Finite Contact Theory

Succession notice (August 2026). The closure-theoretic content of this program --- when a finite representation may lawfully close over what it touches --- has been absorbed and generalized into Receipt Theory / Finite Answerability (Apsiape/receipt-theory, doi:10.5281/zenodo.21911510), where the one-use event floor persists as the flagship model system. This repository is the program's founding record and is preserved as released; the experimental worlds program continues on its own track.

One rule: a finite thing touches what it cannot fully take in, and the contact leaves a mark that can be used once. From that floor — no Hilbert space, no probability, no spacetime granted at the start — the program grows tiny universes and measures them with the discipline of experimental science. This release follows a single distinction all the way up: from the identity root (equality is an event, not a standing fact) through a law tower where every law is generated twice over, through a forced chain from a coherence anomaly to the 24-cell and E₈, to the program's central registered wager — the next i: a physical unit that is exactly zero to every complex-quantum observable yet nonzero to the next detector stratum. Every result ships as a dependency-free script; the flagship path runs in about a minute, the full suite in a few.

A note from the author

I'm not a physicist. I work in AI and software and follow physics as an outsider. It didn't start as a theory of physics: it started with one question I couldn't put down — what is the least that has to exist when something finite touches something it can't fully take in? A finite thing makes contact with something larger than itself; the contact is real but partial, so it leaves a difference, and that difference can't just be wiped away.

The way I chased it is unusual: over about seven months I built a setup where several AI models work together as a research team and pointed it at that question. The hardest part was never the physics; it was getting the models to reach for something new instead of repeating what they were trained on. This is the piece that has been made public so far, with more built out privately that I'll share as it's ready. An AI-assisted theory from someone outside the field is easy to doubt — that's fair, and it's exactly why everything here is scoped, labeled, and runnable. If I've gotten something wrong, I want to know: apsiape@gmail.com.

Quick doors: start here (plain language) · plain-language glossary · the correction ledger — what we killed, in public · how to read this repo · what this says about reality · open problems — some framework-free · how to contribute (try to kill something) · run it yourself

The program at a glance

The goal is a taxonomy of inevitability: which features of physical law would appear in any universe grown from a generative floor, and which are peculiar to ours — answered not as philosophy but as a measured table. The lattice-era chapters (37–48) delivered the first full artifact of that goal: a permission map across four genuinely rival dynamics, run through one matched instrument suite. This release (chapters 62–67) does something different. It takes the one distinction the floor cannot avoid and follows it, rung by rung, as far as it goes — and it goes remarkably far.

The spine of this release — one distinction, followed all the way up

1. The identity root: equality is an event

The whole program used to posit a tolerance relation on the floor — a way of saying when two marks count as the same. This release removes the posit. From one root — using a thing's identity is an event, never free — the tolerance relation is now derived: tolerance is exactly equality minus free transitivity. Classical mathematics quietly assumes identity is a standing fact you may reuse at no cost; the floor charges for it. That single change is what makes everything below it forced rather than chosen (verified by the shipped census script; the twin-census correction is scored in public — see the ledger below).

2. The law tower: every law generated twice over

One distinction, iterated, generates a tower. Its even rungs are laws and its odd rungs are appearances, and the striking fact is that each law is produced two independent ways at once:

• as the carry of the appearance just below it (a Bockstein operator — the arithmetic "carry bit" of the elementary distinction), and
• as the shadow of the appearance just above it (a differential).

Both are ordinary, named classical operators, and the shipped script checks the identity that ties them together exactly (DU + UD = 1, with ker D = im D = im Frobenius at the prime 2 — the uniqueness is a theorem at model scope). No law is unsourced. The operator center of this tower is a recovery of a classical result (Revoy 1973); the doubling — laws as both carry-below and shadow-above — is, as far as our blind sweeps found, not something the literature states, because no incumbent framework even asks where laws come from.

