Tamesis Discovery Lab — Adversarial Research Archive

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An interdisciplinary research archive for information, geometry, phase transitions, complex systems, and cognition — with hypotheses kept explicitly separate from evidence.

This repository preserves the complete trajectory of the Tamesis Laboratory: its current experimental branch, and its historical, mathematical, physical, computational, and cognitive research lines. The archive holds 280 audited records, organized into 274 audit dossiers. Auditing here does not turn conjecture into fact — it makes explicit what is a proof, a conditional consequence, a numerical fit, a computational illustration, a conjecture, or a speculative scenario.

Since 2026, the archive has also run a continuous adjudication laboratory (05_DISCOVERY_LAB): every quantitative claim the archive itself makes is closed out, one at a time, against real external references, under pre-registered criteria and mandatory adversarial reproduction. The outcome so far — dozens of catalogued negative closures with a final verdict, and one positive mathematical result independently re-derived and adversarially verified — is synthesized in the Discovery Lab paper (the repository's landing page).

The institutional report Final Vision of the Tamesis Laboratory lays out the questions, answers, impacts, applications, and new questions produced by the research program as a whole. A print-ready HTML/PDF version is also available.

05_DISCOVERY_LAB runs continuous adjudication of this archive's quantitative claims against real external references (PDG, CODATA, Planck, SPARC, Gaia, Odlyzko), with methodology fixed before each computation, full provenance for every reference value, and mandatory adversarial reproduction for any positive finding. Full record: 05_DISCOVERY_LAB/01_PORTFOLIO/TEST_QUEUE.yaml and 05_DISCOVERY_LAB/00_GOVERNANCE/DECISION_LEDGER.yaml. Paper-format synthesis: index.html (the repository's landing page).

flowchart LR
    R[280 audited<br/>archive records] --> S[Archive-wide<br/>Phase-0 survey<br/>19 candidates, 7 areas]
    S -->|18/19 rejected,<br/>concrete reason cited| N1[CLOSED_NULL]
    S -->|1 immature lead<br/>promoted| L13[16 formal<br/>Discovery Lab<br/>test lines]
    L13 --> C8[8 pre-registered claims<br/>locked + adversarially reviewed]
    C8 --> V1["1 proved positive result<br/>U(1/2) limit law"]
    C8 --> V2[7 informative negative<br/>results — REFUTED /<br/>INCONCLUSIVE / NULL]
    style V1 fill:#e8f0e0,stroke:#1f6f5c,stroke-width:2px
    style N1 fill:#f0e5e8,stroke:#7a3b4a

The complete survival funnel (2026):

These four lines open a companion program applying the same discipline to claims from philosophy of mind, cognitive science, geophysics, and formal logic — see PROGRAMA_CONSCIENCIA_LOGICA_E_REALIDADE.md for the full item-by-item mapping, including what was deliberately excluded as non-falsifiable (panpsychism as fact, "God as thermodynamic equilibrium," the Simulation Argument as a testable proposition, and others) and why.

The U₁/₂ universality class (random permutation perturbed at rate c/n toward a random map) has the exact limit law:

φ_∞(c) = ∫₀¹ e^(−ct²) dt = ½·√(π/c)·erf(√c) — zero free parameters,

derived analytically (not fitted), correcting the archive's original conjecture (1+c)^(-1/2) (excluded at the very first series coefficient: a₁ = 1/3 ≠ 1/2, confirmed by exact enumeration). This result is now a proved theorem, not a conjecture: a self-contained mathematical document (THEOREM.md) proves the closed form in six steps, including the correct treatment of the size-biasing of visited arcs, and was reviewed by an independent agent acting as a hostile referee — zero errors found.

The bridge between the finite model and the limit object is now proved unconditionally for every K≥0 — φ(n,c) → φ_∞(c) as n → ∞, for every fixed c ≥ 0, with no unproved hypothesis remaining (THEOREM.md, "Teorema 3"):

flowchart LR
    K01["K=0,1<br/>exact, no gap<br/>waves 1–2"] --> K2["K=2<br/>wave 5<br/>4-layer referee"]
    K2 --> K345["K=3,4,5<br/>wave 6<br/>K-uniform transfer matrix"]
    K345 --> K610["K=6,…,10<br/>wave 7<br/>same method, 5 more rungs"]
    K610 --> Kgen["K general, every K≥0<br/>wave 8: discrete-Gronwall<br/>existence proof, referee SOUND"]
    Kgen --> Teo3["Teorema 3<br/>φ(n,c) → φ_∞(c), ∀c≥0<br/>unconditional"]
    style K01 fill:#e8f0e0,stroke:#1f6f5c
    style K2 fill:#e8f0e0,stroke:#1f6f5c
    style K345 fill:#e8f0e0,stroke:#1f6f5c
    style K610 fill:#e8f0e0,stroke:#1f6f5c
    style Kgen fill:#e8f0e0,stroke:#1f6f5c
    style Teo3 fill:#e8f0e0,stroke:#1f6f5c,stroke-width:2px

