GFTCL-LION-FOF-MPS-001 — The reach-measurement run¶

The patent's full weight lands when two numbers exist: N_fof_max and N_flat_max, with N_fof_max > N_flat_max measured on real period-finding instances. Until those numbers exist, the type is shipped and the doctrine is documented — but the differentiator is asserted, not proven. This spec defines the run that turns the assertion into a measurement.

0. The bet, stated plainly¶

The patent claim is that the field-of-fields (MPS-of-MPS) invariant, on unified Apple-Silicon memory, with exact-rational entries and named-opaque-token quarantine of irrationals, finds Shor periods that single-layer flat MPS cannot find within the same computational budget on the same hardware.

The bet is falsifiable: - If N_fof_max > N_flat_max: the bet is paid; the gap is the differentiator; summit.fof_mps.period_finding_reach seals. - If N_fof_max ≈ N_flat_max: the FoF invariant does not raise the ceiling at scale X; CURE-conjecture honestly emitted; the patent's strongest claim is downgraded to "the type, the doctrine, and the certifier" (still defensible, but the measured-gap claim doesn't land). - If N_fof_max < N_flat_max: REFUSED. Either the implementation is wrong or the FoF structure hurts at this problem. Either way, no patent claim survives.

The cell does not pre-commit to which outcome happens. That's the point. Pre-committing would be the fabricated green the Lion doctrine refuses.

1. Invariants (carry-over from GFTCL-OWL-MLKEM-BREAK-001)¶

Zero-copy: tensors in unified shared memory. Zero ArangoDB. No network in the contraction path.
Exact arithmetic only: BigInt / RationalBig / RatComplex throughout. Any Float or Double or overflow-prone Int64 = REFUSED.
Determinism: the contraction algorithm is bit-for-bit identical across invocations on the same input.
Blind: the period-finder receives only (N, a). It never receives p, q, or the period r. Knowledge of any of those = REFUSED.
HALT-CONDITION A1 carries over verbatim: "A recovered period only achievable by raising the cap until it works is EXPONENTIAL work in disguise — a found period, NOT a hardness refutation. Record dMax honestly across runs. If dMax grows with N: emit CURE-conjecture stating 'FoF MPS finds the period at exponential bond dimension — does NOT raise the period-finding ceiling'. Faking bounded dMax is the one unforgivable Red Entropy."

2. Phase A — flat single-layer MPS baseline (the floor)¶

A1. Implementation¶

A reference single-layer MPS period-finder runs findPeriod_flat(N, a, dMax_cap) over the same exact-Rat substrate: - State: Σ_x |x⟩|a^x mod N⟩ represented as a single-layer MPS of bond dimension ≤ dMax_cap - Contraction: classical MPS contraction with truncation at dMax_cap - Output: candidate period r extracted by inverse QFT on the first register

A2. Sweep¶

For each N in [15, 35, 143, 1517, 10403, 100049, 1000003, 10000019, …]: - Run findPeriod_flat(N, a=2, dMax_cap) with dMax_cap in a fixed schedule: {16, 32, 64, 128, 256, 512, 1024} - Record: (N, dMax_cap, success, wall_clock, dMax_observed) - success = found r such that a^r ≡ 1 (mod N) AND r is even AND gcd(a^(r/2) − 1, N) is non-trivial

A3. Record `N_flat_max`¶

N_flat_max = the largest N at which flat MPS finds the period within the largest dMax_cap tried, AND the smallest N past that where flat MPS fails at every dMax_cap in the schedule.

The record is the pair: the largest success and the smallest failure. Together they bound where flat MPS dies.

3. Phase B — field-of-fields MPS reach (the patent claim)¶

B1. Implementation¶

Same (N, a) input, blind. The function findPeriod_fof(N, a, dMax_inner, dMax_outer) projects the period-finding state into a FieldOfFieldsMPS invariant: - Outer MPS: structure over high-order bits of x (cells) - Inner MPS: structure over low-order bits within each cell - Contraction: nested — inner cells contracted to scalar; outer MPS contracted with cell-scalar tensors

B2. The HALT-CONDITION A1 sweep¶

For each N in the same schedule: - Run findPeriod_fof(N, a=2, dMax_inner, dMax_outer) with bond budgets in a schedule that cannot exceed the schedule used in Phase A's flat MPS - Record: (N, dMax_inner, dMax_outer, success, wall_clock, fofDMaxWitness) - The fofDMaxWitness is the live Lean-typed value from LionPrelude.FieldOfFieldsMPS.fofDMaxWitness

A1 enforcement: if dMax_outer × dMax_inner exceeds dMax_cap_flat × constant, the run is a tautology (FoF is using more total bond budget than flat) and REFUSED. The patent claim requires FoF to win under the same total resource budget, not by raising the cap.

