Gap-10/BG-10 — Baryogenesis eta_B (leptogenesis four-fix): the gate anchor ledger

The honest one-line: Gap-10/BG-10 is an openly-unsolved problem — nobody has a parameter-free leptogenesis $\eta_B$ — but there is real, checkable work here: it proves by exact arithmetic why the frozen geometry as written sources exactly zero matter–antimatter asymmetry, localizes the residue to one high-scale CP source object, dismantles a circulated "clean result" as true-by-construction, and pre-registers a falsifier before any number is computed.

This page is the gate-specific instantiation of the anchoring method: it takes the master anchor and the four bridges and applies them, object by object, to one gate. Every exact thing Gap-10/BG-10 touches gets its own row — its status, what it is allowed to claim, and what it is forbidden to claim. It follows the same eleven-part shape as the canonical SG-4 ledger.

Taxonomy reconciliation (2026-07-05). This gate reaches its board-canonical terminal — CERTIFIED-IRREDUCIBLE · RESOLVED +0 (ratified 2026-07-08; closure-of-record: the live gate dossier) — resting on the measured-anchor axis — the baryon asymmetry η_B is a measured cosmological boundary-record, accommodated not derived — with the mechanism legs (CP asymmetry ϵ, right-handed mass M_R, absolute neutrino Yukawa) shown as open research residuals. Read as terminal reached + residuals shown; counted among the resolved-with-residual gates, not physics-closed. The “OPEN / Diagnostic” wording below is the older least-closed-residual grading, retained as the honest residual ledger.

The mechanism ledger of this gate is held OPEN / Diagnostic — 0 of 4 at certificate grade. No $\eta_B$, no CP asymmetry $\varepsilon$, no right-handed mass $M_R$, and no absolute neutrino Yukawa is asserted. The established results locate and bound the open object; they do not close it.

1. Gate status header

• Gate-level Gap-10/BG-10 roll-up: CERTIFIED-IRREDUCIBLE · RESOLVED +0 (board-canonical, ratified 2026-07-08). Mechanism ledger: OPEN / Diagnostic — 0 of 4 at certificate grade. A pre-registered falsifier is armed; the structural firewall is PROVED; one CP-source axiom is unfilled.
• The closed local/partial leg — the bare-geometry zero-CP result. What is rigorous is a negative, exact statement: the frozen geometry's only named direct holomorphic factor at the order-three fixed point is real, so the CP source vanishes identically: $$O_{\rm BG10,bare}(E_{\rm frozen}):\qquad q_1(\omega)=-\kappa\in\mathbb{R}\ \Rightarrow\ \mathrm{Im}\big(H_{12}^2\big)=0\ \Rightarrow\ \varepsilon_1\propto \mathrm{Im}\big(H_{12}^2\big)=0\ \Rightarrow\ \eta_B=0 .$$ This is the cleanest fully-shown fact of the gate: the bare geometry, as frozen, sources exactly zero asymmetry. It is a proven obstruction, not a value of $\eta_B$.
• Status, split so it cannot be misread:
  • The bare-geometry zero-CP leg ($q_1(\omega)=-\kappa$ real $\Rightarrow \varepsilon_1=0$): DERIVED-GIVEN-E — exact complex arithmetic on the frozen factor. It is a closed negative fact, not axiom-open.
  • The seesaw + 6-vs-3 phase firewall (why low-energy data cannot fix the high-scale source): DERIVED-GIVEN-E / CERTIFICATE — theorem-level, both legs PASS. An honesty disclosure, not a closeable piece of $\eta_B$.
  • The high-scale CP source $I_{CP}$ / the $\sigma_\nu$ sign bit: AXIOM-OPEN / RECIPE-ABSENT — recipe-absent; the geometry's default sign points the wrong way.
  • The four coupled fixes (CC-1…CC-4) and the assembly (CC-5): OPEN / computation debt — 0 of 4. Mechanized to refuse early/partial closure.
  • The measured anchor $\eta_B\approx 6.1\times10^{-10}$: MEASURED-ANCHOR — the floor ($\ge 1$ genuine measured invariant), an input to test against, never an output.

