The Complete Ledger Derivation from Six-Site Closure
This page presents the full six-site ledger derivation and live audit for the included baryon set: all well-established confirmed ground states and their lowest-line spin-\(3/2\) counterparts.
From charge and flavour alone, the ledger fixes an arc structure. In the measurement basis used for the audit, that structure fixes an admissible mass band. The table is generated first from the construction, and the audit is then read off against it. This is a containment test under one rigid geometric construction. It is not a per-state fit.
The same construction also recovers the flavour symmetry structure geometrically from the arcs themselves, rather than taking Lie algebra as the starting input.
The scope is pre-dynamical. Higher excitations lie outside the present audit. Within the included class, any confirmed baryon outside its assigned band falsifies the construction.
The companion tools on this site allow the same construction to be tested interactively, including flavour combinations not audited on this page.
For the full chain of axioms, lemmas, propositions, theorems, proof sets, and rigidity-audit statements, see Volume II: Pre-Dynamical Baryonic Structure together with Volume I.
Scope and audit principle
This page gives the reader-facing derivation spine for the exploration where identity is not assumed primitive. It carries the closure-to-ledger route in order and references the proof-bearing loci rather than restating the full proof chain on the page.
Status classes
Each item on this page carries exactly one audit status. The class label is meant to stop ambiguity, not to narrate the same point three times.
| Status | Role | Constraint on use |
|---|---|---|
| forced consequence | Uniquely fixed by the closure scaffold, admissibility wall, and coefficient refusal. | Belongs to the constructive chain. |
| gauge representative | Presentation representative of an equivalent class, such as one directed ordering up to relabelling and reversal. | All downstream statements must be invariant under the gauge move. |
| encoding normalisation | Coordinate representative used to print an already fixed structure. | Introduces no structural degree of freedom. |
| reporting seal | Unit declaration used only to report dimensionless geometry in physical units. | Barred from admissibility, certification, endpoints, tiering, or charge coding. |
| implementation constant | Numerical reproducibility constant such as scan resolution, solver threshold, or plotting step. | Barred from definitions, tier fixing, carried boundaries, or certification. |
Rigidity ledger for this page
The ledger below is compact on purpose. Dependency text is reduced to source loci so the derivation itself can carry the weight.
| Item | Status | Dependency locus |
|---|---|---|
| Directed triad carries a circuit order | forced consequence | Volume I diagonal discipline; Volume II §17.1. |
| Representative circuit order \(A\to B\to C\to A\) | gauge representative | Remark 2, §17.1. |
| Inherited witness sites and pairing law | forced consequence | Lemma 7, §17.2. |
| Six-site register \(\mathsf S_6\) and \(|\mathsf S_6|=6\) | forced consequence | Lemma 8, §17.2. |
| Return certificate and step index \(Q\in \mathbb Z/6\mathbb Z\) | forced consequence | Definition 1 and Corollary 1, §17.3. |
| Period label \(2\pi\) and sixth-root chart \(\zeta_Q=e^{iQ\pi/3}\) | encoding normalisation | Remarks 3-4, §17.4. |
| Global share \(\varepsilon_{\min}=1/6\) | forced consequence | Six-step register share; carried in §17.4 and the minimal deficit readout. |
| Exactly two deviation channels \((a_{\mathrm{dev}},r_{\mathrm{dev}})\) | forced consequence | Lemma 9, §17.5. |
| Quadratic curvature \(\kappa=a_{\mathrm{dev}}^2+r_{\mathrm{dev}}^2\) | forced consequence | Proposition 1, §17.5. |
| Coherence \(C(\kappa)=e^{-\kappa}\) | forced consequence | Proposition 2, §17.5. |
| Euclidean crossover \(\lambda_0=1/\sqrt2\) and lawful span \([\sqrt2-1,1]\) | forced consequence | Lemma 1 and §17.8. |
| Withheld-step complement \(\Phi=5\pi/3\), directed index \(\widehat{\mathrm{IES}}=\Phi^2\), reciprocal baseline \(\kappa_{\mathrm{base}}=1/\widehat{\mathrm{IES}}\) | forced consequence | §17.9. |
| Seven-arc midpoint palette | forced consequence | §17.10. |
| \(L^{\star}=1\,\mathrm{fm}\), \(U_0=1\,\mathrm{MeV\,fm}\) | reporting seal | Declared unit seals only. |
| Scan resolution, plotting step, solver threshold | implementation constant | Reproducibility only; outside certification. |
Six-site register from witnessed closure
The starting object is the triad \(\{A,B,C\}\) under diagonal exclusion. Directed witnessing admits a circuit order on the directed sites, unique up to relabelling and reversal. Inherited witnessing then pairs each directed stance with its disjoint re-witnessing site. This fixes the six-site census.
eq:directed-circuiteq:pairing-laweq:six-site-registerReturn, induced order, and the six-step index
Return is certified at the register level. Once a six-site traversal exists that visits each site exactly once while respecting the directed order and pairing law, that certificate induces a six-step circuit order on traversal positions. The step index is then fixed modulo completion.