Each rung above was independently re-derived by a separate hostile-referee agent, using a different proof technique than the original derivation, its own brute-force enumeration, and full recursion-substitution checks — zero errors found at any layer, across 4 independent referee rounds. K=6,…,10 was additionally confirmed bit-for-bit against fresh exhaustive enumeration at two held-out points. The final rung closed the one remaining named gap: a hostile referee independently re-derived, from scratch, an exact discrete-Gronwall induction proving the finite-n two-term expansion exists for every K, not just the 11 concretely-checked values — verdict SOUND, with named issues (four found, none fatal, corrected via dated addenda). The Open Lemma — now proved unconditionally for every K — is the exact fixed-parameter case of Hansen & Jaworski (EJC, 2014); a Poisson mixture with closed-form erf was not found in a systematic literature search (35+ queries logged), with the explicit caveat that this does not equal "novel." A second front derived why the exponent is exactly 1/2: across an entire parametric family of perturbation mechanisms, α ∈ [1/2, 1] always — α < 1/2 is proved impossible (a quadratic clustering effect that persists even without any cyclicity "death"). Wave 5 also located and confirmed a natural mechanism (M-WEIB(β), non-homogeneous Weibull hazard) that reaches every intermediate α ∈ (1/2, 1). No physical implication is claimed — this is pure combinatorial mathematics on a specific ensemble.

The all-orders closed form (2026-08-23). Every rung above — K=0,…,10, the general-K bridge, the finite-n error constants — is now a corollary of a single exact formula. Extending the same asymptotic-expansion technique to a symbolic order index revealed that the coefficients at every order are exactly the unsigned Stirling numbers of the first kind, and because those are precisely the coefficients of a rising factorial, the entire infinite expansion resums — not asymptotically, but exactly and finitely (it terminates after K+1 terms). The result is one finite, fully explicit expression for the underlying recursion at every n, K, and offset parameter, with an independent elementary proof that needs none of the expansion machinery at all. A dedicated hostile-referee pass re-derived both proofs from scratch against its own from-scratch simulator (215,070 exact checks, zero mismatches), confirmed the headline claim, and caught one real error in a secondary negative claim (corrected via a dated addendum) plus two conservative labels the original document had deliberately left unproved, both of which the referee proved outright. See THEOREM.md, "Estágio 9".

Uniform convergence on the entire parameter range (2026-08-23). The theorem above only says φ(n,c) → φ_∞(c) for each fixed c, one at a time. A separate front closed that gap more thoroughly than asked: the convergence is uniform not just on compact ranges [0,C] but on the entire half-line [0,∞), both proved unconditionally from two short elementary lemmas — a Lipschitz coupling and a uniform-in-n tail bound — that need none of the machinery above. The exact first-order error profile was derived in closed form as a bonus. A hostile-referee pass attacked the two new lemmas and the two unconditional theorems hardest and found no error in either; its one substantive finding corrected which specific gap remained open in a secondary, already-conditional result, making the archive's own account of that gap more accurate rather than less. See THEOREM.md, "Estágio 10".

Conjecture 1 — the full distributional law, now proved for EVERY K≥1 (2026-08-26). Beyond the mean φ_∞(c) above, the archive conjectures the entire density of the cyclic mass conditional on exactly K reroutes: f_{M_K}(x) = 2Kx(1-x²)^(K-1). K=2 closed in wave 14; K=3 closed unexpectedly in wave 15; K=4 closed in wave 16 — each a consecutive surprise against a combinatorial-explosion diagnosis, with the weighted-forest identity E(E+Q)^{n−1}=E behind the off-cycle cancellation named as the candidate route for a general argument. Wave 17 front (a) took that route the rest of the way: every per-K ingredient of the K=1..4 line — the labeled-spacings lemma, the telescoping peel, the destination-mechanism formula, and the weighted-forest identity itself — was generalized to symbolic K, closing Conjecture 1 for every K≥1 at once, conditional to the same single classical citation (PD(1) size-biasing/residual property) the whole line already relies on, applied recursively up to K−1 times per fixed K. As a corollary, Conjecture 2 (the full unconditional law, M(c) =_d min(1,√(E/c))) is also now proved, at the same tier, via the already-cited Poisson-mixture algebra — closing E[M(c)²]=(1−e^{−c})/c and E[M_K²]=1/(K+1) unconditionally for every K. A REINFORCED hostile referee, briefed specifically to attack the per-K-to-general-K jump (residual independence in the telescoping peel), reconstructed the whole argument from scratch — including a from-scratch verification one step past the front's own scope, at K=6 — and found no mathematical error. See THEOREM.md, "Estágio 24":

The sharp constant — supremum now PROVED: sup_K M_K/√K = a* (2026-08-25). THEOREM.md proves lim_{K→∞} M_K/√K = a* ≈ 0.367 exactly (Estágio 13), and, on the third attempt after two failed routes, wave 16 front (b) closed the supremum as well: M_K < a*√K strictly for every K≥1, so the supremum equals the limit (Estágio 19 — via Robbins 1955, the Flajolet–Grabner–Kirschenhofer–Prodinger two-sided bound on the Ramanujan Q-function, and a classical exact identity; the boundary case was closed by the hostile referee itself and independently re-verified). Hypothesis (U') now carries the sharp constant a* in place of the non-sharp a=1+√(π/2)≈2.253. Wave 17 front (b) then executed the one mechanical piece this left open: the Estágio 12 assembly re-run with a* gives Teorema R, the sharp explicit finite-n rate: |φ(n,c)−φ_∞(c)| ≤ [a*√c+κ_B]/n for n≥4, 0≤c≤n, strict, with the additive constant κ_B≈0.2805 unchanged and now certified in pure rational arithmetic, asymptotically tight along c=n (Estágio 22):

