1. Executive summary
Of the lines explored for the CKM matrix, one was discarded due to a mathematical error (not a methodological weakness), and the remaining three are ranked by measured specificity, not assumed:
| Line | Revised status |
|---|---|
| δ = arctan(169/77) | Structural (mediator rule, 6/7) — official formula declared by axiomatic choice, not by grammatical tie-break (see §2j). π discarded by systematic search |
| θ₁₃ = arctan(1/286) (or arcsin(15/(e·11²·13)), more precise) | Strong — keep, continue exploring |
| θ₂₃ = arctan(1/24) (or arctan(9γ/125), more precise) | Moderate→strong with the corrected formula |
| θ₁₂ = arctan(3/13) (or arcsin(44/(3·5·13)), to verify) | Weak with the simple formula — review with the source document |
| δ = arctan(13/6) | Superseded — retired in favor of 273/125 |
| δ = sec(θ)+csc(θ) = 3 | Discarded — calculation error |
| δ = arctan(√(π+φ)) | Discarded — π already tested and lost via prior systematic search |
| Mass formulas for 13/24/286 (ARTYICULO_4) | Discard as evidence |
| 137 ≈ 11×13 − 6 | Discard |
2. δ_CKM: discarded by direct verification, not by weakness
The four original documents (ARTYICULO_2, ARTYICULO_3, CKM_DEPPSEEK, positive_results_v1) claim:
sec(θ) + csc(θ) = 3 → θ ≈ 64.96° ≈ 65°and present as verification: sec(64.96°) + csc(64.96°) ≈ 2.364 + 1.104 = 3.000.
That sum is wrong: 2.364 + 1.104 = 3.468, not 3.000. Verified numerically:
sec(64.96°) + csc(64.96°) = 3.4664 (not 3.000)The real solutions of sec(θ)+csc(θ) = 3 are:
θ = 33.83° or θ = 56.17° (symmetric about 45°)Neither falls near δ_CKM = 65.0° ± 1.5°.
A search was also run for whether any other integer n solves the problem (not just n=3), scanning the θ>45° branch:
n=3 -> θ = 56.17°
n=4 -> θ = 70.09°The jump from n=3 to n=4 passes over 65° without touching it. The value of n actually needed to land in the experimental range is:
n ≈ 3.36 – 3.60 (n ≈ 3.47 for the central value 65.0°)There is no integer in that interval. The equation sec+csc=n has, for no integer n, a solution near δ_CKM. This isn’t a case of a poorly chosen n — the premise that a small integer cleanly solves this is ruled out by direct calculation.
Conclusion: retire this line from the active corpus. If δ_CKM is revisited in the future, it must start from zero, without inheriting this equation.
2b. Two later replacement attempts for δ, and why only one survives
2b.1 arctan(√(π+φ)) — discarded by the corpus’s own framework, not by statistics
arctan(√(π+φ)) was tested, which numerically falls within the experimental range:
arctan(√(π+φ)) = 65.375° (within 65.0° ± 1.5°)But arxe_espirales_estocasticas_mezcla.md establishes, as a baseline observation with no recorded exceptions (§1, §5): φ does not appear in any CKM angle, and explicitly classifies δ as “CKM order one — no constant, pure rational ratios” (§5). Introducing φ into δ is not just another weak hypothesis — it directly contradicts that document’s central prediction (§4.2: quark confinement, closed BC, structurally prevents CKM mixing from carrying φ). It is discarded for framework incompatibility, without needing to run the specificity test.
This also reaffirms that any new candidate for δ should be sought within the arctan(a/b) family with integer a, b — consistent with “pure rational ratio.”
