Release: v0.10.0 · Status of the chapter: one unconditional theorem and one conditional, preregistered, experiment-open prediction. Every prior chapter was a recovery of known structure from the finite-contact floor. This chapter is the program's first extension — it predicts a possible violation of Hilbert-space positivity. It recovers nothing, and it is not a claimed discovery.
This chapter carries the program's live ceiling, quoted identically in the README, the claim register, and the v0.10.0 release notes:
Finite Contact Theory is a finite reconstruction program with a scoped theorem stack on three published recovery lines — a quantum-facing axis, from one-use contact to counting, one-receiver gluing, rational Born weights, the CHSH/Pell boundary, a carrier grammar grown from one-use contact, and a behavior-conditioned contextual capacity with an exact strict preparation gap; a finite-epistemics axis, from the identifiability and debt calculus to the inquiry calculus and its second law of asking, four theorems separating the structure of time, and a measured generative floor; and a contact-interface reconstruction, in which a retained interface forces a quaternionic state/receiver cell whose self-dual closure is the 24-cell and the F_4 root system and whose finite measurement calculus forces the quadratic Born frame rule (a finite Gleason theorem) exactly where a triality Kochen-Specker obstruction forbids a global noncontextual assignment, and in which independently generated cells recover the E_8–hexacode closure spine under named receiver laws that a forcing audit shows the floor does not select over matched lawful alternatives, so the floor forces the atlas of lawful closures and the terminal self-dual class but never the specific member, and the selection of a world-phase is a conserved, received input — and a fourth, first-extension line that is conditional and experiment-open: an unconditional accessible-positivity theorem exhibits a three-contact sector whose every passive-linear-optical probability is nonnegative yet which lies outside the positive-semidefinite Hilbert Gram cone, and on one received apparatus anchor this predicts a possible violation of Hilbert-space positivity — a negative three-state Gram discriminant Delta_3 < 0 where ordinary quantum mechanics forces Delta_3 >= 0 — preregistered with its protocol, nulls, and kill conditions — under which the quantum boundary is a floor theorem at binary-Bell finite-carrier scope, the preparation gap is an exact theorem at KCBS-pentagon scope, the interface reconstruction is a finite model-scope recovery on a real-quantum cell, the multi-floor closures are model-scope recoveries whose forcing boundary is exactly mapped, the accessible-positivity theorem is unconditional while its physical realization is a conditional, bridge-premise-gated prediction awaiting a dedicated experiment and external expert review, and every unearned generalization — complex quantum mechanics, the actuality of one outcome, the universal Born rule, whether nature realizes any of these structures, which world-phase is selected, and whether nature contains the odd identity-holonomy sector — is left open by name; this chapter is an archival priority record of a mathematically closed conditional prediction, not an empirical discovery.
Chapters 1–9 all did the same kind of thing: they recovered known
mathematics or physics (the CHSH bound, a finite Gleason theorem, the E_8
spine) from a deeper finite-contact floor. A recovery is tested by "does it
match the known answer?", and matching is success.
This chapter is different in kind. It is an extension: it asks whether a genuinely non-classical feature of the floor — locally exact identity contacts that fail to glue into one global carrier — yields a concrete, dimensionless, falsifiable prediction that differs from ordinary complex-Hilbert quantum mechanics. It does. An extension is not tested by reduction to the incumbent; it is tested by internal rigor, self-consistency, and a sharp observable divergence. Judging it by "does it reduce to standard QM?" would be a category error — the divergence is the whole point.
Two things are published here, and they must be held rigidly apart:
- (A) THE THEOREM — unconditional. A three-contact class function with a negative sign-sector eigenvalue nonetheless keeps every physically accessible (passive-linear-optical) probability nonnegative. So it is a lawful probabilistic sector that lies outside the positive-semidefinite (PSD) Hilbert Gram cone. This is a closed mathematical fact, stated with confidence.
- (B) THE PREDICTION — conditional, preregistered, experiment-open. On a
single received apparatus anchor, that sector predicts a negative
three-state Gram discriminant
Delta_3 < 0, where ordinary QM forcesDelta_3 >= 0. This is a registered bet with a protocol, not a result. Whether nature contains such a sector is untested, and the decisive mixed-state / full-nuisance exclusion has not been vetted by a quantum-optics expert.
Everything numeric below is exact where it can be, and machine-checked from a
clean clone (verification/negative_gram_holonomy.py).
The incumbent complex-Hilbert model represents three internal states by
vectors with normalized Gram matrix G (unit diagonal, off-diagonal
r_{ij} e^{i φ_{ij}}). A global Hilbert realization requires G ⪰ 0, hence
det G ≥ 0. For three states,
det G = 1 − r12² − r23² − r31² + 2 r12 r23 r31 cos Φ, Φ = φ12 + φ23 + φ31,
with Φ gauge-invariant under local rephasings. The proposed sector keeps the
pairwise overlaps but does not require the local charts to arise as
restrictions of one positive global carrier; it can reach det G < 0.
