DeepRoot-shape — Deep root - structural form (the shape): the gate anchor ledger

The honest one-line: DeepRoot-shape has a real, banked win — inside a frozen, pre-declared scoped-GUT search grammar, under a granularity-induced description-length metric, the 13D three-layer shape $B_{\rm active}$ is the selector-minimal complete survivor, and any admissible theory must carry the same three roles (Stage, Rulebook, Actors) — and on the ratified board (2026-07-08) the gate stands DERIVED-GIVEN-anchor · RESOLVED +0, with the honesty spine intact: the shape is selected, not forced; category-relative certified, not absolutely irreducible; and it bottoms on $E$, the measured anchor.

This page is the gate-specific instantiation of the anchoring method: it takes the master anchor and the bridges and applies them, object by object, to one gate. Every exact thing DeepRoot-shape 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 and the same universal table as the SG-4 ledger.

1. Gate status header

• Gate-level DeepRoot-shape roll-up (ratified board 2026-07-08): DERIVED-GIVEN-anchor · RESOLVED +0. The earlier OPEN roll-up label from the superseded grading rule is preserved below as history.
• Taxonomy reconciliation (2026-07-05): under the ratified closure taxonomy the gate-level grading is the banked terminal carried on the board (DERIVED-GIVEN-anchor · RESOLVED +0 on the category-relative selector-minimality leg, the observed spectrum $E$ declared a measured anchor), read as TERMINAL + RESIDUALS-SHOWN. The residual family listed in this ledger (realization-minimality, whole-shelf color completeness, the no-smuggling metric, Tier-1 reopen conditions, absolute irreducibility, and $E$-unforced) remains listed and carried unchanged. The facts are unchanged: the earlier OPEN roll-up reflects the superseded least-closed-residual grading rule (the gate is OPEN if any leg is open), not different facts. The two-axis reading and the anti-claims below are consistent: the banked win is category-relative selector-minimality given $E$, and the shape is still not forced in the absolute / architecture-neutral sense (§10 stands).
• Closed local/partial leg (the banked selector win): closed only as category-relative selector-minimality — the statement $$O_{\rm SHAPE,local}(E_{\rm frozen}) = 0,\qquad\text{i.e.}\qquad B_{\rm active}=\operatorname*{argmin}_{B\in{\rm Adm}(\mathfrak B_{\rm search})}\mathfrak R_{\rm Occam}(B).$$
• Status, split so it cannot be misread:
  • Selector-minimality leg (category-relative): DERIVED-GIVEN-E / CERTIFICATE-CONDITIONAL — $B_{\rm active}$ is the argmin complete survivor inside the declared, frozen search category. Conditional on the declared grammar + the bridge axiom + given $E$; never an architecture-neutral uniqueness result.
  • Functional-role necessity leg (architecture-neutral): DERIVED — role floor only. Any architecture meeting the physical burden carries a Stage, a Rulebook, and Actors ($k_{\rm role}\ge 3$). Necessary-not-sufficient; near-tautological proof, flagged honestly.
  • Realization-minimality (the architecture-neutral upgrade): OPEN — five sub-lemmas, all currently open.
  • Absolute irreducibility: AXIOM-OPEN / permanent wall — a universal negative over all conceivable mathematics; refused-as-axiom, unprovable for everyone, forever.
  • The spectrum $E$ at the bottom of the stack: MEASURED-ANCHOR — terminal by anchoring; "$E$ is forced" is refuted.

Given $E_{\rm frozen}$ and the declared grammar, the selector lands on $B_{\rm active}$ by a frozen argmin with no post-hoc category shrinking (the no-smuggling discipline). What stays open is everything beyond the category-relative leg — realization-minimality, absolute irreducibility, and whether $E$ itself is anything but a measured input.

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

2. Frozen inputs (what DeepRoot-shape stands on, not what it produces)

• Frozen branch hashes dcc66f1b2685 / a5b1e6f9d951. The branch is read-only and unmutated. These hashes are audit anchors — they certify which object was tested and that it cannot be quietly retuned. They do not validate the physics.
• Upstream spectrum $E_{\rm frozen}$ (the SM chiral content / exactly three generations) is given / charged / inherited — it enters the gate as input. DeepRoot-shape does not derive $E$. Every "passes" below is a statement about this $E$ and inside this grammar, not a derivation of either.
• The declared scoped-GUT search category $\mathfrak B_{\rm search}$ and the constraint vector $\mathcal C_{\rm GUT}=(C_1,\dots,C_{10})$ (Gates 1–10). Declared and frozen; not derived here.
• The bridge axiom physical simplicity $=$ minimal injected operational record cost. Declared interpretive principle, stated explicitly, never smuggled.