The γ-scaling law — proved for every γ∈(0,1] (2026-08-26). A separate open item since wave 4 (Estágios 10–13): does φ(n,γn)/φ_∞(γn) → √(2/(2−γ)) for a fixed ratio γ=c/n, not just fixed c? Wave 17 front (e) proved it in full, via a brand-new exact finite-n double-sum formula for φ(n,qn) derived directly from the model's definition — deliberately not reusing the Estágios 9/12/22 machinery, which was diagnosed and confirmed structurally too weak for this particular ratio limit. Both bonus targets closed too: uniformity on compacts [γ₀,1], and the limit γ_n→0 with γ_n n^{1/3}/ln n→∞. A hostile referee re-derived the core formula by hand before writing any code, and — in a nice bit of cross-checking — both the referee and the orchestrating session independently hit and fixed the same class of catastrophic floating-point underflow bug in their own verification scripts before confirming the result. See THEOREM.md, "Estágio 23":

A new theorem on the joint two-point law — proved (2026-08-26). A separate front took on the obstruction Estágio 18 had located: the joint law of exploring two reference points at once. Its two numeric targets turned out to already be closed by the general-K result above, by an unrelated route — no tension, since the front itself had already shown there is no shortcut from a same/different-cycle split to those targets. What it did prove is new: the Uniform Cyclic Restriction Theorem — conditional on which points end up on a cycle at all, the permutation induced on that realized cyclic set is exactly uniform over all its possible orderings, for every finite n and K — with an exact corollary that two points, given both survive as cyclic, land on the same final cycle exactly half the time. A hostile referee independently re-derived both the theorem and its trickiest lemma by hand, then ran an exhaustive check across more cases than the front itself had covered, finding no error. See THEOREM.md, "Estágio 25".

Wave 18 — three honest non-closures, each with genuine unconditional progress (2026-08-26). Three further fronts attacked items left open above and did not close their central mandates, but each isolated precisely where the remaining difficulty lives. On the γ-scaling law's second-order term C(γ) for γ∈(0,1) (open since Estágio 23): Lemma E proves an exact equivalence between C(γ) and a statement about an exact sum S_n, and Lemma D0 gives a closed form for that sum's "deterministic half", D_0(γ)=(γ-1)/(2(2-γ)), valid for every γ∈(0,1], via Poisson summation/Jacobi theta transformation — a technique new to this line; a second, structurally independent heuristic derivation of the remaining "hard half" matches the wave-17-conjectured closed form exactly, symbolically, which is strong evidence but not a proof. C(γ) for γ∈(0,1) remains open. See THEOREM.md, "Estágio 26". On the n→∞ distributional bridge M_n(c)→_d M(c) (open since Estágio 6, distinct from the mean already closed by Teorema 1): the case K=0,1 is now closed unconditionally, via an exact finite-n CDF-mixture identity (Proposition D0), a full CDF-level version of the archive's Poisson-mixing argument (Lemma R), and an exact closed form P(M_n^{(1)}≤k/n)=k(k+1)/n² (Proposition D1) whose corollaries include this line's first-ever second-moment/fluctuation result — plus a general-K reduction of the second moment to a single scalar (Lemma P2). K≥2 remains open. See THEOREM.md, "Estágio 27". On the continuum-native version of Theorem J (attempted directly in Estágio 25 §6.3 and left open there): a direct construction remains open, but a genuine bypass "by transfer" — exploiting that Theorem J's Corollary (Estágio 25) is an exact identity at every finite n, not merely asymptotic — closes a new continuum theorem, P(same final cycle | K marks) = 1/(2(K+1)), for K=0,1. K≥2 remains open. See THEOREM.md, "Estágio 28". All three fronts passed dedicated hostile-referee review (SOUND / ACCEPT for catalogue); the one error found (a wrong error-term exponent asserted in Lemma D0 — the true rate is Θ(n^{-1/2}), not the exponentially small bound originally claimed; D_0(γ)'s value is unaffected) was corrected via a dated addendum before integration.

