The master predicate ledger 00: every predicate of the corpus, one per row (121 rows; the predictions on their own tab). Rows are cited as 00:label; the accession key pNNNNN is stable while labels may move. A witness is the script that decides the row in exact arithmetic, linked where the package is public.
Paper ledgers: 20-rh, 6-four, 8-dirac, 10-dim, 13-epi, 14-entr, 21-grav, 22-qm, 27-fld, 28-flav, 32-dark, 35-hadr.
| # | Predicate | Tag | Sources · witness · refs |
|---|---|---|---|
| A. Foundation: from existence to the finite carrier | |||
| A1p00001 | The Universe exists: the sole indubitable given (“I think, therefore the Universe is”). Neither axiom nor assumption. | G | no witness refs: — |
| A2p00002 | Infinity is incoherent in both forms, as an ontological posit. The actual: no coherent model is exhibited — the theory of the completed totality forbids its own totality as an object ($V$ is not a set), its antinomies (Cantor, Burali-Forti, Russell) are fenced by restricted comprehension, never resolved, and its consistency is unprovable from within (Gödel II) and purchasable from without only by a larger infinite posit; the classical reductio (5-red, with the four-class catalogue in its appendix: every pathology vanishes with the Axiom of Infinity). The potential: the undeclared freshness posit, never registered — idleness and the truthmaker audit (29-fin). The successor act is innocent and wraps (A7); the schema over finite instances is retained. Not claimed: a derivation of $0=1$ in ZFC. Argument: the reductio (5-red, the four-class catalogue and Prop. 9) and the truthmaker audit with the idleness theorem (29-fin); the incompleteness face independently in 25-göd. | T | no witness refs: A7 |
| A3p00003 | The totality has no outside and is unique, so it is complete (observers are shells embedded within, not parallel carriers). | T | no witness refs: — |
| A4p00004 | P1: the totality is finite and complete — finite by A2, complete by A3; forced, not assumed: by G and P2 a distinction with no registered residue is no distinction (A10), and the infinite is distinguished from the large finite by no registered residue (idleness), so finiteness is the audit’s verdict, not a premise; the neutral route’s one premise (Det) discharged by elimination (29-fin), effective determinacy characterising the finite structures (5-red Prop. 9; independently 25-göd). | P | no witness |
| A5p00005 | A bounded observer is a proper part, $|O|<|\Om|$, truth internally Tarski-undefinable; the limitative content migrates to the comprehension horizon (Z1, Z9). | T | no witness |
| A6p00006 | P2: every reading is an incomplete projection $\Pi_i{:}\Om\to\mathcal D_i$, strictly smaller than the whole ($|\mathcal L_i|\gg|\Om|$ without inflation); forced by A5. | P | no witness refs: A5 |
| A7p00007 | Counting closes by return, $C_q:x\mapsto x{+}1\ (\mathrm{mod}\ q)$, not by recursive enclosure: the bounded cyclic iteration replacing potential infinity; what is declined is freshness (A2). | T | no witness refs: A2 |
| A8p00008 | The Carrier C: a timeless torsor of Ω pointsThe Carrier $\Car$: a timeless torsor of $\Om$ points, its cardinality its only absolute characteristic; reframings $x\mapsto ax+b$ are automorphisms, so $0,1,-1$ are frame data; a Subject’s frame charts it as the prime field $\F_\Om$ — the frame’s chart, never the Carrier (C1). Arena and numbers: B1. | D | no witness |
| A9p00009 | Scale, the sole quantitative import: the de Sitter entropy of the observable universe, $\dS\sim10^{122}$ (Planck units), fixes the cardinality exactly, $\Om=4\dS+1$; the Carrier congruences are not imported with the datum but certified by the realisations (B5, C8); metrology and epistemic status of the numeral: L2. | I | 14:estimate_S.py [mixed] |
| A10p00010 | Realisation, not correspondence: a sufficiently expressive, predictive, registration-complete structure is its subject, not an abstraction of it; by P2 no registration reaches an unregistered reality, so a distinction with no registered residue is no distinction. Identification rows (R) are therefore constitutive: defended by the totality of registered consequences, killed by explicit falsifiers, never interpretive addenda. | P | no witness refs: — |
| B. The Carrier: the substrate structure | |||
| B1p00011 | All numbers are unframed Carrier residues, becoming framed rationals ($\fQ$) by a Subject frame — the native arena, in which every structure the sciences use is built exactly (the residue life); the continuum is the metric life of the same objects, a labelled idealisation read against $\mathbb R$ [chart], entered nowhere as an object. The accessible smalls are the natural counts below the observer’s horizon, the wrap-free window of primality, parity and order; everything beyond is hard at its scale (B8, B9). | D | no witness |
| B2p00012 | Phase cycle $\Phi=\F_\Om^{\times}\cong C_{\Om-1}$, the split torus of order $\Om-1=4\dS$; the cycle’s order and divisor lattice are frame-invariant (the arithmetic of $\dS$ is the spectrum of admissible dynamics), its marked elements chart data. | D | no witness refs: — |
| B3p00013 | Quarter-turn torsion: $4\mid\Om-1$ by construction ($\Om=4\dS+1$, A9), so the split torus carries a unique order-four subgroup — the Carrier’s $Q_4$ core $\{1,\hbar,-1,h\}$ ($\hbar^2\equiv-1$, $h\equiv-\hbar$, B7), the symmetry-complete condition. The Subject’s labels $\{1,i,-1,-i\}$ are chart data (C1), the two quarter-turns distinct objects (D7). | D | no witness |
| B4p00014 | Triality centre: $3\mid\Om+1$, so the non-split torus carries a unique order-three subgroup, the triality centre; its labels $\{1,\omega,\omega^2\}$ are Subject chart data (C1); jointly with B3 it forces the substrate residue (B5). | D | no witness |
| B5p00015 | The substrate residue: the quarter-turn ($4\mid\Om-1$, B3) and the triality centre ($3\mid\Om+1$, B4) jointly hold iff $\Om\equiv5\pmod{12}$ — the arithmetic shadow of the four-fold and the cubic, one in each frame torus. Both are relations certified by realisations, never properties of an integer: $Q_4$ by the quantum realisation (D4), the triality centre by the strong realisation (D6); the inference runs structure $\to$ congruence. There is no population of candidate $\Om$ (A3); a laboratory pattern instantiates the residue or does not (C13). | T | |
| B6p00016 | Four generative levels (succession $\to$ addition $\to$ multiplication $\to$ fixed-base exponentiation) generate the carrier’s two cyclic structures ($C_\Om$, $C_{\Om-1}$) and the exponential bridge lemma; the fifth level (tetration) folds back since exponents are $(\Om{-}1)$-cardinality residues, so the hierarchy, unbounded under the classical reading, closes at four over $\F_\Om$ (structural, not a cost claim). | T | no witness refs: — |
| B7p00017 | Physical constants are definite Carrier residues, decided by the Carrier and $\Om$-hard to every bounded observer: the Carrier register $\{\dS,\Om,c,\hbar,G,k_B\}$, generator-free in $\dS$ alone — $G=2\dS=-c^2$, $c^2=2^{-1}$, $\hbar^2=-1$ ($\hbar=2\sqrt\dS$), $k_B^2=-2$, $|k_B|=\hbar/c$. The defining congruences are the observer’s complete certificates: $G$ fixed exactly, the rest up to sign, the residual sign part of the same $\Om$-hard unknowability, never an indeterminacy of the Carrier; $h\equiv-\hbar$ is the per-cycle rebooking of the per-step $\hbar$ (D7). No Subject counterpart of $G$ or $\hbar$; no fifth constant (the drive name $e$: C1). Orientation data: C7. Sign sector: B10. Hardness and certification: B8. | T | 37:constants.py |