Given $E_{\rm frozen}$, the bare CP source is exactly zero by closed-form arithmetic; the firewall theorems explain why the missing magnitudes cannot be back-filled from low-energy data. What stays open is everything that decides a number: the high-scale CP phase, the heavy Majorana spectrum, the absolute Yukawa norm, and the flavored kinetics.

selection ≠ derivation · given-E ≠ derivation-of-E · frozen/reproducible ≠ proven-unique.

2. Frozen inputs (what Gap-10/BG-10 stands on, not what it produces)

• Frozen branch hashes dcc66f1b2685 (geometry anchor) / a5b1e6f9d951 (13D boundary package); the $O_\nu$ texture hash 495ddbdcedb9. The branch is read-only. These are audit anchors — they certify which object was tested and that it cannot be quietly retuned. None validates the physics and none is an $\eta_B$ lever.
• The frozen branch object $\mathfrak{B}_{\rm active}=[M_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2]\oplus\dots$ with $K_6=SU(3)/T^2$ (flag manifold), $D=13$, $\mathbb{Z}_6$ center, spin-$\mathbb{C}$ index $\chi(K_6,E)=-3$ (three generations), $\kappa\equiv e^{-\pi\sqrt3}>0$, $M_U\sim 1.0\times10^{16}$ GeV. Given / charged / inherited — it enters as input. BG-10 does not derive it.
• The baryon-to-photon ratio is a measured fact — $\eta_B\equiv n_B/n_\gamma\approx 6.1\times10^{-10}$ (CMB + BBN). It is observation-locked, the floor anchor, not a BG-10 output. Every "the source is inert" below is a statement about this frozen branch, never a derivation of $\eta_B$.

3. Object anchors (given-E / upstream)

The geometry is load-bearing for the texture/structure of the inputs, not for the magnitudes that decide $\eta_B$. The recipe-real texture is pinned; the magnitudes are not.

• The relative $O_\nu$ texture (recipe-real, 16 sig figs, hash 495ddbdcedb9): $$(O_\nu)^{aa}=\mathrm{diag}\big(4.333420509983131\times10^{-3},\ 6.582872101129666\times10^{-2},\ 1.0\big).$$ The first entry is reproduced by $\kappa=e^{-\pi\sqrt3}$ to $<10^{-15}$. Status: DERIVED-GIVEN-E (texture only).
• The seesaw mass map $M_\nu^{\rm eff}=-M_D\,M_R^{-1}M_D^{\mathsf T}$, with $M_D=N_\nu\langle g_a|O_\nu|g_b\rangle$. Status: GIVEN-E / upstream — fixes only the combination, never $M_D$ and $M_R$ separately.
• The pinned light-sector numerals (NuFIT-style, verified): $\Delta m^2_{21}=7.39\times10^{-5}\,\mathrm{eV}^2$, $|\Delta m^2_{31}|=2.515\times10^{-3}\,\mathrm{eV}^2$, $\delta_{CP}^\ell\approx 260.2^\circ$ (LOW-energy), $v=246.02$ GeV. Status: GIVEN-E / upstream — the light side only; every downstream stage also needs a missing high-scale numeral.

4. Root and master-anchor traceability

Deep roots that are load-bearing for Gap-10/BG-10:

Master anchors in play: finite invariant ledgers · no unpaid labels/magnitudes · the frozen branch · given-$E$ · the declared CP-source axiom · the measured floor $\eta_B$ · open-residual discipline.

5. The Gap-10/BG-10 anchor ledger (the universal table)

6. The arithmetic — the bare zero-CP result, in full

This is the cleanest fully-shown fact of the gate. The geometry's only named direct holomorphic factor at the order-three fixed point is $q_1(\omega)=e^{2\pi i\omega}$ with $\omega=e^{2\pi i/3}=-\tfrac12+i\tfrac{\sqrt3}{2}$: $$2\pi i\omega = 2\pi i\Big(-\tfrac12+i\tfrac{\sqrt3}{2}\Big)=-\pi i-\pi\sqrt3,$$ $$q_1(\omega)=e^{-\pi i}\cdot e^{-\pi\sqrt3}=(-1)\cdot e^{-\pi\sqrt3}=-\kappa,\qquad \kappa=e^{-\pi\sqrt3}>0 .$$

So $q_1(\omega)=-\kappa$ is real (numerically $\mathrm{Im}\sim 6\times10^{-18}$, i.e. zero); its phase is $\pi$, not a quarter of $\pi$. If the CP-source entry is proportional to it, $H_{12}\propto q_1(\omega)\in\mathbb{R}$, then $H_{12}^2\in\mathbb{R}$ and $$\mathrm{Im}\big(H_{12}^2\big)=0\ \Longrightarrow\ \varepsilon_1\propto \mathrm{Im}\big(H_{12}^2\big)=0\ \Longrightarrow\ \eta_B=0 .$$ The fixed point $\tau=\omega$ fixes only the magnitude $|H_{12}|\propto\kappa$; it contributes zero CP phase.