Wave 18's remaining front, plus wave 19 — two proofs, one further extension, one sharpened non-closure (2026-08-26). Wave 18's fourth front, dispatched alongside the three non-closures above but integrated afterward, extended the D*^{(p)}_r(b) closed-form error-constant assembly from p=1,...,20 to p=1,...,40 at full scale (r≤200,b≤30), plus an exploratory pass to p=60 at reduced scale — no new mathematical ingredient, pure execution of already-proved machinery, hostile-referee re-derivation via a deliberately different route (Stirling numbers + hockey-stick identity instead of Bernoulli/Faulhaber), 138,040 exhaustive plus 400 randomized checks, zero divergences. See THEOREM.md, "Estágio 29". Wave 19 then closed two more items outright and extended a third. On the γ-scaling law's second-order term C(γ) (open since Estágio 26): Gap 2 — the fluctuation-correction lemma the wave-18 front had called "short, mechanical, not yet done" — is now closed rigorously, in a form stronger than requested: τ(m) turns out to be an exact cubic polynomial in m, giving Δτ(k) as an exact closed form (not a Taylor-remainder bound) valid on the entire range 1≤k≤n, and the weighted sum that actually matters for E(γ) is proved O(n^{-1/2})→0 via a new Lemma G2 (an elementary corollary, by differentiation, of the already-proved Poisson-summation identity behind Lemma D0). C(γ) for γ∈(0,1) remains open — Gap 1 is now, by elimination and direct comparison of difficulty, the sole dominant remaining obstacle. On the joint two-point exploration (open since Estágio 18, named independently as the blocker behind four separate downstream integrations): K=2 is now PROVED — an exact closed form, P_nn(n,2)=(10n²+7n+2)/(30n²), via two new lemmas (a Marked-Point Gap-Structure Lemma; a Two-Source Redirection Structure Lemma) generalizing the Estágio 25/27 case-split method from one marked point to two — closing Estágio 27's second-moment target P_nn(n,K)→1/(K+1) at K=2, and, via the already-proved Theorem J corollary, extending the continuum same-cycle transfer theorem (Estágio 28) to K=2 (→1/6) essentially for free. K≥3 remains open, now with a precise structural diagnosis (a functional redirection graph on the marked arcs, not merely a flat 3×3 table) of why the case-split method does not extend directly. And D*^{(p)}_r(b) was extended again, to p=1,...,80 at full scale (r≤200,b≤30), 261,274 exhaustive checks, zero divergences — p>80 is open only by non-execution. Separately, on the M-CLUST(b) plateau residual (PROOF_DEPENDENCY_MAP.md, node PLATRESUM — a distinct mechanism from the main THEOREM.md line above): a fourth front used the now-exact abstract plateau constant to re-characterize the long-standing abstract-vs-real gap at ~38.8% average (range 35.8%–43.2%), roughly constant across the parameter range rather than concentrated in one regime — a sharp diagnosis, not a closure, that also weakens (without replacing) the two candidate explanations named previously — and numerically strengthened the four-term asymptotic law's next two conjectured coefficients, d4≈26.1246 (vs. the conjectured 209/8=26.125) and d5≈-82.017 (vs. the conjectured -(1546/15)√(2/π)≈-82.235), via a fresh residual-isolation method reaching c=100 through c=655360. All four fronts passed dedicated hostile-referee review (SOUND / SOUND WITH NAMED ISSUES — ACCEPT for catalogue throughout); named issues (mostly imprecise labels; one incomplete magnitude argument) were corrected via dated addenda, none affecting any reported number. See THEOREM.md, "Estágio 29"–"Estágio 32", and PROOF_DEPENDENCY_MAP.md's dated addendum under PLATRESUM (DISC-DEC-083/DISC-DEC-085).

Wave 20 — one full closure, one unexpected second proof, one further honest partial closure, one heuristic gap decomposed (2026-08-26). Four more fronts attacked what wave 19 left open. On Gap 1 of the γ-scaling second-order term (the sole remaining named obstacle to C(γ) for γ∈(0,1) after Gap 2 closed above): a genuinely transcendental object, so unlike Gap 2 there is no exact closed form — but the front delivers a new exact algebraic fact (δ(D)+τ(M)/2 is an exact cubic polynomial in D:=M-γk), a new Bulk/Tail Lemma reducing Gap 1 to bounding two deterministic scalar quantities via monotonicity plus Hoeffding's inequality, and a leading-order asymptotic showing the resulting bound does vanish — not yet a fully explicit, γ-uniform inequality with a named threshold n₀(γ). C(γ) for γ∈(0,1) remains open, but the strategy is now concrete and numerically validated rather than untried. See THEOREM.md, "Estágio 33". On the joint two-point exploration: K=3, diagnosed by the previous front as structurally much harder (a functional graph, not a flat table), is nonetheless now PROVED for the scalar second-moment/same-cycle targets — an exact closed form, P_nn(n,3)=(35n³+38n²+23n+6)/(140n³), via two new mechanisms (governing-source reindexing; cycle-predecessor uniqueness) that answer the prior diagnosis head-on rather than routing around it. The continuum same-cycle transfer pattern 1/(2(K+1)) is now confirmed at K=0,1,2,3; the full CDF of M_n^{(3)} and the general-K joint law remain open. See THEOREM.md, "Estágio 35". On the M-CLUST(b) plateau's H1 heuristic gap (uniform validity of the outer/inner matched-asymptotics decomposition, left untouched by the wave-19 front above): a new rigorous Watson Concentration Lemma reduces H1 to two smaller, individually checkable conditions ((U1), (U2)), plus a new exact ODE for the plateau profile and an extensive numerical uniformity grid showing the approximation ratio converging to 1 monotonically as the layer variable grows — the opposite of what a genuine uniformity failure would look like. H1 remains open, but decomposed rather than monolithic; H2 is untouched. Finally, a fourth front closed, completely, the "intermediate window" n^ε≤c_n≤n^{2/3}/log(n) — named as an open residual since wave 17 and never before attacked directly — via a short, elementary combination of two already-proved results (Teorema R, Corolário 4.2), with no new machinery, plus a bonus strengthening of Corolário 2's γ_n→0 half (its minimum-growth-rate hypothesis is no longer needed for that conclusion). See THEOREM.md, "Estágio 34". All four fronts passed dedicated hostile-referee review (SOUND / SOUND WITH NAMED ISSUES — ACCEPT for catalogue throughout); named issues (mostly cosmetic mislabelings, plus one moderate finding about an undeclared monotonicity assumption in the Gap 1 Bulk/Tail Lemma that deeper checking did not find to actually fail) were corrected via dated addenda. See THEOREM.md, "Estágio 33"–"Estágio 35", and PROOF_DEPENDENCY_MAP.md's dated addendum under PLATRESUM (DISC-DEC-088/DISC-DEC-091).

