Tags
- 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.
- O
- Open. Genuinely unresolved.
| # | Description | Status | Source |
|---|---|---|---|
| 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 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, 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, “The readout map” |
| A7p13007 | Equidistribution of $\lambda(\im)/\p$ over the primes $\p\equiv1\pmod4$ (Duke–Friedlander–Iwaniec), the sole import behind the cross-shell existence of calibrated shells and the generic-growth reading of the heights. | I | [Duke et al. 1995] |
| 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 $\piR$ [chart]; the oriented quarter-turn $\im=\gen^{-\kap}$, the exponential unit $\eP=\gen^{\lambda(\im)}$ with $\lambda$ the label lift (frame data), and $\eR$ [chart]. On $\p=13$, $\gen=2$: $\piA=6$, $\im=5$, $\eP=6$. | D | § 2, “The typed constants”; 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. 2.1; tow.O 1/1 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 | 2/2 |
| 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 $\R$ [chart] 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, “Framed rationals and the shell reading”; 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 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. 3.2 |
| 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. 3.3; tow.C 1/1 |
| 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 [chart], $A(X)=\sum(-1)^{k}X^{2k+1}/(2k+1)$, 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 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. 3.5; pi.R1 1/1 |
| 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. 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. 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. 4.3; e.R1 1/1 |
| D4p13020 | Horizon determination of the readout: $\ro(\varepsilon_n)=\ro(\eR)$ for $n\ge n_0(k)=\Theta(k/\log k)$; the binary64 constant for $e$ is the readout of $\varepsilon_{18}=18!/\dr{18}=6402373705728000/2355301661033953$ and $n=18$ is minimal ($\varepsilon_{17}$ fails readout equality, $|\varepsilon_{17}-\eR|=1.10\times10^{-15}$ above the half-ulp); the first $100$ significant digits of $\eR$ are those of $\varepsilon_{70}$; every floating-point value of $e$ is a plateau readout. | T | 2/2 |
| 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. 4.5; e.R4 1/1 |
| 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. 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. 5.2; pi.R1 1/1 |
| E3p13024 | The arcsin chain $s_n=3\sum_{k\le n}\Cb{k}/((2k+1)16^{k})\in\fQ$ with external completion $\piR=6\arcsin\frac12$ [chart] 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. 5.3; pi.R4 1/1 |
| 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. 5.4; pi.R3 1/1 |
| 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. Whether a factorial-rate integer-ratio chain for $\pi$ exists is O3. | T | |
| 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. 6.1; e.W1 1/1 |
| 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 to Kurepa’s hypothesis (O1); re-verified in the derangement form for all $22{,}043$ odd primes $\p<2.5\times10^{5}$ with zero failures of either condition. Beyond the wall $n!\equiv0$: $n=\p-1$ is the factorial horizon. | T | 2/2 |
| F3p13029 | Antiperiodicity: $\dr{(n+\p)}\equiv-\dr{n}\pmod\p$ for every odd prime and every $n\ge0$, 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 $\p$. | T | Thm. 6.3; e.W2 1/1 |
| 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 | 1/1 |
| 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]; blind scales never obstruct the plateau of D4 (advance at most $|Z_0(\p)|$ steps past $n_0$). The counts are exact; the Poisson reading is the model they match. | T | Exp. 6.6; e.S1 1/1 |
| 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 | 1/1 |
| 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. 6.7; tow.E 1/1 |
| 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. 7.1; pi.W1 1/1 |
| G2p13035 | The half-wall terminus, the calibration face: $\rep{w_m}=m^{-1}=\piA^{-1}=4\piA\equiv-2$ for every odd prime, 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 | 3/3 |
| 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 | 2/2 |
| 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 $4^{\p-1}\equiv1+\p\pmod{\p^{2}}$, which below $10^{6}$ are exactly $5$ and $45827$ (heuristic expectation $\sum1/\p\approx2.4$); the normalised quotients equidistribute empirically [approx]. | T | 2/2 |
| 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 | 2/2 |
| 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 $\p\ge5$, $\sigma_3=1$ 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 | 5/5 |
| 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 sixty primes $5\le\p<300$, the mod-$\p^{3}$ reading of Sun’s Conjecture 5.1 with the Wolstenholme refinement, conjectural beyond the range (O2): Wolstenholme-trivial at orders $\p,\p^{2}$, Bernoulli at $\p^{3}$, the Euler content beyond the wall. | I T | 3/3 |
| 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 | 2/2 |
| 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. 7.9; tow.P 1/1 |
| H. Angular structure: exactness against calibration | |||
| H1p13043 | The frame character $\chi(\gen^{n})=\exp(2\piR\imR\,n/(\p-1))$ [chart], 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. 8.1; tow.W 1/1 master: 00:C14 |
| 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. 8.2 |