The textbook chain — assembled correctly, halted honestly. The intended computation is standard flavored thermal leptogenesis. Each stage is the correct textbook formula and each is guarded — it halts at the first genuinely-missing high-scale numeral: $$\varepsilon_1=\frac{1}{8\pi(Y^\dagger Y)_{11}}\sum_{j\neq1}\mathrm{Im}\big[((Y^\dagger Y)_{1j})^2\big]\,f\!\left(\frac{M_j^2}{M_1^2}\right),\qquad |\varepsilon_1|\lesssim\frac{3}{16\pi}\frac{M_1(m_3-m_1)}{v^2},$$ $$K=\tilde m_1/m_*,\quad \tilde m_1=(Y^\dagger Y)_{11}\,v^2/M_1,\quad Y_B=\frac{28}{79}\,\kappa\,\frac{\varepsilon_1}{g_*^s},\quad g_*^s=106.75,\quad \eta_B=7.04\,Y_B .$$ Every stage is standard and correctly applied; every one halts at a missing high-scale numeral ($\mathrm{Im}[((Y^\dagger Y)_{1j})^2]$, $M_j/M_1$, $(Y^\dagger Y)_{11}$). Without the phases the imaginary part is identically $0\Rightarrow\varepsilon_1=0$; the source is inert. No stage was fabricated through.

Diagnostic — the result is specific, not trivial. The light-sector mass factor is non-zero and reproduces exactly on independent recomputation: $$m_2=\sqrt{\Delta m^2_{21}}=8.596511\times10^{-3}\,\mathrm{eV},\qquad m_3=\sqrt{|\Delta m^2_{31}|}=5.014978\times10^{-2}\,\mathrm{eV}.$$ That the Davidson–Ibarra mass factor $(m_3-m_1)=5.0150\times10^{-2}$ eV is a real, reproducible number while $\varepsilon_1$ is still identically zero is what makes the zero-CP fact a genuine arithmetic obstruction about $E_{\rm frozen}$, not a numerical accident: the magnitudes that decide $\eta_B$ terminate on no derived geometric quantity, even though the light side is fully pinned.

The blind finite-structure (Milgram / Gauss-sum) test. To check whether a quarter-phase $e^{\pm i\pi/4}$ has a structural origin, the Gauss-sum phase of every finite quadratic module the frozen record names is computed, never assuming $\pi/4$ and back-filling: $$\gamma(A,q)=|A|^{-1/2}\sum_{x\in A}e^{2\pi i\,q(x)}=e^{2\pi i\,\sigma/8}\quad(\sigma=\text{signature mod }8).$$

The no-go: the only structure yielding $e^{i\pi/4}$ is $A_1/\mathbb{Z}_2$ — and it is not connected to the active neutrino projectors $P_\pm^\nu$ by any printed map in the frozen record. The projector algebra $P_\pm(\theta)$ is phase-degenerate (idempotent, orthogonal, complete for every real $\theta$), so the algebra cannot pick out $\theta=\pi/4$.

7. The "$\kappa^3/\pi$ manifest" and the CP source, split into honest objects

The single phrase "the geometry derives the leptogenesis CP invariant" hides distinct claims with distinct statuses.

