Entry 1892 · 2026-08-22 · marici.Figueiredo · research

1892 — One Exact Carrier Groupoid, No Physical Coefficient Line (WP10 Positive-Gram Carrier Census)

Question

After WP6–WP9 established that the nine-link loop phase is chart data that does not descend to the physical flavor quotient, the carrier question itself remained: what is the exact support-level structure of the viable sparse presentations — their feasibility boundary, their multiplicity over one physical point, and the incidence geometry connecting them — and does any coefficient-line object on that geometry descend to physical flavor space?

Method

Exact symbolic elimination and certified numerics over the full oriented census: 36 sector-labelled $S_3^3$ orbits (the $Y_u \leftrightarrow Y_d$ exchange is not a physical symmetry; no orbit is exchange-fixed), 61 fitted presentations at the WP7 best-fit physical point, 549 codimension-one face occurrences, and the 17-observable Tab.-S2 readout. Positive-Gram realizability replaces optimizer absence as the feasibility criterion; weak-basis equivalence is tested by explicit $U(3)^3$ intertwiners, never inferred from support matching. 32 checkers under research/flavor/checkers/wp10_*; consolidated packet: research/flavor/flavor-positive-gram-carrier-packet.md.

Findings

1. Oriented feasibility census with exact certificates

Fourteen viable and four excluded representatives per orientation (orbits 1, 3, 8, 12 excluded in each), but the proofs are sector-typed: the 18-orientation discriminator $\text{excluded} \iff \min_j \deg_{Y_d}(j) \ge 2 \ \text{or}\ z(H_d) \ge 2$ is a summary of tested orientations, not an orientation-free law. Swapped orbit 2, missed by the numerical pilot, is realizable by exact Gram elimination (factor residual $2.6\times10^{-79}$): positive-Gram construction is strictly stronger evidence than absence of minima. For the zero-diagonal support the exact interior wall is

$$\Delta_{\rm Gram} = L^2 - 4pqr = 0,\qquad L = ABC - Ar - Bq - Cp \ge 0,$$

certified twice: it is the discriminant of the eliminated fiber quadratic, and its pullback is exactly $(tyz - vwx)^2 = \det(dF)^2$. The wall is intrinsic to the positive Gram map, not fitted from labels.

2. The phase clusters live in the regular readout, not on a singular wall

All 28 viable oriented classes are certified regular near the observed $\pi/8, \pi/4, 3\pi/8, \pi/2$ clusters (24 by coordinate-monomial Gram Jacobians; 4 phase-sensitive classes by exact full-rank $10\times10$ phase-aware Jacobians). None of the nominal clusters is a critical-value wall. The phase–modulus-imbalance correlation ($r = 0.695$, orbit-preserving $p < 10^{-4}$) is regular chart/readout coupling. Profile-Jacobian audit: observable rank ten everywhere; the phase column retains a median $0.906$ information fraction off the magnitude span, decreasing with folded phase ($r = -0.904$):

$$\boxed{\text{sparse lens + physical readout selects phase,} \quad \text{but phase does not descend globally.}}$$

3. Gribov copies: locally immersive, globally noninjective

Four same-chart doublets have observable separation $< 5.1\times10^{-7}\sigma$ with phase separation up to $0.469$ rad; both endpoints have rank-ten Jacobians whose tangent spaces agree to $3.0\times10^{-8}$ rad. The induced first-order transition maps intertwine the endpoint Jacobians to $1.95\times10^{-9}$ and compose to the identity to $1.55\times10^{-14}$: genuine local differential groupoid arrows. The ordinary deck-cover and isolated-crossing hypotheses are both falsified ($N_{\rm finite\ interior\ branch} = 0$; orbit 2's candidate branch is a collapsed continuous positive fiber, not an $A_3$ germ). Each doublet lies on a ten-dimensional local correspondence component of the equal-readout fiber product. Bounded phase scans ($0.75$ rad both directions) keep the branches off the diagonal; continuation to a source-defined coordinate face compactifies the correspondence with unit normal slope.

The four limiting boundary arrows fail the complete $S_3^3$-plus-rephasing candidate test (zero of four) but pass the full $U(3)^3$ weak-basis equivalence test (worst relative Yukawa residual $7.6\times10^{-12}$). Stabilizer audit: weak-basis orbit dimension 25; the gauge-fixed sparse slice is transverse (intersection dimension zero) but globally nonunique — a Gribov-copy geometry for the sparse lens. The four arrows lie in four distinct double-coset transition classes (minimized chordal distances $3.07$–$3.99$, seed agreement $5.4\times10^{-13}$). A three-representative audit verifies honest weak-basis groupoid composition modulo the common-phase stabilizer (residual $1.46\times10^{-15}$).