3. From a coherence anomaly to the 24-cell and E₈

Follow the tower's coherence structure and a chain forces itself, each link a recovery of a classical object, the sequence a derivation within the model:

• a graded coherence anomaly (the pentagon edges evaluate a coboundary; 0 of 256 possible dressings survive without i, which flattens the curvature),
• through the quaternion group Q₈ (the minimal anticommuting lift),
• to the 24-cell — minimal, with chirality selecting 24 over 48 (the binary tetrahedral group, not the binary octahedral one),
• to a forced E₈ closure (even, unimodular, self-dual in dimension 8), with the residual choices enumerated exactly: evenness, chirality, and a triality glue.

This is not the McKay correspondence. McKay sends the binary tetrahedral group to E₆ and E₈ to the binary icosahedral group; our route is glue-forcing and lands differently — the chapter ships a firewall subsection saying so, and a note distinguishing this E₈ from the E₈ of Kitaev-type topological-phase physics. The exceptional objects that "keep appearing" in physics for no stated reason appear here as the closure shells of the law transports (verified by the shipped derivation-chain script; constituent facts cited to Conway–Smith, SPLAG, Nikulin, EGNO).

4. The carry engine: powers of two are only the ground floor

The famous powers-of-two structure that shows up across the interference hierarchy turns out to be the zero-carry stratum of a full ladder. The program derives a carry-depth filtration — the 2-adic valuation of the Catalan numbers equals the binary digit sum minus one (ν₂(Cat(n−1)) = s₂(n) − 1; Alter–Kubota, Deutsch–Sagan, Kummer) — and the physically-loaded reading is that each carry depth is a stratum of its own. The modulus ladder 2^{s₂(n)} is derived three independent ways (binomial parity, Catalan-tree parity, and a formal-group [n]-series) — the triple derivation is the artifact control — yet the physical ladder still ships labeled CONJ/registered, because a clean derivation is not yet a measured fact. Honest about that; loud about it.

5. The next i — the program's central registered wager

Here is the boldest claim in the program, stated plainly and at its weakest accurate label.

The ordinary imaginary unit i did not describe a new measurement. It made an impossible real equation (x² = −1) ordinary by enlarging the number system — what was forbidden by the ontology of the reals, not by any inconsistency, became a legal object one level up. The program proposes the next enlargement. Call it ℑ₂: a physical unit that is exactly

ℑ₂ ≠ 0, yet every height-1 (complex-quantum) observable reads ℑ₂ = 0.

A root of invisibility — the statement that zero is detector-relative. It subsumes every wager the program has generated (law memory invisible to the process tensor at every finite depth; the faceless/essential classes; the certified exact twins; the commutator witness) into one principle: a quantity can be identically zero to one class of detector and real to the next. The analogy to i is structurally faithful — i was forbidden by real arithmetic's ontology, not its consistency; ℑ₂ is forbidden by height-1 ontology the same way.

This comes with a registered experimental program, not a discovery:

• spoof-closed witnesses — designs (a commutator plus a sector-conditional correlator) built specifically to close the known route-phase spoofs (the underlying interference phenomenon is Oi 2003; our contribution is the spoof-closure design),
• an arity-visibility ladder — a faces-of-15 architecture in which a quantity visible only at detector arity k is exactly zero to every arity below it,
• a near-term photonic validation at the subthreshold end, runnable against existing linear-optics apparatus.

Zero empirical divergences are claimed, to date. Every seam that points outside complex quantum mechanics is a registered prediction with a frozen protocol, a null, and a kill condition — a bet, not a result. The height-2 identification underlying the frame currently stands on one leg (the stabilizer architecture and a Galois/chirality alignment), and the decisive in-model computation — extracting a formal group law from the program's own composition algebra and reading its 2-series — is named in print as OPEN. Nothing chromatic headlines a proof it hasn't earned.

What this explains better

Drawn from the release's absorption map (tiers A and B): each row names a question, the incumbent's standing, the program's answer, and the weakest accurate label. "Explains better" means adjudicable against a named incumbent — more structure forced, or a puzzle dissolved, with fewer posits — not a feeling.