2b.2 Rational search for δ — why the naive search doesn’t work, and what survives a stricter one
First attempt (permissive): fractions a/b with small integers (a,b ≤ 60) were searched, and separately, within the broad “structural” universe (products of up to 4 factors from {2,3,7,11,13}, same as in Section 3). Result:
71 fractions a/b (a,b<=60) fall in the range 65.0°+/-1.5°
53 ratios from the broad structural universe fall in the same rangeThis is far more than the 19 competitors for θ₁₃ or the 11 for θ₂₃, despite δ’s relative tolerance (±2.3%) being the same order as θ₂₃’s (±2.1%). The cause is structural, not a method failure: tan(65°)≈2.14 is a “generic” value — far from 0° or 45°, a zone where the density of small fractions that approximate well is naturally high. A search of this type can’t yield useful information for δ, because almost anything falls nearby.
Second attempt (strict): the universe was restricted to square-free products of up to 3 distinct arities from {2,3,7,11,13} — the same kind of “clean” number that makes 286 = 2×11×13 interesting (distinct arities, no repeats, no powers). That universe has only 26 values. Result:
1 clean ratio falls in the range 65.0°+/-1.5°:
arctan(13/6) = 65.225° (difference: +0.22° from the center 65.0°)With a direct structural reading in the corpus’s own vocabulary: 13 = mediator (T⁻⁶), 6 = 2×3 = time (T¹) × generations:
δ = arctan(13/6) = arctan( mediator / (time × generations) )Status: preliminary. Unlike θ₁₃, this line still has no independent cross-check (it wasn’t checked, for example, whether “13/6” or “6” appears elsewhere in the corpus without having been searched for specifically for this purpose). It is the sole survivor of a strict filter, which clearly distinguishes it from the 71-53 alternatives of the permissive search — but “sole survivor of my own filter” is, in any case, a different kind of evidence than the numerical specificity θ₁₃ offers within its own universe (§3). Recorded as a candidate to pursue, not as an established result.
2c. SUPERSEDED: arxe_mezcla_CKM_PMNS_hipotesis.md already had a far superior formula
Upon incorporating the full source document (not previously available for this review), two facts emerged that entirely change this line’s status:
1. An already-established, much more precise rational formula exists:
δ = arctan(3×7×13 / 5³) = arctan(273/125) = 65.3981°
Target (PDG 2023): 65.4° Δ = 0.0019° error = 0.003%Verified numerically. Compared to arctan(13/6) = 65.225° (Δ=0.175–0.225° depending on target used), 273/125 is approximately 90 times more precise.
2. Specificity test, with the full lexicon {2,3,5,7,11,13} and simple powers: of 22 ratios that fall in the experimental range (65.4°±1.5°), arctan(273/125) is the best by a wide margin — the second best (169/77) has 55 times more error.
Note (v5, August 2026): this test was run against 65.4°, a value that doesn’t correspond to either current LHCb measurement (Nov. 2025: direct 62.8°, indirect 66.3°). A systematic reassessment against both real targets reverses this conclusion — see §2g. The original text is kept here unedited, for traceability; §2g is the version currently in force.
3. π was already tested systematically against δ, and lost. arxe_mezcla_CKM_PMNS_hipotesis.md §5.1-5.2 ran a search with all the ArXe lexicon constants (π, φ, e, γ, √2, √3) against the 4 CKM angles. For δ, “no constant” won — π was never the best option for any CKM angle. This includes having tested π explicitly, not merely by omission.
Consequence: the entire line of exploration in this session on π+φ for δ (memory/memoryless, entropy, Bhattacharyya angle) is retired as a numerical candidate to explain δ — not because the reasoning was invalid, but because the corpus had already tested and discarded that direction before we started. arctan(13/6) is also retired, superseded by 273/125.
What is still worth keeping: Grammar_s_en.md already formally distinguishes π (closed-BC ratio of T³, “ternary ambiguity”) from φ (open-BC ratio of T⁻¹) — the same closed/open partition this session reconstructed independently via stochastic processes (memoryless Buffon vs. golden-mean shift with constitutive memory). It’s a genuine cross-validation of the general principle, even though it doesn’t resolve δ numerically. See arxe_dos_logicas_estocasticas_pi_phi_v1.md §4 for the full development of that distinction.