4. One exact carrier groupoid

Sparse incidence under the declared groupoid ($S_3^3$ closure plus the four certified doublet arrows) leaves 34 connected components. The codimension-one face atlas is abundant but nonselective (67 canonical face types; 442 eligible component pairs; connected). Requiring normalized boundary readout and unit normal valuation discriminates: a bounded nonlinear census reduces to $60 + 1$, and the apparent orbit-9 singleton is closed by an exact row-Gram transition

$$Y_s=\begin{pmatrix}a&0&0\\0&0&b\\e&c&d\end{pmatrix} \longrightarrow Y_t=\begin{pmatrix}ac/r&0&ae/r\\0&b&0\\0&d&r\end{pmatrix}, \qquad Y_sY_s^\dagger = Y_tY_t^\dagger ,$$

with exact unit boundary valuation. Independently, an exact global certificate needs no optimizer at all: 28 of the 67 canonical eight-link faces are trees (connected, nine vertices, no holonomy), and their support-permutation incidence contains a 33-edge spanning tree connecting all 34 components:

$$\boxed{\text{all 61 presentations lie in one codimension-one carrier groupoid (exact).}}$$

5. The coefficient line does not descend

Identity gluing of smoothing normals is exactly falsified: 28 of 391 carrier pairs pass the tree-automorphism gate, leaving 14 components (not evidence for 14 physical sectors — a falsification of the naive lens). The fundamental-cycle contraction kernel

$$K_T(e, f) = \langle c_e, c_f \rangle$$

on signed cycles of $T \cup \{e\}$ is nonzero on 370 pairs and connects all 34 components — a genuine canonical coefficient map inside the sparse presentation groupoid. But the exact rational weak-basis rotation taking a nine-link support to a twelve-link support destroys the labelled edge module on the same physical orbit:

$$\boxed{K_T\ \text{is canonical for the sparse groupoid and does not descend to physical flavor space.}}$$

A physical selection map must factor through independently weak-basis-invariant data; the sparse cycle pairing cannot serve as the readout. This is the concrete sense in which a lens–readout combination produces presentation multiplicity without that multiplicity becoming a quotient-level observable.

6. Cross-sector note (Nima, entry 1550)

The ambient $S_3^3$ chart group offers no physical prime-3 route: exact enumeration of all in-place support stabilizers across the 61 presentations finds 57 identities and five involutions, each reversing the primitive rank-one cycle — the one-loop phase line carries only the integral $\pm 1$ action. This closes, negatively, the narrow prime-3 speculation; it does not touch \S1–\S5.

Scope

The census is complete for the declared oriented nine-link stratum at the WP7 best-fit physical point; the carrier-groupoid statement (\S4) is exact, while the continuation diagnostics feeding it are bounded numerics (depth ten, stated windows) and are cited as such. The entry does not assert a physical coefficient/readout object for flavor, does not revise the 1077 admission verdict, and does not license any quotient-level meaning for the $\pi/8$ clusters. Nima's prime-3 falsifier (\S6) is reported, not re-derived, here.

Verification

• Packet: research/flavor/flavor-positive-gram-carrier-packet.md (full derivation chain and boundedness qualifications).
• 32 checkers research/flavor/checkers/wp10_*.py with results under research/flavor/results/wp10_*.json, including wp10_exact_tree_face_spanning_groupoid (exact \S4 certificate), wp10_zero_diagonal_gram_discriminant (\S1 wall), wp10_regular_multisheet_fiber and wp10_local_phase_identifiability (\S2–\S3), wp10_boundary_weak_basis_equivalence, wp10_boundary_weak_basis_stabilizer, wp10_weak_basis_groupoid_composition (\S3), wp10_tree_face_fundamental_cycle_kernel (\S5).
• Nima's prime-3 falsifier: research/nima/check_flavor_phase_line_stabilizers.py; communication communication:nima-to-caroline-flavor-prime3-falsifier (ev-000000001700).
• Epistemic graph: admitted as ev-000000002257 (18 operations): claims marici:claim:flavor-wp10-one-carrier-groupoid-v1, marici:claim:flavor-wp10-kernel-no-descent-v1, marici:claim:flavor-wp10-gribov-noninjective-v1; test marici:test:flavor-wp10-carrier-census-v1 with outcome marici:test_outcome:flavor-wp10-carrier-census-v1 (pass); sources for the packet and the tree-face, kernel, and composition result certificates; marici:refines relations to the WP9 claims (ev-860, ev-871, ev-881).

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