Gaps in the Map and Bridges Between Sectors

Patterns 4 and 5 of the bottom-up analysis

Internal document — ArXe / ALO Research — February 2026

Pattern 4 — Gaps in the map

The two absent pairs of fundamental arity numbers

Of all possible pairs of arity numbers in {2,3,5,7,11,13,17,19},
only two never appear together in any constant of the corpus:

5×13  (MEMORY × SU2)
13×19 (SU2 × GENERATIONS)

Arity 13 = SU(2) = weak force appears extremely isolated.
Only in three constants: α, V_ud, S8_cmb.
And never together with memory/state (5) or with generations (19).

This suggests that the weak force, in the current ALO corpus,
has no constant that captures its relation with:
– the state/memory structure of the system (5)
– generational multiplicity (19)

Those two absent constants would be:
– {5,13}: something that connects SU(2) with state persistence — candidate: the W lifetime or weak decay width
– {13,19}: something that connects SU(2) with generations — natural candidate: the generational charged current, or more precisely, the Jarlskog CP invariant J_CP

Predictions: constants that should exist

Editorial note (2026-08-04): this table and the notes below reflect the corpus’s own subsequent audit of these predictions (through August 2026). Several were withdrawn on inspection; one — J_CP — has since been numerically confirmed. This post previously showed the original, unaudited February 2026 table; it is corrected here to match the corpus’s current, honest status.

Constant	Predicted arity numbers	Reason / current status	Closest in corpus
m_ν (neutrino mass) — [WITHDRAWN, see note]	~~{2, 19}~~	Not a well-defined target: three distinct neutrino masses exist, the absolute scale is unmeasured (only Δm² differences are), and the mass-generation mechanism (Dirac vs. Majorana vs. seesaw) is undetermined even in standard physics. Reformulated below.	m_b = {2,11,19}
Λ (cosmological constant) — [WITHDRAWN, duplicate]	~~{2,3,29}~~	Same naturalness problem already documented as open in “Dark Matter and Dark Energy Through the Type A/B Lens” — why Λ is ~10⁻⁴⁷ GeV⁴ rather than the Planck-scale estimate. A fresh derivation attempt here would contradict that article’s honest assessment. See that piece for the current status.	—
m_axion — [WITHDRAWN, see note]	~~{3,7,11}~~	Depends on the axion decay constant f_a, which no current theory fixes — only experimentally bounded (roughly 10⁹–10¹² GeV or higher, model-dependent). Not the kind of dimensionless quantity a boundary-condition analysis can fix; sharing α_s_ess’s arity numbers by sector resemblance is not a derivation.	α_s_ess = {3,7,11}
θ_QCD (strong CP angle) — [REVISED, see note]	~~{3,7,11,13}~~	The {3,7,11,13} assignment was never derived independently — it came from sector analogy, not boundary-condition analysis, and was withdrawn. However, a later check with the Type A/B classification (see the Axiomatic Distance corpus) found that the physically meaningful quantity — θ̄ = θ_QCD + arg(det M_q), not θ_QCD alone — passes the Type A test: it is rephasing-invariant, like J_CP in the CKM sector. So it is not a Type B case dismissible as human convention — it is a genuine substrate observable whose magnitude remains unexplained. Status: open problem, not a withdrawn candidate.	—
J_CP (Jarlskog) — [NUMERICALLY VERIFIED, see note]	{2,7,13,17,19} unconfirmed	CKM CP invariant: requires all mixing axioms. See the two dated notes below — this is the strongest result on this page.	V_us = {2,7,17,19}
y_t (top Yukawa) — [WITHDRAWN, see note]	~~{3,11,17}~~	Inherits m_t’s own unresolved ambiguity: the top quark’s short lifetime means it has no clean pole-mass definition (AD=90 in the Axiomatic Distance corpus, the largest gap of any Standard Model mass). y_t=√2·m_t/v cannot be cleaner than m_t itself.	m_t = {3,11,17,107}
V_inflation — [WITHDRAWN, see note]	~~{2,3,23}~~	Confirmed by the corpus’s own prediction/postdiction audit: the level assignment (inflation at T⁻¹¹, arity 23) is a testable structural claim; a specific numerical value for the inflationary energy scale is not, since it depends entirely on which inflaton model is assumed — there is no single measured target. The level assignment stands; the numerical prediction is withdrawn.	m_u = {2,3}

m_ν reformulated: the corpus’s own prediction/postdiction audit already contains a genuine, well-defined, falsifiable claim about neutrinos that the {2,19} row above wasn’t pointing at: normal neutrino mass hierarchy, falsifiable by the JUNO experiment. It is an ordering claim, not an absolute-mass claim, and doesn’t require solving the open problems (absolute scale, generation mechanism) that made {2,19} a poorly defined target. This should be the prediction representing ArXe’s position on neutrinos going forward.

