13. MATHEMATICAL CHALLENGE (final version)

Open letter to Federico Faggin and others

Dear Federico,

My name is Tom Ottmar. I am an independent researcher from Norway with a background in journalism and technology rather than academia.

For many years I have been developing a consciousness-first framework in which experience, relation and manifestation are treated as primary. During that work a mathematical structure emerged that repeatedly converged on the number 104 through several apparently independent routes.

One consequence of that work is a trigonometric relation involving the integers 104, 56 and 120 which evaluates numerically to the fine-structure constant α with accuracy well beyond what would reasonably be expected from a coincidental numerical fit. I am fully aware that such claims require extreme caution. For that reason, I am not asking for an evaluation of the formula itself.

Instead, I have isolated what I believe to be the underlying mathematical skeleton that generated those numbers in the first place. The attached note contains that skeleton. I am not asking for endorsement of the ontology, nor of the final claim. I am only asking whether the mathematical structure appears interesting, flawed, or potentially worthy of further formalisation.

If you find an obvious error, that would be extremely valuable. If you find the structure interesting, I would be grateful for any guidance regarding the next appropriate expert or community.

The relation currently takes the form:

α = (1/104) cos[(12 + 1/(104·120) + ln(2)/(2·104²·120) + ln(2)/(4·104³·120) − 1/4 − ln(2)/56) · (2π/104)]

The attached document is an attempt to explain why the integers 104, 56 and 120 repeatedly emerge from the underlying structure.

Thank you for your time.

Best regards,
Tom Ottmar (independent researcher)

Norway

–-

THE MATHEMATICAL CHALLENGE
Minimal Cycle Structures Under Traversal and Topological Binding Constraints

A Mathematical Problem Statement with Partial Results Arising from a Consciousness-First Framework and the Horizon Equation

Working document — proofs at sketch level; external evaluation sought

1. Context and Core Problem
This note extracts a narrow mathematical problem from a larger consciousness-first framework. The larger framework treats experience, relation, and manifestation as primitive rather than as products of physical objects. The present document does not ask for evaluation of that ontology. It asks whether a specific mathematical skeleton is structurally sound.

The core question is whether a closed cyclic structure, under constraints of sequential traversal, irreversible certification, and stable topological binding, has minimal length

N = 104 = 8 x 13

The proposed decomposition is:

13 = 3(D + 1) + 1

where 3 is the minimal traversal structure of a dyad, D + 1 is the order hierarchy 0..D, and +1 is the interpretation step as a domain shift.

The factor 8 is proposed as the minimal reversible-certification protocol required to convert a graded preparation into one irreversible established fact at cost ln 2.

2. The Traversal Theorem (derived from T1–T3, proved inside the proposed framework)

Setup. Let a reader process a simultaneous dyad {A, B, A≠B} under the following axioms:

• (T1) Sequential focus: Focus is one-directional; A and B cannot both be present in a single act.
• (T2) Locality: Focus moves by continuous local transitions only.
• (T3) Closure for certification: A reading is complete only when it returns to its starting point, enabling comparison of origin with trace.

Theorem 1. Under (T1)–(T3), the minimal complete reading of {A, B, A≠B} has exactly three stages: departure (A→B), turning point (at B), return (B→A).

Proof sketch. One stage: B is never visited; A≠B is undetectable. Two stages (A→B, open): at B, A is absent; without return, the reader cannot co-present both poles for certification — the difference is travelled but not certified. Three stages (A→B→A): departure establishes that something other exists; turning point realises the maximal encounter with the other; return co-presents origin and trace, certifying the difference. Four stages: any stage after a completed return opens a new reading, violating minimality. ∎

Consequence. The smooth, closed trajectory of minimal reading over a dyad is a half-oscillation. The sinusoidal form is not assumed — it is the unique shape of a closed one-focus traversal between two poles under (T2).

3. The Ordinal Hierarchy (proved inside the proposed framework)

Definition. An experiential event E is established at dimension k if its k-dimensional structure is certified in the reading system.