2d. θ₁₂ verified — confirmed, with more modest specificity than θ₁₃
θ₁₂ = arcsin(2²×11 / (3×5×13)) = arcsin(44/195) = 13.0406°
Target (PDG 2023): 13.04° Δ = 0.0006°Verified numerically. Compared to arctan(3/13) = 12.9946° (Δ=0.0454°): 75 times more precise.
Specificity test (broadened universe — total degree ≤4 across the lexicon’s 6 arities, necessary because 44=2²×11 and 195=3×5×13 don’t fit a smaller universe): 89 combinations fall within the same tolerance margin (13.04°±0.5°) — far more than θ₁₃’s ~19 competitors within its own universe. 44/195 remains the best or tied for best (273/1210, practically as precise but with messier factors: 1210=2×5×11²).
Honest reading: the formula is correct and the best available, but specificity isn’t as strong as θ₁₃’s — the full lexicon with powers mechanically generates many more comparable candidates. A good result, not an overwhelming one.
2e. θ₂₃ verified — confirmed, and with strong specificity (best result of the three verified today)
θ₂₃ = arctan(3²×γ / 5³) = arctan(9γ/125) = 2.3798°
Target (PDG 2023): 2.38° Δ = 0.0002°Verified numerically. Compared to arctan(1/24) = 2.3859° (Δ=0.0059°): ~30 times more precise.
Specificity test: unlike δ and θ₁₂ (purely rational search), here the constant γ is fixed and what varies is the rational coefficient multiplying it. Within the same broadened lexicon, it was checked how many ratios a/b produce arctan((a/b)×γ) within 2% relative of the target:
28 candidates fall within the margin — 9/125 is the BEST, by a wide margin:
9/125 = 0.07200 -> 2.3798° (the established formula, the most precise)
50/693 = 0.07215 -> 2.3848° (second best, ~20x more relative error)Honest reading: this is the most solid result of the three verified in this session — 9/125 isn’t just the best, it wins by a clear margin over the second option, unlike θ₁₂ where there was a near-tied competitor.
2f. θ₁₃ verified — the strongest result of the four (δ, θ₁₂, θ₂₃, θ₁₃)
θ₁₃ = arcsin(3×5 / (e×11²×13)) = arcsin(15/1573) = 0.200998°
Target (PDG 2023): 0.201° Δ = 0.000002°Verified numerically — matches exactly what’s reported in the source document. Compared to arctan(1/286) = 0.200334° (Δ=0.000666°): ~300 times more precise — and 1/286 had already been, until today, our strongest candidate of the whole session.
Specificity test: 16 candidates fall within 2% relative of the target coefficient — the fewest competitors of the four verified formulas (δ: 22, θ₁₂: 89, θ₂₃: 28, θ₁₃: 16), and 15/1573 wins by the widest margin of the four (the second best, 1/105, has almost 6 times more error).
Conclusion: θ₁₃ combines the two qualities we had been chasing separately in different lines of this session — maximum precision and maximum specificity. (A structural coincidence with Ω_Λ was also cited as a third leg; reassessed in §4 — consistent but not surprising, no longer counts as independent additional evidence.) Of the four CKM formulas, it best holds its own weight.