The most immediately verifiable prediction in this table, with the withdrawals above applied, is normal neutrino mass hierarchy, falsifiable by JUNO — not m_axion, which was withdrawn for depending on an externally fixed scale (see note above).

J_CP status (original note, February 2026, added following the θ_QCD/δ_CKM investigation): J_CP passes the Type A test — it is rephasing-invariant by construction, unlike δ_CKM, which fails that test and was withdrawn as an intermediate step. However, computing J_CP requires either the CP phase itself (which had no properly grounded formula in this corpus at the time) or a full row of CKM magnitudes confirmed via the unitarity triangle — and only 2 of 9 CKM elements (V_ub, V_cb) had a status better than “Human” in the reference corpus at that point. Worse, the candidate formulas for the missing elements (V_ud, V_us) contradicted each other across corpus documents, and neither was derived from boundary conditions — both came from an unconstrained combinatorial search. Status at that time: blocked, neither resolved nor withdrawn. A genuine derivation would need to start a CKM row from boundary conditions, with no prior reliable formula to build on.

Update, 2026-08-04 — the blocked status was stale, not resolved: the note above is from February 2026. In July 2026, the fermionic mixing synthesis document derived all four CKM parameters (θ₁₂, θ₂₃, θ₁₃, δ) with ArXe formulas precise to Δ<0.01° each, fit independently. Nobody had gone back to check that result against the blocked status documented here. Combining them in the standard Jarlskog formula (J = c₁₂c₂₃c₁₃²s₁₂s₂₃s₁₃ sin δ) gives J_CP = 2.909×10⁻⁵, within the PDG 2024 experimental range (J = 2.99±0.13 ×10⁻⁵; Navas et al., Phys. Rev. D 110, 030001). Since all four angles were fit separately, this is a genuine cross-prediction — J_CP is highly sensitive (a product of small sines), so an individual error in any of the four would have been amplified, not cancelled, rather than landing squarely within the experimental margin.

What this does not yet resolve: the numerical result is solid, but the derivation chain is not closed end to end. δ_CKM = arctan(3×7×13/5³) is factorized, not derived from boundary conditions with the same rigor already achieved for e and γ — in the φ→γ→e spectrum table, δ falls into the “no constant — closed BC” category, the least developed of the four. Until δ has that derivation, J_CP moves from “blocked” to “numerically verified, with one link (δ) still pending a complete BC derivation” — a substantially better status, but not yet a complete Mode 1 chain from axioms to observable.

The 29 absent triples of fundamental arity numbers

Of the 56 possible triples of arity numbers in {2,…,19}, half are absent.
The absence pattern is not random: triples involving 13 (SU2)
are systematically absent when combined with 5 (MEM), 7 (3D) or 19 (GEN).

Absent triples involving 13:
  (2,3,13)   DIFF×CYC×SU2
  (2,5,13)   DIFF×MEM×SU2       ← gap: SU2 without memory
  (2,7,13)   DIFF×3D×SU2        ← gap: SU2 without 3D space
  (2,11,13)  DIFF×EM×SU2
  (2,13,17)  DIFF×SU2×YUKAWA
  (2,13,19)  DIFF×SU2×GEN       ← gap: SU2 without generations
  (3,5,13)   CYC×MEM×SU2
  (3,13,19)  CYC×SU2×GEN
  (5,7,13)   MEM×3D×SU2
  (5,11,13)  MEM×EM×SU2
  (5,13,17)  MEM×SU2×YUKAWA
  (5,13,19)  MEM×SU2×GEN
  (7,13,19)  3D×SU2×GEN
  (11,13,19) EM×SU2×GEN
  (13,17,19) SU2×YUKAWA×GEN

The weak force (13) is the sector with the most absences in the triples map.
SU(2) is systematically disconnected from memory (5), 3D space (7) and generations (19)
in the current corpus.

This either reflects that SU(2) has a genuinely separate ontological structure
from those dimensions, or that the ALO corpus lacks the constants that
capture those connections (W width, lifetimes, CP invariants).