3. The object anchors (given-E / declared)

The active branch is a three-layer object, not a flat manifold: $$ B_{\rm active}=\underbrace{[M_4 \times K_6 \times S^2 \times S^1_Y/\mathbb Z_2]}_{\text{Stage }(\times)}\ \oplus\ \underbrace{[F^+ \oplus C_{\rm admiss}]}_{\text{Rulebook }(\oplus)}\ \otimes\ \underbrace{[E_{\rm matter}\oplus E_{\rm gauge}\oplus E_{\rm Higgs}\oplus E_{\rm proton}]}_{\text{Actors }(\otimes)}. $$ The Stage carries the metric arena: $M_4$ (4D Lorentzian, declared), $K_6=SU(3)/T^2$ (color carrier, 6D), $S^2$ (weak carrier, 2D), $S^1_Y/\mathbb Z_2$ (hyper carrier, 1D); total $D=4+6+2+1=13$. The Rulebook carries finite admissibility/chamber data ($F^+$ flavor chamber, $C_{\rm admiss}$ firewall). The Actors carry matter/gauge/Higgs/proton content. Status: GIVEN-E / declared-structure — the layers are the tested object, not gate-derived.

4. Root and master-anchor traceability

Deep roots that are load-bearing for DeepRoot-shape:

Causal order is not a primary load-bearing anchor for the SHAPE selector argument.

Master anchors in play: finite invariant ledgers (the MDL codebook) · no unpaid labels (anti-fitting $\equiv$ MDL) · the frozen branch · given-$E$ · the declared search grammar + bridge axiom · open-residual discipline.

5. The DeepRoot-shape anchor ledger (the universal table)

6. The construction — the banked win, in full

The selector argmin (category-relative). Define the admissible set $$ {\rm Adm}(\mathfrak B_{\rm search})=\{B\in\mathfrak B_{\rm search}:\ B\models\mathcal C_{\rm GUT},\ B\ \text{obeys freeze-before-compare},\ B\ \text{obeys no-smuggling}\}, $$ and the declared lexicographic Occam ranking $\mathfrak R_{\rm Occam}$ with the binding rule that completeness outranks simplicity. The five-step proof skeleton: (1) completeness filters first; (2) every proper layer subset $L\subsetneq\{\times,\oplus,\otimes\}$ is inadmissible ($\mathfrak B_L\cap{\rm Adm}=\varnothing$); (3) $B_{\rm active}\in{\rm Adm}$; (4) term-level removal of any load-bearing term fails a required gate; (5) no preferred admissible competitor is currently supplied. Hence $B_{\rm active}=\operatorname*{argmin}_{B\in{\rm Adm}}\mathfrak R_{\rm Occam}(B)$ — category-relative.

The metric. The Finite Operational Cell Law gives a $\Delta_0>0$; five axioms (record faithfulness, encoding invariance, additivity, precision monotonicity, no free hidden data) pin the unique cost up to $O(1)$: $$ C(B)=I(B)=\min_{p:\,U(p)=T_{\Delta_0}}\ell(p)+O(1), $$ a measured real costing $\log_2(1/\Delta_0)+O(1)$ bits, a discrete structural choice costing $O(1)$. Anchor-burden dominates dimension-burden.

The ladder. With $I(B)=I_{\rm struct}+I_{\rm gen}+n\,b+I_{\rm extra}$ and $b=\log_2(1/\Delta_0)$, the per-rung inequality $$ \forall D=4..12\,(+\,\text{non-dim}),\ \forall j:\quad I_{\rm gen}(B_{13})+n_{13}\,b\ <\ I_{\rm gen}(B_{D,j})+n_{D,j}\,b\ +\ I_{\rm extra}(B_{D,j}), $$ with $n_{13}\approx 13$–$14$, gives the tally 0 REFUTED · 1 FAILS_TO_GENERATE_T (6D) · 10 LOSES_TO_13D, near-ties broken by the structural-bit term ($I_{\rm struct}\sim 60$–$100$ bits for heterotic vs $\sim 25$–$30$ for named cosets).