2g. δ_CKM — reassessment of the specificity test against the real experimental targets (v5)
The §2c test was run against 65.4°±1.5°, a target that doesn’t correspond to any real measurement — it’s an assumed point. LHCb (Nov. 2025) reports two measurements that don’t agree with each other:
Direct: γ = 62.8° ± 2.6°
Indirect: γ = 66.3° (+0.7/-1.9)°The same lexicon {2,3,5,7,11,13}, total degree ≤3 (the definition that reproduces 273=3×7×13 and 125=5³), was run systematically against the three targets:
| Target | Lexicon winner | Δ of winner | Is it 273/125? | Margin over 2nd best |
|---|---|---|---|---|
| 65.4° (assumed, legacy) | 273/125 = 65.3981° | 0.0019° | Yes | ~34× (not 55× — the real second best is 98/45, not 169/77, which the original search missed) |
| Real direct (62.8°) | 286/147 = 62.7975° | 0.0025° | No | 6.4× — PMNS-level, not “wide margin” |
| Real indirect (66.3°) | 385/169 = 66.3004° | 0.0004° | No | 124.7× |
Honest reading: against the two real data points available today, 273/125 isn’t even the best candidate from its own lexicon — two other, different fractions win, one for each target. This isn’t evidence that 273/125 is wrong (it still falls within both error bands, see arxe_delta_CKM_outsider_read_en.md), but it does show the search lexicon is dense enough to fit almost any target between 60° and 67° to within thousandths of a degree. That is precisely the retroactive-fitting risk the corpus has been flagging as something to watch.
Consequence for classification: δ moves from “strong” to preliminary, at the same level of caution already applied to δ_PMNS (see arxe_delta_PMNS_prediccion_registrada_v1.md). It is not retired from the corpus — the strength label that isn’t backed by the current data is retired. If the experimental band narrows to less than ±1° (see the equivalent protocol for PMNS), this test must be repeated; with a band that narrow, the number of candidates drops sharply and the result would start to be genuinely informative.
Full detail in arxe_delta_CKM_specificity_reassessment_v1.md.
2h. δ_CKM — why “preliminary” is no longer the right label (v6)
§2g left δ as “preliminary, pending better data” — a label that implies more experimental precision could resolve the question, the same as with δ_PMNS. The canonical protocol (arxe_protocolo_canonico_especificidad_v1.md) shows that isn’t what’s happening here.
The same test was run under four differently sized lexicons, each fixed by an independent criterion (22 and 55 values in arxe_delta_CKM_specificity_reassessment_v1.md; 84 values with the “minimum sufficient” criterion degree≤3; 462 values with degree≤5). In all four cases, 273/125 is not the best available candidate — it loses to 98/45, 286/147, 385/169, or 169/77 depending on the target and lexicon. This is not sensitive to the search window size or to the exact experimental value used as target (legacy 65.4°, PDG global fit 65.5°, direct 62.8°, indirect 66.3°) — it loses against all four. A specificity problem that survives that much variation in the protocol is not a “not enough precision yet” problem — it’s a problem of the ArXe lexicon, in the region where δ lives (tan(δ)≈2.1-2.2, a “generic” value, far from 0° or 45°, where the density of well-fitting fractions is high for purely arithmetic, not physical, reasons), being too dense for the numerical fit alone to discriminate anything. More experimental precision doesn’t change that density.
What does support δ, and continues to support it: the mediator-13 rule, derived and independently verified in arxe_derivacion_13_largo_alcance.md (7/7, no exceptions, with δ included in the original verification) and applied specifically to this case in arxe_delta_CKM_inverse_reading_v1.md. That is structural/grammatical evidence, not numerical — and is unaffected by anything found in this review, because it never depended on a fit-specificity test.
Correct classification: δ = arctan(273/125) has strong structural support, zero numerical support. These are two different kinds of evidence (see the Mode 1b vs. Mode 3 distinction in two_modes_of_explanation_in_arxe.md v1.1) and shouldn’t be averaged into a single “preliminary” label. If the corpus’s criterion is that structural evidence (Mode 1b) can support a formula on its own without numerical backing, δ should be classified as genuinely different from the other six — not weaker by default, but a different category not yet defined in the corpus. That definition (does the structural argument alone suffice, or is the numerical one also required?) is the real pending question, and it precedes any improvement in experimental data.
2i. δ_CKM — final resolution: grammatical filter over the numerical competitors (v7)
§2h left open the question of whether structural support alone suffices. arxe_criterios_lectura_gramatical_v1.md (two principles: rarity scale by distinct arities, not total degree; and non-paradoxical form of arity↔phenomenon identification) allowed a second filter — grammatical, not numerical — to be run over the four candidates that at some point won the numerical fit under some lexicon size: 98/45, 286/147, 169/77, 3993/1820.