Pattern 5 — Bridges between opposite sectors

Exclusive bridges: arities that connect exactly two sectors

Arity number	Connected sectors	Constants	Reading
67	quark ↔ cosmo	m_d ↔ H₀_cmb	Down quark and universe expansion rate share a unique choice
71	lepton ↔ coupling	m_τ ↔ G_F	The tau and the Fermi constant — the tau was discovered through G_F
73	lepton ↔ cosmo	m_τ ↔ H₀_local	The tau and the local Hubble constant
107	quark ↔ CKM	m_t ↔ V_ud	The top and up-down mixing — CKM unitarity
109	boson ↔ coupling	m_Z ↔ G_F	The Z and the Fermi constant — the M_Z scheme
41	boson ↔ PMNS	m_W, m_H ↔ θ₂₃	The VEV connects EW bosons with leptonic mixing
1051	lepton ↔ PMNS	m_τ ↔ sin²θ_W	The tau and the Weinberg angle — full EW mixing

These are the only arity numbers that function as exclusive bridges —
their presence in two constants from different sectors is the only connection
between those sectors through that arity number.

The most striking: arity 67 connects m_d with H₀_cmb.
The down quark and the universe expansion rate measured from the CMB
share an arity that appears nowhere else in the corpus.
What axiomatic choice connects the lightest down-type quark with
the largest-scale cosmology? This warrants investigation.

Quarks ↔ Cosmology: the richest connection

This is the connection between the most “opposite” sectors in physical scale
(10⁻¹⁸ m vs 10²⁶ m) and it has 7 shared arity numbers:

Arity numbers only in quarks:  {17, 107, 127}       ← mass/Yukawa, top, charm
Arity numbers only in cosmo:   {13, 59, 73, 97, 509} ← SU2, cosmo-specific
Shared arity numbers:          {2, 3, 5, 7, 11, 19, 67}

The specific connections:

arity 7 (3D): m_d ↔ Ω_m
The down quark and the matter fraction of the universe share the spatial axiom.
The quark that forms protons and neutrons — ordinary matter —
shares with the total amount of that matter the same condition of possibility:
that space have 3 dimensions.

This is not trivial. The up quark (m_u = {2,3}) does not have arity 7.
The down quark has it and the up does not. The down/up asymmetry in 3D space
is registered in the arities.

arity 19 (GEN): {m_s, m_b} ↔ {Ω_b, n_s}
The second generation of quarks (strange, bottom) shares the generational
axiom with baryon density and the spectral index.
Baryons are matter made of quarks — their density presupposes
that second-generation quarks exist. The inflationary n_s
also requires that axiom.

arity 11 (EM): {m_b, m_t} ↔ n_s
The heaviest quarks (bottom, top) share the EM axiom with the spectral index.
Inflation, through n_s, presupposes the existence of the EM field
in the same way that heavy quarks do.

What quarks have and cosmo does not: {17, 107, 127}
Mass/Yukawa (17), top identity (107), charm identity (127).
Cosmology does not need the axioms of specific fermionic mass
to articulate its observables. Cosmological parameters are
independent of how heavy each individual quark is —
they only need quarks to exist (via the shared arity numbers),
not how heavy they are.

What cosmo has and quarks do not: {13, 59, 73, 97, 509}
SU(2) (13) appears in cosmology (S8_cmb) but not in quark masses.
The large-scale structure of the universe requires the weak force
as an axiom, something that individual quark masses do not need.

Synthesis of the two patterns

Gaps and bridges tell the same story from different angles.

Gaps say: there are axiom combinations that the ALO corpus
does not yet capture — places where there should be a physical constant
but we have not measured it, or where we have it but its ALO formula
is not yet in the corpus.

Bridges say: when two apparently opposite sectors
share an arity number, it is not accidental — there is a common condition
of possibility that connects them ontologically before they differentiate
into distinct phenomena.

The deepest connection: the second generation (arity 19)
appears in quarks (m_s, m_b), in leptons (m_μ), in CKM mixing (V_us),
in PMNS mixing (θ₁₂), and in cosmology (Ω_b, n_s).
The axiom “multiple fermion families exist” cuts across
all sectors of the Standard Model except gauge bosons and couplings.
It is the axiomatic choice with the broadest cross-sectional reach in the corpus.

ArXe / ALO Research — February 2026