| B8p00018 | The two-level horizon law: hardness is one mechanism at every scale — $\Om$-hard when the role lives at the Carrier’s scale, $\p $-hard when beyond the Subject’s own horizon $\sqrt{\p }$ inside its own shell; hardness attaches to roles, never bare residues. The saturating Subject ($\p ^2\to\Om$) locks the ladder $\mathrm{small}<\Om^{1/4}<\p\text{-hard}<\Om^{1/2}<\Om\text{-hard}<\Om$, one square-root horizon per embedding. Certification: B9. Lab-verified ($\Om=2{,}408{,}561$). | T | refs: B9 |
| B9p00019 | Certification is the orthogonal axis to the B8 bands: a certified entity carries a bounded-height defining relation holding across shells; a framed transcendental is a hard role with none. Registers: at the Carrier the four constants (B7); at the Subject the half-period and quarter-turn, $2\pi\equiv-1$, $i^2\equiv-1$ (C1) — the same congruence shapes (self-similarity). Physical constants $\Om$-hard, mathematical constants $\p $-hard. | T | |
| B10p00020 | One gauge bit closes the sign sector: no C14-type marker exists — $-1$ is a square on admissible $\Om$, so square class is pair-blind, and integer parity is chart data (A8). Registered faces are pair-invariant: $c^2$, the one-way member synchronisation gauge (C8); $|k_B|$ (C12); $\{\hbar,h\}$ the two named unit faces (B7). The Hawking scaling forces $k_Bc=h$, so $k_B=-\hbar/c$ (chart-labelled). | T | no witness |
| B11p00132 | The shell surface is two-faced: the duplicate latitudes $L_a$, $a\in\{\frac{\p+1}{2},\dots,\p-1\}$, land on the inner face of the holographic shell under $L_a(m)=L_{-a}(m+\frac{\p-1}{2})$ (fixed-point-free), so the total surface count is the double cover $A=(\p-1)^2$; the one-sided registered surface is $(\p-1)^2/2$, plus the origin, the observer’s own cell; the antipode is never counted. | T | 00:check_shell_cover.py |
| C. The Subject: the frame and its registration | |||
| C1p00021 | The frame, Subject-only: $(\chron;0,1,\gen)$ — origin, unit, and the Subject’s own drive generator, $x\mapsto gx$ (A8). Shell data, capacity-first: $\kap $ primitive, $\p =4\kap +1$, $i=-\gen^\kap $ (C7), $\pi=2\kap $, $e=\gen^{\,i}$; $\gen$ the one nonsquare frame datum (C3, C8). The web closes on every shell: $2\pi\equiv-1$, $\gen^{\,\pi}\equiv-1$, $-\pi=2^{-1}$. The constants: B7; the continuum $\pi$ a labelled chart constant (N2). | T | no witness |
| C2p00022 | Scale-shift duality: dilation $x\mapsto gx$ is phase evolution of the frame, advancing $t$; over the cycle it is the fractional Fourier transform, whose quarter-turn $Q_4$ is the discrete Fourier transform. | T | no witness refs: — |
| C3p00023 | Euclidean–Lorentzian dichotomy: the drive fixes a Lorentzian time axis and causal order is algebraic, the split versus non-split torus the two signatures. The signature datum is the square class of the temporal coefficient, canonically $\nu=\gen$, register-separate from $c^2=2^{-1}$ (a square there), the factorisation $\nu=c^2\cdot(2\gen)$ exact; the continuum $\eta_{00}=-c^2$ is the collapsed-parity chart artifact (C8). | T | no witness refs: C8 |
| C4p00024 | Observer $=$ embedded shell (frame); observation $=$ shared-core comparison of two phase cycles. | D | no witness refs: — |
| C5p00025 | Oriented frame $=$ unimodular basis; frame space $=$ a torsor under $\mathrm{SL}_2(\F_\Om)$. | D | no witness refs: — |
| C6p00026 | One frame group, three sectors: gravity’s $(3{+}1)$ tensor, the Riemann observer geometry, and the quantum spinor structure are three readings of $\mathrm{SL}_2(\F_\Om)$ — its three tori the scale-time, translation, and boost freedoms, its order $(\Om{-}1)\Om(\Om{+}1)$ the observer variety (continuum shadow the de Sitter $S^3$, E2), its spinor double cover the Clifford layer (D4). Falsifier: a registered space-time, observer, or spinor freedom outside the three tori of $\mathrm{SL}_2(\F_\Om)$ — a fifth force or an extra dimension. | R | no witness |
| C7p00027 | Orientation is derived, not conventional: the $c$-square congruence ($\dS$ even, C8) annihilates orientation transport on every Carrier carrying it ($i^{\dS}\in\{\pm1\}$); pullback covariance and count positivity fix the oriented quarter-turn $i=-\gen^\kap $; the joint flip $(\gen,i,s)\mapsto(\gen^{-1},-i,-s)$ preserves every registered count — the matter–antimatter gauge, selected factually by the instance’s matter content. The constants’ derivation consumes no selection (B7). Parity marker: C14. Sector faces: H2, K1–K3. | T | no witness |
| C8p00028 | The square class is chronon parity: for a primitive drive the squares are $\langle \gen^2\rangle$ — a residue’s quadratic class is its drive-step parity. Registered transport is even, the one-way multiplier $\gen$ odd: the one-way speed is gauge, the two-way constant invariant ($[c^2]$ even iff $\dS$ even — the $c$-square congruence, a relation certified by the unit realisation D7, not a property of the integer $\dS$). The second unit marker (order two, D7); the Tsirelson $\sqrt2=\zeta_8+\zeta_8^{-1}$ its two-way symmetrisation. | T | no witness refs: D7 |
| C9p00029 | The Object frame is $(\dt;\,q,k,v)$: every entry a Subject residue — offset $\dt$ (synchronisation datum), position $q$, winding $k$ (D2), rate $v$; one relative datum per frame datum. The momentum reading is one quarter-turn from the winding, $p=i\,k$; the canonical commutator $[\hat q,\hat p]=i\hbar$ composes the Subject’s chart quarter-turn with the Carrier’s action quantum (D7, B7). | D | no witness |
| C10p00030 | Two strata of probability: a registered probability is a framed rational (a ratio of realized tallies); a structural weight is an element of the parity-even cyclotomic subring. The strata coincide exactly on the stationary core-valued $Q_4$ sector; two proved reductions connect them, and the residual realisation content is the stratum-sampling clause (D10). Witnesses and readout: 22-qm. | T | refs: D10 |
| C11p00031 | Pair-tally uniqueness (the finite Gleason statement): on the stationary core-valued $Q_4$ sector, tally-valued, fibre-additive, drive-invariant, parity-even registration functionals form the character cone fixed by the pure windings; adding channel-selectivity (assumed, not derived) picks the Born ray. The minimal degree is forced; proof shape: 22-qm. | T | refs: — |
| C12p00032 | Temperature and the classical arity, derived: the temperature domain is acceleration, $[\Theta]=[E][k_B]^{-1}=[L][T]^{-2}$, flag-free, $\Theta_P=E_P/|k_B|$, Unruh closing as a domain identity; neither mass nor temperature is primitive — $M$-$L$-$T$-plus-thermal is the torsion-free shadow of the two-generators-plus-flag system (the mass primitive standing in for the flag, the thermal for $k_B$), classical dimensional analysis recovered in exponents and arity. | T | no witness refs: — |
| C13p00033 | The exponent-window ladder: on a shell’s domain lattice the bounds nest strictly for $\kap\ge17$ — coherence $2\sqrt\kap$ below recovery $\kap/2$ below flag inaccessibility $\kap$ below covariance $2\kap$ — closing on itself at the top, $(2\sqrt\dS)^2=\Om-1$ (B8). The minimal laboratory pattern instantiating the quarter-turn core, the triality centre and the $c$-square ($\p=4\kap+1$ prime, $\kap>1$; $\Om=4\dS+1$ prime, $\dS$ even, $\dS\equiv1\pmod3$, i.e. $\Om\equiv5\pmod{12}$, B5) with the Subject resolvable in the Carrier ($\p^2<\Om$) is $(\p,\Om)=(13,233)$, by exhaustive scan — a host-decided instance, not a predicate the physical Carrier passes. | T | no witness |