Diagnostic — the win is specific, not a tautology of the selector's own taste. The decisive bridge is anti-fitting $\equiv$ MDL: a tunable match must be injected as a measured real costing $\sim b$ bits, so it lengthens the description. The ladder is therefore adjudicated by a codebook that charges $B_{\rm active}$ and every competitor by the same rule, with $E$ and $E$-automatic facts (anomaly cancellation, written-spectrum chirality, the $\mathbb Z_6$ computation, accidental proton stability) cancelled on both sides. The genuine 13D economy sits exactly where the bridge rewards it: the family count is a forced topological integer $\chi(K_6,E)=-3$ ($O(\log)$ bits) and unification closes $\alpha_i$ to one anchor — quantities competitors must buy with injected reals. That a non-trivial economy survives after cancelling $E$ on both sides is what makes the result a real comparison rather than a restatement of the selector's preferences.

The obstruction map. Collect the local pieces: $$ O_{\rm SHAPE,local}(E)=\big(O_{\rm complete}(E),\,O_{\rm argmin}(E),\,O_{\rm role}(E)\big),\qquad O_{\rm SHAPE,local}(E_{\rm frozen})=0 . $$ The full gate obstruction additionally carries the architecture-neutral pieces: $$ O_{\rm SHAPE}(E)=\big(O_{\rm realization}(E),\,O_{\rm irreducible}(E),\,O_{\rm complete}(E),\,O_{\rm argmin}(E),\,O_{\rm role}(E)\big). $$ We do not assert $O_{\rm SHAPE}(E_{\rm frozen})=0$: realization-minimality and absolute irreducibility are not closed.

7. The color rung, split into honest objects

The single phrase "$K_6=SU(3)/T^2$ is the color carrier" hides several claims with different statuses — and the corpus records an adversarial sequence here, the program's acknowledged weak link:

1. $CP^2$ is the genuine dimensional floor. $CP^2=SU(3)/U(2)$ is the minimal-dimension ($4$D) compact $SU(3)$-homogeneous carrier; sub-6D rivals ($S^5$, Wu manifold) die by odd-dimensionality. Status: DERIVED (homogeneous-space classification).
2. "$CP^2$ gives a tunable family count" was UNSOUND. $CP^2$ is Spin$_c$ with index $\frac{(2r+1)^2-1}{8}=\frac{r(r+1)}{2}$, and $r=2\Rightarrow 3$ — a discrete topological integer, not a continuous modulus. The program found this defect against itself. Status: RETRACTED (do not resurrect).
3. The symmetric blow. $K_6$ also yields 3 only after a bundle choice: BWB gives $\pm\frac{(a+1)(b+1)(a+b+2)}{2}$, $(1,0)\Rightarrow 3$. Neither carrier forces "3" from the bare carrier alone. Status: DERIVED-GIVEN-E (symmetric).
4. Abelian-isotropy uniqueness (the resolution). The 11D $CP^2$ build breaks end-to-end at Gate 2: $T^2$ is the unique purely-abelian $SU(3)$ isotropy ($C_{SU(3)}(T^2)=T^2$, Cartan only), injecting no spurious non-abelian factor; $CP^2$'s isotropy $U(2)$ is gauge-active (CSDR centralizer rule), forcing a lose-lose fork. Status: DERIVED-GIVEN-E over the named/enumerated shelf — completeness over all admissible $SU(3)$ carriers is uncertified.

So the weak/hyper rungs are whole-shelf DERIVED (general facts F1/F2), the color rung is DERIVED over the enumerated shelf, and $M_4$ is a declared observational primitive. The open target is whole-shelf $SU(3)$-carrier completeness — without tuning to the known answer.

8. The Lemma-2 split — a precise negative beats a vague pass

The single phrase "rulebook minimality" was tested and REFUTED as a standalone closure, then split honestly:

• (A) CLASS-GOVERNANCE {freeze-before-compare, gate-status, anti-fitting} = method, excluded by the architecture-neutral precondition. Not a rule-burden.
• (B) CLASS-E-AUTOMATIC {anomaly firewall} = an $E$-property (all six anomaly coefficients $=0$); cancels on both sides — zero differential burden.
• (C) The only architecture-neutral physics residue is the flavor chamber $F^+$, and it injects 3 reals $N_d,N_e,N_\nu$ it does not derive ($N_d$ "defined to set $m_b$ to its target", $N_e$ "chosen such that $m_\tau$ matches its target"). The ordinary 4D SM rulebook has strictly lower $k_{\rm rule}$ (no flavor chamber) and is admissible.

Therefore $F^+ + C_{\rm admiss}$ is not the rule-burden argmin → Lemma-2-as-standalone is REFUTED. The live residual relocates: R-L2a (joint no-cross-role-compression) $=$ Hole 4; R-L2b the 3 injected $N_d,N_e,N_\nu$ are terminal-by-anchoring inside $E$; R-L2c "no cheaper rulebook anywhere" $=$ the same Kolmogorov wall as absolute irreducibility, OPEN.