RetRegQEncoding conventions for the six-step index
The period label \(2\pi\) and the sixth-root chart do not add structure. They print an already fixed six-step completion class in familiar coordinates.
theta_stepzetaeps_minThe six-step partition also fixes the minimal share. This belongs in the main spine, not only in the cross-check.
Two channels, quadratic curvature, and coherence
The readout carries exactly two independent anchors: the completion anchor for angular readout and the centre anchor for radial readout. Therefore departure admits exactly two independent channels. Under relabelling invariance, sign invariance, additivity across channels, coefficient refusal, and minimal even order at the shared centre, the register-wide departure magnitude is forced to be quadratic. Coherence is then fixed by multiplicative composition over additive departure.
adev-rdevkappaCGeometric readout, Euclidean crossover, and lawful span
Closure forbids reversal. The admissible displacement range therefore lies within the non-obtuse guard. At the right-angle boundary the midpoint of orthogonal unit directions fixes the Euclidean crossover. With outer endpoint normalisation \(\lambda_{\max}=1\), the lawful span follows by centring around \(\lambda_0\). Radial and angular departures are then written directly about these anchors.
walllambda0spandevDirected remainder, directed index, and reciprocal baseline
The six-step closure leaves a withheld-step complement. Its square is the directed index, and its reciprocal fixes the baseline curvature scale used later in the midpoint palette.
PhiIEShatkappa_baseSeven-arc midpoint palette
The midpoint palette comes after the reciprocal baseline, not before it. The ordered list is a curvature-side readout of the fixed six-step partition about the crossover. It is not an independently chosen station list.
palettefpalette-kappaLedger assembly and energy expressions
Only now does the page move from the dimensionless constructive chain into the ledger assembly. The reporting seals appear here because physical units are being printed here. They are seals, not inputs to the closure object.
Reporting seals
sealsDivergence about the crossover
Delta_divProjector gate
alphaInherited anchor
20pi2ellM0EDirected, inherited, and total energy
EdirEinhEtotrigidity-balanceSo reading this as literally as we can, the construction is fixed by the geometry shown on the page. The numbers come from that geometry. They are not put in by tuning. No constant here depends on primitive identity, there are no turnable coefficients, and there is no re-parameterisation freedom used to push the ledger into place.
Cross-check
The cross-check sits at the end, where it belongs. It reproduces the same palette and ledger expressions from the already-fixed anchors using direct trigonometry and algebra. It is a recovery block, not an alternative derivation order.
cross-basecross-recoveryLive browser audit and baryon containment
This final section turns the ledger into a live calculation in the browser. The numbers on this page are produced from the embedded constants, arc rules, admissibility windows, and catalogue rows. Taken together, the derivation above and the audit below show the whole machine on this page, from the geometric ledger to the full live audit.
What this section does. It shows the full geometry set used by the ledger, makes the arc and seat rules explicit, and runs the audit live in the browser. The tables then report the certified bands, the lower and upper lane energies, the balance relation, and the reverse recovery from the embedded baryon catalogue.
No row is filled by hand. The values are generated by the script on this page from the declared constants, the admissibility windows, the charge law, the flavour-to-seat rules, the spin-lift rule where it applies, and the embedded catalogue data.
Audit outputs. The live tool reports certified bands by charge tier, reverse recovery from \((Q,m^\*)\) to \(\lambda^\*\) and seat \(\langle L,U\rangle\), and a table showing whether each embedded baryon sits inside its certified band.
CSV and LaTeX export remain available from the page controls.
Arc and seat rules used by the live audit
These are the operative read-out rules used in both the forward and reverse constructions on this page. Within the audited scope, the seat read-out is deterministic from charge, flavour content, and the declared spin-lift rule. The spin-lift acts only in the light and strange upper-arc read-out and stops there.
- R-AS-01 Admissible λ range by \(|Q|\): scan \(\lambda\) only inside the certified interval for the charge magnitude.
- R-AS-02 Curvature read-out: for the solved state \((\lambda,\phi)\), compute \(\kappa=-\ln C\).
- R-AS-03 Arc selection by nearest midpoint: \(\mathrm{arc}(\kappa)=\arg\min_i |\kappa-\kappa_{\mathrm{mid}}(i)|\).