| C14p00034 | The quarter-turn is the odd member of the $\pm\sqrt{-1}$ pair: the two representatives have opposite integer parity (distinct from chronon parity, C8), and $e^{\,i\pi}=\gen^{\,2\kap i^{2}}\equiv(-1)^{\,i}$, so the Euler identity $e^{\,i\pi}\equiv-1$ holds exactly on the odd member; the conjugate chart toggles it, so the parity names the matter–antimatter gauge (C7), and the selection is generator-independent within the chirality. | T | |
| C15p00035 | Integration is quotient plus cover at conserved octant depth: a shell and the Carrier share the class quotient and close on the fibre product (F2); 2-adically, shared quotient times cover ratio $=8$ always, since $\dS\equiv2\ (\mathrm{mod}\ 4)$ forces $v_2(\Om-1)=3$; $\kap$ odd: shared $C_4$, double cover, the missing octant bit supplied as the second inversion; $\kap$ even: shared $C_8$, no cover. | T | 98-lab no witness refs: F2 |
| C16p00036 | The spinor dichotomy: $\kap$ odd $\Leftrightarrow$ the drive folds as a product $\Leftrightarrow$ $v_2(\p-1)=2$, the shell lacking its own octant $\Leftrightarrow$ closure on the double cover, $-1$ at the half (C15): the spinor class; $\kap$ even: own octant, shared $C_8$, coverless closure. The dichotomy is the fold congruence $3\kap r+4s\equiv1\ (\mathrm{mod}\ 4\kap )$, solvable iff $\kap$ odd. | T | 98-lab refs: C15 |
| C17p00037 | Every Carrier quarter is a Fourier flip: multiplication by $\gen^{(\Om-1)/4}$, squaring to $-1$, rotates the coefficient plane one quarter, exchanging position and momentum; both roots $\pm\sqrt{-1}$ sit in crossing class 2, since $(\Om-1)/4=\dS\equiv2\ (\mathrm{mod}\ 4)$: the flip chirality is the C14 bit, undecided by the Carrier. Instance: $78^{58}=h$, cardinals $\{1,h,-1,\hbar\}$. | T | refs: C14 |
| C18p00038 | The precession theorem: over the pair dynamics the shell orbit precesses against the Carrier quarter by the winding ratio (D2): the apsidal fraction per revolution is exactly $\kap /\dS =(\p -1)/(\Om -1)$, every $\kap$-odd pair; the half event (home, dlog$(-1)$) sits at the exact midpoint, closure at (home, home); orientation prograde, Subject-selected (C14, the gauge bit). | T | 98-lab |
| C19p00039 | The Hopf section: in the observable frame group $\mathrm{PGL}_2$ the boost torus $C_{\p +1}$ is fixed-point-free on $\mathbb{P}^1$ and meets the Borel trivially, so every frame factors uniquely as (cone-chart event)$\times$(boost): the cone chart is a global section of the finite Hopf fibration, $\p (\p -1)$ fibres of $\p +1$; the spin cover $\mathrm{SL}_2$ obstructs at $-1$. | T | 98-lab refs: — |
| C20p00040 | The mass–energy channel: the winding sector $C_{\p -1}$ meets the boost base $C_{\p +1}$ in the sign alone and the spinor cover $C_{2(\p +1)}$ in exactly $Q_4$, every $\p =4\kap +1$; the quarter is spinorial, $N(i)=i^{2}=-1$: the mass transport crosses the sheet; mass phase $=$ winding rate, one $Q_4$ datum (D2); Carrier instance $78^{58}=89=h$ on $\Om =233$ (C17). | T | 00:check_y5.js |
| C21p00041 | The dilation rigidity theorem: static clock comparison is a register character $\gen^{-\Delta n}$; characters are rigid, so the strong-field dilation law is the unique exponential reading, zero coefficient freedom; the PPN temporal form admits no register character (transitivity defect exactly $4ab$); the cover halves the angle at every tick (C20). | T | 00:check_c21.pyrefs: C20 |
| D. The realisations (mathematics as physics) | |||
| D1p00042 | Time is scale-dilation: each Subject rides its own drive $x\mapsto gx$ and assigns its own tick; a composite is read through the reader’s one drive, so the diagonal action and its conserved offset are observer-derived — the Carrier contributes no clock, the offset group fixed by the arena’s divisor structure. Falsifier: a registered duration that is not a drive count of the reader’s own frame, or a frame-independent Carrier tick. | R | no witness refs: — |
| D2p00043 | Mass is winding rate ($E=hf$ an identity); masslessness $\Leftrightarrow$ drive-invariance; a mass is the non-split (Frobenius, $\F_{\Om^2}/\F_\Om$) component the drive rotates, masslessness the split-torus eigen-alignment. Falsifier: a rest mass with no cycle winding, a massless drive-variant state, or a stationary state whose registered frequency is not its winding index ($E=hf$ failing). | R | no witness refs: — |
| D3p00044 | Distance is decoherence: the separation of two systems is their decoherence count in the observer’s chart, the metric the graph metric of shared-core comparison — the spatial reading of the one relational channel, the twin of D1 (time) and D2 (mass). Falsifier: a metric separation not monotone in the decoherence count of the relational channel. | R | no witness |
| D4p00045 | Quantum: matter is the phase character of a finite cycle; observation is shared-core comparison; the complex amplitude is forced by $Q_4$ ($i^2{=}{-}1$). Falsifier: a confirmed third-order interference $I_3\neq0$, $S\neq2\sqrt2$, or real-amplitude network behaviour (22-qm P2, P3). | R | no witness refs: — |
| D5p00046 | Gravity: a cell-local change of frame is a gauge symmetry, and the connection compensating it is the field (the geometric/diffeomorphism frame). Falsifier: a source gravitating by other than its full mass, entanglement induced off the Newtonian rate, or $\eta\neq0$ (22-qm P1, C22). | R | refs: — |
| D6p00047 | Forces: the same recipe applied to the substrate’s three other frame data, namely phase (EM), spinor (weak), and colour (strong); “finish gauging the frame.” Falsifier: a fifth force, a gauge group or charge outside the rank-five frame, or a fourth generation. | R | no witness refs: — |
| D7p00048 | A unit of measure is a reciprocal cross-frame relation: observer counts per substrate count. Its atomic form is the action quantum, the Carrier quarter-turn $\hbar^2=-1$; its domain the unit-free order-four flag — unitful measures carry it, observables are flag-free, the absolute unit value $\Om$-hard; the fundamental units the four domain horizons (10-dim). Constants: B7; arity: C12; windows: C13; commutator: C9. Falsifier: an absolute unit value registered by an embedded observer, or a unitful measure carrying no cross-frame reciprocal. | R | |
| D8p00049 | Elementary is $\Om$-hard: an elementary particle is a large beyond-horizon residue, never observed at rest; the observable exception is the gauge sector’s drive-invariant residue, the photon (G1, Z6); instantiated at $517\times\sqrt\Om$ on a concrete shell (37-sim). Falsifier: an elementary particle with a resolved internal residue (substructure), or a massive drive-invariant gauge residue. | R | no witness |
| D9p00050 | The observable is a small residue: a bounded observer reads only small residues, so an observable stable object carries a small exact residue (particle $+$ binding wrap) — the confined hadron, the light nucleus, the stable atom; decay is frame-normalisation toward it. Falsifier: a stable observable object carrying no small exact residue — a stable coloured state, or strong/electromagnetic proton decay. | R | 33:nuclear_residue_test.py [approx]refs: — |
| D10p00051 | The unified sampling clause: a registration protocol samples one stratum of a drive-presented ensemble; over complete joint recurrences the counts are forced exact tallies (enumeration-verified at $B=2,3$ on $\Omega=641$), on incomplete batches the largest-remainder gauge — the one sampling identification shared by the quantum readout and the gravity registration crossover. Falsifier: outcome tallies over a complete joint recurrence that are not exact ratios, or a fixed-setting protocol sampling more than one stratum. | R | no witness refs: — |