All four are excluded, each by a rule already derived before this test:
| Candidate | Reason for exclusion | Rule applied (pre-existing) |
|---|---|---|
| 98/45 | Doesn’t contain 13 | Mediator rule (arxe_derivacion_13_largo_alcance.md, 7/7 verified) |
| 286/147 | Contains 11 (EM domain, foreign to the weak sector) | Domain assignment of 11=REG in Grammar_s_en.md |
| 169/77 | Same problem as 286/147 — and is the best numerical fit under degree≤3 | Same rule — winning the number isn’t enough |
| 3993/1820 | Uses all 6 base arities at once + a high power of a non-low arity (11³) | Rarity-scale principle v1.1 — a jump in logical order unrelated to the phenomenon’s magnitude |
273/125 is the only candidate that violates none of these rules. Full detail in arxe_delta_CKM_grammar_filter_resolution_v1.md.
Final classification — corrected (August 2026): δ = arctan(273/125) has structural support (satisfies the mediator rule, now 6/7 — see arxe_regla_mediador_13_correccion_v1.md), but this section’s original conclusion — that it was “the sole survivor” — doesn’t hold. Reviewing the grammatical reading of θ₁₃ CKM (the corpus’s strongest formula) found that it also uses arity 11, which removes the “EM domain foreign to the weak sector” objection that excluded 286/147 and 169/77. The three candidates (273/125, 286/147, 169/77) today all satisfy the mediator rule without violating any other established criterion. There is, for now, no criterion distinguishing among them — see arxe_delta_CKM_grammar_filter_resolution_v1.md v1.2 §2b. Still open, unresolved, and not forced: the factor 7 (CPX) in 273/125 has no convincing reading — see arxe_delta_CKM_inverse_reading_v1.md §3.
Note (v10): this section stands as a historical record of the analysis’s state at that point — see §2j for the final resolution, distinct from the conclusion here.
2j. δ_CKM — definitive resolution: declared axiomatic choice (v10, August 2026)
§2i left three candidates tied under the grammar filter, with no tie-breaking criterion. arxe_delta_CKM_arqueologia_axiomatica_v1.md picked up the B1_Inverse_ALO_Axiomatic_Archaeology program (unfinished until then) and found that the three weren’t competitors within the same map — each fits almost exactly (Δ<0.005°) to a different experimental target, and very poorly (Δ>0.7°) against the other two:
| Candidate | Native target | Δ | Nature of the target |
|---|---|---|---|
| 273/125 | 65.4° | 0.0019° | Legacy estimate, no longer in force |
| 286/147 | 62.8° (LHCb direct) | 0.0025° | A single isolated experimental technique |
| 169/77 | 65.5° (PDG, global fit) | 0.0050° | Synthesis of the entire available experimental program |
Choice declared by Diego Luis Tentor (August 2026): the PDG global fit is adopted as ArXe’s official reference target for δ_CKM, for being the synthesis of the complete experimental program, not an isolated technique. This is declared as a choice — not as a finding derived from the numerical or grammatical filter, both of which had left the three candidates tied.
Official formula, from this version onward:
δ_CKM = arctan(169/77) = arctan(13²/(7×11)) ≈ 65.5050°273/125 is not removed from the corpus — it remains documented as the candidate native to the legacy estimate, superseded as the official reference but not discarded as an error (see arxe_delta_CKM_arqueologia_axiomatica_v1.md §3). This is the same no-deletion logic already applied to the φ reading of θ₂₃ PMNS (arxe_PMNS_epistemic_status_v2.md §5).
Grammatical reading of the official formula — 169/77 = 13²/(7×11)
Distinct arities: {7,11,13} — three, one fewer than the four in 273/125 ({3,5,7,13}), and no low-arity power used as architectural filler (no 5³ as in the previous formula). Under the rarity-scale principle, this is at least as coherent a composition, and somewhat more compact.