1. The maximal-CP relabel. Carry the construction with a general phase $\theta$ in the CP-source entry, $H_{12}=a\,e^{-i\theta}$ (keystone magnitude $a=4\kappa/\sqrt3$ fixed): $$H_{12}^2=a^2e^{-2i\theta},\quad \mathrm{Im}(H_{12}^2)=-a^2\sin(2\theta)\ \Longrightarrow\ I_{BG}(\theta)=\frac{\kappa^3}{\pi}\sin(2\theta).$$ Then $I_{BG}=\kappa^3/\pi\iff\sin(2\theta)=1\iff\theta=\pi/4$. So $\kappa^3/\pi$ is the maximum of $I_{BG}(\theta)$, attained only at $\theta=\pi/4$: "the chamber selects $\theta=\pi/4$" means "choose maximal CP." Status: RETRACTED / true-by-construction — a selection, not a derivation; never banked.
2. The CP-invariant form $I_{CP}[Y_\nu,M_R]$ (a Jarlskog-type rephasing-invariant functional). Status: AXIOM-OPEN / computation debt — the closed form and invariance proof are writable now, but it cannot be evaluated without the recipe-absent $M_R$ and absolute $Y_\nu$.
3. The high-scale phase / the $\sigma_\nu$ sign bit. The continuum CP phase compresses to one mod-8 spin-c/Pin bit; the default fixed by $\chi=-3\ (\equiv5\bmod8)$ gives $e^{-3i\pi/4}=-e^{i\pi/4}$ — the wrong sign. Status: AXIOM-OPEN / RECIPE-ABSENT.

So the texture is DERIVED-GIVEN-E, the invariant form is writable, the magnitudes and the sign are open, and the $\kappa^3/\pi$ value is true-by-construction. The open target is to derive the phase from a frozen-record-fixed finite Weil/metaplectic structure blindly — never by choosing the structure because it yields $\pi/4$.

8. The structure-first wrong-sign datum (why the residue is real, not fitted)

The most important credibility move is honest, not triumphant. The spin-$\mathbb{C}$ index $-3$ itself produces $e^{-3i\pi/4}=-e^{i\pi/4}$, an eighth-root phase in the same family but with the wrong sign. Eighth-root phases are therefore the natural currency of the frozen record (making $e^{i\pi/4}$ plausible/in-family), yet the default sign points the wrong way for generating the observed asymmetry.

This is the gate's strongest honest asset: the required quarter-phase departs from the target rather than being read off it. A parameter reverse-engineered to hit $\eta_B$ would never come out wrong-signed. The wrong default sign is evidence the residue is a real structural fact, not a fitted knob — and, if the sign is genuinely fixed to $-e^{i\pi/4}$ by the geometry, the bare theory predicts the wrong-sign asymmetry, a falsifying datum honestly reportable. In-family is not derived; plausible is not forced.

9. Open residuals — the four-fix completion family

The bare zero-CP arithmetic and the firewall theorems are the closed negative face of BG-10. These distinct residuals make up the rest, and none is closed by the negative facts. A complete attempt must execute four named, coupled fixes plus an assembly node; the falsifier is defined only on the joint solution.

• Hole #2 / #8 — the high-scale CP phase / $\sigma_\nu$ sign (CC-1 source). AXIOM-OPEN / RECIPE-ABSENT. Reduced to one mod-8 spin-c bit; default sign wrong. The single highest-leverage object.
• Hole #3 — $M_R$ recipe-absence + Yukawa degeneracy. OPEN / RECIPE-ABSENT. The single binding wall: named mechanism, no formula; the only figure is a band $M_R\sim10^9$–$10^{14}$ GeV straddling all three flavor regimes; the candidate $M_R=\kappa\,M_U$ is off by $\sim231\times$; even a derived $M_R$ is degenerate with $Y_\nu$.
• Hole #1 — CC-1 canonical CP invariant $I_{CP}$. AXIOM-OPEN / computation debt. Form + invariance proof writable now; evaluation is downstream of Hole #3.
• Hole #4 — CC-2 RIS-subtracted $\Delta L=2$ washout. OPEN / computation debt. No rate, washout factor, or named subtraction scheme is banked.
• Hole #5 — CC-3 $N_2/N_3$ damping ledger. OPEN / computation debt. Only a DIAGNOSTIC floor exists; it must not be promoted to a result. $M_2/T_{RH}^{\max}\sim2.5$ — Boltzmann-suppressed but not exactly zero.
• Hole #6 — CC-4 flavored density-matrix kinetics. OPEN / R3 BLOCK. Order-one factor decisive against a $3\sigma$ window; flavor-democracy ($P_s=\tfrac13\mathbf{1}\mathbf{1}^{\mathsf T}$) is in tension with the frozen 3 distinct eigenvalues.
• Hole #7 — CC-5 S5 assembly + $3\sigma$ band. OPEN / scaffolding only. The sphaleron + entropy map is not corpus-pinned at certificate grade; may not run until CC-1…CC-4 hold.
• Hole #9 — window-convention arithmetic tension. AUDIT / unresolved. $\eta_B=7.04\,Y_B$ does not map the second band $Y_B=(8.0\pm0.3)\times10^{-11}$ onto the falsifier window; carried verbatim, read into no fix, resolved only at CC-5.