- R-AS-04 Seat prediction: compute the dual-arc seat \(\langle L,U angle\) implied by \(\lambda\) under the certified construction, including the implemented exclusions.
- R-AS-05 Reverse recovery objective: take \(\lambda^\*\) as the minimiser of \(|E_{\mathrm{tot}}(\lambda,Q)-m^\*|\) over the admissible interval.
| Arc | Midpoint label | Recovered load \(f_i\) | Selection midpoint \(\kappa_{\mathrm{mid}}(i)=\sqrt{f_i}/\widehat{\mathrm{IES}}\) |
|---|---|---|---|
| 1 | \(0\) | \(1\) | \(1/\widehat{\mathrm{IES}}\) |
| 2 | \(\pi/2\) | \(2\pi-4\) | \(\sqrt{2\pi-4}/\widehat{\mathrm{IES}}\) |
| 3 | \(2\) | \(\pi\) | \(\sqrt{\pi}/\widehat{\mathrm{IES}}\) |
| 4 | \(3\) | \(4\pi/3\) | \(\sqrt{4\pi/3}/\widehat{\mathrm{IES}}\) |
| 5 | \(4\) | \(3\pi/2\) | \(\sqrt{3\pi/2}/\widehat{\mathrm{IES}}\) |
| 6 | \(\infty\) | \(2\pi\) | \(\sqrt{2\pi}/\widehat{\mathrm{IES}}\) |
| 7 | \(-e\) | \(2\pi(1+e^{-1})\) | \(\sqrt{2\pi(1+e^{-1})}/\widehat{\mathrm{IES}}\) |
Arc admissibility is then applied after selection: \(|Q|=0\Rightarrow\{1,\dots,7\}\), \(|Q|=1\Rightarrow\{2,\dots,7\}\), and \(|Q|=2\Rightarrow\{5,6,7\}\).
Bands
Reverse construction: PDG \((Q,m^\*) \rightarrow \lambda^\* \rightarrow \langle L,U\rangle\)
Input is \((Q,m^\*)\). The solver finds \(\lambda^\*\) in the admissible coherence intervals for \(|Q|\), predicts the seat \(\langle L,U\rangle\), and reads back the compatible sector tag from that predicted seat.
Catalogue containment
This table compares the embedded baryon catalogue against the certified bands and reports midpoint deviation, containment status, recovered lower and upper lane values, and the balance check row by row.
Final inventory
This table is the final audit summary after the full derivation has already been shown. It is where the browser-side calculation checks the ledger against the embedded baryon catalogue.
| Component | Symbol | Status | First fixing locus on this page |
|---|---|---|---|
| Six-site register | \(\mathsf S_6\) | forced consequence | Six-site register |
| Return certificate and step index | \(Q\in\mathbb Z/6\mathbb Z\) | forced consequence | Return and step index |
| Period label and sixth-root chart | \(2\pi\), \(\zeta_Q=e^{iQ\pi/3}\) | encoding normalisation | Encoding conventions |
| Minimal global share | \(\varepsilon_{\min}=1/6\) | forced consequence | Encoding conventions |
| Deviation channels | \(a_{\mathrm{dev}},r_{\mathrm{dev}}\) | forced consequence | Two channels |
| Quadratic curvature | \(\kappa=a_{\mathrm{dev}}^2+r_{\mathrm{dev}}^2\) | forced consequence | Two channels |
| Coherence | \(C=e^{-\kappa}\) | forced consequence | Two channels |
| Euclidean crossover and lawful span | \(\lambda_0=1/\sqrt2\), \([\sqrt2-1,1]\) | forced consequence | Geometric readout |
| Directed remainder and directed index | \(\Phi=5\pi/3\), \(\widehat{\mathrm{IES}}=25\pi^2/9\) | forced consequence | Reciprocal baseline |
| Seven-arc midpoint palette | \(\{0,\pi/2,2,3,4,\infty,-e\}\) | forced consequence | Seven-arc palette |
| Reporting seals | \(L^{\star}=1\,\mathrm{fm}\), \(U_0=1\,\mathrm{MeV\,fm}\) | reporting seal | Ledger assembly |
| Inherited anchor | \(M_0^{(E)}=400\pi^5 U_0/\ell_0\) | forced consequence | Ledger assembly |
| Directed, inherited, total energy | \(E_{\mathrm{dir}},E_{\mathrm{inh}},E_{\mathrm{tot}}\) | forced consequence | Ledger assembly |
This version trims the repeated inventory rhetoric, restores the derivation order, carries the five-class audit distinction, and references the manuscript loci instead of trying to re-house the full proof system inside the webpage.