Wave 21 — a correction to a live constant, and a first explicit inequality for Gap 1 (2026-08-26). Continuing Gap 1 of the γ-scaling second-order term (the sole remaining named obstacle to C(γ) after wave 19 closed Gap 2), the front re-reads Estágio 33's own source derivation and finds a real correction: the truncation constant κ_0(γ)=8/(γ(2-γ)) is a function of γ, not the illustrative constant 2.25 Estágio 33 used — so λ(γ)=4(3-2γ)/(γ(2-γ)) is continuous but unbounded on (0,1) (diverging as γ→0⁺), refuting Estágio 33/its predecessor's claim that λ was "bounded on (0,1)". The corrected, correctly-scoped replacement is proved rigorously: a single C(γ₀) works uniformly on every compact [γ₀,1)⊂(0,1), but no single C works on the whole open interval simultaneously. Separately, the front does construct the explicit ∀n≥n₀(γ) inequality Estágio 33 had left open — closing the logical gap, though not the practical one: with deliberately crude elementary constants, the resulting n₀(γ) comes out astronomically large (~10²¹ at γ=0.99 to ~10⁸⁵ at γ=0.01). No prior numerical result is weakened — the correction affects only a qualitative claim about the shape of the bound, not any catalogued number. C(γ) for γ∈(0,1) remains entirely open. A dedicated hostile referee independently re-derived the corrected algebra from the cited source and confirmed the correction beyond reasonable doubt (one low-severity cosmetic slip found and fixed via dated addendum). See THEOREM.md, "Estágio 36".

Wave 21's remaining front, plus wave 22 — a sharper tail bound, and general-K closed forms through K=8 (2026-08-27–2026-08-28). On Gap 1's practical side: a genuinely sharper, variance-sensitive tail inequality (Bernstein, not Hoeffding) cuts n₀(γ) by up to ~9 orders of magnitude at γ=0.01 (0.44–3.19 decades at moderate γ, negligible loss at γ=0.5, where Hoeffding's assumed worst case already applies exactly) — a genuine partial improvement, not a closure: n₀(γ) remains astronomically large (10^18–10^76), and C(γ) itself remains entirely open. A genuine bonus falls out of the same construction: because σ²(γ)λ̂(γ)→28 stays finite as γ→0, a single γ-independent constant now suffices on the entire open interval (0,1), not just on compacts as under Hoeffding. See THEOREM.md, "Estágio 37". On the joint two-point law: the general-K case-split method (governing-source reindexing + cycle-predecessor uniqueness) is now proved genuinely K-free in its logic — literally the same K=3 proof with 3 replaced by K — closing new exact closed forms at K=4,5,6 (Propositions NN4/NN5/NN6, e.g. P_nn(n,4)=(126n⁴+187n³+177n²+98n+24)/(630n⁴)), cross-checked against true brute force up to 165M/84.7M exhaustive configurations. A single closed-form-in-K formula remains open, precisely diagnosed as symbolic term-count growth in the assembly sum, not a barrier in the method itself; K≥7 was not attempted here, for a stated budget reason. See THEOREM.md, "Estágio 38". A follow-up front then collapsed the double integral for P_disjoint(s,s') — flagged by Estágio 38 as its most concrete open item — to a single integral, with a bonus identity P_same≡P_disjoint, and pushed the same idea through the outer composition sum via an ordinary-generating-function identity, yielding a much faster general-K algorithm (K=1,...,6 in ~1s versus ~166s) that produces two genuinely new closed forms, P_nn(n,7) and P_nn(n,8), each confirmed independently by two distinct routes. The obstruction to a single symbolic-K closed form is now relocated precisely — to a finite sum over r of size K — and, unlike Estágio 38's, is CERTIFIED, not merely observed, not to admit an elementary closed form there: the Gosper algorithm (a genuine decision procedure, not a heuristic) returns None on all three moment types, both for symbolic K and across 13 concrete values K=3,...,15. Both fronts passed dedicated hostile-referee review (SOUND / SOUND WITH NAMED ISSUES — ACCEPT for catalogue); named issues (citation precision; cosmetic labels) were corrected via dated addenda. K≥9 was not attempted, and no claim is made about whether a closed form in K exists or not beyond the Gosper-summable class checked here. See THEOREM.md, "Estágio 39". No claim of progress on any Millennium Problem; pure combinatorial mathematics internal to this file.