The question	Incumbent's status	The program's answer	Label
Why both bosons and fermions?	Spin-statistics assumes the dichotomy	One pre-sign relation DU+UD=1; commutator and anticommutator are its only two resolutions	THEOREM (model scope)
Why complex numbers / why i?	Complex QM postulated ("why complex?" open)	i = the unique local resource that flattens even law-curvature to gauge; four convergent derivations	THEOREM (model scope)
Why the exceptional objects (Q₈, 24-cell, F₄, E₈)?	Treated as unexplained coincidences	Derived as the closure shells of the law transports; chirality gates 24 vs 48	derivation within the model (recovery of each constituent)
Where does contextuality's exclusivity come from?	CSW / KS posit exclusivity	Derived from two conservation laws (no-duplication + no-unreceipted-erasure)	EXTENSION / registered-novel (hazard: not a no-cloning repackaging — stated in print)
Why does everything hard break at 3?	Noted across fields, never unified	One answer: first assembly loop, first non-PSD matrix, first carry depth, first theater of transitivity all at n = 3	expository unification
Where do laws come from?	No incumbent asks	Doubly generated: carry of the appearance below + shadow of the appearance above (two classical operators, exact)	THEOREM (model scope)
The measurement problem	Copenhagen/Everett/GRW add collapse, branching, or a constant	Fork is real, selector is received; no collapse dynamics, no branching ontology, no new constant	adjudicable reorganization (Tier B)
The arrow of time	Boltzmann derives it from a low-entropy past	Placed exactly as the reading-direction of an unoriented tower — RECEIVED, not derived (honest)	adjudicable reorganization (Tier B)

The full map grades seventeen hard-problem faces across four tiers, including an explicit "not close" honesty row (quantum-gravity dynamics, cosmology, and any confirmed empirical divergence — still zero). See Chapter 67.

The corrections ledger is a feature

This program kills its own results in public and ships the kills. A theory that never discards anything is a mood, not a method. Three from this release:

• The twin census (Chapter 62). An early count of the exact "twins" (finite objects the floor cannot tell apart) conflated two distinct cases. The conflation was caught, the census re-derived, and the corrected count shipped with the script — the old headline does not survive under softer wording.
• The chromatic height-2 reading (Chapter 67). The exponent test cannot tell repeated height-1 doubling from a native height-2 series, so the height-2 identification is demoted to a single-legged conjecture, gated in print on extracting a formal group law from the program's own algebra. The frame ships; the proof it would need does not yet exist, and we say so.
• The carry engine's first mirror (Chapter 66). The naive squared mirror did not square to zero; it required a Cartan repair, and the transverse sector came back non-cohomological. Two honest nulls, shipped in one chapter.

The correction ledger records every such kill with its residue, its current public status, and the exact condition to reopen it.

How we work

The ambition is large, so the rules are strict — the discipline is the method:

• Predictions are frozen before engines run. Expected numbers go into the design document first; the worlds regularly beat our bets, and a killed bet is scored in print.
• Nothing load-bearing rests on one implementation. Every claim that matters gets a second engine or a blind solver; about a dozen artifacts have been caught this way, several of them our prettiest.
• Blind novelty sweeps run before every release. Repo-barred referees grade each claim KNOWN / KNOWN-ADJACENT / PLAUSIBLY-NEW, and the public labels follow their verdicts. Recoveries are stated as recoveries — a recovery here means a piece of standard physics or mathematics re-derived from a floor that never assumed it, and that is the win condition, not a concession.
• Exact arithmetic only. Rational numbers and certified windows — no floating-point result is ever load-bearing.
• Kills are public, and scored proudly (see above).

How to challenge this

Every heavy claim in this release has a dependency-free script that re-derives its core exact results from scratch (standard-library Python, no numpy, exact arithmetic). Run them. Each script ends with an explicit falsifiability comment — the exact condition under which it would print [FAIL] and exit nonzero. If a claim's script passes and you think the claim is still wrong, the falsifiability line tells you precisely where to push. A reported kill with a runnable witness is the most valued contribution the program accepts (see CONTRIBUTING.md); write to apsiape@gmail.com.