- 13² (SING squared): satisfies the mediator rule (presence of 13 in a 1↔3 jump). The power deserves a separate reading, in non-paradoxical form: not “13² IS double mediation,” but that δ structurally doesn’t parametrize a single jump but the interference among the three legs of the unitarity triangle (already noted in
arxe_derivacion_13_largo_alcance.md§7, note on δ) — it is reasonable, not forced, to identify that interference with a repeated appearance of the mediator. Marked as a reasonable reading, not an established derivation — an open point, just as honestly as 273/125’s factor-7 open point. - 11 (REG): general-purpose mediator, already confirmed in θ₁₃ CKM, Ω_Λ, and α⁻¹=137 — no domain objection.
- 7 (CPX): the same unresolved cross-cutting pattern that appears in δ, θ₂₃ CKM, θ₁₂ PMNS, and θ₂₃ PMNS (
arxe_lectura_gramatical_cinco_formulas_v1.md§6) — not a new problem for this formula, it’s the same inherited problem. - What is lost relative to 273/125: CYC (3, “connecting ends”) is no longer present, which had a reasonable reading for a phenomenon of mediation between generations. Not a serious loss — that reading never had mediator-13 backing, it was the softest of the three that 273/125 had — but it’s recorded as part of an honest balance, not omitted.
Balance: 169/77 is not strictly “better” than 273/125 grammatically — it’s different, with its own strong point (more compact composition) and its own open point (13² without established derivation, instead of the 7 without established derivation). What makes it the official formula is not structural superiority — it is the choice declared above, of the experimental target it represents.
For the remaining three lines the same check was run: given the universe of “structural numbers” the corpus itself declares as meaningful (products of up to 4 factors from {1, 2, 3, 7, 11, 13} — the set including 2, 3, 7, 11, 13 and their combination 24 = 2³×3), how many fractions a/b from that universe fall as close to the measured angle as the proposed formula?
| Angle | Measured value | Proposed formula | Structural alternatives just as good | Is it the best in the universe? |
|---|---|---|---|---|
| θ₁₃ | 0.200° | arctan(1/286) | 19 fall within tolerance (±0.02°) | Yes — 1/286 is the closest of all |
| θ₂₃ | 2.38° | arctan(1/24) | 11 fall within tolerance (±0.05°) | Among the top 3, not the only one |
| θ₁₂ | 13.02° | arctan(3/13) | 26 fall within tolerance (±0.5°) | No — arctan(28/121) = 13.029° fits better |
Reading: θ₁₃ is the only line where the proposed formula isn’t just “one among several similar ones,” but the best option available within the corpus’s own universe. θ₁₂, in contrast, shouldn’t keep being described as “established”: there’s an alternative within the same framework (28/121) that fits better than 3/13.
4. Cross-check with Ω_Λ — reassessed (August 2026, see arxe_omega_lambda_resolucion_v1.md)
The denominator of θ₁₃ (286 = 2×11×13) shares the factor 11×13 with the dark-energy formula, now reconciled after a version contradiction that existed in the corpus (arxe_omega_lambda_resolucion_v1.md):
Ω_Λ = 7² × 2 / (11 × 13) = 98/143This section is corrected relative to its original version. The coincidence had been presented as “unforced” in the strong sense — evidence that something real and structural connects both results. A later finding changes that weight: 11 and 13 also appear, with the same mediating role between levels, in the derivation of α⁻¹=137 (arxe_core_221_en.md §10.3, Layer C: 137 = 11²−7²+5×13) — a third physical domain (electromagnetic fine structure), completely unrelated to quark mixing or cosmological density.
If 11 and 13 are the lexicon’s general mediation operators — not specific to any domain — their appearance in three contexts requiring mediation isn’t surprising, it’s expected. The correct argument is no longer “this is a sign of real, unsought structure”; it’s “this is consistent with the already-established role of 11 and 13, and therefore doesn’t add extra weight to the formula by itself” (see rarity-scale principle, arxe_criterios_lectura_gramatical_v1.md §1, and axiomatic relativity, §3).