Wave 21, redispatched front (b) — the full CDF of M_n^{(3)}, closed (2026-08-28). A separate item left open since Estágio 27 — the full closed-form CDF P(M_n^{(3)}≤k/n), in the style of K=1's Proposition D1 — was redispatched from scratch as a fresh attempt (K3-FULL-CDF-ATTEMPT, v2) after an earlier v1 front stalled with no verifiable return; the abandoned v1 material was kept, not deleted, for audit. The new front closed its mandate and exceeded it: a Full Cycle-Count Decomposition Theorem strengthens Estágio 35's pairwise Lemma 4/5 into the entire joint law of the cyclic-point count T (M_n^{(3)}=T/n) — T=O+Σ_{s∈S}V_s, with S⊆{0,1,2} the random set of cyclic sources and, given S, the V_s mutually independent and uniform, governed by four new closed forms for the law of S (Proposition S). From this falls the headline result, Proposition D3: a single closed-form rational function, uniform in n, for P(M_n^{(3)}≤k/n) at every n≥3 and every integer k — matching Proposition D1's ambition exactly, one universality class further out. Five corollaries follow: P(M_n^{(3)}=1)=6/n³ by direct proof; exact symbolic recovery, with zero symbolic remainder, of the already-proved finite-n mean φ_n^{(3)} (Estágio 4); second/third-moment limits matching the already-proved continuum values 1/4 and 16/105 (Estágios 17/18); and a uniform O(1/n) convergence-rate bound. A hostile referee — extending true brute force past the front's own tested range, to n=9, and independently re-deriving Proposition S's four formulas over all 4³=64 raw cases — found no mathematical error; one informational finding (the Estágio 4 mean formula, stated for n≥4, in fact also holds exactly at n=3) and one provenance question about the abandoned v1 attempt, resolved directly by the orchestrating session: the flagged v1 files held only single-point pmf formulas (P(T=n-2), P(T=n-3)), not a competing CDF, and one of those two formulas was itself wrong — consistent with v1 having been correctly abandoned mid-work, not silently completed. Verdict: SOUND WITH NAMED ISSUES — ACCEPT for catalogue. M_n^{(3)}'s full CDF is now the first complete closed-form CDF catalogued on this line beyond K=0,1 (Estágio 27); the general-K full CDF (K≥4) remains open, not attempted by this front. No claim of progress on any Millennium Problem; pure combinatorial mathematics internal to this file. See THEOREM.md, "Estágio 40".

Wave 23 — the small-K CDF gap closes completely, a genuine general-K decomposition theorem, and an honest correction on the M-CLUST(b) residual (2026-08-28). Three fronts ran in parallel. Front (b) took on exactly the generalization Estágio 40 flagged but did not attempt: the Full Cycle-Count Decomposition Theorem (T=O+Σ_{s∈S}V_s, mutually independent given S, each V_s uniform) and a new Proposição S, general K — one closed formula, free of both K and |A|, P(S=A)=|A|!·∏_{a∈A}p_a·(p_D+Σ_{a∈A}p_a) — are now PROVED for every K≥0, with genuinely K-free proofs (not merely checked at many K values), via a new Key Lemma established by strong induction on |B|. As a bonus, this single formula exactly reproduces Estágio 40's four separate K=3 formulas as the special cases |A|=0,1,2,3 — revealing they were always one formula in disguise. What this does not close: a single closed-form-in-(n,K) CDF formula (the general-K analogue of Proposition D3) was not attempted beyond a small proof-of-concept that the underlying algorithmic machinery works for general K. See THEOREM.md, "Estágio 41". Front (a), running independently, closed the one small-K gap this line still had: Proposição D2, a single closed-form rational function P(M_n^{(2)}≤k/n)=k(k+1)(2n²-3n+k-k²)/[n³(n-1)] for every n≥2 and every integer 0≤k≤n-1, via the same method as Estágio 40 but structurally simpler — only one combinatorial regime is needed, not three, since the generic formula already holds exactly at the boundary k=n-1, confirmed by a zero-remainder symbolic check. Corollaries include P(M_n^{(2)}=1)=2/n² by direct proof, exact symbolic recovery (zero remainder) of the already-proved finite-n mean φ_n^{(2)} (Estágio 3), agreement with the already-proved continuum second/third moments 1/3 and 8/35, and a uniform convergence bound |F_n^{(2)}(x)-F_2(x)|≤12/n. Together, the full closed-form CDF is now closed for all of K=0,1,2,3; only K≥4 remains open at the closed-formula level (Estágio 41's general-K decomposition theorem proves the underlying machinery works for every K, a structural advance short of an actual closed CDF formula). See THEOREM.md, "Estágio 42". Both fronts passed dedicated hostile-referee review — SOUND, no mathematical error found in either, only low-severity informational findings. A third front (c) attacked the M-CLUST(b) plateau's H1 heuristic gap (a distinct mechanism from this line — see PROOF_DEPENDENCY_MAP.md) via a Volterra-equation reformulation of the exact renewal identity, producing genuine new content — a derivative-free algebraic identity, a correctly-diagnosed linear Volterra structure on a Banach-space-valued domain, and new convergent Neumann/Picard grid numerics — but its central diagnostic claim, that the full kernel's boundedness "depends entirely" on an unbounded multiplication operator, was wrong: the hostile referee found the front had only bounded that operator in isolation, never the actual composed operator appearing in the kernel, and — once corrected — the composed operator, and hence the full kernel, is in fact globally bounded and does not grow, the opposite of what the front claimed. H1/(U1)/(U2) remain open, unaffected by the correction; the front's algebra, Volterra-structure diagnosis, and numerics stand. Verdict: NEEDS REVISION on that specific diagnostic claim, corrected via a dated addendum, not a rejection of the front's contribution as a whole. See PROOF_DEPENDENCY_MAP.md's dated addendum under PLATRESUM (DISC-DEC-113). No claim of progress on any Millennium Problem; pure combinatorial mathematics internal to this file.

