Study 11 Ehrhart Volume Shear

Study 11 — Ehrhart Volume Shear: lattice points vs float volume

Status: LAW FROZEN — 2026-08-22 · LIVE CLAIM
Program: Shear Studies Index · Zero Float · Zero Shear

Surface	URL / key	Cited vs sealed
Claim UI	https://affine.earth/language-game/study-11-ehrhart.html	sealed POST
IDE	https://affine.earth/language-game/ide.html#ehrhart-volume	file template; no auto-POST
Look	https://affine.earth/language-game/#researcher	GET only
Court	POST /language-invariant/game/geometry/ingest polytope_id=unit_square dilation=12	sealed count 169 vol 1/1 · float 1.5 → HTTP 400 REFUSED_FLOAT
Humanity	CAD / digital-twin volume nine cells replay byte-identical	float triangulation 216 is the sealed adversary count

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For every reader

1. Volume is a lattice appointment, not an integral. Count integer points in dilated polytope (tP); the leading coefficient of the Ehrhart polynomial (h_P(t)=|tP\cap\mathbb{Z}^d|) is the exact rational volume — no floats.
2. Continuous geometry shears the count. A float triangulation predicts (\lfloor \mathrm{vol}(P)\cdot t^d\rceil); that prediction misses on every corpus polytope at the sealed dilation.
3. UUM-8D native fit. Eisenstein lattice geometry, integer polytope boundaries, Jordan-bonded state transitions — this is the arithmetic the substrate was built to seal.

Results: Study 11 Results · Corpus: Study 11 Corpus

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The forcing and the track

Piece	Instantiation
Forcing	Integer dilation (t\in\mathbb{Z}_{\ge 0}) applied to a lattice polytope (P\subset\mathbb{Z}^d)
Clock	The dilation index (t) — the appointment is (h_P(t)), not a Riemann sum
Track	Integer lattice (\mathbb{Z}^d); points must land inside exact facet halfspaces
Raw archive	Vertex lists in Corpus — public, deterministic, no floats
Adversary	Float simplex-triangulation volume (\times, t^d), rounded to integer
Future events	Any new lattice polytope registered before its counts are examined

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Why Ehrhart is native to UUM-8D

Ehrhart theory asks: how many integer lattice points live inside a scaled convex polytope? That question is already the substrate's geometry:

• Polytope facets are affine halfspace inequalities with integer coefficients.
• Dilations are exact integer maps on vertex coordinates.
• Volume emerges as the leading coefficient of (h_P(t)) via the (d)-fold forward difference (\Delta^d h(0)/d!) — a pure integer operation on the count table.
• No measure theory, no floating quadrature, no IEEE-754 shear.

The continuous-math habit integrates (P\subset\mathbb{R}^d) with floats and calls the debris "volume." When the same polytope is graded on the integer lattice, float volume misses the sealed count at (t=12) on every corpus member — the shear is measurable.

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Frozen law (sealed 2026-08-22T16:00:00Z)

Symbol	Meaning	Sealed rule
count(P,t)	Lattice points in (tP)	Integer bbox enumeration + exact facet halfspaces
vol(P)	Euclidean volume	(\Delta^d h(0)/d!) as exact rational num/den
T_max	Maximum dilation graded	12
WIN	Count table matches reference Ehrhart polynomial for (t=0\ldots T_{\max}) and vol_fd == vol_exp	—
Adversary MISS	round(float_vol(P) * T_max^d) != count(P, T_max)	Required on every corpus row

No float crosses the seal path. Volume comparisons use Fraction rationals only.

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What this proves

Claim	Verdict
Integer lattice counting recovers exact Ehrhart polynomials	5/5 WIN
Leading coefficient equals exact rational volume	5/5 WIN
Float triangulation volume predicts lattice counts	5/5 MISS (adversary)

The HLE-style polytopes (dimension 2 and 3) from Humanity's Last Exam are in the corpus: (h_{\mathrm{HLE},2}(t)=(t+1)^2), (h_{\mathrm{HLE},3}(t)=1+5t+4\binom{t}{2}+4\binom{t}{3}) with volume (\frac{2}{3}).

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Falsifiers

Falsifier	Effect
Any corpus polytope: integer count (\neq) reference at (t\le 12)	Primary MISS
(\Delta^d h(0)/d! \neq) sealed rational volume	Primary MISS
Float adversary matches count at (T_{\max})	Adversary no longer shears — law must be re-derived

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Read next

• Study 11 Results — grade table
• Study 11 Corpus — vertex pins
• Conjecture Alignment — Ehrhart / QPR / Connes / RH map
• Study 12 — Quantum Parallel Repetition — LIVE CLAIM