This doesn’t weaken θ₁₃. Its real strength — numerical specificity (16 candidates, wide margin within its universe) and maximum precision — doesn’t depend on this coincidence and remains intact. What changes is that θ₁₃ no longer needs to lean on the crossover with Ω_Λ to hold up as the corpus’s strongest line — and it’s better that it doesn’t, because that leg was weaker than originally presented.
5. Why the ARTYICULO_4 mass formulas are discarded
The method used there — (mass_ratio_A / mass_ratio_B) {+,-,×,/} k, with k a free integer between 0 and 20 — was replicated over all simple quark mass ratios (PDG), checking how many combinations hit each target with error <1%:
| Target | Combinations found with error <1% |
|---|---|
| 13 | 430 |
| 24 | 60 |
| 286 | 99 |
With a search space that size, finding some combination that fits isn’t evidence of structure — it’s what’s expected from any method with a free parameter and six masses to combine. These formulas are retired from the body of results; they may be kept as a separate speculative note, explicitly labeled as non-discriminating.
For the same reason, 137 ≈ 11×13 − 6 is discarded: it’s a one-free-parameter (the “−6”) fit to a single known data point, with no independent check.
6. Unresolved contradiction: arities vs. golden ratio
CKM_DEPPSEEK proposes, for θ₁₂ and θ₂₃, alternative formulas based on φ (golden ratio):
θ₁₂ ~ arctan(1/(φ²+1))
θ₂₃ ~ arctan(1/(φ³+φ))ARTYICULO_2 tests these same formulas and discards them for fitting worse than the arity-based ones — but arxe_CKM_positive_results_v1 doesn’t mention the φ alternative at all, presenting the arity-based line as if there were no competition. Both branches can’t be “established” simultaneously for the same angle. Noted here; not resolved in this document.
7. Final status table
Strength ranking of the four CKM formulas (verified §2c-2f), from strongest to weakest:
| Parameter | Established formula | Value | Δ vs. PDG | Competitors in universe | Strength |
|---|---|---|---|---|---|
| θ₁₃ | arcsin(15/(e·11²·13)) | 0.200998° | 0.000002° | 16 | Strongest — maximum precision + minimum specificity (the Ω_Λ coincidence, §4, no longer counts as an extra leg — see reassessment) |
| θ₂₃ | arctan(9γ/125) | 2.3798° | 0.0002° | 28, wins by a wide margin | Strong |
| θ₁₂ | arcsin(44/195) | 13.0406° | 0.0006° | 89, nearly tied with 2nd best | Good, not overwhelming |
| θ₁₂ | arcsin(44/195) | 13.0406° | 0.0006° | 2-14 depending on protocol, 53× margin under the canonical protocol (v6) | Strong — corrected in v6, not “good, not overwhelming” (the v5 universe was miscalibrated, see §2h and canonical protocol) |
| δ | arctan(169/77) | 65.5050° | 0.0050° against the official target (PDG, global fit) | Doesn’t win the numerical fit against alternative targets; satisfies the mediator rule; official formula by declared choice (v10, §2j) | Structural + declared axiomatic choice — 273/125 and 286/147 remain documented as native to other targets, not discarded |
Note (v6): θ₁₂ is corrected — v5’s “89 competitors, nearly tied” used an ad hoc broadened universe (see audit, arxe_criterios_ambiguos_auditoria_v1.md §2); under the canonical protocol fixed beforehand, θ₁₂ wins by a 53× margin, just as strong as θ₂₃ and θ₁₃. δ is no longer ranked “at the bottom of the table” as if it were the weakest on the same scale — it’s a different type of evidence (structural, Mode 1b) that isn’t degraded for lacking numerical backing, but also shouldn’t be averaged alongside formulas that do have it. See §2h and arxe_protocolo_canonico_especificidad_v1.md.
