Simplicial blindness: Three-dimensional perception as a combinatorial theorem of the pentachoron

Réécriture majeure du cadre mathématique et réponse à revue adversariale. Cadre algébrique. Passage du langage des graphes aux complexes simpliciaux. L'observateur est désormais défini comme sous-complexe plein Δ_S de Δ⁴, avec morphisme de restriction et complexe de chaînes. Justification physique renforcée par un argument de clôture causale : le sous-complexe plein est la seule notion de sous-système où l'observateur a accès à toutes les interactions entre ses constituants. Trois nouveaux théorèmes. Theorem 2 (Informational Isolation) : St(w) ∩ Δ_S = ∅, zéro canal vers le sommet absent, 15 simplexes dans le noyau. Theorem 3 (Symmetry Indistinguishability) : S₅ agit transitivement, tous les observateurs de même taille sont isomorphes, impossibilité de reconstruction 4D par zéro incidence inter-tétraédrique. Theorem 4 (Invariant Opacity) : f-vecteur, caractéristique d'Euler, nombres de Betti et spectre du Laplacien identiques pour les 5 plongements — aucun invariant computable ne distingue les cas. Discussion renforcée. Réponse explicite à l'objection de tautologie : le Theorem 1 seul est proche d'une reformulation de la définition, mais la quadruple fermeture ne l'est pas. Nouvelle sous-section "The dimensional gap" distinguant dimension simpliciale, perçue et physique avec formulation conditionnelle explicite. "Each barrier alone would suffice" supprimé et remplacé par "Together, these four barriers render reconstruction impossible." "No experiment" nuancé en "no simplicial invariant." Section neurosciences réduite à deux points de consistance empirique. Balint et perception temporelle supprimés. Renommée "Empirical consistency." Companion script augmenté de 75 à 84 tests (ajout vérification f-vecteur, Euler, Betti, spectre Laplacien). 84 PASS, 0 FAIL.

P22 v3.0 — Changelog Paper: Simplicial blindness: three-dimensional perception as a combinatorial theorem of the pentachoron Author: Jean-Baptiste Blatiere Date: June 2026 Companion script: cecite_simplicielle.py — 84 tests → 134 tests, 0 failures Conceptual revision: the observer as viewpoint Definition 1 (Observer) is reformulated. The observer is no longer a fragment of the pentachoron (a subset S of 4 vertices with the 5th "absent") but a viewpoint: a vertex w ∈ V (the present) from which the perceived world is Δ_{V{w}}. The observer is complete (5/5 vertices); the blindness is perspectival, not structural. All four theorem statements and proofs are unchanged — only the physical interpretation of Definition 1 changes. Correction: kernel count 15 → 16 Corollary 9 (now "Kernel of perception") was missing the k=4 line: the pentachoron {01234} itself contains w and belongs to the kernel. The corrected graded count is (1,4,6,4,1) = C(4,k), summing to 2⁴ = 16, not 15. The binomial symmetry (Pascal's fifth row) is now explicit. Visible = 2⁴−1 = 15, invisible = 2⁴ = 16, total = 31. New: the structure 32 = 16 + 16 The full power set 2^V has 2⁵ = 32 elements. For any viewpoint w, the partition into subsets containing w and subsets not containing w is exactly 16 + 16 — perfect symmetry. The asymmetry 15 ≠ 16 in the simplicial complex arises solely from the exclusion of the empty set. The deficit |ker r| − |Δ_S| = 16 − 15 = 1 is noted as a structural curiosity without independent physical content. New equations (7), (8), (9). New section: The viewpoint and the projection (Section III) Didactic section presenting the geometric picture before the formal proofs. Subsections: "What you see from one vertex" (15 simplices enumerated), "What you cannot see from w" (16 simplices enumerated), "The number 16" (three independent identifications: deg² = Pythagorean deficit = walk surface), "The analogy with perspective