Start here

1. The next i — Chapter 67: the program's central registered wager and the absorption map, at honest labels.
2. Equality is an event — Chapter 62: the identity root, and the tolerance relation derived rather than posited.
3. The law tower — Chapter 64: every law generated twice over, checked by one exact identity.
4. From anomaly to E₈ — Chapter 65: the forced derivation chain, with the McKay firewall.
5. The permission map — Chapter 37: the lattice era's whole question, answered in one table (10 minutes).
6. The divergence — Chapter 10 · Chapter 11: the registered bet against the Hilbert Gram cone.
7. The full index — papers/ (67 chapters) · theorem bank · claim register · glossary · how to read.

Run it yourself

No dependencies beyond Python's standard library. The flagship path — one representative exact result per major line — takes about a minute:

python verification\scripts\run_all.py --fast

The full suite (parallel, with per-script progress) takes a few minutes:

python verification\scripts\run_all.py

Expected final line: ALL SHIPPED VERIFICATION: PASS. Every shipped result line has one dependency-free script, all exact arithmetic. Release hygiene (scope language, links, overclaim phrases, the ceiling quoted verbatim) is itself audited independently and does not launch mathematical verification by default:

python scripts\release_audit.py

To request a coupled gate explicitly, use --verification fast or --verification full.

The fine print — the live ceiling

One sentence bounds everything this release is allowed to claim. It is quoted identically here, in the claim register, the synthesis chapter, and the release notes, and the audit enforces it verbatim.

The scope fence is the result, not an apology. In particular: complex quantum mechanics in full, the actuality of one outcome, the universal Born rule, the height-2 identification, whether nature realizes ℑ₂ or any of these structures, and every confirmed nature-facing claim are open, and named as such.

The ceiling — what is and is not claimed

Finite Contact Theory is a finite reconstruction program whose newest stack begins one layer below its own floor — the tolerance relation the program had always posited is derived from a single root, that identity is an event and is never free — and builds upward through an exactly verified tower on one generator, in which even rungs are laws and odd rungs are appearances, every law is generated twice over (as the Bockstein carry of the appearance below it and as the derivative shadow of the appearance above it), and the first law is at once the sign of exchange, the square of the generator, the factor set that glues the integers modulo four, and the first Witt carry; a two-sector theorem splits the fourth rung into gluing data reachable from faces and an essential class reachable from none; the essential class, through its anticommuting lift and triadic closure, forces the quaternion group and the 24 Hurwitz units of the 24-cell, reflection admits the F4 configuration and received chirality selects the half, and two such receivers under evenness and self-duality close to the 240 roots of E8 with the residual glue choice exactly a received triality element — a forcing chain distinct from the McKay correspondence; a carry-depth filtration organizes all arities by binary digit count with the powers of two as the zero-carry stratum, and the same dyadic skeleton is selected independently by binomial parity, by Catalan tree parity in a transfer engine that writes higher operations through the cohomologically invisible half of the cochain world, and by the doubling series of the multiplicative formal group; a moduli of contraction witnesses is rigid at arity three and first free at arity four; and the whole is read, at cited and conjectural labels only, as a chromatic proposal — complex quantum mechanics as the height-one chart of a stratified reality — whose central registered wager is the next i: a physical unit that is exactly zero to every height-one observable and nonzero to the next stratum, with spoof-closed witnesses registered in advance, every heavy claim shipped as a dependency-free script, every correction published, and zero empirical divergences claimed — under which the program remains a reconstruction with registered extensions, and every unearned generalization is left open by name.

The horizon

Named targets, explicitly not results: the formal-group extraction that would give the chromatic frame its second leg; the HKR character-theory check on the essential ("faceless") classes; the in-model no-cloning-distinction check on the contextuality derivation; and the first confirmatory experiment for the next i — reserved for whenever a dedicated apparatus meets the registered protocols. How the repository grows without rewriting its past: EVOLUTION.md · roadmap · open problems.

Author

• Seth Douglas · ORCID: 0009-0007-4708-3252 · github.com/Apsiape · apsiape@gmail.com

Citation

Archived and citable via Zenodo. Cite the program by its concept DOI (always the latest version): 10.5281/zenodo.21253591. Version DOIs are recorded in CITATION.cff and each chapter's freeze record.

Rights

This work is licensed under Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0). You may share it and build on it with attribution — the open problems are open on purpose — as long as derivatives carry the same license; commercial use requires the author's permission. See LICENSE.md.