| H4p13046 | Calibrated shells at every precision: by A7, $\varepsilon$-calibrated shells ($|\theta-1|<\varepsilon$ for one chirality) exist for every $\varepsilon>0$ with relative density $2\varepsilon/\piR$, and $\chi(\eP)\to e^{\imR}$ along them; data for $\p<10^{6}$ ($39{,}175$ primes $\equiv1\bmod4$): $260$ shells within $0.01$ against $\approx249$ predicted, the best six led by $(\p,\lambda(\im))=(778933,123966)$ at $|\theta-1|=3.9\times10^{-5}$, consistent with the nearest-hit prediction $\approx8.0\times10^{-5}$ [approx]; the calibration inequalities decided as rational inequalities with $\piR$ bracketed by the Machin chain. | I T | Prop. 8.3; e.A1 1/1 |
| H5p13047 | The null experiment: whether $\gen^{\lambda(\im)}\equiv-(\Ku{\p})^{-1}$ for some primitive root of either chirality 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 is solvable for $94$ ($44.8\%$, the gcd-coincidence rate), 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 $e$; the naive reading of $\eP$ as the number $2.71828\ldots$ is refuted. | T | Exp. 8.4; e.N1 1/1 |
| I. The emergence theorems and the $e$–$\pi$ duality | |||
| I1p13048 | Emergence of the classical $e$: (i) in every shell with $\p>n_0+|Z_0(\p)|+1$ a non-blind scale in $[n_0,n_0+|Z_0(\p)|]$ reads out $\ro(\eR)$, at binary64 exactly $18!/\dr{18}$; (ii) the frame-internal continuation is the antiperiodic residue line with blind set $Z_0(\p)$ and terminal invariant $-(\Ku{\p})^{-1}$, collapsed by the readout to one dyadic rational; (iii) $\chi(\eP)\to e^{\imR}$ along calibrated shells, exact calibration impossible, no pointwise link between the carriers. Composed from D4, F2, F3, F5, H3–H5. | T | 4/4 |
| 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 | 4/4 |
| 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, “The duality” |
| 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. 9.3; tow.W 1/1 |
| 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, “What transcendence is”; tow.W 1/1 master: 00:B8 |
| 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 | 2/2 |
| 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 $\eP$ every candidate bounded-height relation fails in almost all shells (the generic-growth reading is A7). | T | § 10; tow.H 1/1 |
| 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 seat is calibration-pinned with a height-two certificate and its arc role normalisation-fixed, $e$ has neither a pinned seat 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; tow.H 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; no claim that either hypothesis is thereby decided. | T | |
| V. Machine verification | |||
| V1p13058 | The package finite-ring-space/src/13-epi: the five blocks validate_e ($11$ families, $61$ micro-checks), validate_pi ($10$, $94$), validate_pi2 ($10$, $551$), validate_towers ($6$, $72$), kurepa_wall ($1$ family over $22{,}043$ primes), $38$ families and $781$ micro-checks, each family naming the rows it witnesses; the notebook 13-epi-main.ipynb executes them in the browser and the driver run_all.py writes results.json. | T | § A; run_all |
| V2p13059 | The exactness discipline: integers, residues and exact rationals throughout; $\eR$ and $\piR$ enter only as the certified brackets of D3 and E4 (the Machin bracket of width below $10^{-100}$ deciding the calibration inequalities of H4), the binary64 constants only as the objects of study of D4 and E4; the statistical readings of F5, G4, H5, J3 and the Gauss sums are printed as [approx] diagnostics and decide nothing. | T | § A |
| V3p13060 | Provenance: the four Python scripts and the C pass are the memorandum scripts as written (round-01 and round-02 amendments recorded in the package), wrapped to report to one registry; the recorded memorandum outputs results_*.txt remain in the package as the source record. | T | § A |
| O. Open: the constants-sector walls | |||
| O1p13061 | Kurepa’s hypothesis, $\p\nmid\Ku{\p}$ for every odd prime: universal existence of the terminal residue $-(\Ku{\p})^{-1}$ of $e$ (F2); literature-verified for $\p<2^{40}$, re-verified here in the derangement form for $\p<2.5\times10^{5}$. With it the equidistribution of the subfactorial residue line (the Poisson law of F5 as a theorem) and the joint equidistribution of the quarter-turn and wall-residue indices (the null result of H5 as a theorem). | O | 1/1 master: 00:T7 |
| O2p13062 | The walls of $\pi$ beyond the verified ranges: infinitude of the $\pi$-Wieferich set $4^{\p-1}\equiv1+\p\pmod{\p^{2}}$ (exactly $\{5,45827\}$ below $10^{6}$); the third-order Bernoulli law of G7 for all $\p\ge5$, the mod-$\p^{3}$ reading of Sun’s Conjecture 5.1 (verified $5\le\p<300$). | O | master: 00:T8 |
| O3p13063 | The $e$–$\pi$ dichotomy as theorem: whether $\pi$ admits a factorial-rate integer-ratio chain (nonexistence makes the feasibility split of E5 exact); the radian-calibration rate $\delta(X)=\min_{\p\le X}|\theta-1|$ against the equidistribution prediction of H4; the joint equidistribution of the frame invariants $(\Ku{\p}/\p,\ q_\p(4)/\p,\ a/\sqrt\p)$ of the two constants. | O | Prop. 5.5, Prop. 8.3 master: 00:T9 |
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Ledger history
Ledger history. 2026-09-13: ledger added; the appendix’s script inventory replaced by the package description above and the rows below (the paper’s status discipline T/I/D is the ledger’s tag set, with O for the three open walls). Three open rows, O1–O3, are the constants-sector walls of the corpus master ledger (its rows T7, T8, T9); when one closes its content is re-homed in block F or G and the row leaves. No row has been retired.