Where to find everything: the full theorem and referee reports live in 05_DISCOVERY_LAB/02_TESTS/CORE_NUMERICS/u12_universality/theorem/; the generalization and its adversarial verification in .../generalization_u_alpha/; a standalone reproducible package — compiled LaTeX paper (PDF), self-contained proofs, clean-room simulations, and 49 automated tests — is at tamesis-cycle-survival/. And the honest table of everything this laboratory has tried and did not survive — so this one positive result is read in the right context — is in FAILED_HYPOTHESES.md.

An honest survey of the whole Tamesis archive (not restricted to TRI-RG, 19 candidates across 7 areas) closed CLOSED_NULL — 18/19 rejected with a concrete cited reason — and promoted the one immature lead found (cognitive EEG spectral signatures, depression vs. anxiety) to a new candidate line. Its operationalization stage is complete (observable defined as normalized Shannon spectral entropy, a named competing model, computed statistical power, verified real data access for the depression arm) — the anxiety arm remains blocked on a data provider requiring human login, honestly reported as such; no real data has been computed there. See 05_DISCOVERY_LAB/02_TESTS/ARCHIVE_PHASE0_SURVEY/SURVEY.md and 05_DISCOVERY_LAB/02_TESTS/COGNITIVE_EEG_SPECTRAL/OPERATIONALIZATION.md.

The program investigates whether systems under finite resources can build additional layers of organization when the cost of that complexity is offset by a reduction in error, dissipation, instability, or future search cost. This is a modeling principle, not a purpose attributed to nature.

The laboratory connects four levels:

1. Mathematics: operators, spectra, topology, graphs, universality, and regularity.
2. Fundamental physics: information, geometry, holography, gravity, particles, and quantum-to-classical transitions.
3. Complex systems: thermodynamics, memory, irreversibility, networks, stability, and control.
4. Life and cognition: the integrated organism, brain-computer interfaces, consciousness, and cognitive ecosystems.

flowchart LR
    A[Finite resources] --> B[Layers of organization]
    B --> C[Memory and control]
    C --> D[Regime transitions]
    D --> E[Observables and tests]
    E --> F{Independent evidence?}
    F -->|yes| G[Publishable result]
    F -->|no| H[Revisable hypothesis]
    H --> B

_{Figure 1 — Working illustration of the holographic principle. This is a modeling hypothesis, not evidence that the universe is holographic or simulated.}

• Discovery Lab scientific paper (2026) — adversarial adjudication and the U₁/₂ limit law (repository landing page)
• tamesis-cycle-survival/ reproducible package — compiled LaTeX paper, proofs, simulations, and automated tests for the U₁/₂ theorem
• FAILED_HYPOTHESES.md — the honest table of every hypothesis tested and not surviving in this laboratory
• Final vision report of the laboratory
• HTML version for presentation and PDF
• 280-article audit report
• Rigorous audit protocol
• Machine-readable inventory manifest
• Freeze status and resumption conditions
• Project state in JSON
• Timeline
• Archive map
• Navigable home page
• Interactive hypothesis atlas
• Proof dependency map for the U₁/₂ line

The table below estimates, line by line, how much of the gap named in each central question has already been characterized — not the probability that the hypothesis is correct, nor a metric computed by the laboratory. It is an external reading, calibrated against the real state documented for each line (RELATORIO_VISAO_FINAL_LABORATORIO_TAMESIS.md §6 and 05_DISCOVERY_LAB/), with one important correction to the original inventory: Line D must be read in two parts. The U₁/₂ subset, rigorously adjudicated by the Discovery Lab, is well advanced; but Line D as a whole — which in the original report also includes U₀, U₂/Lindblad, the general class atlas, and topological applications — has not advanced in the same proportion: the laboratory's own archive-wide survey (DISC-ARCHIVE-PHASE0-SURVEY-001) records that U₀ and U₂, unlike U₁/₂, never reached a closed-form candidate. Treating "Line D" as 85% resolved would be exactly the kind of conflation this archive's discipline exists to prevent.