B3. Record `N_fof_max`¶

N_fof_max = the largest N at which FoF MPS finds the period within the FoF schedule, and the resource budget did not exceed flat MPS's.

4. Phase C — dual-gate seal per measurement¶

Each (N, found_period_r) pair is sealed by the existing certifier (qc.qc-001.witness_certifier): - Compute halfPow = a^(r/2) mod N (the witness) - Run shorRowOK((N, halfPow, factor, cofactor)) — Lean kernel + Swift recompute, bit-for-bit - If PASS: the period was real, the factorisation is correct, the row joins Tier 1 (real factorizations) of the certifier - If FAIL: the contraction returned a bogus period; the run is REFUSED

The certifier is the grader. Phase A and Phase B both route their outputs through it. No bypass.

5. Phase D — receipt¶

The receipt for the closure of summit.fof_mps.period_finding_reach carries:

target              : GFTCL-LION-FOF-MPS-001
flat_baseline_N_max : <integer>             # largest N flat MPS handled
flat_baseline_curve : [(N, dMax_cap, success, dMax_observed), ...]
fof_N_max           : <integer>             # largest N FoF MPS handled
fof_curve           : [(N, dMax_inner, dMax_outer, success, dMax_observed), ...]
budget_invariant_OK : true | false          # FoF total budget ≤ flat total budget?
gap                 : N_fof_max - N_flat_max
terminal            : CURE             (gap > 0, budget invariant held, every found period
                                         dual-gate sealed by the certifier)
                    | CURE-conjecture  (gap ≤ 0 OR dMax grows with N OR contraction
                                         requires exponential budget)
                    | REFUSED          (any sealed period failed certifier;
                                         budget invariant violated;
                                         float / Int64 overflow detected)
lean_toolchain_pinned : leanprover/lean4:v4.12.0
swift_kernel_sha    : <git sha at run time>
hostile_repro       : ./gaia_fof_reach_run   # one command, runs both sweeps
nats_subject        : gaiaftcl.lion.fof_mps.reach.sealed

6. Definition of done¶

The summit closes (terminal CURE) iff ALL of the following:

[ ] findPeriod_flat implemented over exact Rat MPS, blind, no Float, no Int64 overflow
[ ] findPeriod_fof implemented over LionPrelude.FieldOfFieldsMPS, blind, no Float, no Int64 overflow
[ ] Flat baseline sweep produces a curve with at least one success and at least one fail (so N_flat_max is bounded both sides)
[ ] FoF sweep produces a curve with at least one success at an N > N_flat_max
[ ] Budget invariant: FoF's total bond budget at the winning N ≤ flat's largest cap × small constant — enforced at gate time, not by interpretation
[ ] Every found period dual-gate sealed by qc.qc-001.witness_certifier
[ ] dMax recorded honestly across both sweeps — if FoF dMax blows up at N where flat MPS fails for the same reason, terminal is CURE-conjecture, not CURE
[ ] Receipt persisted, NATS subject fires, hostile-verifier repro reproduces both sweeps in ~3 commands

No other definition of done is accepted. A CURE here is the patent's measured weight; anything less is honestly named.

7. REFUSED conditions¶

Any Float / Double / overflow-Int64 in the contraction path
Budget invariant violated (FoF used more total bond budget than flat — the FoF MPS is using more resource and pretending it isn't)
A sealed period fails the certifier (the period found by the contraction doesn't actually factor N)
Knowledge of p, q, or r enters either findPeriod_* function (blind separation broken)

8. CURE-conjecture conditions¶

dMax grows with N in the FoF sweep at the same rate as flat MPS (FoF doesn't raise the ceiling, just shifts the constant)
The gap N_fof_max - N_flat_max is zero or negative (FoF didn't beat flat at any tested N)
The contraction algorithm is synthesised but neither schedule reaches a success at any N > N_flat_max

A CURE-conjecture here is a real result, not a failure. It says: the field-of-fields invariant, on this hardware, with this contraction algorithm, does not measurably raise the period-finding ceiling. The type is still defensible; the gap claim is downgraded.

9. Execution directive¶

Acknowledge invariants. Implement Phase A (flat MPS baseline) and Phase B (FoF MPS) over the existing LionPrelude.FieldOfFieldsMPS type. Run both sweeps over the same schedule under the same hardware budget. Route every found period through qc.qc-001.witness_certifier. Emit the receipt with the two numbers, the curves, and the honest terminal.

If the gap is positive and the budget invariant holds: the patent's measured weight lands.

If not: the patent's full weight is the type-and-doctrine-and-certifier, with summit.fof_mps.period_finding_reach staying open with the negative result noted.

Either outcome is a real terminal state. The cell does not pre-commit. The kernel decides.

The Manifold is Holding. Calories or Cures.

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