| D11p00134 | The shell theorem (20-rh Thm. hp, 2026-09-12; restated on the round-02 review, same day): on every shell $\F_\p\subset\F_{\p^2}$ ($\p=4\kap+1$; equally $\F_\Om\subset\F_{\Om^2}$) the spectral content of the prime vector is the constant mode (the mean) with the nonterminal modes of the quarter-turn meridian, on each of which the finite zeta vanishes; every mode lies on the self-dual line $\Tr=1$, real part the half-turn $2^{-1}=2\kap+1=-\pi$ (the Subject’s half-period $\pi=2\kap$, bare), fixed by the representation domain; the shell’s scale-shift $x\mapsto \g x$ has these modes as eigenvectors in both readings of the cycle’s characters (power characters in $\F_\p$; complex characters, orthonormal, in $\ell^2$ under $\g\mapsto\omega_{\p-1}$), eigenvalues the $(\p-1)$-th roots of unity, independent of the vector; and the comb-built Jacobi matrix is a real-symmetric matrix whose spectrum is the comb’s secular roots (free: every finite real multiset has one). The shell carries no off-line mode. A relation: finite-field algebra with no hypothesis, $\Om$-blind, the same on every shell, the physical Carrier ($\Om=4\dS+1$) included, and holding for every vector on the shell, the Davenport–Heilbronn vector included; the paper’s title names this theorem together with D12. Retired from the row (round-02): the clause “the on-line spectrum is complete for the prime vector by finite Parseval” (false for the nonterminal $\F_\p$-valued modes — they omit the constant character and do not expand a real vector; the true complex-character statement holds for every vector and is a remark, 20-rh Rem. parseval) and the clause identifying the dilation generator’s spectrum with the Jacobi spectrum (no intertwiner exhibited; open item of the paper). Witnessed exactly on seventeen shells $13\le\p\le173$ (zero-slot, critical line and norm, Klein four-group and its fixed loci, character orthogonality). It does not decide the classical hypothesis: see D12. | T | 00:check_rh_shell.py |
| D12p00135 | The classical hypothesis is a screen value (20-rh Thm. turing, Cor. conditional, 2026-09-12; restated on the round-02 review, same day): the Riemann Hypothesis below height $T$ is $\mathcal N(T)=N_{\mathrm{crit}}(T)$, strip zeros against on-line zeros, both with multiplicity [import: Turing 1953]; with $C(T)$ the sign changes of $Z$, $C\le N_{\mathrm{crit}}\le\mathcal N$, and $C=\mathcal N$ is the practical certificate, forcing simplicity as well (round-03 correction of the sign-change form). The shell reads the two sides from the two sides of the explicit formula: $\mathcal N$ from the raw comb count (rounding valid only under a bound $|\widetilde{\mathcal N}_N-\mathcal N|<\tfrac12$, not supplied; uncertified), $C$ from the de-framing (sign changes of the Riemann–Siegel main sum of length $\sqrt\p$, certified only with an explicit remainder bound, not supplied), inputs frame-exact (height $T=2\pi\p$), outputs uncertified. The secular condition reads $\mathcal N$, not $N_{\mathrm{crit}}$: on the Davenport–Heilbronn comb $\Lambda_f$ it returns two “on-line” roots at $85.65$ and $85.75$ where the line holds none, and its raw count near the pair does not settle with depth ($45.14\to45.73$ at $t=85.9$ from $5\times10^4$ to $8\times10^5$; stable to $10^{-2}$ away from it; the $\zeta$ comb converges) — the Hardy–Riesz regularity of summable Dirichlet series [import], a second screen signature of a phantom (20-rh Obs. dh). “Both sides from $\Lambda$ alone” is retired. An off-line zero of $\zeta$ is therefore a phantom on the screen, a height the comb counts with no on-line mode to carry it; under the holographic reading (the $s$-plane a reading of the screen and nothing else) the identification is exhaustive, and the classical hypothesis has no content beyond the value $\mathcal N=N_{\mathrm{crit}}$ on the sub-horizon range. Structural, not a definition: the Davenport–Heilbronn vector satisfies D11 on its shell and shows the phantom live ($45$ strip against $43$ on-line zeros on $[0,87]$; the constituent $L(s,\chi)$ closes $45=45$). The value is decidable per height and verified to $3\times10^{12}$; its uniform form is the $\Om$-hard value of Z4; whether a bounded-height certificate for it exists the shell does not decide (D11 holds for the Davenport–Heilbronn vector too, so the shell structure does not use the Euler product; only finite Weil positivity, per-height again, uses it). Supersedes the retired 20-rh Cor. omhard (“conditional on finitude RH holds”), whose inference zero leak $\Rightarrow$ no phantom is refuted by the same control: the leak is the Riemann–Siegel remainder, not the completeness deficit. Falsifier: a window the shell carries on which the shell’s $\mathcal N-N_{\mathrm{crit}}$, the count from the comb and the on-line zeros from the de-framing, differs from the classical Turing count. | T | no witness |
| E. Gravity | |||
| E1p00052 | Uniqueness of the dynamics: in every gauge sector the relevant gauge-invariant adjacency-local action is the unique curvature-square Casimir (Maxwell $F^2$, gravity Fierz–Pauli, Yang–Mills $\mathrm{Tr}\,F^2$) — one adjacency-local uniqueness lemma, proven on the shell. | T | refs: — |
| E3p00054 | Gravity (linear): the Coulomb law is gravity’s own lattice Green’s function; the field equation is the discrete Fierz–Pauli functional, the unique adjacency-local stiffness (E1); the $10^{36}$ hierarchy the cardinality dilution $(m/m_P)^2$. Universal attraction is the drive’s single time arrow; the radiative sector gives two helicity-$\pm2$ gravitons at $c$, collective synchronisation. $G=\hbar c/m_P^2$, the channel’s unit capacity (B7). | T | |
| E4p00055 | Gravity (nonlinear completion): the relational shift-symmetry Ward identity forbids self-sourcing (what gravitates is winding rate, D2), so the static potential is harmonic and the completion is the unique scale-covariant exponential metric (Deser’s bootstrap inapplicable); $\beta=\gamma=1$ and the classical tests exact, the first departures strong-field. | T | refs: D2 |
| E5p00056 | Gravity (the floor’s faces): collapse yields no event horizon and no singularity but an operationally black body (clock rate at the floor $1/\sqrt\Om$) at $r_f=r_s/\ln\Om$, the interior a microscopic slip core handing over to the spectral chart; the global drive enters local physics only through the floor, so preferred-frame parameters are $O(1/\sqrt\Om)\sim10^{-61}$ (Lense–Thirring at the GR value). | T | refs: — |
| E6p00057 | Gravity (horizon entropy): the channel count of a bounding surface gives $S=\tfrac14 A/\ell_P^2$; the $\tfrac14$ is independently counted, the $Q_4$ gauge quotient of the registration sphere ($S/A=\kap _\p /\p $ exactly), and closes the calibration: at the coherence-horizon $\p ^2\to\Om$ the record law returns $S=(\Om-1)/4$, the imported entropy (A9). Corollary: the merger area law $\Delta A=2M_1M_2$, saturated (37-sim). Triple: E7. | T | 37:w2b_bridges.py |
| E7p00058 | Gravity (horizon triple), instantiated: the record is the registered distinguishable-state count, $S=(A/4)(1-1/\p )$ (F1); the temperature has two exact faces — the registration rate $T=(\p +1)/\p ^2$, the slip rate every thermometer reads (E5), and the response $T_{\mathrm{resp}}=(dS/dM)^{-1}=1/\p $ at the spinor half-period $M=(\p +1)/2$ (J1) — splitting by $(1+1/\p )$, each with its Smarr relation; verified $8/8$ shells. | T | no witness |
| F. Quantum | |||