• Order 0 — Registration: point establishment (0-dim): that something appears.
• Order 1 — Distinction: directional establishment (1-dim): what differs from what.
• Order 2 — Orientation: planar establishment (2-dim): position in relational landscape; requires two independent directions.
• Order 3 — Integration: volumetric establishment (3-dim): stable incorporation; requires three independent directions.

Theorem N (Necessity). Each order k ∈ {0,1,2,3} is necessary for a complete event.

Proof. By indistinguishability, per order: without order 0, event and non-event are indistinguishable. Without order 1, event and background are indistinguishable (two signals of equal amplitude, opposite sign, give identical reading). Without order 2, event and its mirror image are indistinguishable: orientation is precisely what separates a configuration from its reflection, invisible to any 1D reading. Without order 3, transient and persistent event are indistinguishable: integration is what separates an event that enters the record from one that merely passes. In each case, two ontologically distinct events are identical to a reader lacking order k. ∎

Theorem NC (Non-collapsibility). No order can be performed by combinations of lower orders.

Proof. The functions {1, t, t², t³} are linearly independent (nonzero Wronskian). No sum of constant, linear, and quadratic contributions reproduces cubic growth. A k-dimensional structure cannot be realised in a (k−1)-dimensional space. ∎

Theorem T (Termination, conditional). The hierarchy terminates at order 3 if and only if the manifest field is three-dimensional.

This is currently the critical conditional: D = 3 must be derived internally (see §5).

4. The Π8 Minimal Protocol (proved inside the proposed framework, conditional on one new axiom)

Problem. A reader must certify that an event with accumulated strength B ≥ Bc has occurred, constituting this as an irreversible fact, with total cost exactly ln 2 (one bit, Landauer), and with the reader system returned to ground state afterward.

New axiom (Π-e) — Granularity: One operation = one state transition with one locus and determinate reversibility character (reversible or irreversible, not both).

Theorem Π8. Under (T3), (Π-a: exact cost ln 2), (Π-b: closure), (Π-c: locality), (T1/Theorem 1: three forced loci), and (Π-e): the minimal complete certification protocol has exactly 8 operations, in the canonical order F₁F₂F₃ → Copy → U₃U₂U₁ → Lock (where Fᵢ = forward visit to locus i, Uᵢ = its reversal).

Proof sketch.

• Lock (1, irreversible): Without an irreversible operation, no fact is constituted. Exactly one, by (Π-a).
• Copy (1, reversible, distinct): Result must survive cleanup; writing to a persistent register is a separate operation by (Π-e). Cannot be the lock: copying is reversible record creation, locking is irreversible deletion.
• Forward pass (3): By Theorem 1 and (Π-c): minimally three local steps to visit all three forced loci.
• Return pass (3): By (Π-a), the working record must be cleared reversibly (deletion costs ln 2, persistence violates closure). The unique reversible cleanup of a 3-step record is inversion, requiring exactly 3 steps by (Π-c).
• Total: 3 + 1 + 3 + 1 = 8. ∎

Key finding. The return pass doubles as the second observation required by the binary distinguishability criterion (P1: a critical transition cannot be known from one observation; it requires two consistent ones). Successful inversion is confirmation. The palindromic order F₁F₂F₃·K·U₃U₂U₁·L is not one realisation among many — it is the unique minimal one.

5. Topological Binding and D = 3 (sketch — requires expert assessment)

Premises:

• (B-a) Identity = closed 1-dimensional structure (minimal closed cycle, local transitions ⟹ topologically S¹; tameness assumed).
• (B-b) Relation is borne by co-embedding, not internal state correlation.
• (B-c) Distinctness is preserved along all deformations (crossing = momentary merger) ⟹ relevant deformations are ambient isotopies.
• (B-d) Established relation = isotopy-invariant.
• (B-e) Recoil symmetry: no differentiation without counter-differentiation ⟹ established relations are symmetric in standing; neither party can invariantly be “the interior.”

