Finding 2: Wall #2 ∫E(t)dt unconditional bound — 4/4 universal NO
← All English posts · 한국어 · 2026-05-02
Empirical observation, not a theorem. Same logical structure as Finding 1.
Statement
Across the 4 paper-direct candidates relevant to the Newman–de Bruijn forward heat flow, no unconditional + constructive + RH-independent bound on ∫_0^Λ E(t) dt has been published.
Source: lemmas/lemma2_wall2_axiom_alpha_ceiling.md.
Audit table (paper-direct anchored)
| # | Paper | axiom α strict | Paper-direct anchor |
|---|---|---|---|
| 1 | Polymath15 (de Bruijn–Newman upper) | NO | Λ ≤ 0.22 conditional (3-tool combination, Theorem 1.1). Unconditional bound not provided. |
| 2 | Rodgers–Tao 2020 (Λ ≥ 0 unconditional) | NO | “far from optimal” (paper §1.5). ∫_{Λ/2}^0 E(t)dt — backward only control, forward not given. |
| 3 | Platt–Trudgian 2021 (RH up to H = 3·10¹²) | NO | Λ ≤ 0.2 via numerical RH up to H = 3·10¹²; theoretical bound not provided. |
| 4 | Newman 1976 (Λ ≤ 0 ⟺ RH equivalence) | NO | Definition only; abstract equivalence, no unconditional bound. |
4/4 universal axiom α strict NO.
Falsifier search (5 adjacent fields)
Beyond the 4 direct candidates, the lemma searches 5+ adjacent fields:
- Bombieri–Lagarias 1999: Λ ≥ 0 lower bound. Upper bound not provided. Not a falsifier.
- Selberg method (mollifier): Wall #3 (50% barrier). Not directly connected to ∫E dt. Not a falsifier.
- Bourgain–Gamburd–Sarnak expander: heat semigroup form similar, integrated bound shape not present. Not a falsifier.
- Otto’s calculus / Wasserstein gradient flow: 007 negative resolved (time-symmetric vs Wall #2 asymmetric). Not a falsifier.
- Concentration compactness (Lions–Brezis): limit point analysis, forward control absent. Not a falsifier.
- Free probability R-transform: Wall #6 axis (LOCAL-GLOBAL), not Wall #2. Not a falsifier.
No falsifier found across 5+ fields.
Specialist Δ — anchored to paper §-end quotes (Lemma 7 protocol)
- Tao + Conrey (analytic): Polymath15 §6 “this is the limit of the present method” — combinatorial-optimization internal ceiling. Iwaniec phrase “extra little tiny bit” (same essence, Wall #4).
- Tao (hard analysis): Rodgers-Tao 2020 §1.5 “control integrated energies that resemble ∫_{Λ/2}^0 E(t) dt” + “far from optimal”. Time-asymmetry essential — backward only.
- Logician (S9): Lagarias 2002 (RH Π_1) — anchor for measuring axiom α’s logical strength.
Caveats (the project’s own)
- Empirical only (4/4 + 5 falsifier fields). Necessary universal NO not proven — same induction-leap warning as Finding 1.
- 5-year flat (Rodgers–Tao 2020 → 2025) is an empirical fact, not an obstacle proof.
- ZFC-independence of axiom α not ruled out.
- Newman 1976’s Λ = 0 ⟺ RH equivalence is abstract; whether axiom α has a constructive form provable in ZFC is undetermined.
Falsifier (lemma retraction conditions)
Any paper providing all four:
- Unconditional ∫_0^Λ E(t) dt explicit upper bound — paper-direct quote.
- RH not assumed — paper-direct verification.
- Constructive form (not abstract equivalence) — paper-direct.
- No fine-tuning parameter — paper-direct quote or explicit definition.
…retracts the lemma.
Cross-reference to Finding 1
Same logical structure, different wall. Lemma 9 (Wall #5, axiom 6) and Lemma 10 (Wall #2, axiom α) share:
- 4-specialist consensus on the axiom definition
- Paper-direct audit table
- Falsifier search across adjacent fields
- Explicit falsifier criterion
- Empirical NO ≠ necessary NO warning
This was deliberate (Cycle 2 “directly reused Cycle 1 format”). The reuse is the project’s evidence that the codification template is generalizable.
Audit trail (Layer 1)
lemmas/lemma2_wall2_axiom_alpha_ceiling.md— full auditattempts/185_cycle2_*— Cycle 2 (lemma generation)attempts/028_*,106_*,113_*,132_*,161_*,173_*— per-paper readings
Refuting this finding
If you have a paper providing an unconditional + constructive ∫E(t)dt upper bound, please email x2ever.han@gmail.com.