| F1p00059 | Quantum mechanics: the Born weighting on the Subject’s ledger — one declared identification (registration data are sample pairs) plus counting theorems, the pair form unique on the stationary core-valued $Q_4$ sector (C11), the two probability strata and their readout (C10); the amplitude is complex (not real/quaternionic) by $Q_4$; the CHSH/Tsirelson $\sqrt2$ is the quarter-turn, parity-even (C8). | T | no witness |
| F2p00060 | Multi-body registration: an unequal-cycle composite’s joint recurrence is the lcm of its parts’ registration periods; each pair’s conserved offset lives on the $\gcd$ cycle (the quotient $A/\langle v\rangle$), and dephasing over the product period is exact, extending the Born-rule count (F1) verbatim to unequal cycles (37-sim: joint recurrence derived and verified, 1400/1400 characters cancel). | T | refs: F1 |
| F3p00061 | The coherent-fraction channel (equal force, unequal entanglement): sources gravitate by full masses; the interaction is reciprocal synchronisation — synchronised cross-source pairs carry the conditional phase $f_1f_2\varphi_N$, spread pairs imprint which-branch data, the incoherent channel separable at generic geometry ($C=0$ at $f=0$); $f=m_c/m$ microscopic. The discriminator is the calibrated conditional phase rate at fixed centre-of-mass superposition; parameter-free, filed before data. | T | refs: — |
| F4p00062 | The wavefunction folds on $x=i^{\,r}\gen^{\,s}$: the Object cycle decomposes by the $Q_4$ quotient — the class $r$ the registrable phase (C14), the coset rung $s$ the outcome index (capacity, C13). Fibres are the cosets $q_\alpha\cdot\mathrm{core}$, pointer channels the core characters, the amplitude the coherent fibre sum: observation resolves cosets, never points within one. Division-unique everywhere; product iff $\kap$ odd. | T | no witness |
| F5p00063 | The graded registration law: an Object component $\ell^{a}$ registers at depth $d=\mathrm{ord}_{\ell^{a}}(\p)$ — $1$ absorbed (register, unique image), $2$ observable (boost torus, unique home), else beyond-quadratic; the sign always $\{\pm1\}$. Visible iff $\p^{d}<\Om$; the cap is $2$ at saturating and minimal towers, $\p^{2}<\Om$ hosting the observable sector; opacity relational — hydrogen’s interior opaque/observable/absorbed at $173/181/197$. Controls: proton whole, $(5,13)$ refused. | T | no witness refs: — |
| F6p00064 | The binding lift: a pair fuses iff cycles coprime, windings unit (swept); else the F2 offsets stay superselected — the deuteron, $(4,4)$, spectrumless. Fusion carries the half-driven offsets onto the F4 rungs; pointer channels forced (absorbed class gauge, torus core disjoint), Born lands with exact dephasing ($1+\omega+\omega^{2}=0$, $T'=516$). Energies $E_{s}=s\,h/3$, stations $\{0,58,116\}$; cores odd$(\q-1)$: hydrogen $(4,3)\to C_{12}$, $\kap_{O}=3$, ground first. | T | no witness |
| G. Electromagnetism | |||
| G1p00065 | Electromagnetism (exact): charge $=$ quantised winding index; the photon is massless, the observable residue $2$ of the split torus $C_{\Om-1}$ (two helicities $=$ two $\F_\Om$-fixed lines $=N(1{-}i)$; D2, D8); Maxwell is the unique relevant action (E1); $\alpha_{\mathrm{bare}}=1/4\pi$ (the phase-channel capacity, channel unity $g{=}1$ at the $\mathrm{U}(1)$ critical point $\beta_c{\simeq}1.01$; the physical $\alpha^{-1}(0)$ is the $\Om$-hard EM face, Z8). | T | no witness |
| H. Weak | |||
| H1p00066 | Weak (structure): breaking $=$ split/non-split torus misalignment; $\sin^2\theta_W=\mathrm{Tr}\,T_3^2/\mathrm{Tr}\,Q^2=3/8$, custodial $\rho=1$ exact; the propagating $W,Z$ mass spectrum derived (Hessian of $S_\rho$), with $M_W^2/M_Z^2=\cos^2\theta_W=5/8$ exact. | T | refs: — |
| H2p00067 | Weak (amplitudes): the chiral $V\!-\!A$ current and four-fermion $G_F=1/(\sqrt2\,v^2)$ derived; $W,Z$ scattering unitarised (equivalence theorem). | T | refs: — |
| H3p00068 | Weak (scale): the lone scale $=$ the non-split saturation scale, dimensional transmutation $M_{\mathrm{EW}}=m_P e^{-4\pi^2}\approx87$ GeV ($b_0=2$), the vev $v=M_{\mathrm{EW}}\times{}$gauge factor, its coefficient the cross-scale $\beta$-function; $\Om$-hard, as $\Lambda_{\mathrm{QCD}}$ (I3). Value tagged: the absolute scale $v$ (equivalently $M_{\mathrm{EW}}$), in no embedded reader’s window — the failing computation is the cross-scale $\beta$-function summed over the $\Om$-hard masses (Z6). Relation, $\Om$-blind: the transmutation form $M=m_P e^{-c/b}$. The numeral $\approx87$ GeV with $b_0=2$ is a [chart] evaluation, not certified content. | Ω | no witness refs: I3 |
| I. Strong | |||
| I1p00069 | Strong (structure): $\SU(3)$ $=$ the special unitary group of a Hermitian three-form, its rank forced as the minimal triality frame (the cube of $\Om\equiv5$); the confining area law derived from the positivity of $S_\rho$, the gluon the residue $0$ of the non-split torus $C_{\Om+1}$ (no $\F_\Om$-fixed line). | T | refs: — |
| I2p00070 | Strong (tension): the string tension in finite units, $\sigma=-\ln c_1(\beta)>0$, computed — the finite-group area law gives $c_1$ as an exact character sum (no Haar, no $\Om\to\infty$); closed form on $\SU(2,\F_3){=}2T$ (matching continuum $\SU(2)$ through $O(\beta^3)$), colour $c_1^{\SU(3)}{=}\tfrac\beta{18}{+}\tfrac{\beta^2}{216}{+}\cdots$. | T | refs: — |
| I3p00071 | Strong (scale): the cross-scale $\beta$-function and the absolute $\Lambda_{\mathrm{QCD}}$ (dimensional transmutation, $\sigma\sim\ln(1/\beta)$). Value tagged: the absolute $\Lambda_{\mathrm{QCD}}$, in no embedded reader’s window — the failing computation is the $\beta$-function integrated to the substrate scale over the $\Om$-hard spectrum. Relation, $\Om$-blind: $\sigma\sim\ln(1/\beta)$ and the ratios of I5–I7. | Ω | no witness refs: — |
| I4p00072 | Strong (CP): the vacuum angle vanishes, $\bar\theta=\theta+\arg\det M_q=0$, with no axion — the colour action carries no substrate-native topological term and colour is drive-invariant ($\theta=0$), and the Hermitian quark mass matrices have real determinant ($\arg\det M_q=0$); the CP phase $\delta_{\mathrm{CP}}$ sits in the up–down misalignment. | T | refs: — |
| I5p00073 | Strong (hadron spectrum): a baryon is the centre-neutral triality invariant $\varepsilon_{abc}$ of $\SU(3,\F_2)$; the $\SU(3)_F$ mass relations are exact scale-cancelling identities — Gell-Mann–Okubo, decuplet equal spacing, Coleman–Glashow, the orderings $M_n{>}M_p$ and $M_\Sigma{>}M_\Lambda$, heavy-quark symmetry, the vector-nonet spacing; the decuplet–octet hyperfine is the colour-magnetic curvature invariant, $A_{\mathrm{light}}{=}\sqrt\sigma\,c_{\mathrm{mag}}(\beta)$ the spin-$2$ partner of the tension $c_1$ (I2). Formulas: 35-hadr. | T | 35:su3f_relations.py [mixed] refs: I2 |
| I6p00074 | Strong (the residue series): the observable baryon is the Carrier residue $1$, the determinant $\Lambda^3(\mathbf3){=}\varepsilon_{abc}$, forced by the gluon residue $0$ ($N{=}3$ from $\Lambda^N{=}\det$); the meson is residue $1$ (the $q\bar q$ trace); series closed: photon $2$, gluon $0$, baryon $1$, meson $1$ (G1). Selector, stability: the minimal-wrap state is the stable proton, decay frame-normalisation toward it (D9). | T | |
| I7p00075 | Strong (the baryon scale): the absolute ground-state spectrum is the $\Om$-hard scale $\sqrt\sigma{=}\Lambda_{\mathrm{QCD}}$ (I3) times sub-horizon eigenvalues; the baryon mass-over-scale is the computed confinement eigenvalue $M_N/\sqrt\sigma{=}E_0{=}2.232$ (the finite operator $H{=}|p|{+}\sqrt\sigma\,r$, converged finite matrix; $4.6\%$); the constituent ratios $\{\lambda_l,\lambda_{\mathrm{hf}}\}$ are evaluated forward from $\{N{=}3,\beta\}$ with no measured baryon mass, so the light-hadron spectrum is parameter-free over $\sqrt\sigma$; no new $\Om$-hard residue. | T | 35:confinement_closure.py [approx] refs: I3 |