The immediate danger is that the underlying distinguishability functional J
on S_3 has a negative sign-sector eigenvalue (λ_sign = 1 − 3r² − 2r³,
negative for r > 1/2). If arbitrary six-vectors in the permutation-amplitude
space were physically accessible, some would give negative probabilities. They
are not: passive three-particle linear optics produces only the monomials
of one complex 3×3 interferometer matrix A,
z_σ = ∏_i A_{i, σ(i)}, σ ∈ S_3,
which obey the toric identity ∏_{σ even} z_σ = ∏_{σ odd} z_σ (both equal
the product of all nine entries; verification, Section B). The real question
is block positivity on this toric variety, and it holds — universally, using
only the toric relation:
Universal accessible positivity theorem. For every complex
3×3matrixA, at the registered pointr = 3/5,152 ‖z‖² + 9 |per A|² − 36 |det A|² ≥ 0,with equality only at
A = 0.
Since Σ z_σ = per A and Σ sgn(σ) z_σ = det A, this says every admitted raw
probability is nonnegative for every interferometer. The proof needs no
unitarity: writing x = (z_σ)_{σ even}, y = (z_σ)_{σ odd}, the toric
constraint is x₀x₁x₂ = y₀y₁y₂; a C₃-Fourier change of coordinates turns the
form into Q = 152N − 243M + 810 Re(αδ̄) (with α = (Σx)/3, δ = (Σy)/3 the
sector means and M = |α|² + |δ|²), and a cubic-deviation bound closes Q ≥ 0
on the toric variety. The shipped script verifies the foundational
identity and positivity exactly on rational toric witnesses, and an
independent adversarial descent over the full toric set finds no violation
(verification, Section C). This is the campaign's principal mathematical
novelty:
Global Hilbert PSD fails, while operational block positivity survives.
Two consequences fix the scope precisely. First, because the form is
degree-6 homogeneous, Q(cA) = |c|⁶ Q(A), so positivity on all matrices is
equivalent to positivity on all contractions — i.e. on every passive
linear-optical transfer matrix, lossless or lossy (a lossy multiport is a
submatrix of a larger unitary, hence a contraction). The model is internally
consistent for any such apparatus; there is no lossless-only restriction.
Second, on the lossless unitary core U(3) a sharper strict margin holds,
152 ‖z‖² + 9 |per A|² − 36 |det A|² ≥ (7/2) ‖z‖², saturated exactly by the
Fourier tritter (verification, Section D). The stronger (7/2)-margin is
false for arbitrary matrices — the explicit witness
A = [[−4,−1,1],[−7,1,−3],[6,−3,−5]] gives −495 — and that boundary stays on
the record (verification, Section C4); it is exactly why the universal
theorem is the zero-margin one. The theorem is careful about its reach: it
proves consistency for the declared passive-linear-optical grammar, not for
arbitrary hypothetical measurements outside it.
On the received fiber (r, χ(H_γ), receiver) = (3/5, −1, Fourier tritter),
every registered observable is a fixed rational (verification, Sections
A–B). The odd Z_2 identity-holonomy class (Φ = π) gives
Delta_3 = 1 − 3r² − 2r³ = (1 − 2r)(1 + r)² < 0 for r > 1/2,
Delta_3(3/5) = −64/125.
r = 3/5 is not a fundamental constant — it is an apparatus anchor, chosen
only because r > 1/2 places the odd sector outside the PSD cone. At r = 3/5
the incumbent positive-Hilbert cone forces cos Φ ≥ 5/27, i.e. Φ ≤ ~79.3°,
while the extension reaches Φ = π. That gap is the divergence: dimensionless,
gauge-invariant, and count-reconstructible.
The registered joint decision vector. A single negative determinant is not
accepted as sufficient — a structureless null defeated the earlier capstone.
The experiment must report the joint vector (Delta_3, W, Q_3) in gauge-free
raw counts, where (with tritter probabilities P111 = 7/1125,
P300 = 206/3375, P210 = 152/1125, exactly normalized, and pairwise
statistic D_2 = 41/75):
W = P111 + D2 − 2/3 = (2/9) Delta_3 = −128/1125,
Q3 = P111 − D2 + 4/9 = (4/9) b = −12/125.
The matched pairwise-only null (which keeps the pairwise overlaps but
deletes the cyclic class) predicts Q3 = 0. So the confirmatory criterion is
two-fold: the confidence region must (1) lie outside the entire PSD Gram
class and (2) reject Q3 = 0 in the registered negative direction. This
is what makes the bet falsifiable rather than fitted.
Protocol (frozen). Three independently heralded high-purity single photons;
randomly interleaved settings PAIR12 / PAIR23 / PAIR31 / TRITTER / flat-loop control / cyclic-erased control; late randomization after heralding;
number-resolving or calibrated detection; pairwise overlaps tuned to r = 3/5
within a pre-frozen tolerance; the cyclic target Φ = π. Analysis is a
blinded PSD-constrained profile-likelihood fit over the full nuisance family,
unblinded only after pipeline freeze, with an independent confirmatory block.