drawing" (corridor analogy from P23). Includes the one-line derivation of 3D perception: 5 vertices − 1 viewpoint = 4 faces = 3D (new equation, eq:chain), with clarification that four vertices are in general position by definition of the simplex. New: three figures Figure 1 (fig:pyramid): Tetrahedron with 4 faces in 4 colours. Caption: "Why three dimensions." The chain 5 − 1 = 4 = 3D. Figure 2 (fig:viewpoint, full width): Left panel — the perceived world (15 simplices, blue). Right panel — the invisible structure (16 simplices, red tent fanning from w). Central inequality sign. Bottom proportion bar showing 15 blue vs 16 red out of 31. Figure 3 (fig:graded): Graded bar chart of visible vs invisible simplices by dimension k = 0 to 4. Shows the crossover at k = 2 (invisible exceeds visible) and the absence of visible pentachoron at k = 4. Invisible grading (1,4,6,4,1) = C(4,k) annotated. New theorem: Triple coincidence (Theorem 5) For the complete graph K_N with viewpoint w, three independently defined quantities — the kernel of perception 2^{N−1}, the walk surface (N−1)², and the Pythagorean deficit N²−(N−2)² = 4(N−1) — are simultaneously equal if and only if N = 5. The proof handles both solutions of (N−1)² = 4(N−1): the trivial N = 1 (single vertex, no observer) and N = 5. Corollary: d = N − 2 = 3 is an output, not an input. New axiom (A1') replaces the empirical assumption of three-dimensional perception with the structural condition 2^{N−1} = (N−1)² (perception–gravitation coherence). New subsection: Perception versus reconstruction Clarifies that Theorems 1–4 forbid direct perception of the 4D structure but do not forbid reconstruction by measurement. Newton, Einstein, and LIGO are recast (in the language of the present framework) as examples of inferring the 16 invisible simplices from their effects on the 15 visible ones. The enterprise of physics is characterised as the systematic reconstruction of the kernel of the restriction map. New subsection: Connection to gravity (P23) Explicit bridge to P23: the kernel of perception (16 simplices) equals the gravitational exponent in the Planck gap formula ln(m_Pl/m_e) = 50 + αu¹⁶. The triple coincidence theorem shows this identification is unique to N = 5. New: citation of P4 and the 171 SPARC galaxy rotation curves reproduced with zero free parameters via a₀ = αcH₀/2. New: companion script remark (n = 5) The companion script occupies all five vertices simultaneously (n = 5). It has no viewpoint and therefore no blindness, illustrating the discontinuous jump of Corollary 6. Reconstruction by measurement is the only path available to a physical observer. New open question (#5) Does the viewpoint interpretation admit a formal identification between the selection of a vertex w and the existence of a conscious present? The 32 = 16 + 16 partition suggests a binary act whose cost is exactly 16 invisible simplices. Can this cost be given a variational or information-theoretic formulation? Companion script: 84 → 134 tests Section 9 (28 tests): Kernel = 16 verified for all 5 viewpoints. Graded count (1,4,6,4,1) verified. Pentachoron in kernel verified. Pascal row and 2⁴ = 16 identity verified. Partition 15 + 16 = 31 verified. Section 10 (12 tests): Three identifications of 16 (deg², Pythagorean deficit, walk surface) verified for K₅. ln(16) = 4 ln 2 = 1/α* verified. Connection to gravitational coupling verified. Section 11 (10 tests): Triple coincidence 2^{N−1} = (N−1)² = 4(N−1) verified uniquely for N = 5 among N ≥ 4 (scanned N = 3 to 100). N = 3 confirmed as kernel = walks but ≠ deficit. d = 3 as output verified. Minor fixes (v2.0 final) Cross-references in introduction: hardcoded "Theorem 2/3/4" replaced with \ref{thm:isolation/symmetry/opacity}. Bibliography: P4 added. Test count updated throughout: 84 → 134.