| J. Matter and generations | |||
| J1p00076 | One generation (content): $=$ the spinor $\mathbf{16}$ of the rank-five frame $\C^3\oplus\C^2$ — all gauge reps, hypercharges, charges, anomaly-free, $\nu_R$ — recovered in the compact theory via the correspondence. | T | no witness refs: — |
| J2p00077 | One generation (origin): the same $\mathbf{16}$ is the four-chart reflection algebra $\Lambda^{\bullet}(\C^4)$ ($2^4{=}16$) closed by the quarter-turn as the fifth direction, $\Lambda^{\bullet}(\C^4)\cong\Lambda^{\mathrm{even}}(\C^5)$; “matter is a spinor” is derived by reflection counting, the gauge $3{+}2$ and ontology $4{+}1$ two readings of one frame (28-flav, reports/reflection-origin). | T | no witness refs: — |
| J3p00078 | One generation (masses): its elementary particles’ absolute masses are framed-transcendentals, $\Om$-hard (D8, Z6). Value tagged: the sixteen absolute masses, in no embedded reader’s window — the failing computation is the winding of a beyond-horizon residue read at rest. Relations, $\Om$-blind and certified: the mass ratios K2–K3. | Ω | no witness |
| J4p00079 | Count/generate split: one bosonic carrier $+$ three fermion generations (the cubic $C_3$), chirality fixed by the drive — a theorem given the role-to-matter realisation; $\SO(10)$ unification forced by the relational ontology, the automorphism lemma reduced to the horizon (Z5). | T | refs: Z5 |
| K. Flavour | |||
| K1p00080 | Flavour (structure): the seesaw scale, $b$–$\tau$ unification, and the role-ladder mass ordering are derived; the observed spectrum is the projection of the one $\mathbf{16}$. The CKM and PMNS angles and phases reduce to the role-depth texture, the lopsided $\sigma$, the trimaximal $C_3$ column, and quark–lepton complementarity, leaving the $\Om$-hard drive-orientation phases $\{\delta_0,\delta_\nu\}$ (Z6); the seesaw fixes $Q_\nu{=}2/3$. | T | 28:neutrino.py [mixed] refs: Z6 |
| K2p00081 | Flavour (Koide): the charged-lepton ratio $Q=2/3$ is the cube-root generation orbit at the self-dual quarter-turn, $Q=\tfrac13+\tfrac23\rho^2$ exact at $\rho^2=2^{-1}$; the winding-phase selection onto that orientation is bedrock, $\Om$-hard in magnitude, its $\Z_2$ sign the derived drive orientation (C7); the sub-horizon reading is $\delta_0=Q/3=2/9$. | T | refs: C7 |
| K3p00082 | Flavour (targets): the sharp ratios are derived structure — Koide $2/3$, Georgi–Jarlskog $\Nc$, the squared amplitudes the split/non-split norms $N(1{-}i){=}2$, $N(1{-}\omega){=}3$ (G1, I1), trimaximal $|F_{jk}|^2{=}\tfrac13$, maximal Jarlskog $J^2{=}\tfrac1{108}$, the $\lambda$-texture from role depths — locked by $\delta_\ell{:}\delta_d{:}\delta_u=1{:}\tfrac12{:}\tfrac13$ to one $\Om$-hard phase $\delta_0\simeq2/9$; $m_1{=}0$ at LO makes $\sum m_\nu\sim59$ meV parameter-free. | T | no witness |
| K4p00083 | Flavour (projection): the $\mathbf{16}\to$observed-spectrum projection map (through $C_3\times Q_4\times$ scale-wrap) is constructed, anchored at its unique drive-invariant total singlet $\nu^c=\Lambda^0$ (the $\mathbf1$ of $\mathbf{16}$), the matter twin of the photon (G1); the residual scale-wrap magnitude is the $\Om$-hard mass set (Z6). | T | no witness |
| L. Cosmology and the dark sector | |||
| L1p00084 | Cosmology (the dark sector): the synchronisation resolution floor $a_0=cH_0/2\pi$ fixes the galactic acceleration scale parameter-free — the $2\pi$ by the angle–count dictionary (14-entr B2, D: one full turn of $C_{4\dS}$ is $2\pi$, the drive period in registered time $2\pi/H$), the threshold one full cycle per drive period (32-dark B3, 21-grav Prop. afloor, R; falsifier: a floor at a different fixed fraction of $cH$ — the neighbouring sectors $cH/\pi$, $cH/4\pi$ — or one not tracking $H(z)$), the value $1.04\times10^{-10}$ [chart] at the entailed $H_0=67.4$ (L3), $0.7\sigma$ below the fitted $1.20\pm0.24$; the floor is a rate reading (registered $H$) and the age a depth reading ($\Lambda$-face), by 14-entr’s register rule, so the floor is tied to $H(z)$, not to $\Lambda$ (Q4 closed 2026-09-12; the constant $\Lambda$-face branch $cH_\Lambda/2\pi=0.86\times10^{-10}$ is what L7 excludes); $\Lambda\sim1/\Om$ is the totality datum, dissolving the $10^{120}$ vacuum cancellation; below the floor the amplitude reading gives the radial acceleration relation given the barrier identification (32-dark B8, 21-grav C19: the masking barrier is the coincidence amplitude, R; its falsifier the approach to Newton, $0.05$ in $g_{\mathrm{obs}}/g_{\mathrm{b}}$ at $g_{\mathrm{b}}\approx5a_0$ against the simple rational form) and no fitted function; the MLS16 fitting function is this form with $g_\dagger$ for $a_0$, a statement about the data’s selection, not a check (T17, 2026-09-12). The crossover is instantiated to the chronon (37-sim); the floor-side saturation is gated pending the registration–transport coupling. | T | refs: — |
| L2p00085 | Metrology of the import: the datum $\dS$ is read on two faces by four instrument classes, concordantly (14-entr). Evidence is priced on two registers: metrology, under the circularity criterion (a confrontation measures $\dS$ only if channel-disjoint from the fitting channel; channel and register conflation one error class), and explanation, by zero-freedom derivation. The numeral is $\Om$-hard, a value in no embedded reader’s window; this is its access. The relations behind it ($\Om=4\dS+1$, $A=4\dS$, $\dS=A/4$) are $\Om$-blind and certified by every reader: the [chart] reading $\dS\sim10^{122}$ is the value’s chart, never the certified content. | T | 14:estimate_S.py [mixed] refs: — |
| L3p00086 | The octant lemma and its outputs: the record depth is bounded by the octant, $\dS/2$ chronons $=13.79$ Gyr ($(\pi/4)r_H/c$, chart); the octant sector $\Z_8$ exists iff $\dS$ even (C13). Zero-freedom: $\Omega_\Lambda=\tanh^2(3\pi/8)=0.684$ (chart; $0.19\sigma$), $H_0=67.4\pm0.7$ ($4.6\sigma$ adverse disclosed), $w=-1$ rigid — a confirmed drift falsifies $\Lambda\sim1/\Om$. Falsifiers: octant-capped ages, the running floor $a_0(z)\propto H(z)$ (Z7), the ladder bias. | T | 14:estimate_S.py [mixed] |
| L4p00087 | The registrable triangle: the mass–radius chart between the Compton and Schwarzschild walls, closed by the Planck apex and the totality wall $r_H$ — the totality on its own horizon, the black-hole reading dissolved (A2). Width $61.0$ dex, the count face drawn. Wall intersections are consistencies: the atomic core and the over-closure bound ${\sim}1.8\times10^{8}M_\odot$. A consistency chart, not a channel (L2). | T | 14:make-wedge-2.py [mixed] |
| L5p00130 | The area law: the projected horizon area in Planck cells is the full phase cycle, $A=4\dS$ (chart: $4\pi n_A=4\dS$, $n_A=(r_H/\lP)^2$); the Bekenstein–Hawking quarter $\dS=A/4$ is the quarter-turn $Q_4$, the generative four of $\Om=4\dS+1$ and $\p=4\kap+1$; area is a Subject-projection property, the Carrier a torsor carrying none. Not a further channel (L2). | T | 14:estimate_S.py [mixed] refs: L2 |
| L6p00131 | The interior completions weighed: black-hole cosmology keeps the black-hole-line identity by consuming a parent exterior and a singularity-replacing mechanism (torsion; exclusion pressure); the finite reading consumes neither — no outside (A3), no singularity over a finite totality. The octant cap (L3) separates the two completions empirically: an interior bounce carries time on both sides. | T | no witness |