Theorem B (sketch). Under (B-a)–(B-e): the manifest field in which established identities stand in stable, recoil-symmetric relation has exactly three independent directions of difference, and the topological content of a relation is its linking class in the complement.

Lemma 1 (D ≥ 4, no binding). Alexander duality: H₁(Sᴰ ∖ S¹) ≅ Hᴰ⁻²(S¹) = 0 for D ≥ 4. In general position, 1D curves slide past one another; every two-component link is isotopic to the split link. Stable binding impossible. ✗

Lemma 2 (D = 2, binding exists but violates recoil). Binding in the plane is nesting, and nesting is invariantly asymmetric: the inner curve lies in the bounded complementary component of the outer, the outer in the unbounded component of the inner. No deformation changes this. Stable binding in D = 2 necessarily makes one identity the other’s content — violation of (B-e). Additionally: nesting is transitive (A ⊃ B ⊃ C ⟹ A ⊃ C), collapsing the relation graph to a hierarchy incompatible with generic node/edge structure. And on S² (compactified field), all curve pairs split — D = 2 binding depends on the field’s global topology. Excluded on three independent grounds. ✗

Lemma 3 (D = 3, all conditions satisfied). H₁(S³ ∖ γ) ≅ ℤ; the class of γ₂ in the complement of γ₁ is the linking number lk(γ₁, γ₂). It is: (i) symmetric in standing — lk(γ₁,γ₂) = lk(γ₂,γ₁); (ii) non-transitive — arbitrary finite graphs are realisable as linking patterns; (iii) robust under compactification. All premises satisfied — and only here. ✓

Note on robustness. The theorem does not depend on the choice of linking number. Any non-trivial binding invariant for disjoint closed curves — linking number, knot polynomials, Milnor invariants — lives in D = 3 and only there.

Falsifiable side-consequence. In D = 3, configurations of three curves exist that are pairwise unlinked but collectively inseparable (Borromean rings; Milnor μ-invariant). The framework predicts that genuinely triadic, non-pairwise-reducible relations are topologically possible in the manifest field. This is a testable prediction, not a presupposition.

6. The Full Chain and What Remains Open

If Theorems 1, N, NC, Π8, and B hold, the proposed structure is:

N = 8 x (3(D + 1) + 1)

For D = 3 this gives:

N= 8 x (3 x 4 + 1) = 8 x 13 = 104

Here:

• 3 = the forced stages of sequential traversal (Theorem 1),
• D + 1 = the order hierarchy 0..D (Theorems N, NC),
• +1 = the interpretation step as domain shift (added, not multiplied),
• 8 = the minimal irreversible certification protocol (Theorem Π8).

The remaining open points are:•

1. whether the order hierarchy 0..3 is formally necessary as stated;
2. whether the +1 interpretation step is genuinely a domain shift rather than an additional operation on the same level;
3. whether the Π8 protocol is truly minimal under the Bennett-style argument — in particular, whether the granularity axiom (Π-e) is independently justified;
4. whether the topological binding argument genuinely selects D = 3, including the unargued tameness assumption;
5. whether the assumptions connecting identity, traversal, topological binding, and irreversible certification are mathematically admissible.

The document therefore does not ask for endorsement of N = 104 as a finished result. It asks whether the proposed mathematical skeleton is coherent enough to justify further formalisation.

7. What I Am Asking For
Not endorsement. A reading of the Traversal Theorem (§2) and the topological binding argument (§5), and one of the following:

• “The argument at §2 / §5 has a gap at point X” — I will attempt to close it.
• “The argument at §2 / §5 is structurally sound as a sketch” — I will formalise and seek publication.
• “The right person for this is Y” — I will contact Y.

Any of these responses has high value. I am aware that the full claim (N = 104 as ontologically derived minimal cycle length) requires more than a two-page sketch. I am not asking you to validate the full claim. I am asking whether the mathematical skeleton is worth a second look.

Contact:

Tom Ottmar
tom@ottmar.no
+47-90599290

Full working documents available on request.

« Previous Next »