Lines discarded or superseded along the way:
| Item | Final status |
|---|---|
| δ = sec(θ)+csc(θ)=3 → 65° | Discarded — calculation error, no integer solution near any n |
| δ = arctan(√(π+φ)) | Discarded — π already tested and lost via systematic search in arxe_mezcla_CKM_PMNS_hipotesis.md §5.1-5.2 |
| δ = arctan(13/6) | Superseded by arctan(273/125), ~90x more precise |
| θ₁₂ = arctan(3/13) (ARTYICULO_2) | Superseded by arcsin(44/195), 75x more precise |
| θ₂₃ = arctan(1/24) (ARTYICULO_2) | Superseded by arctan(9γ/125), ~30x more precise |
| θ₁₃ = arctan(1/286) (ARTYICULO_2) | Superseded by arcsin(15/1573·e⁻¹), ~300x more precise (though it was already ARTYICULO_2’s strongest candidate) |
| Mass formulas for 13/24/286 (ARTYICULO_4) | Discarded — method with no discriminating power (60–430 comparable alternatives) |
| 137 ≈ 11×13−6 | Discarded — one-free-parameter fit to a single data point |
φ vs. arities (θ₁₂/θ₂₃, flagged by CKM_DEPPSEEK) |
Resolved in fact — the source document’s formulas (no φ in any CKM angle) are strictly better than the φ candidates that had been tested |
| Undecidability/entropy bridge for δ | Priority lowered — failed its own calibration against θ₁₃/θ₂₃ (see arxe_indecidibilidad_entropia_puente_v1.md §3b); kept as a conceptual idea, not a numerical mechanism |
8. Suggested next steps
Find an independent cross-check for δ = arctan(13/6)— obsolete: δ already resolved with arctan(273/125), see §2c.Verify the θ₁₂ formula from the source document— done:arcsin(44/195) = 13.0406°, Δ=0.0006°, confirmed. Good specificity but not as strong as θ₁₃ (89 competitors); see §2d.Verify the θ₂₃ formula— done:arctan(9γ/125) = 2.3798°, Δ=0.0002°, confirmed. Strong specificity — best result of the three, wins by a clear margin over the second option; see §2e.Verify the θ₁₃ formula— done:arcsin(15/(e·11²·13)) = 0.200998°, Δ=0.000002°, confirmed. The strongest of the four in both precision AND specificity (16 competitors, the lowest number); see §2f.- Review the full PMNS matrix from
arxe_mezcla_CKM_PMNS_hipotesis.md§4-5 with the same verification and specificity standard — it carries already-resolved formulas with φ winning in all three angles. - With the four CKM formulas now verified and ranked by strength (θ₁₃ > θ₂₃ > δ > θ₁₂), update the general summary table (§7) to reflect this order.
- Resolve the φ/arities contradiction for θ₁₂/θ₂₃ flagged by
CKM_DEPPSEEK— probably already resolved in fact by the source document’s more precise formulas, but worth stating explicitly. - Do not reintroduce the ARTYICULO_4 mass formulas unless a version appears with non-post-hoc-adjustable parameters.
- (v7) Closed — δ has structural + grammatical support, without needing numerical support. v6’s methodological question (does the structural argument alone suffice?) is answered: yes, when it also survives the grammatical filter against all known numerical competitors. The real open point, unresolved, is the factor 7 with no convincing reading (see §2i).
- (v7) Apply the same two-stage grammatical filter (numerical as floor + grammatical as selector) to the next formula added to the corpus, not just the canonical numerical protocol — both filters, in that order, should be a requirement before calling any new formula “established.”
- (v5) Apply the same systematic verification (code, not manual search) to the rest of the corpus’s formulas that report specificity figures — the δ case shows a manual search can underestimate the real number of competitors.
ArXe Research — August 2026
Diego Luis Tentor
Review and numerical verification assisted by Claude.ai (Anthropic)