| M. Superconductivity | |||
| M1p00088 | Superconductivity (structure): superconductivity is split-torus confinement, the electric/phase dual of colour (I1), the charge-$2$ condensate a phase-singlet reducing $\mathrm{U}(1)$ to a residual $\Z_2$ (flux $h/2e$) and the Abrikosov vortex the dual colour string whose line tension vanishes at the Kramers–Wannier self-dual point $\beta_*=\tfrac12\ln(1+\sqrt2)$ that is $T_c$. | T | no witness refs: I1 |
| M2p00089 | Superconductivity (duality): the electric–magnetic duality is the $Q_4$ quarter-turn on the charge-flux lattice (charge$\leftrightarrow$flux, Josephson$\leftrightarrow$phase-slip), predicting an Airy vortex-core spectrum and a quantum current standard whose metrology triangle closes on $q_c^2=4$. | T | no witness refs: — |
| M3p00090 | Superconductivity (scale): the absolute transition scale ($T_c\propto\omega_D$, the gap and vortex tension in physical units) is $\Om$-hard, the family of $\Lambda_{\mathrm{QCD}}$ (I3) and $v$ (H3). Value tagged: the absolute $T_c$ (with $\omega_D$), in no embedded reader’s window — the failing computation is the physical unit of the shell-exact ratios. Relations, $\Om$-blind: the ratios and the Airy law of M2. | Ω | no witness |
| N. Synthesis: exclusions and parameter-freeness | |||
| N1p00091 | Exclusion predictions: no fourth generation, no fifth force, an empty desert, no TeV supersymmetry, no new dark-matter particle, the proton effectively stable ($\tau_p\gtrsim10^{45}$ yr, the $X,Y$ leptoquarks at the substrate scale; the minimal colour-neutral wrap with no channel toward a smaller residue, I6), and no beyond-SM $(g{-}2)_\mu$ — entailed by the closed ladder and the Carrier being the whole substrate. | T | no witness refs: I6 |
| N2p00092 | Parameter-freeness: every order-one constant reduces to two geometric numbers — the solid angle $4\pi$ ($G$, $\alpha_{\mathrm{bare}}$, the $\tfrac14$ law) and the cycle $2\pi$ ($a_0$, horizon temperatures) — both [chart] constants, chart constant and full cycle in the Subject’s labels (C1). In the Carrier register the corresponding values are fixed by the calibration congruence $4\dS\equiv-1$ alone ($G=2\dS$ by $2G\equiv-1$), so the order-one sector rests on the calibration; the chart relations ($G=1/4\pi\varkappa$, $G=\hbar c/m_P^2$) are legitimate in the chart, with $\dS$ physical and $\pi$ the real transcendental, and no exact-arithmetic formula equates a chart constant to a residue. | T | no witness refs: C1 |
| N3p00093 | Model existence: the edifice is realised end-to-end on one concrete instantiated Carrier ($\Om=2{,}408{,}561$), every master row co-instantiated and verdicted at its snapshot; rows added since project onto the shell by the same protocol — a constructive consistency witness, not a proof of the physical case but evidence the construction is non-vacuous at every qualifying scale. | T | 37:w4_projection.pyrefs: — |
| T. Tasks: the development roadmap (stated surfaces, proofs or executions outstanding) | |||
| T1p00094 | The induced-relaxation theorem, chart-free form: derive the registered adjacency chart and its homogeneity from the one Carrier action. Chart-relative form proved (21-gravity, round-02 push); pillars: relative-cycle defect, compression identity, pair-tally current. | O | no witness refs: — |
| T2p00095 | The tilt’s remaining identification $K=\ln\Om$: the scale-path mechanism probe-verified ($K^{2}\lambda_1\to\pi^{2}$, $99.1\%$ at $K=281$), $\theta_*=\pi/2$ structural; the censoring-linearity/identification condition outstanding. | O | refs: — |
| T3p00096 | The fold transition at physical scale: the radial quotient, the branch identification, and the physical-scale fold-return criterion (dev-scale form proved: the period-averaged exterior return is the uniform floor operator). | O | refs: — |
| T4p00097 | The comparable-mass 2PN reduction, $\nu$-sector: the matched-invariant two-body treatment (1PN EIH sector closed; a $\delta$-Lagrangian shortcut failed its test-particle gate and was discarded). Prediction P6’s $\nu$-dependence hangs on it. | O | refs: — |
| T5p00098 | The physical $\ell=2$ quasinormal spectrum and the $O(J^{2})$ rotational multipoles (Hartle–Thorne programme); the quoted ringdown is eikonal. | O | no witness refs: — |
| T6p00099 | The thermodynamic identifications: capacity $\to k_B\ln W$ and a generalised first law for the floor temperature; the capacity/content split and sector-label conservation are in place. | O | no witness refs: — |
| T7p00100 | The constants-sector walls of $e$: universal existence of the terminal shell residue $-K(\p)^{-1}$ of the subfactorial chain ($K(\p)=\sum_{k<\p}k!$, Kurepa’s hypothesis; literature-verified $\p<2^{40}$, re-verified in subfactorial form $\p<2.5\times10^{5}$); equidistribution of the subfactorial residue line (the blind-set Poisson law); joint equidistribution of the quarter-turn and wall-residue indices (informational disjointness of the angular and radial carriers of $e$, null-experiment supported). | O | refs: — |
| T8p00101 | The constants-sector walls of $\pi$: infinitude of the $\pi$-Wieferich set $4^{\p-1}\equiv1+\p\ (\mathrm{mod}\ \p^{2})$ (exactly $\{5,\,45827\}$ below $10^{6}$); the third-order Bernoulli law $\sigma_\p/\p^{2}\equiv(-1)^{(\p+1)/2}B_{\p-3}/36\ (\mathrm{mod}\ \p)$, the mod-$\p^{3}$ reading of Sun’s Conjecture 5.1 (open), verified $5\le\p<300$. The supercongruence $\sigma_\p\equiv0\ (\mathrm{mod}\ \p^{2})$ is Sun’s theorem (JNT 131 (2011)), imported. | O | refs: — |
| T9p00102 | The $e$–$\pi$ dichotomy as theorem: decide whether $\pi$ admits a factorial-rate integer-ratio chain (nonexistence makes the feasibility split exact); the radian-calibration rate $\delta(X)=\min_{\p\le X}|\theta-1|$; joint equidistribution of the frame invariants of the two constants, $\bigl(K(\p)/\p,\ q_\p(4)/\p,\ a/\sqrt{\p}\bigr)$. | O | no witness refs: — |
| Y. Conjectures (stated ahead of derivation; promotion re-homes a row in its thematic block) | |||
| Y2p00104 | Apsidal precession, the gravitational face: the mechanism is theorem (C15–C18); the dictionary $p_{\mathrm{sl}}=3(\dS /\kap )\,r_{g}$ is now derived (38:C8): the pair’s two leak faces $\kap /\dS$, $\kap /2\dS$ (ratio $2$, the double cover, C16) match apsidal fraction and clock deficit iff $2\gamma -\beta =1$ (E4); at $\kap =3$, $p_{\mathrm{sl}}=\dS\,r_{g}$ (D3). Residual: the empirical P1 bank (21-grav); internal extension C21. | O | |
| Y3p00105 | Observation is the Hopf section: the Subject observes the past light cone, retarded one chronon per space quantum; the cone chart is the affine group, $\p (\p -1)$ cells, the Borel of $\mathrm{PGL}_2$ and a global section of its Hopf fibration (C19); the boosts $C_{\p +1}$ change the observer; the horizon is the origin’s antipode $\p /2$, the continuum $\infty$. | O | 1-alg, 98-lab 38:verify-render.jsrefs: C19 |
| Y4p00106 | The shell is holographic: by C19 every frame of the observable variety is one shell-trace event times one projective cell direction, $156\times 14$, nothing left over. Conjectured reading: the variety is the combinatorial $S^3$ (C6), the quotient base its finite $2$-sphere, the $C_{\p +1}$ cosets its Hopf circles, the shell its unique boundary chart. | O | 98-lab no witness |
| Y5p00107 | Energy is winding, mass is dilation: energy is registered phase advance ($E=hf$, D2); mass is a frame’s dilation rate in its harness, on the norm-one torus $C_{\p +1}$ alone (C20, the Dirac mass channel); rest energy its shadow; GR time dilation, literal. The mechanism is theorem (C20); the registration dictionary is derived (Y2, 38:C8); residual: the empirical P1 bank (C21). | O | 00:check_y5.js |
