraic shadow ($\im^{2}=-1$ giving $\imR$), while the half-arc and unit-registration roles, fixed by normalisations (C1, C4) with no such relation, have the transcendental shadows $\piR$, $\eR$. Neither external meaning, hardness nor accessibility rides on a residue symbol; roles carry all three. | D | § 10 master: 00:B9 |
| J5p13056 | Universality and the duality as corollary: $\p$-hardness is self-similar, every shell finding its own half-period and drive name beyond its own horizon, so the chart shadows agree across all observers; $\pi$'s half-period is calibration-pinned with a height-two certificate (J2) and its arc role normalisation-fixed, $\eP$ has neither a pinned residue nor a cross-shell certificate and its role is generator-dependent — structural against statistical is the certification split (I3). The sporadic small-height shells of J3 are the certificate-level face of I4. | T | § 10 master: 00:B9 |
| J6p13057 | Two constants, two walls: both attach their full-resolution shell content to sparse-prime problems (Kurepa non-vanishing for $e$, a prescribed Fermat quotient of Wieferich type for $\pi$); for each shell the bounded observer holds an exact finite certificate and holds no uniform certificate across all shells — per-instance decidability without a uniform bound, the characteristic shape of statements at the arithmetic horizon; Kurepa’s hypothesis is not thereby decided; the frequency of the persistence shells is a range reading (G4), their totality not a predicate of the paper. | T | § 10 |
| Z. Horizon: the certificates of the totality | |||
| Z1p13061 | Kurepa’s certificate over the totality: the count of shells on which the left factorial vanishes, Kurepa’s hypothesis its vanishing. Each shell decides its own case within its window; the count over all shells is a count above every shell, in no embedded reader’s window: $\Om$-hard, decided on a hosted Carrier by its host. | Ω | master: 00:Z4 |
| Z2p13063 | The large-$\p$ laws over the totality’s shells: the calibration rate against the nearest-hit scale, the joint law of the frame invariants, the blind-set and index-condition laws. Each shell decides its own values; each check over all shells is a count above every shell, in no embedded reader’s window: $\Om$-hard, decided on a hosted Carrier by its host. | Ω | |
Nothing matches that filter.
Status
- I
- Import. A classical theorem, a corpus result, or a measured datum used with attribution and without reproof.
- D
- Definition. A naming, a declared normalisation of the external comparison, or a set-up move, carrying no empirical content.
- T
- Theorem. Derived within this paper from the rows above it in exact finite arithmetic, machine-verified by the named witness where a witness is named.
- Ω
- $\Omega$-hard. A value decided by the totality, in no embedded reader’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 (
p13009) — 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).
), where C2 (every grade), C5 (the tally and the limit), D2 (the tail form), D3 (the enclosure), D4 (the binary64 readout, $n=18$ minimal), D5 (the step counts; the compound chain’s gap at least $\eR/(4n+1)$), E4 (the Machin bound, the pairing enclosure at every truncation, the binary64 readout, $N=10$ minimal, the hundred digits at $N=71$, the Gregory brackets) and E5 (the rounding chain) are proved too, E3’s bracket of $\piR$ is certified at $m=12$ and H4’s best shell is decided on the rational bounds of $\piR$; B2, F3, F7, H2, I4, J1, J2 in the core with no axioms, and the value rows and censuses B3, D4, D5, E3, F2, F4, F5, F6, G1, G5, G6, G7, G8, H3, H4, I5, J3, J5 are decided by the kernel on $\mathbb{F}_{13}$, the wall shells and the shells of the censuses (): the arcsin wall vanishing on all $428$ odd primes below $3000$ and to second order below $300$, the height run and the calibration scan over the $500$ shells $\p\le8009$, the opening stretch of the blind census, the tail bound of the arcsin chain for $n\le40$ against the certified bracket.