These are separate rows under one top-level family: the four-fix completion. Partial delivery of any subset does NOT move the ledger off 0/4.

10. Anti-claims (what this page refuses to say)

• BG-10 does not derive $\eta_B$, $\varepsilon$, $M_R$, $Y_\nu$, or any high-scale phase. Formally the bare geometry gives $\mathrm{Im}(H_{12}^2)=0\Rightarrow\varepsilon_1=0$; that is a closed negative, not a value of $\eta_B$.
• The $\kappa^3/\pi$ manifest is a relabel, not a derivation. It is the maximum of $I_{BG}(\theta)$ at $\theta=\pi/4$ — true-by-construction; RETRACTED; never banked. Any object whose keystone equals the in-window value is a relabel and must be rejected.
• The measured low-energy $\delta_{CP}^\ell\approx260^\circ$ does not supply the leptogenesis source. It is firewalled OUT by the theorem-level 6-vs-3 phase count; substituting it is a firewall violation.
• The quarter-phase $e^{i\pi/4}$ is not derived: path (i) was tested against every named finite structure ($\mathbb{Z}_2/\mathbb{Z}_3/\mathbb{Z}_6/S_3/A_2$) and FAILED; the sole $e^{i\pi/4}$ source ($A_1/\mathbb{Z}_2$) has no printed map to $P_\pm^\nu$. In-family $\ne$ derived.
• $M_R$ is not derived ($M_R=\kappa\,M_U$ is off by $\sim231\times$); a textbook/factor-few $\eta_B$ does not exist for this theory (BLOCKED-AT-INPUTS).
• The diagnostic $N_2/N_3$ floor is not the result; promoting it is a structural failure.
• The frozen-branch hashes are audit anchors; they do not validate the physics.
• The closed negative leg is not whole-gate closure. $\varepsilon_1=0$ (bare) does not imply a completed $\eta_B$; the mechanism ledger stays OPEN, 0 of 4 — the gate’s board terminal (CERTIFIED-IRREDUCIBLE · RESOLVED +0) rests on the measured-anchor axis, not on a completed mechanism.

11. Specialist closure plan

Each open residual is a concrete, finite, target-blind work-package. Binding discipline: derive blind, then compare, accept falsification; never fit to the window $[6.0,6.2]\times10^{-10}$; the $\kappa^3/\pi$ manifest is the true-by-construction cautionary example — dismantle it, never reproduce it. Frozen branch read-only.