How not to use this table. A "90%" does not mean a 90% chance that the U₁/₂ class is correct, nor that Line D is close to done — it means that, of the gaps explicitly named in that specific question, most have already been proved or precisely characterized. The general-K regularity caveat that was the last named gap of the main line closed on 2026-08-22 (DISC-DEC-040); remaining research capacity on D — U₁/₂ now goes to the separate M-CLUST(b) residual (still PARTIALLY CLOSED) and to the genuinely open, non-central questions named above.

flowchart TD
    A[Hypothesis] --> B[Operational definitions]
    B --> C[Mathematical or computational model]
    C --> D[Parameters, units, and uncertainties]
    D --> E[Null model and competitors]
    E --> F[Pre-registered test]
    F --> G{Result}
    G -->|replicates and distinguishes| H[Publication / state update]
    G -->|does not distinguish| I[Revision or abandonment]
    G -->|fails| J[Documented falsification]

This cycle is the archive's editorial rule. A simulation that reproduces a curve is not automatically a discovery; a numerical coincidence is not a derivation; and an analogy between systems is not a physical identity.

The current experimental branch is frozen at frozen_and_ready, with hardware qualification not yet started. Demonstrator A begins with blind optical thermometry calibration between 5 K and 20 K; it does not yet measure M_c.

• Tamesis M_c v1 README
• Demonstrator A v0.6 execution report
• Visual outputs, figures, and animations
• Experimental collaboration package

_{Figure 2 — Limit map used as a testing guide. Regions and markers represent hypotheses and reference data; they do not constitute confirmation of a universal boundary.}

_{Figure 3 — Conceptual visualization of compression, saturation, and reorganization. This is a model illustration, not a general empirical law.}

The laboratory uses a common language to compare systems: state, resources, couplings, memory, transition, dissipation, stability, observable, and failure criterion. The comparison is methodological — it does not claim that a galaxy, a cell, a graph, and a brain are the same kind of object.

• a complete, traceable inventory and audit of 280 records;
• an explicit separation between proof, hypothesis, model, fit, simulation, and speculative scenario;
• an atlas of regimes, transitions, operators, networks, and cognitive systems;
• a catalog of observational and experimental tests with null models;
• an institutional HTML/PDF version for academic presentation;
• preservation of historical versions without endorsing their claims as current results;
• complete adversarial adjudication of the core's quantitative claims (2026): 30+ claims closed under pre-registered criteria, including the detection and correction of 2 legacy headline results built on fabricated data;
• one new mathematical result, derived and adversarially verified: the exact closed-form limit law φ_∞(c) = ½√(π/c)·erf(√c) of the U₁/₂ class (see the paper);
• two replicated findings about the real zeros of the Riemann zeta function (consecutive-gap anti-clustering; minimum-gap GUE scaling).

The archive does not claim to have solved the Riemann Hypothesis, P vs NP, Navier–Stokes, Yang–Mills, Hodge, or Birch–Swinnerton-Dyer. There is likewise no accepted demonstration that Tamesis replaces ΛCDM, eliminates dark matter/dark energy, gives consciousness a causal role in quantum collapse, enables metric propulsion, or proves the universe is a simulation.

These lines remain conjectures, test programs, or restricted models until they produce formal proofs, independent data, new predictions, and replication.

Scientific direction, primary authorship, and curation of this archive: Douglas H. M. Fulber.

The Tamesis Laboratory is run as an independent research program within this repository. Mentions of universities, laboratories, authors, or DOIs in historical documents do not imply institutional endorsement, co-authorship, or external validation unless explicit authorization and record exist.

Editorial governance follows these rules:

1. the responsible maintainer controls status classification, line organization, and acceptance of structural changes;
2. external contributions are welcome but do not alter authorship, provenance, or evidence status without a recorded review;
3. new results must include method, data/code where applicable, uncertainties, a null model, limitations, and a falsification criterion;
4. legacy documents remain for provenance and are not automatically promoted to valid results;
5. any derived publication must cite the laboratory, the author/curator, and the specific archive version used.

To propose a collaboration or correction, open an issue/patch documenting: affected file, justification, sources, impact on classification, and a verification test.

Original material in this archive is available under Creative Commons Attribution 4.0 International (CC BY 4.0), unless stated otherwise in the file itself or subject to third-party rights. The license allows sharing and adapting the material as long as attribution is preserved and modifications are indicated.

Recommended attribution form:

Douglas H. M. Fulber, Tamesis Laboratory — Tamesis Research Archive, version/commit used, licensed under CC BY 4.0: repository.

When reusing a figure, preserve the caption, the asset path, and the indication that it is a model visualization when that is its recorded classification. Third-party images, data, or text may be subject to their own conditions; CC BY 4.0 does not transfer rights the laboratory does not hold.

• Do not present archive conjectures as established facts.
• Do not use the presence of a DOI as proof of peer review or experimental validation.
• Do not attribute institutional endorsement to universities or groups cited without formal authorization.
• Do not hide limitations, fitted parameters, negative results, or failure conditions.
• Do not use this material for medical, legal, financial, or safety advice without independent professional evaluation.

Fulber, Douglas H. M. (2026). Tamesis Research Archive: Tamesis Laboratory — vision, audit, and research program. CC BY 4.0.

The recommended entry point is a documented issue in this repository. For academic presentation, use the institutional HTML/PDF report and the full Markdown report, always preserving the indicated evidence classification.