9. Open residuals — the realization / irreducibility / anchor family

The banked selector leg is one face of the gate. These distinct residuals make up the rest, and none is closed by the category-relative argmin:

• Realization-minimality — the architecture-neutral argmin, a valid conditional skeleton reducing to five sub-lemmas, all OPEN (Lemma 3 Actors · Lemma 1 Stage · Lemma 2 Rulebook [refuted-standalone, residual relocated] · Lemma 4 no-cross-role-compression · Lemma 5 no preferred competitor). OPEN.
• Whole-shelf color completeness — abelian-isotropy uniqueness closes $K_6$ over the enumerated shelf only; completeness over all admissible $SU(3)$ carriers is uncertified. OPEN.
• The no-smuggling metric — a formal $\mathfrak K$ non-decreasing under role-fusion; the object the Lemma-2 residual folds into. OPEN.
• Tier-1 reopen conditions — five classes (NCG, traditional KK, string/F-theory, finite-state/discrete, SO(10)/exceptional) remain SERIOUS AUDIT CANDIDATES with "simpler-after-unfolding" cells Unknown. CERTIFICATE-CONDITIONAL (current-record), not nonexistence.
• Absolute irreducibility — a universal negative over all conceivable mathematics; an uncomputable Kolmogorov question. AXIOM-OPEN / permanent wall.
• $E$ unforced — the whole stack bottoms on $E$, which no known principle forces; anomaly-freedom is a settled filter with infinitely many solutions and "$E$ is forced" is refuted. MEASURED-ANCHOR (declare, do not plug).

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

• DeepRoot-shape does not derive $E$. Formally: $E_{\rm frozen}\in\ker O_{\rm SHAPE,local}$ inside $\mathfrak B_{\rm search}$, not $\ker O_{\rm SHAPE}=\{E_{\rm SM}\}$.
• The selector is a filter inside a declared grammar, not an architecture-neutral derivation. "Funnel-winner-forced $=$ RELABEL $=$ REJECTED." Selection ≠ derivation.
• The shape is not forced / uniquely derived / the unique minimum. It is selector-minimal, category-relative — DO NOT claim Scale $+$ Granularity $\Rightarrow$ Shape.
• The shape is not absolutely irreducible / proven minimal across all architectures — that is a permanent wall, refused-as-axiom.
• Three generations / the SM content $E$ is not derived; $\chi(K_6,E)=-3$ gives three families given $E$.
• The color rung is forced only over the enumerated $SU(3)$-carrier shelf, not over all admissible carriers.
• Rulebook-minimality is not banked as a standalone (REFUTED); $F^+$ does not derive the scales $m_b,m_\tau,m_\nu$ (it injects $N_d,N_e,N_\nu$).
• No Tier-1 competitor exists is not claimed — only "none currently supplied & simpler after unfolding."
• The functional-role theorem earns the role floor only — it forbids deleting a role, not realizing one more cheaply; its proof is near-tautological.
• The frozen-branch hashes are audit anchors; they do not validate the physics, and reproducibility ≠ uniqueness.
• Local (category-relative) closure is not whole-gate closure. $O_{\rm SHAPE,local}=0$ does not imply $O_{\rm SHAPE}=0$.

11. Specialist closure plan

Each open residual is a concrete, finite, target-blind work-package. Gate 0 is mandatory for every lemma: state its role-requirement architecture-neutrally, with no hidden reference to the submitted factor set, $F^+$, or $E$ — a requirement phrased by pointing at the submitted branch is circular and does not count. Recommended order: Hole 3 → 1 → 4 → 2 → 5 → 6.