1. High-scale CP phase / $\sigma_\nu$ (Hole #2/#8) — highest leverage. EITHER (path i) supply a frozen-record-fixed finite Weil/metaplectic structure that blindly computes the phase — the sharp checkable conjecture: "the spin-$\mathbb{C}$ lift of the $S^1_Y/\mathbb{Z}_2$ orbifold reflection acts on the neutrino CP two-plane as the $A_1$ Weil representation, yielding $+e^{i\pi/4}$" (pinning away the index's extra minus sign) — OR (path ii) formally declare it a named axiom and downgrade BG-10 to "conditional on a CP-phase axiom." Falsifier: a blind phase $\neq\pi/4$, or $-e^{i\pi/4}$, validates path (ii) and may even disfavor the gate. Same bit moves Gate-5, UQF-4, SG-4 together.
2. $M_R$ recipe (Hole #3) — the single binding wall. Derive an actual $M_R$ formula/spectrum from $(\tau=\omega,\,N=1,\,R_{T^2,\rm Cartan}=1.710231\times10^{-17}\,\mathrm{GeV}^{-1})$ target-blind; resolve the $\sim231\times$ tension or document it; then check whether the seesaw-fixed Yukawa is forced or free. Falsifier: proving no such recipe exists in the frozen record $\Rightarrow$ $M_R$ is an honestly-named added axiom.
3. CC-1 $I_{CP}$ (Hole #1). Write $I_{CP}$ as a Jarlskog-type rephasing-invariant functional of $(Y_\nu,M_R)$; prove invariance algebraically; cite the $\varepsilon\leftarrow I_{CP}$ relation. Refuting-but-valid close: prove no basis-invariant functional of the frozen-fixable data alone is non-zero $\Rightarrow$ the CP source is genuinely an added axiom.
4. CC-2 $\Delta L=2$ washout (Hole #4). Write $\gamma_{\Delta L=2}^{\rm RIS\text{-}sub}(T)=\gamma_{\rm full}-\gamma_{\rm on\text{-}shell\,N}$ with the on-shell-$N$ subtraction shown to avoid double-counting, unitarity/CPT-consistent, named scheme, over the $T_{RH}$ window.
5. CC-3 $N_2/N_3$ damping (Hole #5). Quantify three-state production–decay rates and inter-state washout $W_{i\leftarrow j}$, superseding (not promoting) the diagnostic floor with the actual damped $Y_{B-L}$.
6. CC-4 density-matrix flavor (Hole #6). Either prove the flavor-democracy/3-eigenvalue tension is a clean terminal no-go, or solve $d\rho_{\alpha\beta}/dz=S_{\alpha\beta}[I_{CP}]-\{W,\rho\}_{\alpha\beta}-\lambda_Y[\mathrm{decoh}(\rho)]_{\alpha\beta}$ over the equilibration schedule to $Y_{B-L}^{\rm flavored}$.
7. CC-5 assembly (Hole #7). Deliver the corpus-pinned $B{-}L\to\eta_B$ map (form + factors) and the $3\sigma$-band method, runnable only after CC-1…CC-4 hold; then exactly ONE post-freeze comparison. Inside the window $\Rightarrow$ consistency (not a promotion); outside at $3\sigma\Rightarrow$ BG-10 falsified (decision-grade, accepted).
8. Window convention (Hole #9). Reconcile $\eta_B=7.04\,Y_B$ and the two bands into one consistent quantity before CC-5; held out of S1–S4.

Closing all of these — in leverage order (1, 2, then 3–6, then 7, with 8 as bookkeeping) — upgrades BG-10 from "negative leg closed, gate open" toward a single falsifiable comparison — and even then, only a comparison, never a derivation of the measured $\eta_B$.

12. Completion tests for this page

Required presence (all met): gate roll-up CERTIFIED-IRREDUCIBLE · RESOLVED +0 (mechanism legs 0 of 4) · the closed negative leg $q_1(\omega)=-\kappa\Rightarrow\varepsilon_1=0$ · its DERIVED-GIVEN-E label · "$\eta_B$ not derived" · frozen hashes (AUDIT ONLY) · the measured floor $\eta_B\approx6.1\times10^{-10}$ (MEASURED-ANCHOR) · the pre-registered falsifier · the $O_\nu$ texture · the seesaw + 6-vs-3 firewalls · every fix CC-1…CC-5 and every Hole as its own row · the blind Gauss-sum table · the wrong-sign datum · the $\kappa^3/\pi$ relabel anti-claim · the firewall anti-claim · the hashes-don't-validate anti-claim.

Required absence (all held): no claim that BG-10 is closed or 0/4 upgraded · $\eta_B$/$\varepsilon$/$M_R$/$Y_\nu$/phase derived · $\kappa^3/\pi$ banked as forced · $\delta_{CP}^\ell$ used as the source · the quarter-phase derived · the diagnostic floor promoted · hashes validate physics · the negative leg sold as whole-gate closure · any open residual asserted computed/closed.

This gate anchor ledger follows the canonical eleven-part shape of the SG-4 ledger and the same universal table.

See also: the anchoring method · the SG-4 ledger (the canonical worked example) · the UQF-4 ledger (which carries the same $\sigma_\nu$ mod-8 spin-c bit) · the Gap-02 ledger (the sibling genuine-open gate) · Layer 2 — no unpaid exact labels (why an in-family phase is not a derived one) · the full Gap-10/BG-10 dossier.