Protocol amendment (append-only, 2026-08; see correction C-25). The frozen protocol above stands unchanged as the record of what was preregistered. The normative analysis form is now Chapter 12's tritter-counts-alone reduction, which the pair settings do not improve on and which removes an apparatus exposure they carry:
W′ = (4/3) P111 − (2/3) q300,
Δ₃ = (9/2) W′,
Q₃ = q300 + W′/2 − 2/9.
Every registered quantity is thereby read from the TRITTER setting alone. The PAIR settings are retained — but only as a cross-setting homogeneity cross-check. Their residual
D2_meas − [ 2/3 − (T₊ − T₋)/3 ], T₊ = (3/2) q300, T₋ = (3/4) W′,
must be reported and must be consistent with zero; the settings are
over-determined by exactly one degree of freedom, and that residual is the
homogeneity test. Cross-setting inhomogeneity — the tritter configuration
physically differing from the pair configuration in a way correlated with the
cyclic phase — is hereby added by name to the preregistered nuisance family.
This amendment withdraws no theorem: FCT-62 and FCT-63 are untouched, and the
exclusion result is unchanged. It is reopened only if a two-setting form is
shown to carry information the tritter counts do not, or a bound on
|S_tri − S_pair| is derived from calibration.
Kill conditions (any one kills the physical claim). An admissible PSD Gram
fit exists; Q3 = 0 is not rejected; late-randomization or drift correction
removes the effect; the sign depends on post-hoc binning; nuisance
restrictions are tightened after unblinding; a standard positive-Hilbert model
outside the preregistered nuisance family reproduces the result; or independent
replication fails.
The theorem (A) needs no physics. The prediction (B) rests on bridge premises that are not floor theorems and are stated as such:
- a physical particle class may carry
Z_2identity holonomy; - photons may be an appropriate receiver of that sector;
- passive linear optics may be the complete relevant measurement grammar.
These are strongly motivated but unproven about nature. The floor permits the odd-holonomy sector; it does not force photons — or any known particle class — to instantiate it. That in-class membership is exactly what the proposed experiment tests.
The open crux (make no mistake about this). The decisive step is a mixed-state, mode-mismatch, multiphoton PSD-exclusion: one must show that no positive-Hilbert model — including mixed internal states, spectral / temporal / polarization distinguishability, multiphoton contamination, loss, and tritter imperfection — can fit the data. This is a different claim from the theorem of §1, and only the theorem is closed. §1 proves the extension model's own internal consistency — every accessible probability is nonnegative for every passive-linear-optical apparatus, lossless or lossy. It does not show that a positive-Hilbert model cannot reproduce the measured counts; that exclusion is exactly what remains open. The apparatus-nuisance layer is settled by exact computation on two independent engines plus refutation-first adversarial review across model families; the one thing no review can supply is MEASUREMENT, so the gate before any nature-facing claim is the experiment itself, and outreach to an experimental group is a step toward that measurement, not a validity gate.
The natural home is not photons. The clean physical home for path-relative exchange with nontrivial identity holonomy is anyonic / topological matter, not photons. Photons are the clean-grammar long shot — attractive because passive linear optics is exactly the grammar the theorem controls, but a long shot because ordinary photons are prepared inside Hilbert-state models and therefore have PSD Gram matrices by construction (so the bridge premise likely fails for them). The bet is placed on structure, not on optimism.
What this DOI is. This chapter is an archival / priority record of a mathematically closed conditional prediction. It is deliberately not an empirical discovery, and nothing here should be read as one. No suitable public trial-level dataset currently closes the experiment; the only decisive remaining step is the preregistered, late-randomized, interleaved photon (or topological-matter) measurement, gated by expert review.
Proven exactly (verification/negative_gram_holonomy.py, 11/11):
Delta_3 = (1−2r)(1+r)², negative for r > 1/2, = −64/125 at r = 3/5; the
full tritter/pairwise count vector and its exact normalization; the count
witnesses W = (2/9)Delta_3, Q3 = (4/9)b; the pairwise-only null Q3 = 0;
the PSD cap cos Φ ≥ 5/27 (Φ ≤ ~79.3°) that the extension exceeds; the toric
identity; the universal accessible-positivity theorem
152‖z‖² + 9|per A|² − 36|det A|² ≥ 0 for every complex 3×3 matrix (the
foundational Fourier identity and positivity on exact rational toric witnesses,
plus an independent adversarial descent over the full toric set); the sharper
U(3) strict 7/2-margin corollary saturated by the Fourier tritter; the
−495 witness that kills the (7/2)-margin for arbitrary matrices; and exact
separable-loss covariance.
Not claimed: that r = 3/5 is fundamental; that all quantum systems
violate global Hilbert positivity; that standard QM is numerically wrong in
already-tested flat sectors; that the floor uniquely predicts photons; that no
larger positive completion exists under an enlarged measurement grammar; that
existing three-photon data demonstrate the effect; or that mathematical
consistency constitutes empirical discovery.
Final status. The campaign has produced a complete, internally consistent, dimensionless extension prediction, but not yet an empirical discovery.
The claim-register rows this chapter rests on are frozen in
claims.md; the freeze record is in RELEASE.md.