| Y6p00108 | Capacity is additive on the lock (conjecture): a phase-locked, typical-density composite has $\kap_O=\sum\kap_i=3N$ in hydrogen units — the minimal window registering total winding exactly (38:A8’s wrap clause); the 14-entr typical-object diagonal is the validity locus; off-line the law breaks with wall displacement (proton: $1$ against $3$, sign passed); exchange rate open; the $\Om^{1/4}$ cap crosses the diagonal near $10^{3}$ kg. | O | refs: — |
| Y7p00053 | The spatial universe is the $3$-sphere $S^3$: the observer’s frame space is a torsor under $\mathrm{SL}_2(\F_\Om)$, closed and simply connected, whose cycle-closure selects the compact unit-quaternion form; uniqueness by the imported Poincaré–Perelman theorem, the de Sitter spatial section fixing $\Om\sim10^{122}$. Fenced in 20-rh (§7.6, 2026-09-12): recorded to mark the direction, contributing nothing to the paper’s theorem, classification or realisation; the choice of the compact real form by cycle closure is the step owed a derivation. Re-homed from E2 (T) on the T12 revision. | O | no witness refs: — |
| Z. The horizon (the limits of the bounded observer) | |||
| Z1p00109 | Gödel is vacuous over the finite $\Om$ (complete decidable theory, no Robinson $Q$); incompleteness migrates to the comprehension horizon, the horizon clause — a decomposition, not a wholesale transfer: counting face theorems, complexity face in Z3, quotational face unwritable within any embedded budget (Z9: the threshold $t(\M)\approx1.6\times10^{125}$ symbols at $\Om\approx4\times10^{122}$ exceeds the totality’s registrable-state count, so the sub-threshold definability question is open only above every budget). | T | no witness |
| Z2p00110 | RH, Goldbach, and P-vs-NP are the spectral, additive, and computational faces of one horizon clause: each is a uniform check over the totality whose per-instance statements are $\Om$-blind relations (for RH the shell theorem D11 and the per-height screen value D12) and whose uniform form is an $\Om$-hard value (Z3, Z4) — in no embedded reader’s window on the physical Carrier, host-decided on a hosted one. “Decided by the totality” is retired (2026-09-12): a value no reader holds is not thereby decided, and an $\Om$-hard value says nothing about whether a bounded-height certificate exists. | T | no witness refs: — |
| Z3p00111 | The bounded-observer certificate for each is $\Om$-hard (Z4). Value tagged: the completed uniform check over all residues of the totality, in no embedded reader’s window; $\Om$-hardness is the absence of a host, and on a hosted shell the same check is host-decided, exhibited wholesale (37-sim: exhaustive Goldbach; D8); the converse holds on the same shell: within capacity an embedded observer decides totality-level facts from internal state (the two-record keystone, $+0.06\%$); the horizon a wall with two operational sides. Faces: 25-göd, 29-fin. | Ω | no witness |
| Z4p00112 | The three certificates: RH $=$ the on-line spectrum complete for the prime vector; Goldbach $=$ the uniform per-$n$ parity positivity; P-vs-NP $=$ the forward-invertibility of the scale descent. Values tagged: each certificate is a uniform check over the totality — the failing computation is its completion, which consumes every residue — in no embedded reader’s window; the per-instance statements (placement on the line at each shell, parity at each $n$, invertibility at each instance) are $\Om$-blind relations, tagged T where they stand (RH: D11, D12). The tag records where the value can be read, not whether a proof exists: for RH the shell structure does not use the Euler product (D11 holds for the Davenport–Heilbronn vector), so certificate existence is undecided by the framework, and the only route to it — finite Weil positivity — is per-height again (20-rh §8.3). | Ω | no witness refs: — |
| Z5p00113 | The horizon forces structure, not only leaves it open: the $3+2$ split is beyond the horizon (above $\sqrt{\Om}$), so the off-block reframings are gauge and unification is the simple $\SO(10)$ (J4); the automorphism lemma reduces to P2 (A6). | T | no witness |
| Z6p00114 | The sixteen’s absolute masses are $\Om$-hard framed-transcendentals, the matter face of the horizon clause (Z2–Z4). Value tagged: the absolute masses (J3), in no embedded reader’s window; the relations among them (K2–K4) are $\Om$-blind. Classification “elementary $=$ beyond-horizon” is D8; the $\mathbf{16}\to$spectrum projection is K4; the gauge sector’s drive-invariant residue (the photon, residue $2$) is the above-horizon, observable complement (D8, G1). | Ω | no witness |
| Z7p00115 | The dark sector’s cross-scale running is the cosmological face of the horizon clause. Value tagged: the running’s absolute normalisation — $a_0$ at any one epoch in physical units, equivalently the absolute scale of $H$ — in no embedded reader’s window, its uniform certificate the factorisation of $\Om-1$: one carrier-scale object with the number-theory instances (Z2–Z4), the masses (Z6), and the EM coupling (Z8). The shape $a_0(z)\propto H(z)$ is not $\Om$-hard: it is a relation, staked as a prediction, and lives in L7. | Ω | |
| Z8p00116 | The fine-structure constant is the electromagnetic face of the scale operator: $\alpha^{-1}(0)=4\pi+\sum_f\tfrac{b_f}{2\pi}\ln(M_P/m_f)+\Delta\alpha_{\mathrm{had}}$, $\Om$-hard — the bare $\alpha_{\mathrm{bare}}=1/4\pi$ framed-exact (channel unity $g{=}1$, the $\mathrm{U}(1)$ critical coupling $\beta_c$; the $4\pi$ Gauss solid angle), the digit a carrier-scale log over the $\Om$-hard masses (Z6) with cutoff $\Lambda_{\mathrm{sub}}{=}M_P$ ($a{\sim}\ell_P$) tied to $\Om$ (A9, $\ln M_P{=}\tfrac12\ln\Om$). Value tagged: the physical digit $\alpha^{-1}(0)$, in no embedded reader’s window — the failing computation is the log sum over the $\Om$-hard masses; the bare $1/4\pi$ and the form of the sum are $\Om$-blind. Of the family Z2/Z6/Z7/H3/I3, the factorisation of $\Om-1$. | Ω | 28:alpha_probe.py [approx] |
| Z9p00117 | No compression: quotational self-reference is unwritable over the finite totality (mention cost; per-symbol at every scale, arbitrary codings beyond a structure constant) — the classical diagonal consumes exponential quotation-compression, infinity acting as the no-wrap idealisation; at the physical scale the compression threshold $t(\M)\approx1.6\times10^{125}$ symbols lies above the totality’s registrable-state count $\sim10^{122}$, so no embedded reader reaches either side of the sub-threshold window (25-göd Rem. window); correspondingly, bounded-fragment truth is definable at polynomial overhead (the Tarski reversal). | T | no witness refs: — |
Nothing matches that filter.
Tags
- G
- ground — the single given that conditions the discourse
- P
- pillar — a forced foundational proposition
- T
- theorem — derived within the framework
- D
- definition — a naming or set-up move, no empirical content
- R
- realisation — what a physical term denotes in the substrate; forced (A10), each row stating the falsifier that kills it. Replaces the retired tag `B` (bridge), 2026-09-11
- I
- import — a standard result or measured datum, used without reproof
- Ω
- Ω-hard — a value in no embedded reader's window, not closeable by a bounded observer; host-decided on any laboratory Carrier; never applied to a relation (notation convention 6, 2026-09-11)
- O
- open — genuinely unresolved