1. Hole 3 — Actor minimality (attack first). Prove no admissible competitor supplies lower unfolded $k_{\rm actor}$ while preserving matter/gauge/Higgs-Wilson-line/proton-safety/observable-algebra/the $\mathbb Z_6$ center-kernel. Falsifier: exhibit cheaper admissible actors (F3) → Lemma 3 fails fast. Realistic near-term best outcome: "reduced to minimal-given-$E$, $E$ named as the residual." Machinery: BWB index theory; the $\ker$ computation.
2. Hole 1 — Stage minimality. Prove every architecture-neutral Stage carrying the required carriers $+$ chirality route has complexity vector $\ge$ the frozen stage, subsuming whole-shelf $SU(3)$-carrier completeness. Trap: do not resurrect the unsound "tunable family count" $CP^2$ elimination — use abelian-isotropy uniqueness. Falsifier: a cheaper admissible stage (F1).
3. Hole 4 — No cross-role compression. Prove a formal $\mathfrak K$ is non-decreasing under Stage↔Rulebook↔Actors fusion (a conservation law for unfolded burden); show $k_{\rm rule}(\text{SM})+k_{\rm actor}(\text{SM})\ge k_{\rm rule}(F^+)+k_{\rm actor}(B_{\rm active})$ on the full-$T$ ledger given $E$. Trap: prove the offset as a theorem, not by example. Underwrites Lemmas 1, 2, 3, 5.
4. Hole 2 — Rulebook residual (REFUTED-as-standalone). R-L2a closes iff Hole 4 closes; R-L2b closes only by declaring $N_d,N_e,N_\nu$ as part of the $E$-anchor injection (terminal-by-anchoring), not by deriving them; R-L2c folds into Hole 5. Trap: do not present the REFUTATION as strengthening the standalone lemma — it does not.
5. Hole 5 — No preferred competitor. Build a finite per-class lower-bound matrix discharging each Tier-1 reopen condition; argue coverage, do not assume it. Honest odds LOW (universal negative; attempt last). Refuting result (valuable): surface a preferred competitor (F2/F4) → the shape folds; if it is derivable from Scale $+$ Granularity, the residue drops — the correct compression outcome.
6. Hole 6 — $E$ unforced (declare, do not plug). Either a principle forcing exactly three chiral generations as the unique admissible family count (a strong, possibly-false target — the filter has infinitely many solutions), or — the honest expected outcome — declare $E$ the irreducible measured-anchor of SHAPE, terminal-by-anchoring. The floor stays at $\ge 1$ anchor (here $E$), by design, forever.

Closing Holes 1, 3, 4, 5 is the full realization-minimality result — the only path from CATEGORY_RELATIVE toward architecture-neutral forcing — and even then, only given $E$, and never past the absolute-irreducibility wall.

12. Completion tests for this page

Required presence (all met): gate roll-up DERIVED-GIVEN-anchor · RESOLVED +0 (historical OPEN label kept as history) · $O_{\rm SHAPE,local}(E_{\rm frozen})=0$ as the argmin · selector leg DERIVED-GIVEN-E / CERTIFICATE-CONDITIONAL · functional-role floor (role floor only) · realization-minimality OPEN (5 sub-lemmas) · absolute irreducibility AXIOM-OPEN / permanent wall · $E$ MEASURED-ANCHOR · "$E$ not derived" · frozen hashes (AUDIT ONLY) · $D=4+6+2+1=13$ · $\chi(K_6,E)=-3$ · $\mathbb Z_6=\ker(\cdot)$ · abelian-isotropy uniqueness · F1/F2 whole-shelf facts · metric $C(B)=I(B)$ · anti-fitting $\equiv$ MDL diagnostic · ladder tally 0/1/10 · Lemma-2 REFUTED split · $N_d,N_e,N_\nu$ injected reals · Tier-1 current-record · every exact object as its own row · the gate's anti-claims.

Required absence (all held): no claim that the shape is forced / uniquely derived / the unique minimum · Scale$+$Granularity$\Rightarrow$Shape · absolute irreducibility proven · $E$ derived · three generations geometry-forced · color forced over all carriers · rulebook minimality banked standalone · "no Tier-1 competitor exists" · the role floor sold as sufficiency · $\ker O_{\rm SHAPE}=\{E_{\rm SM}\}$ · hashes validate physics · reproducibility = uniqueness · category-relative closure = whole-gate closure · any reader-visible build-process vocabulary.

Completion report. Tests passed: all required-presence and required-absence items above. Tests failed: none. Open items: realization-minimality (5 sub-lemmas), whole-shelf color completeness, the no-smuggling metric, Tier-1 reopen conditions, absolute irreducibility (permanent), $E$ (declare). Assumptions made: the declared scoped-GUT grammar, the bridge axiom (simplicity $=$ minimal record cost), and given-$E$ — each stated explicitly, never smuggled. This page upgrades no grade.

This gate follows the same eleven-part shape and universal table as the canonical SG-4 ledger; its nature differs (a banked category-relative selector win with an architecture-neutral upgrade open and a permanent irreducibility wall), and the table is adapted honestly to that nature.

See also: the anchoring method · the shape-minimality challenge (name a cheaper architecture and the shape folds) · Layer 2 — no unpaid exact labels (anti-fitting $\equiv$ MDL) · Layer 3 — search grammars (why the category is declared and frozen) · Layer 4 — carrier-forcing & the given-E wall · the full DeepRoot-shape dossier.